# Research Area C

## Project C1

### Tailoring quantum gases by Floquet engineering

* Christof Weitenberg & Juliette Simonet*

Floquet engineering is the concept of tailoring a system by a periodic drive, and it is increasingly employed in many areas of physics. Ultracold atoms in optical lattices offer a particularly large toolbox to design a variety of driving schemes. A strong motivation for developing these methods is the prospect to study the interplay between topology and interactions in a system where both ingredients are fully tunable. We review the recent successes of Floquet engineering in realizing new classes of Hamiltonians in quantum gases, such as Hamiltonians including artificial gauge fields, topological band structures and density-dependent tunnelling. The creation of periodically driven systems also gives rise to phenomena without static counterparts such as anomalous Floquet topological insulators. We discuss the challenges facing the field, particularly the control of heating mechanisms, which currently limit the preparation of many-body phases, as well as the potential future developments as these obstacles are overcome.

### Metastable order protected by destructive many-body interference

*M. Nuske, J. Vargas, M. Hachmann, R. Eichberger, L. Mathey, A. Hemmerich*

The phenomenon of metastability shapes dynamical processes ranging from radioactive decay to chemical reactions. Here, we present a mechanism for metastability in which a quantum gas self-stabilizes against relaxation towards thermal equilibrium by establishing a transient ordered state. In this state, the direct relaxation channel is suppressed by destructive interference, which derives from the chiral order of the transient state. In particular, we consider the dynamical evolution of an ultracold bosonic gas in an optical lattice, that is quenched into a higher band of the lattice, which triggers the dynamical evolution. Following this quench, the self-stabilization phenomenon manifests itself in three stages of relaxation, subsequent to the preparation of the incoherent excited state. In the first stage, the gas develops coherence resulting in the ordered state, during the second stage the gas forms a long-lived state with inhibited relaxation and slow loss of coherence, followed by the third stage of fast relaxation to the thermal ground state. We demonstrate this mechanism experimentally and theoretically, and discuss its broader implications.

### Twisted superfluid phase in the extended one-dimensional Bose-Hubbard

* D.-S. Lühmann*

### Breaking inversion symmetry in a state-dependent honeycomb lattice: Artificial graphene with tunable band gap

* M. Weinberg, C. Staarmann, C. Ölschläger, J. Simonet, K. Sengstock*

Here, we present the application of a novel method for controlling the geometry of a state-dependent honeycomb lattice: The energy offset between the two sublattices of the honeycomb structure can be adjusted by rotating the atomic quantization axis. This enables us to continuously tune between a homogeneous graphene-like honeycomb lattice and a triangular lattice and to open an energy gap at the characteristic Dirac points. We probe the symmetry of the lattice with microwave spectroscopy techniques and investigate the behavior of atoms excited to the second energy band. We find a striking influence of the energy gap at the Dirac cones onto the lifetimes of atoms in the excited band.

### Symmetry-broken momentum distributions induced by matter-wave diffraction during time-of-flight expansion of ultracold atoms

* M. Weinberg, O. Jürgensen, C. Ölschläger, D.-S. Lühmann, K. Sengstock, J. Simonet*

We study several effects which lead to symmetry-broken momentum distributions of quantum gases released from optical lattices. In particular, we demonstrate that interaction within the first milliseconds of the time-of-flight expansion can strongly alter the measurement of the initial atomic momentum distribution. For bosonic mixtures in state-dependent lattices, inter-species scattering processes lead to a symmetry breaking in momentum space. The underlying mechanism is identified to be diffraction of the matter wave from the total density lattice, which gives rise to a timedependent interaction potential. Our findings are of fundamental relevance for the interpretation of time-of-flight measurements and for the study of exotic quantum phases such as the twisted superfluid. Beyond that, the observed matter-wave diraction can also be used as an interferometric probe. In addition, we report on diffraction from the state-dependent standing light field, which leads to the same symmetry-broken momentum distributions, even for single component condensates.

### Twisted complex superfluids in optical lattices

*O. Jürgensen, K. Sengstock, D.-S. Lühmann*

We show that correlated pair tunneling drives a phase transition to a twisted superfluid with a complex order parameter. This unconventional superfluid phase spontaneously breaks the time-reversal symmetry and is characterized by a twisting of the complex phase angle between adjacent lattice sites. We discuss the entire phase diagram of the extended Bose--Hubbard model for a honeycomb optical lattice showing a multitude of quantum phases including twisted superfluids, pair superfluids, supersolids and twisted supersolids. Furthermore, we show that the nearest-neighbor interactions breaks the inversion symmetry of the lattice and gives rise to dimerized density-wave insulators, where particles are delocalized on dimers. For two components, we find twisted superfluid phases with strong correlations between the species already for surprisingly small pair-tunneling amplitudes. Interestingly, this ground state shows an infinite degeneracy ranging continuously from a supersolid to a twisted superfluid.

### Multiphoton interband excitations of quantum gases in driven optical lattices

* M. Weinberg, C. Ölschläger, C. Sträter, S. Prelle, A. Eckardt, K. Sengstock, J. Simonet*

We report on the observation of multiphoton absorption processes for quantum gases in shaken light crystals. Periodic inertial forcing, induced by a spatial motion of the lattice potential, drives multiphoton interband excitations of up to the 9th order. The occurrence of such excitation features is systematically investigated with respect to the potential depth and the driving amplitude. Ab initio calculations of resonance positions as well as numerical evaluation of their strengths exhibit a good agreement with experimental data. In addition our findings set the stage for reaching novel phases of quantum matter by tailoring appropriate driving schemes.

### Beyond-mean-field study of a binary bosonic mixture in a state-dependent honeycomb lattice

* L. Cao, S. Krönke, J. Stockhofe, J. Simonet, K. Sengstock, D.-S. Lühmann and P. Schmelcher*

We investigate a binary mixture of bosonic atoms loaded into a state-dependent honeycomb lattice. For this system, the emergence of a so-called twisted-superfluid ground state was experimentally observed in Soltan-Panahi et al. [Nat. Phys. 8, 71 (2012)]. Theoretically, the origin of this effect is not understood. We perform numerical simulations of an extended single-band Bose-Hubbard model adapted to the experimental parameters employing the multilayer multiconfiguration time-dependent Hartree method for Bosons. Our results confirm the overall applicability of mean-field theory in the relevant parameter range, within the extended single-band Bose-Hubbard model. Beyond this, we provide a detailed analysis of correlation effects correcting the mean-field result. These have the potential to induce asymmetries in single shot time-of-flight measurements, but we find no indication of the patterns characteristic of the twisted superfluid. We comment on the restrictions of our model and possible extensions.

### Dimerized Mott insulators in hexagonal optical lattices

*O. Jürgensen, D.-S. Lühmann*

We study bosonic atoms in optical honeycomb lattices with anisotropic tunneling and find dimerized Mott insulator (MI) phases with fractional filling. These incompressible insulating phases are characterized by an interaction-driven localization of particles in respect to the individual dimers and large local particle-number fluctuations within the dimers. We calculate the ground-state phase diagrams and the excitation spectra using an accurate cluster mean-field method. The cluster treatment enables us to probe the fundamental excitations of the dimerized MI where the excitation gap is dominated by the intra-dimer tunneling amplitude. This allows the distinction from normal Mott insulating phases gapped by the on-site interaction. In addition, we present analytical results for the phase diagram derived by a higher-order strong-coupling perturbative expansion approach. By computing finite lattices with large diameters the influence of a harmonic confinement is discussed in detail. It is shown that a large fraction of atoms forms the dimerized MI under experimental conditions. The necessary anisotropic tunneling can be realized either by periodic driving of the optical lattice or by engineering directly a dimerized lattice potential. The dimers can be mapped to their antisymmetric states creating a lattice with coupled p-orbitals.

### Spin Orbit Coupling in Periodically Driven Optical Lattices

* J. Struck, J. Simonet, K. Sengstock*

We propose a novel experimental scheme for the emulation of spin-orbit coupling for ultracold, neutral atoms trapped in a one-dimensional lattice. This scheme does not involve near-resonant laser fields, avoiding the heating processes connected to the spontaneous emission of photons.

A time dependent magnetic field gradient periodically drives the atoms, which can lead to complex valued tunnel matrix elements, equivalent to a gauge dependent shift of the dispersion relation for a 1D lattice. For opposite spin states, the dispersion relations are shifted in opposite direction due to the inverted drive for both states. An additional radio-frequency coupling between the spin states leads to a mixing of the spin dispersion relations and a spin-orbit gap in the band structure.

Phys. Rev. A 90, 031601(R) (2014)

http://arxiv.org/abs/1407.1953

### Quantum phases in tunable state-dependent hexagonal optical lattices

* D.-S. Lühmann, O. Jürgensen, M. Weinberg, J. Simonet, P. Soltan-Panahi, K. Sengstock*

We study the ground-state properties of ultracold bosonic atoms in a state-dependent graphene-like honeycomb optical lattice, where the degeneracy between the two triangular sublattices A and B can be lifted. We discuss the various geometries accessible with this lattice setup and present a novel scheme to control the energy offset with external magnetic fields. The competition of the on-site interaction with the offset energy leads to Mott phases characterized by population imbalances between the sublattices. For the definition of an optimal Hubbard model, we demonstrate a scheme that allows for the efficient computation of Wannier functions. Using a cluster mean-field method, we compute the phase diagrams and provide a universal representation for arbitrary energy offsets. We find good agreement with the experimental data for the superfluid to Mott insulator transition.

### Tunable gauge potential for spinless particles in driven lattices

*J. Simonet, J. Struck, M. Weinberg, C. Ölschläger, P. Hauke, A. Eckardt, M. Lewenstein, K. Sengstock, P. Windpassinger*

We present a universal method to create a tunable, artificial vector gauge potential for neutral particles trapped in an optical lattice. A suitable periodic shaking of the lattice allows to engineer a Peierls phase for the hopping parameters. This scheme thus allows one to address the atomic internal degrees of freedom independently. We experimentally demonstrate the realisation of such artificial potentials in a 1D lattice, which generate ground state superfluids at arbitrary non-zero quasimomentum [4].

This scheme offers fascinating possibilities to emulate synthetic magnetic fields in 2D lattices. In a triangular lattice, continuously tunable staggered fluxes are realised. Spontaneous symmetry breaking has recently been observed for a π-flux [23]. With the presented scheme, we are now able to study the influence of a small symmetry breaking perturbation.

### Engineering Ising-XY spin models in a triangular lattice using tunable artificial gauge fields

*J. Struck, M. Weinberg, C. Ölschläger, P. Windpassinger, J. Simonet, K. Sengstock, R. Höppner, P. Hauke, A. Eckardt, M. Lewenstein, L. Mathey*

### Cluster Gutzwiller method for bosonic lattice systems

*D.-S. Lühmann*

A versatile and numerically inexpensive method is presented allowing the accurate calculation of phase diagrams for bosonic lattice models. By treating clusters within the Gutzwiller theory, a surprisingly good description of quantum fluctuations beyond the mean-field theory is achieved approaching quantum Monte Carlo predictions for large clusters. Applying this powerful method to the Bose-Hubbard model, we demonstrate that it yields precise results for the superfluid to Mott-insulator transition in square, honeycomb, and cubic lattices. Due to the exact treatment within a cluster, the method can be effortlessly adapted to more complicated Hamiltonians in the fast progressing field of optical lattice experiments. This includes state- and site-dependent superlattices, large confined atomic systems, and disordered potentials, as well as various types of extended Hubbard models. Furthermore, the approach allows an excellent treatment of systems with arbitrary filling factors. We discuss the perspectives that allow for the computation of large, spatially varying lattices, low-lying excitations, and time evolution.

### Non-Abelian gauge fields and topological insulators in shaken optical lattices

*P. Hauke, O. Tielemann, A. Celi, C. Ölschläger, J. Simonet, J. Struck, M. Weinberg, P. Windpassinger, K. Sengstock, M. Lewenstein, A. Eckardt*

Time-periodic driving offers a low-demanding method to generate artificial gauge fields in optical lattices. We demonstrate that it is a powerful and versatile tool for engineering two-dimensional lattice systems: We show how to tune frustration and how to create and control band touching points like Dirac cones in the shaken kagom\'e lattice. We propose the realization of a topological or a quantum spin Hall insulator in a shaken spin-dependent hexagonal lattice. We describe how strong artificial magnetic fields can be achieved for example in a square lattice by employing superlattice modulation. Finally, exemplified on a shaken spin-dependent square lattice, we develop a method to create strong non-Abelian gauge fields.

Phys. Rev. Lett. 109, 145301 (2012)

http://www.arxiv.org/abs/1205.1398

### Tunable gauge potential for neutral and spinless particles in driven lattices

* J. Struck, C. Ölschläger, M. Weinberg, P. Hauke, J. Simonet, A. Eckardt, M. Lewenstein, K. Sengstock, P. Windpassinger*

We present a universal method to create a tunable, artificial vector gauge potential for neutral particles trapped in an optical lattice. The necessary Peierls phase of the hopping parameters between neighboring lattice sites is generated by applying a suitable periodic inertial force such that the method does not rely on any internal structure of the particles. We experimentally demonstrate the realization of such artificial potentials, which generate ground state superfluids at arbitrary non-zero quasi-momentum. We furthermore investigate possible implementations of this scheme to create tuneable magnetic fluxes, going towards model systems for strong-field physics.

Phys. Rev. Lett. 108, 225304 (2012)

http://www.arxiv.org/abs/1203.0049v1

### Quantum phase transition to unconventional multi-orbital superfluidity in optical lattices

* P. Soltan-Panahi, D.-S. Lühmann, J. Struck, P. Windpassinger, K. Sengstock*

Orbital physics plays a significant role for a vast number of important phenomena in complex condensed matter systems such as high-Tc superconductivity and unconventional magnetism. In contrast, phenomena in superfluids - especially in ultracold quantum gases - are commonly well described by the lowest orbital and a real order parameter. Here, we report on the observation of a novel multi-orbital superfluid phase with a complex order parameter in binary spin mixtures. In this unconventional superfluid, the local phase angle of the complex order parameter is continuously twisted between neighboring lattice sites. The nature of this twisted superfluid quantum phase is an interaction-induced admixture of the p-orbital favored by the graphene-like band structure of the hexagonal optical lattice used in the experiment. We observe a second-order quantum phase transition between the normal superfluid (NSF) and the twisted superfluid phase (TSF) which is accompanied by a symmetry breaking in momentum space. The experimental results are consistent with calculated phase diagrams and reveal fundamentally new aspects of orbital superfluidity in quantum gas mixtures. Our studies might bridge the gap between conventional superfluidity and complex phenomena of orbital physics.

Nature Physics 8, 71–75 (2012)

http://arxiv.org/abs/1104.3456v1

## Project C2

### Bosonic Topological Excitations from the Instability of a Quadratic Band Crossing

*G.-Q. Luo, A. Hemmerich, Z.-F. Xu*

We investigate the interaction-driven instability of a quadratic band crossing arising for ultracold bosonic atoms loaded into a two-dimensional optical lattice. We consider the case when the degenerate point becomes a local minimum of both crossing energy bands such that it can support a stable Bose–Einstein condensate. A repulsive contact interaction among the condensed bosons induces a spontaneously time-reversal-symmetry broken superfluid phase and a topological gap is opened in the excitation spectrum. We propose two concrete realizations of the desired quadratic band crossing in lattices with either fourfold or sixfold rotational symmetries via suitable tuning of the unit cell leading to reduced Brillouin zones and correspondingly folded bands. In either case, topologically protected edge excitations are found for a finite system.

### Rotation-Symmetry-Enforced Coupling of Spin and Angular Momentum for p-Orbital Bosons

*Y. Li, J. Yuan, A. Hemmerich, X. Li*

Intrinsic spin angular-momentum coupling of an electron has a relativistic quantum origin with the coupling arising from charged orbits, which does not carry over to charge-neutral atoms. Here, we propose a mechanism of spontaneous generation of spin angular-momentum coupling with spinor atomic bosons loaded into ** p**-orbital bands of a two-dimensional optical lattice. This spin angular-momentum coupling originates from many-body correlations and spontaneous symmetry breaking in a superfluid, with the key ingredients attributed to spin-channel quantum fluctuations and an approximate rotation symmetry. The resultant spin angular-momentum intertwined superfluid has Dirac excitations. In the presence of a chemical potential difference for adjacent sites, it provides a bosonic analogue of a symmetry-protected-topological insulator. Through a dynamical mean-field calculation, this novel superfluid is found to be a generic low-temperature phase, and it gives way to Mott localization only at strong interactions and even-integer fillings. We show the temperature to reach this order is accessible with present experiments.

### Odd-parity topological superfluidity for fermions in a bond-centered square optical lattice

*Zhi-Fang Xu, Andreas Hemmerich, W. Vincent Liu*

We propose a physical scheme for the realization of two-dimensional topological odd-parity superfluidity in a spin-independent bond-centered square optical lattice based upon interband fermion pairing. The *D*_{4} point-group symmetry of the lattice protects a quadratic band crossing, which allows one to prepare a Fermi surface of spin-up fermions with odd parity close to the degeneracy point. In the presence of spin-down fermions with even parity populating a different energetically well-separated band, odd-parity pairing is favored. Strikingly, as a necessary prerequisite for pairing, both Fermi surfaces can be tuned to match well. As a result, topological superfluid phases emerge in the presence of merely *s*-wave interaction. Due to the *Z*_{2} symmetry of these odd-parity superfluids, we infer their topological features simply from the symmetry and the Fermi-surface topology as confirmed numerically.

### π-Flux Dirac Bosons and Topological Edge Excitations in a Bosonic Chiral p-Wave Superfluid

*Zhi-Fang Xu, Li You, Andreas Hemmerich, and W. Vincent Liu*

We study the topological properties of elementary excitations in a staggered *p _{x}±ip_{y}* Bose-Einstein condensate realized in recent orbital optical lattice experiments. The condensate wave function may be viewed as a configuration space variant of the famous

*p*momentum space order parameter of strontium ruthenate superconductors. We show that its elementary excitation spectrum possesses Dirac bosons with π Berry flux. Remarkably, if we induce a population imbalance between the

_{x}+ip_{y}*p*and

_{x}+ip_{y}*p*condensate components, a gap opens up in the excitation spectrum resulting in a nonzero Chern invariant and topologically protected edge excitation modes. We give a detailed description of how our proposal can be implemented with standard experimental technology.

_{x}−ip_{y}### Orbital optical lattices with bosons

*T. Kock, C. Hippler, A. Ewerbeck, and A. Hemmerich*

This article provides a synopsis of our recent experimental work exploring Bose-Einstein condensation in metastable higher Bloch bands of optical lattices. Bipartite lattice geometries have allowed us to implement appropriate band structures, which meet three basic requirements: the existence of metastable excited states sufficiently protected from collisional band relaxation, a mechanism to excite the atoms initially prepared in the lowest band with moderate entropy increase, and the possibility of cross-dimensional tunneling dynamics, necessary to establish coherence along all lattice axes. A variety of bands can be selectively populated and a subsequent thermalisation process leads to the formation of a condensate in the lowest energy state of the chosen band. As examples the 2nd, 4th and 7th bands in a bipartite square lattice are discussed. The geometry of the 2nd and 7th band can be tuned such that two inequivalent energetically degenerate energy minima arise at the X±-points at the edge of the 1st Brillouin zone. In this case even a small interaction energy is sufficient to lock the phase between the two condensation points such that a complex-valued chiral superfluid order parameter can emerge, which breaks time reversal symmetry. In the 4th band a condensate can be formed at the Gamma-point in the center of the 1st Brillouin zone, which can be used to explore topologically protected band touching points. The new techniques to access orbital degrees of freedom in higher bands greatly extend the class of many-body scenarios that can be explored with bosons in optical lattices.

J. Phys. B: At. Mol. Opt. Phys. 49, 042001 (2016)

http://arxiv.org/abs/1601.00500

### Observation of chiral superfluid order by matter wave interference

* T. Kock, M. Ölschläger, A. Ewerbeck, W.-M. Huang, L. Mathey, A. Hemmerich*

The breaking of time reversal symmetry via the spontaneous formation of chiral order is ubiquitous in nature. Here, we present an unambiguous demonstration of this phenomenon for atoms Bose-Einstein condensed in the second Bloch band of an optical lattice. As a key tool we use a matter wave interference technique, which lets us directly observe the phase properties of the superfluid order parameter and allows us to reconstruct the spatial geometry of certain low energy excitations, associated with the formation of domains of different chirality. Our work marks a new era of optical lattices where orbital degrees of freedom play an essential role for the formation of exotic quantum matter, similarly as in electronic systems.

Physical Review Letters 114, 115301 (2015)

http://arxiv.org/abs/1411.3483

### Controlling coherence via tuning of the population imbalance in a bipartite optical lattice

* M. Di Liberto, T. Comparin , T. Kock, M. Ölschäger, A. Hemmerich, C. Morais Smith*

The control of transport properties is a key tool at the basis of many technologically relevant effects in condensed matter. The clean and precisely controlled environment of ultracold atoms in optical lattices allows one to prepare simplified but instructive models, which can help to better understand the underlying physical mechanisms. We show that by tuning a structural deformation of the unit cell in a bipartite optical lattice, one can induce a phase transition from a superfluid into various Mott insulating phases forming a shell structure in the superimposed harmonic trap. The Mott shells are identified via characteristic features in the visibility of Bragg maxima in momentum spectra. The experimental findings are explained by Gutzwiller mean-field and quantum Monte Carlo calculations. Our system bears similarities with the loss of coherence in cuprate superconductors, known to be associated with the doping induced buckling of the oxygen octahedra surrounding the copper sites.

### Proposed formation and dynamical signature of a chiral Bose liquid in an optical lattice

* X. Li, A. Paramekanti, A. Hemmerich, W. Vincent Liu*

Recent experiments on p-orbital atomic bosons have suggested the emergence of a spectacular ultracold superfluid with staggered orbital currents in optical lattices. This raises fundamental questions concerning the effects of thermal fluctuations as well as possible ways of directly observing such chiral order. Here we show via Monte Carlo simulations that thermal fluctuations destroy this superfluid in an unexpected two-step process, unveiling an intermediate normal phase with spontaneously broken time-reversal symmetry, dubbed a ‘chiral Bose liquid’. For integer fillings (n≥2) in the chiral Mott regime, thermal fluctuations are captured by an effective orbital Ising model, and Onsager’s powerful exact solution is adopted to determine the transition from this intermediate liquid to the para- orbital normal phase at high temperature. A lattice quench is designed to convert the staggered angular momentum, previously thought by experts difficult to directly probe, into coherent orbital oscillations, providing a time-resolved dynamical signature of chiral order.

Nature Communications 5, 3205 (2014)

http://arxiv.org/abs/1309.0523

### Interaction-induced chiral px ± i py superfluid order of bosons in an optical lattice

*M. Ölschläger, T. Kock, G. Wirth, A. Ewerbeck, C. Morais Smith, A. Hemmerich*

The study of superconductivity with unconventional order is complicated in condensed matter systems by their immense complexity. Optical lattices with their exceptional precision and control allow one to emulate superfluidity avoiding many of the complications of condensed matter. A promising approach to realize unconventional superfluid order is to employ orbital degrees of freedom in higher Bloch bands. In recent work, indications were found that bosons condensed in the second band of an optical chequerboard lattice might exhibit px ± i py order. Here we present experiments, which provide strong evidence for the emergence of px ± i py order driven by the interaction in the local p-orbitals. We compare our observations with a multi-band Hubbard model and find excellent quantitative agreement.

New Journal of Physics 15, 083041 (2013)

http://arxiv.org/abs/1305.1177

### Topologically induced avoided band crossing in an optical chequerboard lattice

* M. Ölschläger, G. Wirth, T. Kock, A. Hemmerich*

We report on the condensation of bosons in the 4th band of an optical chequerboard lattice providing a topologically induced avoided band crossing involving the second, third, and fourth bands. When the condensate is slowly tuned through the avoided crossing, accelerated band relaxation arises and the zero momentum approximately C4-invariant condensate wave function acquires finite momentum order and reduced C2 symmetry. For faster tuning Landau-Zener oscillations between different superfluid orders arise, which are used to characterize the avoided crossing.

Physical Review Letters, 108, 075302 (2012)

http://lanl.arxiv.org/abs/1110.3716

### Topological semimetal in a fermionic optical lattice

* K. Sun, W. V. Liu, A. Hemmerich, S. Das Sarma*

Optical lattices have an important role in advancing our understanding of correlated quantum matter. The recent implementation of orbital degrees of freedom in chequerboard and hexagonal3 optical lattices opens up a new avenue towards discovering novel quantum states of matter that have no prior analogues in solid-state electronic materials. Here, we predict that an exotic topological semimetal emerges as a parity-protected gapless state in the orbital bands of a two-dimensional fermionic optical lattice. This new quantum state is characterized by a parabolic band-degeneracy point with Berry flux 2 Pi, in sharp contrast to the Pi-flux of Dirac points as in graphene.We showthat the appearance of this topological liquid is universal for all latticeswith D4 point-group symmetry, as long as orbitals with opposite parities hybridize strongly with each other and the band degeneracy is protected by odd parity. Turning on inter-particle repulsive interactions, the system undergoes a phase transition to a topological insulator whose experimental signature includes chiral gapless domain-wall modes, reminiscent of quantumHall edge states.

## Project C3

### Interorbital Interactions in an SU(2)xSU(6)-Symmetric Fermi-Fermi Mixture

*Benjamin Abeln, Koen Sponselee, Marcel Diem, Nejira Pintul, Klaus Sengstock, Christoph Becker*

We characterize inter- and intraisotope interorbital interactions between atoms in the 1S0 ground state and the 3P0 metastable state in interacting Fermi-Fermi mixtures of 171Yb and 173Yb. We perform high-precision clock spectroscopy to measure interaction-induced energy shifts in a deep 3D optical lattice and determine the corresponding scattering lengths. We find the elastic interaction of the interisotope mixtures 173Yb_e-171Yb_g and 173Yb_g-171Yb_e to be weakly attractive and very similar, while the corresponding two-body loss coefficients differ by more than two orders of magnitude. By comparing different spin mixtures we experimentally demonstrate the SU(2)xSU(6) symmetry of all elastic and inelastic interactions. Furthermore, we measure the spin-exchange interaction in 171Yb and confirm its previously observed antiferromagnetic nature.* *

### Identifying quantum phase transitions using artificial neural networks on experimental data

*B. S. Rem, N. Käming, M. Tarnowski, L. Asteria, N. Fläschner, C. Becker, K. Sengstock, C. Weitenberg*

Machine-learning techniques such as artificial neural networks are currently revolutionizing many technological areas and have also proven successful in quantum physics applications1,2,3,4. Here, we employ an artificial neural network and deep-learning techniques to identify quantum phase transitions from single-shot experimental momentum-space density images of ultracold quantum gases and obtain results that were not feasible with conventional methods. We map out the complete two-dimensional topological phase diagram of the Haldane model5,6,7 and provide an improved characterization of the superfluid-to-Mott-insulator transition in an inhomogeneous Bose–Hubbard system8,9,10. Our work points the way to unravel complex phase diagrams of general experimental systems, where the Hamiltonian and the order parameters might not be known.

### Dynamics of Ultracold Quantum Gases in the Dissipative Fermi-Hubbard Model

* K. Sponselee, L. Freystatzky, B. Abeln, M. Diem, B. Hundt, A. Kochanke, T. Ponath, B. Santra, L. Mathey, K. Sengstock and C. Becker*

Abstract. We employ metastable ultracold 173-Yb atoms to study dynamics in the 1D dissipative Fermi-Hubbard model experimentally and theoretically, and observe a complete inhibition of two-body losses after initial fast transient dynamics. We attribute the suppression of particle loss to the dynamical generation of a highly entangled Dicke state. For several lattice depths and for two- and six-spin component mixtures we find very similar dynamics, showing that the creation of strongly correlated states is a robust and universal phenomenon. This offers interesting opportunities for precision measurements.

### Observation of Topological Bloch-State Defects and Their Merging Transition

*Matthias Tarnowski, Marlon Nuske, Nick Fläschner, Benno Rem, Dominik Vogel, Lukas Freystatzky, Klaus Sengstock, Ludwig Mathey, and Christof Weitenberg*

Topological defects in Bloch bands, such as Dirac points in graphene, and their resulting Berry phases play an important role for the electronic dynamics in solid state crystals. Such defects can arise in systems with a two-atomic basis due to the momentum-dependent coupling of the two sublattice states, which gives rise to a pseudospin texture. The topological defects appear as vortices in the azimuthal phase of this pseudospin texture. Here, we demonstrate a complete measurement of the azimuthal phase in a hexagonal optical lattice employing a versatile method based on time-of-flight imaging after off-resonant lattice modulation. Furthermore, we map out the merging transition of the two Dirac points induced by beam imbalance. Our work paves the way to accessing geometric properties in optical lattices also with spin-orbit coupling and interactions.

### Split-and-delay unit for FEL interferometry in the XUV spectral range

*S. Usenko, A. Przystawik, L.L. Lazzarino, M.A. Jakob, F. Jacobs, C. Becker, C. Haunhorst, D. Kip, and T. Laarmann*

In this work we present a reflective split-and-delay unit (SDU) developed for interferometric time-resolved experiments utilizing an (extreme ultraviolet) XUV pump–XUV probe scheme with focused free-electron laser beams. The developed SDU overcomes limitations for phase-resolved measurements inherent to conventional two-element split mirrors by a special design using two reflective lamellar gratings. The gratings produce a high-contrast interference signal controlled by the grating displacement in every diffraction order. The orders are separated in the focal plane of the focusing optics, which enables one to avoid phase averaging by spatially selective detection of a single interference state of the two light fields. Interferometry requires a precise relative phase control of the light fields, which presents a challenge at short wavelengths. In our setup the phase delay is determined by an in-vacuum white light interferometer (WLI) that monitors the surface profile of the SDU in real time and thus measures the delay for each laser shot. The precision of the WLI is 1 nm as determined by optical laser interferometry. In the presented experimental geometry it corresponds to a time delay accuracy of 3 as, which enables phase-resolved XUV pump–XUV probe experiments at free-electron laser (FEL) repetition rates up to 60 Hz.

### Relaxation dynamics of a closed high-spin Fermi system far from equilibrium

* U. Ebling, J. S. Krauser, N. Fläschner, K. Sengstock, C. Becker, M. Lewenstein, A. Eckardt*

A fundamental question in many-body physics is how closed quantum systems reach equilibrium. We address this question experimentally and theoretically in an ultracold high-spin Fermi gas where we find a complex interplay between internal and motional degrees of freedom. The fermions are initially prepared far from equilibrium with only a few spin states occupied. The subsequent dynamics leading to redistribution among all spin states is observed experimentally and simulated theoretically using a kinetic Boltzmann equation with full spin coherence. The latter is derived microscopically and provides good agreement with experimental data without any free parameters. We identify several collisional processes, which occur on different time scales. By varying density and magnetic field, we control the relaxation dynamics and are able to continuously tune the character of a subset of spin states from an open to a closed system.

Phys. Rev. X 4, 021011 (2014)

http://arxiv.org/abs/1312.6704

### Detecting quadrupole interactions in ultracold Fermi gases

*M. Lahrz, M. Lemeshko, K. Sengstock, C. Becker, L. Mathey*

### Creation of Quantum-Degenerate Gases of Ytterbium in a Compact 2D-/3D-MOT Setup

*S. Dörscher, A. Thobe, B. Hundt, A. Kochanke, R. Le Targat, P. Windpassinger, C. Becker, K. Sengstock*

*The following article has been accepted by Review of Scientific Instruments. After it is published, it will be found at http://rsi.aip.org*

We report on the first experimental setup based on a 2D-/3D-MOT scheme to create both Bose-Einstein condensates and degenerate Fermi gases of several ytterbium isotopes. Our setup does not require a Zeeman slower and offers the flexibility to simultaneously produce ultracold samples of other atomic species. Furthermore, the extraordinary optical access favors future experiments in optical lattices. A 2D-MOT on the strong 1S0-1P1 transition captures ytterbium directly from a dispenser of atoms and loads a 3D-MOT on the narrow 1S0-3P1 intercombination transition. Subsequently, atoms are transferred to a crossed optical dipole trap and cooled evaporatively to quantum degeneracy.

Review of Scientific Instruments 84, 043109

http://arxiv.org/abs/1303.1105

### Intrinsic Photoconductivity of Ultracold Fermions in Optical Lattices

* J. Heinze, J. S. Krauser, N. Fläschner, B. Hundt, S. Götze, A. Itin, L. Mathey, K. Sengstock, C. Becker*

We report on the first experimental observation of a persistent alternating photocurrent in an ultracold gas of fermionic atoms in an optical lattice. The dynamics is induced and sustained by an external harmonic confinement. We find a counterintuitively momentum-dependent oscillation frequency for excited particles and a fast decay of holes which we attribute to spatial trapping. Lifetime measurements reveal a significant enhancement of particle-hole recombination with increasing interactions.

Phys. Rev. Lett. 110, 085302 (2013)

http://arxiv.org/abs/1208.4020

## Project C4

### Observing the influence of reduced dimensionality on fermionic superfluids

*Lennart Sobirey, Hauke Biss, Niclas Luick, Markus Bohlen, Henning Moritz, and Thomas Lompe*

Understanding the origins of unconventional superconductivity has been a major focus of condensed matter physics for many decades. While many questions remain unanswered, experiments have found the highest critical temperatures in layered two-dimensional materials. However, to what extent the remarkable stability of these strongly correlated 2D superfluids is affected by their reduced dimensionality is still an open question. Here, we use dilute gases of ultracold fermionic atoms as a model system to directly observe the influence of dimensionality on the stability of strongly interacting fermionic superfluids. We find that the superfluid gap follows the same universal function of the interaction strength regardless of dimensionality, which suggests that there is no inherent difference in the stability of two- and three-dimensional fermionic superfluids. Finally, we compare our data to results from solid state systems and find a similar relation between the interaction strength and the gap for a wide range of two- and three-dimensional superconductors.

### Excitation Spectrum and Superfluid Gap of an Ultracold Fermi Gas

*Hauke Biss, Lennart Sobirey, Niclas Luick, Markus Bohlen, Jami J. Kinnunen, Georg M. Bruun, Thomas Lompe, and Henning Moritz*

Ultracold atomic gases are a powerful tool to experimentally study strongly correlated quantum many-body systems. In particular, ultracold Fermi gases with tunable interactions have allowed to realize the famous BEC-BCS crossover from a Bose-Einstein condensate (BEC) of molecules to a Bardeen-Cooper-Schrieffer (BCS) superfluid of weakly bound Cooper pairs. However, large parts of the excitation spectrum of fermionic superfluids in the BEC-BCS crossover are still unexplored. In this work, we use Bragg spectroscopy to measure the full momentum-resolved low-energy excitation spectrum of strongly interacting ultracold Fermi gases. This enables us to directly observe the smooth transformation from a bosonic to a fermionic superfluid that takes place in the BEC-BCS crossover. We also use our spectra to determine the evolution of the superfluid gap and find excellent agreement with previous experiments and self-consistent* T*-matrix calculations both in the BEC and crossover regime. However, toward the BCS regime a calculation that includes the effects of particle-hole correlations shows better agreement with our data.

### Observation of superfluidity in a strongly correlated two-dimensional Fermi gas

*Lennart Sobirey, Niclas Luick, Markus Bohlen, Hauke Biss, Henning Moritz and Thomas Lompe*

Understanding how strongly correlated two-dimensional (2D) systems can give rise to unconventional superconductivity with high critical temperatures is one of the major unsolved problems in condensed matter physics. Ultracold 2D Fermi gases have emerged as clean and controllable model systems to study the interplay of strong correlations and reduced dimensionality, but direct evidence of superfluidity in these systems has been missing. We demonstrate superfluidity in an ultracold 2D Fermi gas by moving a periodic potential through the system and observing no dissipation below a critical velocity vc. We measure vc as a function of interaction strength and find a maximum in the crossover regime between bosonic and fermionic superfluidity. Our measurements enable systematic studies of the influence of reduced dimensionality on fermionic superfluidity.

### Single-atom counting in a two-color magneto-optical trap

*Martin Schlederer, Alexandra Mozdzen, Thomas Lompe, and Henning Moritz*

Recording the fluorescence of a magneto-optical trap (MOT) is a standard tool for measuring atom numbers in experiments with ultracold atoms. When trapping few atoms in a small MOT, the emitted fluorescence increases with the atom number in discrete steps, which allows one to measure the atom number with single-particle resolution. Achieving such single-particle resolution requires stringent minimization of stray light from the MOT beams, which is very difficult to achieve in experimental setups that require in-vacuum components close to the atoms. Here, we present a modified scheme that addresses this issue: Instead of collecting the fluorescence on the MOT (D2) transition, we scatter light on an additional probing (D1) transition and collect this fluorescence with a high-resolution microscope while filtering out the intense MOT light. Using this scheme, we are able to reliably distinguish up to 17 40K atoms with classification fidelities of ∼98% for up to 5 atom numbers and fidelities of more than 85% for up to 17 atoms.

### An ideal Josephson junction in an ultracold two-dimensional Fermi gas

*Niclas Luick, Lennart Sobirey, Markus Bohlen, Vijay Pal, Ludwig Mathey, Thomas Lompe, Henning Moritz*

The role of reduced dimensionality in high-temperature superconductors is still under debate. Recently, ultracold atoms have emerged as an ideal model system to study such strongly correlated two-dimensional (2D) systems. Here, we report on the realization of a Josephson junction in an ultracold 2D Fermi gas. We measure the frequency of Josephson oscillations as a function of the phase difference across the junction and find excellent agreement with the sinusoidal current phase relation of an ideal Josephson junction. Furthermore, we determine the critical current of our junction in the crossover from tightly bound molecules to weakly bound Cooper pairs. Our measurements clearly demonstrate phase coherence and provide strong evidence for superfluidity in a strongly interacting 2D Fermi gas.

### Sound Propagation and Quantum-Limited Damping in a Two-Dimensional Fermi Gas

*Markus Bohlen, Lennart Sobirey, Niclas Luick, Hauke Biss, Tilman Enss, Thomas Lompe, and Henning Moritz*

Strongly interacting two-dimensional Fermi systems are one of the great remaining challenges in many-body physics due to the interplay of strong local correlations and enhanced long-range fluctuations. Here, we probe the thermodynamic and transport properties of a 2D Fermi gas across the BEC-BCS crossover by studying the propagation and damping of sound modes. We excite particle currents by imprinting a phase step onto homogeneous Fermi gases trapped in a box potential and extract the speed of sound from the frequency of the resulting density oscillations. We measure the speed of sound across the BEC-BCS crossover and compare the resulting dynamic measurement of the equation of state both to a static measurement based on recording density profiles and to quantum Monte Carlo calculations and find reasonable agreement between all three. We also measure the damping of the sound mode, which is determined by the shear and bulk viscosities as well as the thermal conductivity of the gas. We find that the damping is minimal in the strongly interacting regime and the diffusivity approaches the universal quantum bound * ℏ/m* of a perfect fluid.

### Detecting Friedel oscillations in ultracold Fermi gases

* Keno Riechers, Klaus Hueck, Niclas Luick, Thomas Lompe, Henning Moritz*

Investigating Friedel oscillations in ultracold gases would complement the studies performed on solid state samples with scanning-tunneling microscopes. In atomic quantum gases interactions and external potentials can be tuned freely and the inherently slower dynamics allow to access non-equilibrium dynamics following a potential or interaction quench. Here, we examine how Friedel oscillations can be observed in current ultracold gas experiments under realistic conditions. To this aim we numerically calculate the amplitude of the Friedel oscillations which a potential barrier provokes in a 1D Fermi gas and compare it to the expected atomic and photonic shot noise in a density measurement. We find that to detect Friedel oscillations the signal from several thousand one-dimensional systems has to be averaged. However, as up to 100 parallel one-dimensional systems can be prepared in a single run with present experiments, averaging over about 100 images is sufficient.

Eur. Phys. J. D 71, 232 (2017)

http://lanl.arxiv.org/abs/1704.06626

### Two-Dimensional Homogeneous Fermi Gases

* Klaus Hueck, Niclas Luick, Lennart Sobirey, Jonas Siegl, Thomas Lompe, Henning Moritz*

We report on the experimental realization of homogeneous two-dimensional (2D) Fermi gases trapped in a box potential. In contrast to harmonically trapped gases, these homogeneous 2D systems are ideally suited to probe local as well as non-local properties of strongly interacting many-body systems. As a first measurement, we use a local probe to extract the equation of state (EOS) of a non-interacting Fermi gas. We then perform matter wave focusing to extract its momentum distribution and directly observe Pauli blocking in a near unity occupation of momentum states. Finally, we measure the momentum distribution of strongly interacting homogeneous 2D gases in the crossover between attractively interacting fermions and deeply-bound bosonic molecules.

PRL 120, 060402 (2018).

http://lanl.arxiv.org/abs/1704.06315

### Calibrating High Intensity Absorption Imaging of Ultracold Atoms

* Klaus Hueck, Niclas Luick, Lennart Sobirey, Jonas Siegl, Thomas Lompe, Henning Moritz, Logan W. Clark, Cheng Chin*

Absorption imaging of ultracold atoms is the foundation for quantitative extraction of information from experiments with ultracold atoms. Due to the limited exposure time available in these systems, the signal-to-noise ratio is largest for high intensity absorption imaging where the intensity of the imaging light is on the order of the saturation intensity. In this case, the absolute value of the intensity of the imaging light enters as an additional parameter making it more sensitive to systematic errors. Here, we present a novel and robust technique to determine the imaging intensity in units of the effective saturation intensity to better than 5%. We do this

by measuring the momentum transferred to the atoms by the imaging light while varying its intensity. We further utilize the method to quantify the purity of the polarization of the imaging light and to determine the correct

imaging detuning.

Opt. Express 25, 8670-8679 (2017)

https://arxiv.org/abs/1702.01943

### Suppression of kHz-Frequency Switching Noise in Digital Micro-Mirror Devices

* Klaus Hueck, Anton Mazurenko, Niclas Luick, Thomas Lompe, Henning Moritz*

igh resolution digital micro-mirror devices (DMD) make it possible to produce nearly arbitrary light fields with high accuracy, reproducibility and low optical aberrations. However, using these devices to trap and manipulate ultracold atomic systems for e.g. quantum simulation is often complicated by the presence of kHz-frequency switching noise. Here we demonstrate a simple hardware extension that solves this problem and makes it possible to produce truly static light fields. This modification leads to a 47 fold increase in the time that we can hold ultracold 6Li atoms in a dipole potential created with the DMD. Finally, we provide reliable and user friendly APIs written in Matlab and Python to control the DMD.

Rev. Sci. Instrum. 88, 016103 (2017)

https://arxiv.org/abs/1611.03397

### Sudden and slow quenches into the antiferromagnetic phase of ultracold fermions

*M. Ojekhile, R. Höppner, H. Moritz, L. Mathey*

### Probing superfluidity of Bose-Einstein condensates via laser stirring

*Vijay Pal Singh, Wolf Weimer, Kai Morgener, Jonas Siegl, Klaus Hueck, Niclas Luick, Henning Moritz, Ludwig Mathey*

We investigate the superfluid behavior of a Bose-Einstein condensate of 6Li molecules. In the experiment by Weimer et al., Phys. Rev. Lett. 114, 095301 (2015) a condensate is stirred by a weak, red-detuned laser beam along a circular path around the trap center. The rate of induced heating increases steeply above a velocity vc, which we define as the critical velocity. Below this velocity, the moving beam creates almost no heating. In this paper, we demonstrate a quantitative understanding of the critical velocity. Using both numerical and analytical methods, we identify the non-zero temperature, the circular motion of the stirrer, and the density profile of the cloud as key factors influencing the magnitude of vc. A direct comparison to the experimental data shows excellent agreement.

Phys. Rev. A 93, 023634 (2016)

http://lanl.arxiv.org/abs/1509.02168

### The critical velocity in the BEC-BCS crossover

* W. Weimer, K. Morgener, V. P. Singh, J. Siegl, K. Hueck, N. Luick, L. Mathey, H. Moritz*

We map out the critical velocity in the crossover from Bose-Einstein condensation (BEC) to Bardeen-Cooper-Schrieffer superfluidity with ultracold 6Li gases. A small attractive potential is dragged along lines of constant column density. The rate of the induced heating increases steeply above a critical velocity vc. In the same samples, we measure the speed of sound vs by exciting density waves and compare the results to the measured values of vc. We perform numerical simulations in the BEC regime and find very good agreement, validating the approach. In the strongly correlated regime, where theoretical predictions only exist for the speed of sound, our measurements of vc provide a testing ground for theoretical approaches.

Phys. Rev. Lett. 114, 095301 (2015)

http://arxiv.org/abs/1408.5239

## Project C5

### Observation of a continuous time crystal

*Phatthamon Kongkhambut, Jim Skulte, Ludwig Mathey, Jayson G. Cosme, Andreas Hemmerich, and Hans Keßler*

Time crystals are classified as discrete or continuous depending on whether they spontaneously break discrete or continuous time translation symmetry. While discrete time crystals have been extensively studied in periodically driven systems, the experimental realization of a continuous time crystal is still pending. We report the observation of a limit cycle phase in a continuously pumped dissipative atom-cavity system, that is characterized by emergent oscillations in the intracavity photon number. The phase of the oscillation found to be random for different realizations, and hence this dynamical many-body state breaks continuous time translation symmetry spontaneously. Furthermore, the observed limit cycles are robust against temporal perturbations and therefore demonstrate the realization of a continuous time crystal.

### Parametrically driven dissipative three-level Dicke model

*Skulte J., Kongkhambut P., Keßler H., Hemmerich A., Mathey L., Cosme J. G.*

We investigate the three-level Dicke model, which describes a fundamental class of light-matter systems. We determine the phase diagram in the presence of dissipation, which we assume to derive from photon loss. Utilizing both analytical and numerical methods we characterize the incommensurate time crystalline, light-induced, and light-enhanced superradiant states in the phase diagram for the parametrically driven system. As a primary application, we demonstrate that a shaken atom-cavity system is naturally approximated via a parametrically driven dissipative three-level Dicke model.

### Realization of a Periodically Driven Open Three-Level Dicke Model

*Kongkhambut P., Keßler H., Skulte J., Mathey L., Cosme J., Hemmerich A.*

A periodically driven open three-level Dicke model is realized by resonantly shaking the pump field in an atom-cavity system. As an unambiguous signature, we demonstrate the emergence of a dynamical phase, in which the atoms periodically localize between the antinodes of the pump lattice, associated with an oscillating net momentum along the pump axis. We observe this dynamical phase through the periodic switching of the relative phase between the pump and cavity fields at a small fraction of the driving frequency, suggesting that it exhibits a time crystalline character.

### Dynamical density wave order in an atom–cavity system

*Georges C., Cosme J. G., Keßler H., Mathey L., Hemmerich A.*

We theoretically and experimentally explore the emergence of a dynamical density wave (DW) order in a driven dissipative atom–cavity system. A Bose–Einstein condensate is placed inside a high finesse optical resonator and pumped sideways by an optical standing wave. The pump strength is chosen to induce a stationary superradiant checkerboard DW order of the atoms stabilized by a strong intracavity light field. We show theoretically that, when the pump is modulated with sufficient strength at a frequency ω_{d} close to a systemic resonance frequency ω_{>}, a dynamical DW order emerges, which oscillates at the two frequencies ω_{>} and ω_{<} = ω_{d} − ω_{>}. This order is associated with a characteristic momentum spectrum, also found in experiments in addition to remnants of the oscillatory dynamics presumably damped by on-site interaction and heating, not included in the calculations. The oscillating density grating, associated with this order, suppresses pump-induced light scattering into the cavity. Similar mechanisms might be conceivable in light-driven electronic matter.

### Observation of a Dissipative Time Crystal

*Keßler H., Kongkhambut P., Georges C., Mathey L., Cosme J., Hemmerich A.*

We present the first experimental realization of a time crystal stabilized by dissipation. The central signature in our implementation in a driven open atom-cavity system is a period doubled switching between distinct checkerboard density wave patterns, induced by the interplay between controlled cavity dissipation, cavity-mediated interactions, and external driving. We demonstrate the robustness of this dynamical phase against system parameter changes and temporal perturbations of the driving.

### Mott transition in a cavity-boson system: A quantitative comparison between theory and experiment

*Lin R., Georges C., Klinder J. S., Molignini P., Büttner M., Lode A. U. J., Chitra R., Hemmerich A., Keßler H.*

The competition between short-range and cavity-mediated infinite-range interactions in a cavity-boson system leads to the existence of a superfluid phase and a Mott-insulator phase within the self-organized regime. In this work, we quantitatively compare the steady-state phase boundaries of this transition measured in experiments and simulated using the Multiconfigurational Time-Dependent Hartree Method for Indistinguishable Particles. To make the problem computationally feasible, we represent the full system by the exact many-body wave function of a two-dimensional four-well potential. We argue that the validity of this representation comes from the nature of both the cavity-atomic system and the Bose-Hubbard physics. Additionally we show that the chosen representation only induces small systematic errors, and that the experimentally measured and theoretically predicted phase boundaries agree reasonably. We thus demonstrate a new approach for the quantitative numerical determination of the superfluid--Mott-insulator phase boundary.

### From a continuous to a discrete time crystal in a dissipative atom-cavity system

*Keßler H., Cosme J. G., Georges C., Mathey L., Hemmerich A.*

We propose the dynamical stabilization of a nonequilibrium order in a driven dissipative system comprised an atomic Bose–Einstein condensate inside a high finesse optical cavity, pumped with an optical standing wave operating in the regime of anomalous dispersion. When the amplitude of the pump field is modulated close to twice the characteristic limit-cycle frequency of the unmodulated system, a stable subharmonic response is found. The dynamical phase diagram shows that this subharmonic response occurs in a region expanded with respect to that where stable limit-cycle dynamics occurs for the unmodulated system. In turning on the modulation we tune the atom-cavity system from a continuous to a discrete time crystal.

### Emergent limit cycles and time crystal dynamics in an atom-cavity system

*Hans Keßler, Jayson G. Cosme, Michal Hemmerling, Ludwig Mathey, and Andreas Hemmerich*

We propose an experimental realization of a time crystal using an atomic Bose-Einstein condensate in a high finesse optical cavity pumped with laser light detuned to the blue side of the relevant atomic resonance. By mapping out the dynamical phase diagram, we identify regions in parameter space showing stable limit cycle dynamics. Since the model describing the system is time independent, the emergence of a limit cycle phase indicates the breaking of continuous time translation symmetry. Employing a semiclassical analysis to demonstrate the robustness of the limit cycles against perturbations and quantum fluctuations, we establish the emergence of a time crystal.

### Nonequilibrium quantum phase transition in a hybrid atom-optomechanical system

*N. Mann, M. Reza Bakhtiari, A. Pelster, M. Thorwart*

We consider a hybrid quantum many-body system formed by a vibrational mode of a nanomembrane, which interacts optomechanically with light in a cavity, and an ultracold atom gas in the optical lattice of the out-coupled light. The adiabatic elimination of the light field yields an effective Hamiltonian which reveals a competition between the force localizing the atoms and the membrane displacement. At a critical atom-membrane interaction, we find a nonequilibrium quantum phase transition from a localized symmetric state of the atom cloud to a shifted symmetry-broken state, the energy of the lowest collective excitation vanishes, and a strong atom-membrane entanglement arises. The effect occurs when the atoms and the membrane are nonresonantly coupled.

### Dynamical Control of Order in a Cavity-BEC System

*Jayson G. Cosme, Christoph Georges, Andreas Hemmerich, and Ludwig Mathey*

We demonstrate dynamical control of the superradiant transition of cavity-BEC system via periodic driving of the pump laser. We show that the dominant density wave order of the superradiant state can be suppressed, and that the subdominant competing order of Bose-Einstein condensation emerges in the steady state. Furthermore, we show that additional, nonequilibrium density wave orders, which do not exist in equilibrium, can be stabilized dynamically. Finally, for strong driving, chaotic dynamics emerge.

### Light-induced coherence in an atom-cavity system

*C. Georges, J. G. Cosme, L. Mathey, A. Hemmerich*

We demonstrate a light-induced formation of coherence in a cold atomic gas system that utilizes the suppression of a competing density wave (DW) order. The condensed atoms are placed in an optical cavity and pumped by an external optical standing wave, which induces a long-range interaction mediated by photon scattering and a resulting DW order above a critical pump strength. We show that the light-induced temporal modulation of the pump wave can suppress this DW order and restore coherence. This establishes a foundational principle of dynamical control of competing orders analogous to a hypothesized mechanism for light-induced superconductivity in high*-T*_{c} cuprates.

### Bloch oscillations of a Bose-Einstein condensate in a cavity-induced optical lattice

*Ch. Georges, J. Vargas, H. Keßler, J. Klinder, A. Hemmerich*

This article complements previous work on the nondestructive observation of Bloch oscillations of a Bose-Einstein condensate in an optical lattice formed inside a high-finesse optical cavity [H. Keßler et al., New J. Phys. 18, 102001 (2016)]. We present measurements showing that the observed Bloch frequency is independent of the atom number and hence the cooperative coupling strength, the intracavity lattice depth, and the detuning between the external pump light and the effective cavity resonance. We find that in agreement with theoretical predictions, despite the atom-cavity dynamics, the value of the Bloch frequency agrees with that expected in conventional optical lattices, where it solely depends on the sizes of the force and the lattice constant. We also show that Bloch oscillations are observed in a self-organized two-dimensional lattice, which is formed if, instead of axially pumping the cavity through one of its mirrors, the Bose-Einstein condensate is irradiated by an optical standing wave oriented perpendicularly with respect to the cavity axis. For this case, however, excessive decoherence prevents a meaningful quantitative assessment.

### Driven Bose-Hubbard Model with a Parametrically Modulated Harmonic Trap

*N. Mann, M. Reza Bakhtiari, F. Massel, A. Pelster, M. Thorwart*

We investigate a one-dimensional Bose–Hubbard model in a parametrically driven global harmonic trap. The delicate interplay of both the local interaction of the atoms in the lattice and the driving of the global trap allows us to control the dynamical stability of the trapped quantum many-body state. The impact of the atomic interaction on the dynamical stability of the driven quantum many-body state is revealed in the regime of weak interaction by analyzing a discretized Gross–Pitaevskii equation within a Gaussian variational ansatz, yielding a Mathieu equation for the condensate width. The parametric resonance condition is shown to be modified by the atom interaction strength. In particular, the effective eigenfrequency is reduced for growing interaction in the mean-field regime. For a stronger interaction, the impact of the global parametric drive is determined by the numerically exact time-evolving block decimation scheme. When the trapped bosons in the lattice are in a Mott insulating state, the absorption of energy from the driving field is suppressed due to the strongly reduced local compressibility of the quantum many-body state. In particular, we find that the width of the local Mott region shows a breathing dynamics. Finally, we observe that the global modulation also induces an effective time-independent inhomogeneous hopping strength for the atoms.

### Topological Varma superfluid in optical lattices

* M. Di Liberto, A. Hemmerich, and C. Morais Smith*

Topological states of matter are peculiar quantum phases showing different edge and bulk transport properties connected by the bulk-boundary correspondence. While non-interacting fermionic topological insulators are well established by now and have been classified according to a ten-fold scheme, the possible realisation of topological states for bosons has not been much explored yet. Furthermore, the role of interactions is far from being understood. Here, we show that a topological state of matter exclusively driven by interactions may occur in the p-band of a Lieb optical lattice filled with ultracold bosons. The single-particle spectrum of the system displays a remarkable parabolic band-touching point, with both bands exhibiting non-negative curvature. Although the system is neither topological at the single-particle level, nor for the interacting ground state, on-site interactions induce an anomalous Hall effect for the excitations, carrying a non-zero Chern number. Our work introduces an experimentally realistic strategy for the formation of interaction-driven topological states of bosons.

Physical Review Letters 117, 163001 (2016)

https://arxiv.org/abs/1604.06055

### In-situ observation of optomechanical Bloch oscillations in an optical cavity

*H. Keßler, J. Klinder, B. Prasanna Venkatesh, Ch. Georges, A. Hemmerich*

It is shown experimentally that a Bose-Einstein condensate inside an optical cavity, operating in the regime of strong cooperative coupling, responds to an external force by an optomechanical Bloch oscillation, which can be directly observed in the light leaking out of the cavity. Previous theoretical work predicts that the frequency of this oscillation matches with that of conventional Bloch oscillations such that its in-situ monitoring may help to increase the data acquisition speed in precision force measurements.

New Journal of Physics 18, 102001 (2016)

https://arxiv.org/abs/1606.08386

### Dynamical phase transition in the open Dicke model

*J. Klinder, H. Keßler, M. Wolke, L. Mathey, A. Hemmerich*

The Dicke model with a weak dissipation channel is realized by coupling a Bose–Einstein condensate to an optical cavity with ultranarrow bandwidth. We explore the dynamical critical properties of the Hepp–Lieb–Dicke phase transition by performing quenches across the phase boundary. We observe hysteresis in the transition between a homogeneous phase and a self-organized collective phase with an enclosed loop area showing power-law scaling with respect to the quench time, which suggests an interpretation within a general framework introduced by Kibble and Zurek. The observed hysteretic dynamics is well reproduced by numerically solving the mean-field equation derived from a generalized Dicke Hamiltonian. Our work promotes the understanding of nonequilibrium physics in open many-body systems with infinite range interactions.

### Observation of a superradiant Mott insulator in the Dicke-Hubbard model

* J. Klinder, H. Keßler, M. Reza Bakhtiari, M. Thorwart, and A. Hemmerich*

It is well known that the bosonic Hubbard model possesses a Mott insulator phase. Likewise, it is known that the Dicke model exhibits a self-organized superradiant phase. By implementing an optical lattice inside of a high finesse optical cavity both models are merged such that an extended Hubbard model with cavity-mediated infinite range interactions arises. In addition to a normal superfluid phase, two superradiant phases are found, one of them coherent and hence superfluid and one incoherent Mott insulating.

Physical Review Letters 115, 230403 (2015)

http://arxiv.org/abs/1511.00850

### Nonequilibrium phase transition of interacting bosons in an intra-cavity optical lattice

* M. R. Bakhtiari, A. Hemmerich, H. Ritsch, M. Thorwart*

We investigate the nonlinear light-matter interaction of a Bose-Einstein condensate trapped in an external periodic potential inside an optical cavity, which is weakly coupled to the vacuum radiation modes and driven by a transverse pump field. Based on a generalized Bose-Hubbard model, which incorporates a single cavity mode, we include the collective back action of the atoms on the cavity light field and determine the nonequilibrium quantum phases within the non-perturbative bosonic dynamical mean-field theory. With the system parameters adapted to recent experiments, we find a quantum phase transition from a normal phase to a self-organized superfluid phase, which is related to the Hepp-Lieb-Dicke phase transition. For even stronger pumping, a self-organized Mott insulator phase arises.

Physical Review Letters 114, 123601 (2015)

http://arxiv.org/abs/1410.5735

### Steering matter wave superradiance with an ultra-narrowband optical cavity

* H. Keßler, J. Klinder, M. Wolke, A. Hemmerich*

A superfluid atomic gas is prepared inside an optical resonator with an ultra-narrow band width on the order of the single photon recoil energy. When a monochromatic off-resonant laser beam irradiates the atoms, above a critical intensity the cavity emits superradiant light pulses with a duration on the order of its photon storage time. The atoms are collectively scattered into coherent superpositions of discrete momentum states, which can be precisely controlled by adjusting the cavity resonance frequency. With appropriate pulse sequences the entire atomic sample can be collectively accelerated or decelerated by multiples of two recoil momenta. The instability boundary for the onset of matter wave superradiance is recorded and its main features are explained by a mean field model.

Physical Review Letters 113, 070404 (2014)

http://arxiv.org/abs/1407.4954

### Optomechanical atom-cavity interaction in the sub-recoil regime

* H. Keßler, J. Klinder, M. Wolke, A. Hemmerich*

We study the optomechanical interaction of a Bose-Einstein condensate with a single longitudinal mode of an ultra-high finesse standing wave optical resonator. As a unique feature the resonator combines three extreme regimes, previously not realized together, i.e., strong cooperative coupling, cavity dominated scattering with a Purcell factor far above unity, and sub-recoil resolution provided by a cavity damping rate smaller than four times the single photon recoil frequency. We present experimental observations in good agreement with a two-mode model predicting highly non-linear dynamics with signatures as bistability, hysteresis, persistent oscillations, and superradiant back-scattering instabilities.

New Journal of Physics 16, 053008 (2014)

http://arxiv.org/abs/1403.3545

### Cavity cooling below the recoil limit

* M. Wolke, J. Klinner, H. Keßler, A. Hemmerich*

Conventional laser cooling relies on repeated electronic excitations by near-resonant light, which constrains its area of application to a selected number of atomic species prepared at moderate particle densities. Optical cavities with sufficiently large Purcell factors allow for laser cooling schemes avoiding these limitations. Here, we report on an atom-cavity system, combining a Purcell factor above 40 with a cavity bandwidth below the recoil frequency associated with the kinetic energy transfer in a single photon scattering event. This lets us access a yet unexplored regime of atom-cavity interactions, in which the atomic motion can be manipulated by targeted dissipation with sub-recoil resolution. We demonstrate cavity-induced heating of a Bose-Einstein condensate and subsequent cooling at particle densities and temperatures incompatible with conventional laser cooling.

## Project C6

### Symmetry effects on the spin switching of adatoms

* C. Hübner, B. Baxevanis, A. A. Khajetoorians, D. Pfannkuche*

Highly symmetric magnetic environments have been suggested to stabilize the magnetic information stored in magnetic adatoms on a surface. Utilized as memory devices such systems are subjected to electron tunneling and external magnetic fields. We analyze theoretically how such perturbations affect the switching probability of a single quantum spin for two characteristic symmetries encountered in recent experiments and suggest a third one that exhibits robust protection against surface-induced spin flips. Further we illuminate how the switching of an adatom spin exhibits characteristic behavior with respect to low energy excitations from which the symmetry of the system can be inferred.

### Isospin correlations in two-partite hexagonal optical lattices

* M. Prada, E.-M. Richter, D. Pfannkuche*

Two-component mixtures in optical lattices reveal a rich variety of different phases. We employ an exact diagonalization method to obtain the relevant correlation functions in hexagonal optical lattices which characterize those phases. We relate the occupation difference of the two species to the magnetic polarization. “Iso” -magnetic correlations disclose the nature of the system, which can be of easy-axis type, bearing phase segregation, or of easy-plane type, corresponding to super-counter-fluidity. In the latter case, the correlations reveal easy-plane segregation, involving a highly entangled state. We identify striking correlated supersolid phases appearing within the superfluid limit.

### Doublon Relaxation in the Bose-Hubbard Model

*A. L. Chudnovskiy, D. M. Gangardt, A. Kamenev*

## Project C7

### Radiofrequency spectroscopy of one-dimensional trapped Bose polarons: crossover from the adiabatic to the diabatic regime

*S.I. Mistakidis, G.M. Koutentakis, F. Grusdt, H.R. Sadeghpour and P. Schmelcher*

We investigate the crossover of the impurity-induced dynamics, in trapped one-dimensional Bose polarons subject to radio frequency (RF) pulses of varying intensity, from an adiabatic to a diabatic regime. Utilizing adiabatic pulses for either weak repulsive or attractive impurity-medium interactions, a multitude of polaronic excitations or mode-couplings of the impurity-bath interaction with the collective breathing motion of the bosonic medium are spectrally resolved. We find that for strongly repulsive impurity-bath interactions, a temporal orthogonality catastrophe manifests in resonances in the excitation spectra where impurity coherence vanishes. When two impurities are introduced, impurity–impurity correlations, for either attractive or strong repulsive couplings, induce a spectral shift of the resonances with respect to the single impurity. For a heavy impurity, the polaronic peak is accompanied by a series of equidistant side-band resonances, related to interference of the impurity spin dynamics and the sound waves of the bath. In all cases, we enter the diabatic transfer regime for an increasing bare Rabi frequency of the RF field with a Lorentzian spectral shape featuring a single polaronic resonance. The findings in this work on the effects of external trap, RF pulse and impurity–impurity interaction should have implications for the new generations of cold-atom experiments.

### Many-Body Collisional Dynamics of Impurities Injected into a Double-Well Trapped Bose-Einstein Condensate

*F. Theel, K. Keiler, S.I. Mistakidis and P. Schmelcher*

We unravel the many-body dynamics of a harmonically trapped impurity colliding with a bosonic medium confined in a double well upon quenching the initially displaced harmonic trap to the center of the double well. We reveal that the emerging correlation dynamics crucially depends on the impurity-medium interaction strength allowing for a classification into different dynamical response regimes. For strong attractive impurity-medium couplings the impurity is bound to the bosonic bath, while for intermediate attractions it undergoes an effective tunneling. In the case of weak attractive or repulsive couplings the impurity penetrates the bosonic bath and performs a dissipative oscillatory motion. Further increasing the impurity-bath repulsion results in the pinning of the impurity between the density peaks of the bosonic medium, a phenomenon that is associated with a strong impurity-medium entanglement. For strong repulsions, the impurity is totally reflected by the bosonic medium. To unravel the underlying microscopic excitation processes accompanying the dynamics, we employ an effective potential picture. We extend our results to the case of two bosonic impurities and demonstrate the existence of a qualitatively similar impurity dynamics.

### Bosonic Quantum Dynamics Following Two Colliding Potential Wells

*F. Köhler and P. Schmelcher*

We employ the multiconfiguration time-dependent Hartree method for bosons in order to investigate the correlated nonequilibrium quantum dynamics of two bosons confined in two colliding and uniformly accelerated Gaussian wells. As the wells approach each other an effective, transient double-well structure is formed. This induces a transient and oscillatory over-barrier transport. We monitor both the amplitude of the intrawell dipole mode in the course of the dynamics as well as the final distribution of the particles between the two wells. For fast collisions we observe an emission process which we attribute to two distinct mechanisms. Energy transfer processes lead to an untrapped fraction of bosons and a resonant enhancement of the deconfinement for certain kinematic configurations can be observed. Despite the comparatively weak interaction strengths employed in this work, we identify strong interparticle correlations by analyzing the corresponding von Neumann entropy.

### Phase diagram, stability and magnetic properties of nonlinear excitations in spinor Bose–Einstein condensates

*G.C. Katsimiga, S.I. Mistakidis, P. Schmelcher and P.G. Kevrekidis*

We present the phase diagram, the underlying stability and magnetic properties as well as the dynamics of nonlinear solitary wave excitations arising in the distinct phases of a harmonically confined spinor *F* = 1 Bose–Einstein condensate. Particularly, it is found that nonlinear excitations in the form of dark–dark–bright solitons exist in the antiferromagnetic and in the easy-axis phase of a spinor gas, being generally unstable in the former while possessing stability intervals in the latter phase. Dark–bright–bright solitons can be realized in the polar and the easy-plane phases as unstable and stable configurations respectively; the latter phase can also feature stable dark–dark–dark solitons. Importantly, the persistence of these types of states upon transitioning, by means of tuning the quadratic Zeeman coefficient from one phase to the other is unravelled. Additionally, the spin-mixing dynamics of stable and unstable matter waves is analyzed, revealing among others the coherent evolution of magnetic dark–bright, nematic dark–bright–bright and dark–dark–dark solitons. Moreover, for the unstable cases unmagnetized or magnetic droplet-like configurations and spin-waves consisting of regular and magnetic solitons are seen to dynamically emerge remaining thereafter robust while propagating for extremely large evolution times. Interestingly, exposing spinorial solitons to finite temperatures, their anti-damping in trap oscillation is showcased. It is found that the latter is suppressed for stronger bright soliton component 'fillings'. Our investigations pave the wave for a systematic production and analysis involving spin transfer processes of such waveforms which have been recently realized in ultracold experiments.

### Pattern formation of correlated impurities subjected to an impurity-medium interaction pulse

G. Bougas, S.I. Mistakidis and P. Schmelcher

We study the correlated dynamics of few interacting bosonic impurities immersed in a one-dimensional harmonically trapped bosonic environment. The mixture is exposed to a time-dependent impurity-medium interaction pulse moving it across the relevant phase-separation boundary. For modulation frequencies smaller than the trapping one, the system successively transits through the miscible and immiscible phases according to the driving of the impurity-medium interactions. For strong modulations, and driving from the miscible to the immiscible regime, a significant fraction of the impurities is expelled to the edges of the bath. They exhibit a strong localization behavior and tend to equilibrate. Following the reverse driving protocol, the impurities perform a breathing motion while featuring a two-body clustering and the bath is split into two incoherent parts. Interestingly, in both driving scenarios, dark-bright solitons are nucleated in the absence of correlations. A localization of the impurities around the trap center for weak impurity-impurity repulsions is revealed, which subsequently disperse into the bath for increasing interactions.

### Asymptotic population imbalance of an ultracold bosonic ensemble in a driven double well

*Jie Chen, Aritra K. Mukhopadhyay, and Peter Schmelcher*

We demonstrate that an ultracold many-body bosonic ensemble confined in a one-dimensional double-well potential exhibits a population imbalance between the two wells at large timescales, when the depth of the wells is modulated by a time-dependent driving force. The specific form of the driving force is shown to break spatial parity and time-reversal symmetries, which leads to such an asymptotic population imbalance (API). The value of the API can be flexibly controlled by changing the phase of the driving force and the total number of particles. While the API is highly sensitive to the initial state in the few-particle regime, this dependence on the initial state is lost as we approach the classical limit of large particle numbers. We perform a Floquet analysis in the few-particle regime and an analysis based on a driven classical nonrigid pendulum in the many-particle regime. Although the obtained API values in the many-particle regime agree very well with those obtained in the classical limit, we show that there exists a significant disagreement in the corresponding real-time population imbalance due to quantum correlations.

### Induced Correlations Between Impurities in a One-Dimensional Quenched Bose Gas

*S.I. Mistakidis, A.G. Volosniev and P. Schmelcher*

We explore the time evolution of two impurities in a trapped one-dimensional Bose gas that follows a change of the boson-impurity interaction. We study the induced impurity-impurity interactions and their effect on the quench dynamics. In particular, we report on the size of the impurity cloud, the impurity-impurity entanglement, and the impurity-impurity correlation function. The presented numerical simulations are based upon the variational multilayer multiconfiguration time-dependent Hartree method for bosons. To analyze and quantify induced impurity-impurity correlations, we employ an effective two-body Hamiltonian with a contact interaction. We show that the effective model consistent with the mean-field attraction of two heavy impurities explains qualitatively our results for weak interactions. Our findings suggest that the quench dynamics in cold-atom systems can be a tool for studying impurity-impurity correlations.

### Many-Body Quantum Dynamics and Induced Correlations of Bose Polarons

*S.I. Mistakidis, G.M. Koutentakis, G.C. Katsimiga, Th. Busch and P. Schmelcher*

We study the ground state properties and non-equilibrium dynamics of two spinor bosonic impurities immersed in a one-dimensional bosonic gas upon applying an interspecies interaction quench. For the ground state of two non-interacting impurities we reveal signatures of attractive induced interactions in both cases of attractive or repulsive interspecies interactions, while a weak impurity–impurity repulsion forces the impurities to stay apart. Turning to the quench dynamics we inspect the time-evolution of the contrast unveiling the existence, dynamical deformation and the orthogonality catastrophe of Bose polarons. We find that for an increasing postquench repulsion the impurities reside in a superposition of two distinct two-body configurations while at strong repulsions their corresponding two-body correlation patterns show a spatially delocalized behavior evincing the involvement of higher excited states. For attractive interspecies couplings, the impurities exhibit a tendency to localize at the origin and remarkably for strong attractions they experience a mutual attraction on the two-body level that is imprinted as a density hump on the bosonic bath.

### Entanglement-assisted tunneling dynamics of impurities in a double well immersed in a bath of lattice trapped bosons

*F. Theel, K. Keiler, S.I. Mistakidis and P. Schmelcher*

We unravel the correlated tunneling dynamics of an impurity trapped in a double well and interacting repulsively with a majority species of lattice trapped bosons. Upon quenching the tilt of the double well it is found that the quench-induced tunneling dynamics depends crucially on the interspecies interaction strength and the presence of entanglement inherent in the system. In particular, for weak couplings the impurity performs a rather irregular tunneling process in the double well. Increasing the interspecies coupling it is possible to control the response of the impurity which undergoes a delayed tunneling while the majority species effectively acts as a material barrier. For very strong interspecies interaction strengths the impurity exhibits a self-trapping behavior. We showcase that a similar tunneling dynamics takes place for two weakly interacting impurities and identify its underlying transport mechanisms in terms of pair and single-particle tunneling processes.

### Pulse- and continuously driven many-body quantum dynamics of bosonic impurities in a Bose-Einstein condensate

*K. Mukherjee, S.I. Mistakidis, S. Majumder and P. Schmelcher*

We unravel the periodically driven dynamics of two repulsively interacting bosonic impurities within a bosonic bath upon considering either the impact of a finite pulse or continuous shaking of the impurities harmonic trap. Following a pulse driving of initially miscible components, we reveal a variety of dynamical response regimes depending on the driving frequency. At resonant drivings, the impurities decouple from their host, while if exposed to a high-frequency driving, they remain trapped in the bosonic gas. For continuous shaking, we showcase that in the resonantly driven regime the impurities oscillate back and forth within and outside the bosonic medium. In all cases, the bosonic bath is perturbed performing a collective dipole motion. Referring to an immiscible initial state, we unveil that for moderate driving frequencies the impurities feature a dispersive behavior while for a high-frequency driving they oscillate around the edges of the Thomas-Fermi background. Energy transfer processes from the impurities to their environment are encountered, especially for large driving frequencies. Additionally, coherence losses develop in the course of the evolution with the impurities predominantly moving as a pair.

### Interaction-induced single-impurity tunneling in a binary mixture of trapped ultracold bosons

*Kevin Keiler and Peter Schmelcher*

We investigate the tunneling dynamics of an ultracold bosonic impurity species which interacts repulsively with a second, larger Bose gas. Both species are held in a finite-sized quasi-one-dimensional box potential. In addition, the impurity bosons experience a periodic potential generated by an optical lattice. We initially prepare our binary mixture in its ground state, such that the impurities and Bose gas are phase separated and the impurities localize pairwise in adjacent sites of the periodic potential, by tuning the interaction strengths and the lattice depth correspondingly. The dynamics is initiated by suddenly lowering the repulsive interspecies interaction strength, thereby entering a different regime in the crossover diagram. For specific postquench interspecies interaction strengths we find that a single impurity tunnels first to the neighboring empty site and depending on the quench strength can further tunnel to the next-neighboring site. Interestingly, this effect is highly sensitive to the presence of the Bose gas and does not occur when the Bose gas does not interact with the impurity species throughout the dynamics. Moreover, we find that the tunneling process is accompanied by strong entanglement between the Bose gas and the impurity species as well as correlations among the impurities.

### Analytical treatment of the interaction quench dynamics of two bosons in a two-dimensional harmonic trap

*G. Bougas, S.I. Mistakidis, and P. Schmelcher*

We investigate the quantum dynamics of two bosons, trapped in a two-dimensional harmonic trap, upon quenching arbitrarily their interaction strength and thereby covering the entire energy spectrum. Utilizing the exact analytical solution of the stationary system, we derive a closed analytical form of the expansion coefficients of the time-evolved two-body wave function, whose dynamics is determined by an expansion over the postquench eigenstates. The emergent dynamical response of the system is analyzed in detail by inspecting several observables such as the fidelity, the reduced one-body densities, the radial probability density of the relative wave function in both real and momentum space, and the Tan contact, which unveils the existence of short range two-body correlations. When the system is initialized in its bound state, it is perturbed in the most efficient manner as compared to any other initial configuration. Moreover, starting from an interacting ground state, the two-boson response is enhanced for quenches toward the noninteracting limit.

### Dissipative Correlated Dynamics of a Moving Bosonic Impurity Immersed in a Bose-Einstein Condensate

*S.I. Mistakidis, F. Grusdt, G.M. Koutentakis and P. Schmelcher*

We unravel the nonequilibrium correlated quantum quench dynamics of an impurity traveling through a harmonically confined Bose–Einstein condensate in one-dimension. For weak repulsive interspecies interactions the impurity oscillates within the bosonic gas. At strong repulsions and depending on its prequench position the impurity moves towards an edge of the bosonic medium and subsequently equilibrates. This equilibration being present independently of the initial velocity, the position and the mass of the impurity is inherently related to the generation of entanglement in the many-body system. Focusing on attractive interactions the impurity performs a damped oscillatory motion within the bosonic bath, a behavior that becomes more evident for stronger attractions. To elucidate our understanding of the dynamics an effective potential picture is constructed. The effective mass of the emergent quasiparticle is measured and found to be generically larger than the bare one, especially for strong attractions. In all cases, a transfer of energy from the impurity to the bosonic medium takes place. Finally, by averaging over a sample of simulated in situ single-shot images we expose how the single-particle density distributions and the two-body interspecies correlations can be probed.

### Correlated quantum dynamics of two quenched fermionic impurities immersed in a Bose-Einstein condensate

*S. I. Mistakidis, L. Hilbig, and P. Schmelcher*

We unravel the nonequilibrium dynamics of two fermionic impurities immersed in a one-dimensional bosonic gas following an interspecies interaction quench. Monitoring the temporal evolution of the single-particle density of each species we reveal the existence of four distinct dynamical regimes. For weak interspecies repulsions both species either perform a breathing motion or the impurity density splits into two parts which interact and disperse within the bosonic cloud. Turning to strong interactions we observe the formation of dark-bright states within the mean-field approximation. However, the correlated dynamics shows that the fermionic density splits into two repelling density peaks which either travel toward the edges of the bosonic cloud where they equilibrate or they approach an almost steady state propagating robustly within the bosonic gas which forms density dips at the same location. For these strong interspecies interactions an energy transfer process from the impurities to their environment occurs at the many-body level, while a periodic energy exchange from the bright states (impurities) to the bosonic species is identified in the absence of correlations. Finally, inspecting the one-body coherence function for strong interactions enables us to draw conclusions on the spatial localization of the quench-induced fermionic density humps.

### Effective approach to impurity dynamics in one-dimensional trapped Bose gases

*S.I. Mistakidis, A.G. Volosniev, N.T. Zinner and P. Schmelcher*

We investigate a temporal evolution of an impurity atom in a one-dimensional trapped Bose gas following a sudden change of the boson-impurity interaction strength. Our focus is on the effects of inhomogeneity due to the harmonic confinement. These effects can be described by an effective one-body model where both the mass and the spring constant are renormalized. This is in contrast to the classic renormalization, which addresses only the mass. We propose an effective single-particle Hamiltonian and apply the multilayer multiconfiguration time-dependent Hartree method for bosons to explore its validity. Numerical results suggest that the effective mass is smaller than the impurity mass, which means that it cannot straightforwardly be extracted from translationally invariant models.

### Quench dynamics and orthogonality catastrophe of Bose polarons

*G.C. Katsimiga, S.I. Mistakidis, G.M. Koutentakis, Th. Busch, and P. Schmelcher*

We monitor the correlated quench induced dynamical dressing of a spinor impurity repulsively interacting with a Bose-Einstein condensate. Inspecting the temporal evolution of the structure factor, three distinct dynamical regions arise upon increasing the interspecies interaction. These regions are found to be related to the segregated nature of the impurity and to the Ohmic character of the bath. It is shown that the impurity dynamics can be described by an effective potential that deforms from a harmonic to a double-well one when crossing the miscibility-immiscibility threshold. In particular, for miscible components the polaron formation is imprinted on the spectral response of the system. We further illustrate that for increasing interaction an orthogonality catastrophe occurs and the polaron picture breaks down. Then a dissipative motion of the impurity takes place leading to a transfer of energy to its environment. This process signals the presence of entanglement in the many-body system.

### Quench dynamics of two one-dimensional harmonically trapped bosons bridging attraction and repulsion

*L. Budewig, S.I. Mistakidis and P. Schmelcher*

We unravel the nonequilibrium quantum dynamics of two harmonically confined bosons in one spatial dimension when performing an interaction quench from finite repulsive to attractive interaction strengths and vice versa. A closed analytical form of the expansion coefficients of the time-evolved two-body wavefunction is derived, while its dynamics is determined in terms of an expansion over the postquench eigenstates. For both quench scenarios the temporal evolution is analysed by inspecting the one- and two-body reduced density matrices and densities, the momentum distribution and the fidelity. Resorting to the fidelity spectrum and the eigenspectrum we identify the dominant eigenstates of the system that govern the dynamics. Monitoring the dynamics of the above-mentioned observables we provide signatures of the energetically higher-lying states triggered by the quench.

### Phase-separation dynamics induced by an interaction quench of a correlated Fermi-Fermi mixture in a double well

*J. Erdmann, S.I. Mistakidis and P. Schmelcher*

We explore the interspecies interaction quench dynamics of ultracold spin-polarized few-body mass-balanced Fermi-Fermi mixtures confined in a double well with an emphasis on the beyond Hartree-Fock correlation effects. It is shown that the ground state of particle-imbalanced mixtures exhibits a symmetry breaking of the single-particle density for strong interactions in the Hartree-Fock limit, which is altered within the many-body approach. Quenching the interspecies repulsion towards the strongly interacting regime, the two species phase separate within the Hartree-Fock approximation while remaining miscible in the many-body treatment. Despite their miscible character on the one-body level, the two species are found to be strongly correlated and exhibit a phase separation on the two-body level that suggests the antiferromagneticlike behavior of the few-body mixture. For particle-balanced mixtures we show that an intrawell fragmentation (filamentation) of the density occurs both for the ground state and upon quenching from weak to strong interactions, a result that is exclusively caused by the presence of strong correlations. By inspecting the two-body correlations, a phase separation of the two species is unveiled, being a precursor towards an antiferromagnetic state. Finally, we simulate in situ single-shot measurements and showcase how our findings can be retrieved by averaging over a sample of single-shot images.

### Correlated tunneling dynamics of an ultracold Fermi-Fermi mixture confined in a double well

*J. Erdmann, S.I. Mistakidis and P. Schmelcher*

We unravel the correlated tunneling dynamics of a mass imbalanced few-body Fermi-Fermi mixture upon quenching the tilt of a double well. The nonequilibrium dynamics of both species changes from Rabi oscillations close to the noninteracting limit to a delayed tunneling dynamics for moderate interspecies repulsions. Considering strong interspecies interactions, the lighter species experiences quantum self-trapping due to the heavier species which acts as an effective material barrier, while performing almost perfect Rabi oscillations. The degree of entanglement, inherent in the system, is analyzed and found to be significant at both moderate and strong repulsions. To relate our findings to possible experimental realizations, we simulate in situ single-shot measurements and discuss how a sampling of such images dictates the observed dynamics. Finally, the dependence of the tunneling behavior on the mass ratio, the particle number in each species, and the height of the barrier of the double well is showcased.

### State engineering of impurities in a lattice by coupling to a Bose gas

*K. Keiler and P. Schmelcher*

We investigate the localization pattern of interacting impurities, which are trapped in a lattice potential and couple to a Bose gas. For small interspecies interaction strengths, the impurities populate the energetically lowest Bloch state or localize separately in different wells with one extra particle being delocalized over all the wells, depending on the lattice depth. In contrast, for large interspecies interaction strengths we find that due to the fractional filling of the lattice and the competition of the repulsive contact interaction between the impurities and the attractive interaction mediated by the Bose gas, the impurities localize either pairwise or completely in a single well. Tuning the lattice depth, the interspecies and intraspecies interaction strength correspondingly allows for a systematic control and engineering of the two localization patterns. The sharpness of the crossover between the two states as well as the broad region of their existence supports the robustness of the engineering. Moreover, we are able to manipulate the ground state's degeneracy in the form of triplets, doublets and singlets by implementing different boundary conditions, such as periodic and hard wall boundary conditions.

### Quench dynamics of finite bosonic ensembles in optical lattices with spatially modulated interactions

*T. Plaßmann, S.I. Mistakidis and P. Schmelcher*

The nonequilibrium quantum dynamics of few boson ensembles which experience a spatially modulated interaction strength and are confined in finite optical lattices is investigated. We utilize a cosinusoidal spatially modulated effective interaction strength which is characterized by its wavevector, inhomogeneity amplitude, interaction offset and a phase. Performing quenches either on the wavevector or the phase of the interaction profile an enhanced imbalance of the interatomic repulsion between distinct spatial regions of the lattice is induced. Following both quench protocols triggers various tunneling channels and a rich excitation dynamics consisting of a breathing and a cradle mode. All modes are shown to be amplified for increasing inhomogeneity amplitude of the interaction strength. Especially the phase quench induces a directional transport enabling us to discern energetically, otherwise, degenerate tunneling pathways. Moreover, a periodic population transfer between distinct momenta for quenches of increasing wavevector is observed, while a directed occupation of higher momenta can be achieved following a phase quench. Finally, during the evolution regions of partial coherence are revealed between the predominantly occupied wells.

### Bunching-antibunching crossover in harmonically trapped few-body Bose-Fermi mixtures

*J. Chen, J. Schurer and P. Schmelcher*

We investigate the ground state of a few-body Bose-Fermi mixture in a one-dimensional harmonic trap with varying interaction strengths and mass ratio. A bunching-antibunching crossover of the bosonic species for increasing interspecies' repulsion is observed within our fully correlated ab initio studies. Interestingly, this crossover is suppressed if the bosonic repulsion exceeds a critical value which strongly depends on the mass ratio. In order to unveil the physical origin of this crossover, we employ different levels of approximations: while a species mean-field approach can account for the antibunching, only the inclusion of the interspecies correlations can lead to the bunching. We show that these correlations effectively create an induced bosonic interaction, which in turn elucidates the occurrence of the bosonic bunching. Finally, we derive a two-site extended Bose-Hubbard model which reveals the low-energy physics of the bosons for the case of much heavier fermions.

### Many-body dissipative flow of a confined scalar Bose-Einstein condensate driven by a Gaussian impurity

*G.C. Katsimiga, S.I. Mistakidis, G.M. Koutentakis, P.G. Kevrekidis and P. Schmelcher*

The many-body dissipative flow induced by a mobile Gaussian impurity harmonically oscillating within a cigar-shaped Bose-Einstein condensate is investigated. For very small and large driving frequencies the superfluid phase is preserved. Dissipation is identified, for intermediate driving frequencies, by the nonzero value of the drag force whose abrupt increase signals the spontaneous downstream emission of an array of gray solitons. After each emission event, typically each of the solitary waves formed decays and splits into two daughter gray solitary waves that are found to be robust propagating in the bosonic background for large evolution times. In particular, a smooth transition toward dissipation is observed, with the *critical* velocity for solitary wave formation depending on both the characteristics of the obstacle, namely its driving frequency and width as well as on the interaction strength. The variance of a sample of single-shot simulations indicates the fragmented nature of the system; here it is found to increase during evolution for driving frequencies where the coherent structure formation becomes significant. Finally, we demonstrate that for fairly large particle numbers *in situ* single-shot images directly capture the gray soliton's decay and splitting.

### Quantum point spread function for imaging trapped few-body systems with a quantum gas microscope

*S. Krönke, M. Pyzh, C. Weitenberg, P. Schmelcher*

Quantum gas microscopes, which image the atomic occupations in an optical lattice, have opened a new avenue to the exploration of many-body lattice systems. Imaging trapped systems after freezing the density distribution by ramping up a pinning lattice leads, however, to a distortion of the original density distribution, especially when its structures are on the scale of the pinning lattice spacing. We show that this dynamics can be described by a filter, which we call in analogy to classical optics a quantum point spread function. Using a machine learning approach, we demonstrate via several experimentally relevant setups that a suitable deconvolution allows for the reconstruction of the original density distribution. These findings are both of fundamental interest for the theory of imaging and of immediate importance for current quantum gas experiments.

### Spectral properties and breathing dynamics of a few-body Bose–Bose mixture in a 1D harmonic trap

*M. Pyzh, S. Krönke, C. Weitenberg, P. Schmelcher*

We investigate a few-body mixture of two bosonic components, each consisting of two particles confined in a quasi one-dimensional harmonic trap. By means of exact diagonalization with a correlated basis approach we obtain the low-energy spectrum and eigenstates for the whole range of repulsive intra- and inter-component interaction strengths. We analyse the eigenvalues as a function of the inter-component coupling, covering hereby all the limiting regimes, and characterize the behaviour in-between these regimes by exploiting the symmetries of the Hamiltonian. Provided with this knowledge we study the breathing dynamics in the linear-response regime by slightly quenching the trap frequency symmetrically for both components. Depending on the choice of interactions strengths, we identify 1 to 3 monopole modes besides the breathing mode of the centre of mass coordinate. For the uncoupled mixture each monopole mode corresponds to the breathing oscillation of a specific relative coordinate. Increasing the inter-component coupling first leads to multi-mode oscillations in each relative coordinate, which turn into single-mode oscillations of the same frequency in the composite-fermionization regime.

### Entanglement Induced Interactions in Binary Mixtures

*J. Chen, J.M. Schurer and P. Schmelcher*

We establish a conceptual framework for the identification and the characterization of induced interactions in binary mixtures and reveal their intricate relation to entanglement between the components or species of the mixture. Exploiting an expansion in terms of the strength of the entanglement among the two species enables us to deduce an effective single-species description. In this way, we naturally incorporate the mutual feedback of the species and obtain induced interactions for both species which are effectively present among the particles of same type. Importantly, our approach incorporates few-body and inhomogeneous systems extending the scope of induced interactions where two particles interact via a bosonic bath-type environment. Employing the example of a one-dimensional ultracold Bose-Fermi mixture, we obtain induced Bose-Bose and Fermi-Fermi interactions with short-range attraction and long-range repulsion. With this, we show how beyond species mean-field physics visible in the two-body correlation functions can be understood via the induced interactions.

### Bosonic quantum dynamics following a linear interaction quench in finite optical lattices of unit filling

*S.I. Mistakidis, G.M. Koutentakis and P. Schmelcher*

The nonequilibrium ultracold bosonic quantum dynamics in finite optical lattices of unit filling following a linear interaction quench from a superfluid to a Mott insulator state and vice versa is investigated. The resulting dynamical response consists of various inter and intraband tunneling modes. We find that the competition between the quench rate and the interparticle repulsion leads to a resonant dynamical response, at moderate ramp times, being related to avoided crossings in the many-body eigenspectrum with varying interaction strength. Crossing the regime of weak to strong interactions several transport pathways are excited. The higher-band excitation dynamics is shown to obey an exponential decay possessing two distinct time scales with varying ramp time. Studying the crossover from shallow to deep lattices we find that for a diabatic quench the excited band fraction decreases, while approaching the adiabatic limit it exhibits a non-linear behavior for increasing height of the potential barrier. The inverse ramping process from strong to weak interactions leads to a melting of the Mott insulator and possesses negligible higher-band excitations which follow an exponential decay for decreasing quench rate. Finally, independently of the direction that the phase boundary is crossed, we observe a significant enhancement of the excited to higher-band fraction for increasing system size.

### Many-body expansion dynamics of a Bose-Fermi mixture confined in an optical lattice

*P. Siegl, S.I. Mistakidis and P. Schmelcher*

We unravel the correlated nonequilibrium dynamics of a mass balanced Bose-Fermi mixture in a one-dimensional optical lattice upon quenching an imposed harmonic trap from strong to weak confinement. Regarding the system's ground state, the competition between the inter- and intraspecies interaction strength gives rise to the immiscible and miscible phases characterized by negligible and complete overlap of the constituting atomic clouds, respectively. The resulting dynamical response depends strongly on the initial phase and consists of an expansion of each cloud and an interwell tunneling dynamics. For varying quench amplitude and referring to a fixed phase, a multitude of response regimes is unveiled, being richer within the immiscible phase, which are described by distinct expansion strengths and tunneling channels.

### Correlation effects in the quench-induced phase separation dynamics of a two species ultracold quantum gas

*S.I. Mistakidis, G.C. Katsimiga, P.G. Kevrekidis and P. Schmelcher*

We explore the quench dynamics of a binary Bose–Einstein condensate crossing the miscibility–immiscibility threshold and vice versa, both within and in particular beyond the mean-field approximation. Increasing the interspecies repulsion leads to the filamentation of the density of each species, involving shorter wavenumbers and longer spatial scales in the many-body (MB) approach. These filaments appear to be strongly correlated and exhibit domain-wall structures. Following the reverse quench process multiple dark–antidark solitary waves are spontaneously generated and subsequently found to decay in the MB scenario. We simulate single-shot images to connect our findings to possible experimental realizations. Finally, the growth rate of the variance of a sample of single-shots probes the degree of entanglement inherent in the system.

### Correlation induced localization of lattice trapped bosons coupled to a Bose–Einstein condensate

*K. Keiler, S. Krönke and P. Schmelcher*

We investigate the ground state properties of a lattice trapped bosonic system coupled to a Lieb–Liniger type gas. Our main goal is the description and in depth exploration and analysis of the two-species many-body quantum system including all relevant correlations beyond the standard mean-field approach. To achieve this, we use the multi-configuration time-dependent Hartree method for mixtures (ML-MCTDHX). Increasing the lattice depth and the interspecies interaction strength, the wave function undergoes a transition from an uncorrelated to a highly correlated state, which manifests itself in the localization of the lattice atoms in the latter regime. For small interspecies couplings, we identify the process responsible for this cross-over in a single-particle-like picture. Moreover, we give a full characterization of the wave function's structure in both regimes, using Bloch and Wannier states of the lowest band, and we find an order parameter, which can be exploited as a corresponding experimental signature. To deepen the understanding, we use an effective Hamiltonian approach, which introduces an induced interaction and is valid for small interspecies interaction. We finally compare the ansatz of the effective Hamiltonian with the results of the ML-MCTDHX simulations.

### Quantum dynamics of two trapped bosons following infinite interaction quenches

*L.M.A. Kehrberger, V.J. Bolsinger and P. Schmelcher*

We investigate the quantum dynamics of two identical bosons in a one-dimensional harmonic trap following an interaction quench from zero to infinite interaction strength and vice versa. For both quench scenarios, closed analytical expressions for the temporal evolution of the wave function as well as the Loschmidt echo are found and the dynamics of the momentum density as well as the reduced single-particle density matrix are analyzed. We observe a crossover of these quantities between bosonic, “symmetrized” fermionic, and fermionic properties. Furthermore, several combined quenches are analyzed as well.

### Quantum dynamical response of ultracold few-boson ensembles in finite optical lattices to multiple interaction quenches

*J. Neuhaus-Steinmetz, S. Mistakidis and P. Schmelcher*

The correlated nonequilibrium quantum dynamics following a multiple interaction quench protocol for few-bosonic ensembles confined in finite optical lattices is investigated. The quenches give rise to an interwell tunneling and excite the cradle and a breathing mode. Several tunneling pathways open during the time interval of increased interactions, while only a few occur when the system is quenched back to its original interaction strength. The cradle mode, however, persists during and in between the quenches, while the breathing mode possesses distinct frequencies. The occupation of excited bands is explored in detail revealing a monotonic behavior with increasing quench amplitude and a nonlinear dependence on the duration of the application of the quenched interaction strength. Finally, a periodic population transfer between momenta for quenches of increasing interaction is observed, with a power-law frequency dependence on the quench amplitude. Our results open the possibility to dynamically manipulate various excited modes of the bosonic system.

### Collective excitations of dipolar gases based on local tunneling in superlattices

*L. Cao, S.I. Mistakidis, X. Deng and P. Schmelcher*

The collective dynamics of a dipolar fermionic quantum gas confined in a one-dimensional double-well superlattice is explored. The fermionic gas resides in a paramagnetic-like ground state in the weak interaction regime, upon which a new type of collective dynamics is found when applying a local perturbation. This dynamics is composed of the local tunneling of fermions in separate supercells, and is a pure quantum effect, with no classical counterpart. Due to the presence of the dipolar interactions the local tunneling transports through the entire superlattice, giving rise to a collective dynamics. A well-defined momentum-energy dispersion relation is identified in the ab-initio simulations demonstrating the phonon-like behavior. The phonon-like characteristic is also confirmed by an analytical description of the dynamics within a semiclassical picture.

### Mode coupling of interaction quenched ultracold few-boson ensembles in periodically driven lattices

*S.I. Mistakidis and P. Schmelcher*

The out-of-equilibrium dynamics of interaction quenched finite ultracold bosonic ensembles in periodically driven one-dimensional optical lattices is investigated. It is shown that periodic driving enforces the bosons in the outer wells of the finite lattice to exhibit out-of-phase dipolelike modes, while in the central well the atomic cloud experiences a local breathing mode. The dynamical behavior is investigated with varying driving frequencies, revealing resonantlike behavior of the intrawell dynamics. An interaction quench in the periodically driven lattice gives rise to admixtures of different excitations in the outer wells, enhanced breathing in the center, and amplification of the tunneling dynamics. We then observe multiple resonances between the inter- and the intrawell dynamics at different quench amplitudes, with the position of the resonances being tunable via the driving frequency. Our results pave the way for future investigations of the use of combined driving protocols in order to excite different inter- and intrawell modes and to subsequently control them.

### Quench-induced resonant tunneling mechanisms of bosons in an optical lattice with harmonic confinement

*G.M. Koutentakis, S.I. Mistakidis and P. Schmelcher*

The nonequilibrium dynamics of small boson ensembles in a one-dimensional optical lattice is explored upon a sudden quench of an additional harmonic trap from strong to weak confinement. We find that the competition between the initial localization and the repulsive interaction leads to a resonant response of the system for intermediate quench amplitudes, corresponding to avoided crossings in the many-body eigenspectrum with varying final trap frequency. In particular, we show that these avoided crossings can be utilized to prepare the system in a desired state. The dynamical response is shown to depend on both the interaction strength as well as the number of atoms manifesting the many-body nature of the tunneling dynamics.

### Resonant quantum dynamics of few ultracold bosons in periodically driven finite lattices

* S.I. Mistakidis, T. Wulf, A. Negretti and P. Schmelcher*

The out-of-equilibrium dynamics of finite ultracold bosonic ensembles in periodically driven one-dimensional optical lattices is investigated. Our study reveals that the driving enforces the bosons in different wells to oscillate in-phase and to exhibit a dipole-like mode. A wide range from weak-to-strong driving frequencies is covered and a resonance-like behavior of the intra-well dynamics is discussed. In the proximity of the resonance a rich intraband excitation spectrum is observed. The single particle excitation mechanisms are studied in the framework of Floquet theory elucidating the role of the driving frequency. The impact of the interatomic repulsive interactions is examined in detail yielding a strong influence on the tunneling period and the excitation probabilities. Finally, the dependence of the resonance upon a variation of the tunable parameters of the optical lattice is examined. Our analysis is based on the ab initio multi-configuration time-dependent Hartree method for bosons.

### Two-body correlations and natural-orbital tomography in ultracold bosonic systems of definite parity

* S. Krönke and P. Schmelcher*

The relationship between natural orbitals, one-body coherences, and two-body correlations is explored for bosonic many-body systems of definite parity with two occupied single-particle states. We show that the strength of local two-body correlations at the parity-symmetry center characterizes the number-state distribution and controls the structure of nonlocal two-body correlations. A recipe for the experimental reconstruction of the natural-orbital densities and quantum depletion is derived. These insights into the structure of the many-body wave function are applied to the predicted quantum-fluctuation-induced decay of dark solitons.

### Magnetic Kink States Emulated with Dipolar Superlattice Gases

* X. Yin, L. Cao. P. Schmelcher*

We propose an effective Ising spin chain constructed with dipolar quantum gases confined in a one-dimensional optical superlattice. Mapping the motional degree of freedom of a single particle in the lattice onto a pseudo-spin results in an effective Ising type chain dressed with transverse and longitudinal magnetic fields. The ground state of this effective Ising chain changes from a paramagnetic to a single-kink state as the dipolar interaction increases. Particularly in the single-kink state this effective chain permits emulations of magnetic kink effects. Being realizable with current experimental techniques, this effective Ising chain presents a unique platform for emulations of Ising physics and enriches the toolbox for quantum emulation of spin models by ultracold quantum gases.

### Correlated quantum dynamics of a single atom collisionally coupled to an ultracold finite bosonic ensemble

* S. Krönke, J. Knörzer and P. Schmelcher*

We explore the correlated quantum dynamics of a single atom, regarded as an open system, with a spatio-temporally localized coupling to a finite bosonic environment. The single atom, initially prepared in a coherent state of low energy, oscillates in a one-dimensional harmonic trap and thereby periodically penetrates an interacting ensemble of NA bosons held in a displaced trap. We show that the inter-species energy transfer accelerates with increasing NA and becomes less complete at the same time. System-environment correlations prove to be significant except for times when the excess energy distribution among the subsystems is highly imbalanced. These correlations result in incoherent energy transfer processes, which accelerate the early energy donation of the single atom and stochastically favour certain energy transfer channels, depending on the instantaneous direction of transfer. Concerning the subsystem states, the energy transfer is mediated by non-coherent states of the single atom and manifests itself in singlet and doublet excitations in the finite bosonic environment. These comprehensive insights into the non-equilibrium quantum dynamics of an open system are gained by ab initio simulations of the total system with the recently developed multi-layer multi-configuration time-dependent Hartree method for bosons.

### Beyond-mean-field study of a binary bosonic mixture in a state-dependent honeycomb lattice

* L. Cao, S. Krönke, J. Stockhofe, J. Simonet, K. Sengstock, D.-S. Lühmann and P. Schmelcher*

We investigate a binary mixture of bosonic atoms loaded into a state-dependent honeycomb lattice. For this system, the emergence of a so-called twisted-superfluid ground state was experimentally observed in Soltan-Panahi et al. [Nat. Phys. 8, 71 (2012)]. Theoretically, the origin of this effect is not understood. We perform numerical simulations of an extended single-band Bose-Hubbard model adapted to the experimental parameters employing the multilayer multiconfiguration time-dependent Hartree method for Bosons. Our results confirm the overall applicability of mean-field theory in the relevant parameter range, within the extended single-band Bose-Hubbard model. Beyond this, we provide a detailed analysis of correlation effects correcting the mean-field result. These have the potential to induce asymmetries in single shot time-of-flight measurements, but we find no indication of the patterns characteristic of the twisted superfluid. We comment on the restrictions of our model and possible extensions.

### Negative-quench-induced excitation dynamics for ultracold bosons in one-dimensional lattices

* S.I. Mistakidis, L. Cao and P. Schmelcher*

The nonequilibrium dynamics following a quench of strongly repulsive bosonic ensembles in one-dimensional finite lattices is investigated by employing interaction quenches and/or a ramp of the lattice potential. Both sudden and time-dependent quenches are analyzed in detail. For the case of interaction quenches we address the transition from the strong repulsive to the weakly interacting regime, suppressing in this manner the heating of the system. The excitation modes such as the cradle process and the local breathing mode are examined via local density observables. In particular, the cradle mode is inherently related to the initial delocalization and, following a negative interaction quench, can be excited only for incommensurate setups with filling larger than unity. Alternatively, a negative quench of the lattice depth which favors the spatial delocalization is used to access the cradle mode for setups with filling smaller than unity. Our results shed light on possible schemes to control the cradle and the breathing modes. Finally, employing the notion of fidelity we study the dynamical response of the system after a diabatic or adiabatic parameter modulation for short and long evolution times. The evolution of the system is obtained numerically using the ab initio multilayer multiconfiguration time-dependent Hartree method for bosons, which permits us to follow nonequilibrium dynamics including the corresponding investigation of higher-band effects.

### Interaction quench induced multimode dynamics of finite atomic ensembles

*S. Mistakidis, L. Cao, P. Schmelcher*

The correlated non-equilibrium dynamics of few-boson systems in one-dimensional finite lattices is investigated. Starting from weak interactions we perform a sudden interaction quench and employ the numerically exact multi-layer multi-configuration time-dependent Hartree method for bosons to obtain the resulting quantum dynamics. Focusing on the low-lying modes of the finite lattice we observe the emergence of density-wave tunneling, breathing and cradle-like processes. In particular, the tunneling induced by the quench leads to a 'global' density-wave oscillation. The resulting breathing and cradle modes are inherent to the local intrawell dynamics and connected to excited-band states. Moreover, the interaction quenches couple the density-wave and the cradle modes allowing for resonance phenomena. These are associated with an avoided-crossing in the respective frequency spectrum and lead to a beating dynamics for the cradle. Finally, complementing the numerical studies, an effective Hamiltonian in terms of the relevant Fock states is derived for the description of the spectral properties and the related resonant dynamics.

### Quantum breathing dynamics of ultracold bosons in one-dimensional harmonic traps: Unraveling the pathway from few- to many-body systems

* R. Schmitz, S. Krönke, L. Cao, P. Schmelcher*

Following a “bottom-up approach” in understanding many-particle effects and dynamics we provide a systematic ab initio study of the dependence of the breathing dynamics of ultracold bosons in a one-dimensional (1D) harmonic trap on the number of bosons ranging from few to many. To this end, we employ the multilayer multiconfiguration time-dependent Hartree method for bosons (ML-MCTDHB) which has been developed very recently [ Krönke, Cao, Vendrell and Schmelcher New J. Phys. 15 063018 (2013)]. The beating behavior for two bosons is found numerically and consequently explained by an analytical approach. Drawing on this, we show how to compute the complete breathing mode spectrum in this case. We examine how the two-mode breathing behavior of two bosons evolves to the single-frequency behavior of the many-particle limit when adding more particles. In the limit of many particles, we numerically study the dependence of the breathing mode frequency on both the interaction strength as well as on the particle number. We provide an estimate for the parameter region where the mean-field description provides a valid approximation.

### The multi-layer multi-configuration time-dependent Hartree method for bosons: Theory, implementation, and applications

* L. Cao, S. Krönke, O. Vendrell, P. Schmelcher*

We develop the multi-layer multi-configuration time-dependent Hartree method for bosons (ML-MCTDHB), a variational numerically exact ab initio method for studying the quantum dynamics and stationary properties of general bosonic systems. ML-MCTDHB takes advantage of the permutation symmetry of identical bosons, which allows for investigations of the quantum dynamics from few to many-body systems. Moreover, the multi-layer feature enables ML-MCTDHB to describe mixed bosonic systems consisting of arbitrary many species. Multi-dimensional as well as mixed-dimensional systems can be accurately and efficiently simulated via the multi-layer expansion scheme. We provide a detailed account of the underlying theory and the corresponding implementation. We also demonstrate the superior performance by applying the method to the tunneling dynamics of bosonic ensembles in a one-dimensional double well potential, where a single-species bosonic ensemble of various correlation strengths and a weakly interacting two-species bosonic ensemble are considered.

### Non-equilibrium quantum dynamics of ultra-cold atomic mixtures: the multi-layer multi-configuration time-dependent Hartree method for bosons

* S. Krönke, L. Cao, O. Vendrell, P. Schmelcher*

We develop and apply the multi-layer multi-configuration time-dependent Hartree method for bosons, which represents an ab initio method for investigating the non-equilibrium quantum dynamics of multi-species bosonic systems. Its multi-layer feature allows for tailoring the wave function ansatz to describe intra- and inter-species correlations accurately and efficiently. To demonstrate the beneficial scaling and efficiency of the method, we explored the correlated tunneling dynamics of two species with repulsive intra- and inter-species interactions, to which a third species with vanishing intra-species interaction was weakly coupled. The population imbalances of the first two species can feature a temporal equilibration and their time evolution significantly depends on the coupling to the third species. Bosons of the first and second species exhibit a bunching tendency, whose strength can be influenced by their coupling to the third species.

### Two-component few-fermion mixtures in a one-dimensional trap: Numerical versus analytical approach

*I. Brouzos, P. Schmelcher*

We explore a few-fermion mixture consisting of two components that are repulsively interacting and confined in a one-dimensional harmonic trap. Different scenarios of population imbalance ranging from the completely imbalanced case where the physics of a single impurity in the Fermi sea is discussed to the partially imbalanced and equal population configurations are investigated. For the numerical calculations the multiconfigurational time-dependent Hartree method is employed, extending its application to few-fermion systems. Apart from numerical calculations we generalize our ansatz for a correlated pair wave function proposed recently [ I. Brouzos and P. Schmelcher Phys. Rev. Lett. 108 045301 (2012)] for bosons to mixtures of fermions. From weak to strong coupling between the components the energies, the densities and the correlation properties of one-dimensional systems change vastly with an upper limit set by fermionization where for infinite repulsion all fermions can be mapped to identical ones. The numerical and analytical treatments are in good agreement with respect to the description of this crossover. We show that for equal populations each pair of different component atoms splits into two single peaks in the density while for partial imbalance additional peaks and plateaus arise for very strong interaction strengths. The case of a single-impurity atom shows rich behavior of the energy and density as we approach fermionization and is directly connected to recent experiments [ G. Zürn et al. Phys. Rev. Lett. 108 075303 (2012)].

### Controlled Excitation and Resonant Acceleration of Ultracold Few-Boson Systems by Driven Interactions in a Harmonic Trap

*I. Brouzos, P. Schmelcher*

## Project C8

### Hydration shell effects in the relaxation dynamics of photoexcited Fe-II complexes in water

*P. Nalbach, A. J. A. Achner, M. Frey, M. Grosser, C. Bressler, M. Thorwart*

We study the relaxation dynamics of photoexcited Fe-II complexes dissolved in water and identify the relaxation pathway which the molecular complex follows in presence of a hydration shell of bound water at the interface between the complex and the solvent. Starting from a low-spin state, the photoexcited complex can reach the high-spin state via a cascade of different possible transitions involving electronic as well as vibrational relaxation processes. By numerically exact path integral calculations for the relaxational dynamics of a continuous solvent model, we find that the vibrational life times of the intermittent states are of the order of a few ps. Since the electronic rearrangement in the complex occurs on the time scale of about 100 fs, we find that the complex first rearranges itself in a high-spin and highly excited vibrational state, before it relaxes its energy to the solvent via vibrational relaxation transitions. By this, the relaxation pathway can be clearly identified. We find that the life time of the vibrational states increases with the size of the complex (within a spherical model), but decreases with the thickness of the hydration shell, indicating that the hydration shell acts as an additional source of fluctuations.

### Photon-assisted confinement-induced resonances for ultracold atoms

*V. Leyton, M. Roghani, V. Peano, M. Thorwart*

We solve the two-particle s-wave scattering for an ultracold-atom gas confined in a quasi-one-dimensional trapping potential which is periodically modulated. The interaction between the atoms is included via Fermi’s pseudopotential. For a modulated isotropic transverse harmonic confinement, the atomic center of mass and relative degrees of freedom decouple and an exact solution is possible. The modulation opens additional photon-assisted resonant scattering channels. Applying the Bethe-Peierls boundary condition, we obtain the general scattering solution of the time-dependent Floquet-Schrödinger equation which is universal at low energies. The effective one-dimensional scattering length can be controlled by the external driving.

### Quantification of non-Markovian effects in the Fenna-Matthews-Olson complex

*C. Mujica-Martinez, P. Nalbach, M. Thorwart*

The excitation energy transfer dynamics in the Fenna-Matthews-Olson complex is quantified in terms of a non-Markovianity measure based on the time evolution of the trace distance of two quantum states. We use a system description derived from experiments and different environmental fluctuation spectral functions, which are obtained either from experimental data or from molecular dynamics simulations. These exhibit, in all cases, a nontrivial structure with several peaks attributed to vibrational modes of the pigment-protein complex. Such a structured environmental spectrum can, in principle, give rise to strong non-Markovian effects. We present numerically exact real-time path-integral calculations for the transfer dynamics and find, in all cases, a monotonic decrease of the trace distance with increasing time which renders a Markovian description valid.

### Nonequilibrium quantum fluctuation relations for harmonic systems in nonthermal environments

*D. Pagel, P. Nalbach, A. Alvermann, H. Fehske, M. Thorwart*

We formulate exact generalized nonequilibrium fluctuation relations for the quantum mechanical harmonic oscillator coupled to multiple harmonic baths. Each of the different baths is prepared in its own individual (in general nonthermal) state. Starting from the exact solution for the oscillator dynamics we study fluctuations of the oscillator position as well as of the energy current through the oscillator under general nonequilibrium conditions. In particular, we formulate a fluctuation–dissipation relation for the oscillator position autocorrelation function that generalizes the standard result for the case of a single bath at thermal equilibrium. Moreover, we show that the generating function for the position operator fulfils a generalized Gallavotti–Cohen-like relation. For the energy transfer through the oscillator, we determine the average energy current together with the current fluctuations. Finally, we discuss the generalization of the cumulant generating function for the energy transfer to nonthermal bath preparations.

### Crossover from coherent to incoherent quantum dynamics due to sub-Ohmic dephasing

*P. Nalbach, M. Thorwart*

We report exact results for the influence of purely sub-Ohmic dephasing on the dynamics of a quantum two-level system. From response functions, we determine a crossover coupling strength between oscillatory coherent and overdamped dynamics. Surprisingly, we find no overdamping even at arbitrary large dephasing for spectra with spectral exponent.

### Organic pi-conjugated copolymers as molecular charge qubits

*C. A. Mujica-Martinez, P. Nalbach, M. Thorwart*

We propose a design for molecular charge qubits based on π-conjugated block copolymers and determine their electronic structure as well as their vibrational active modes. By tuning the length of the oligomers, the tunnel coupling in the charge qubit and its decoherence properties due to molecular vibrations can be chemically engineered. Coherent oscillations result with quality factors of up to 104 at room temperature. In turn, the molecular vibrational spectrum induces strong non-Markovian electronic effects which support the survival of quantum coherence.

### Noise-Induced Förster Resonant Energy Transfer between Orthogonal Dipoles in Photoexcited Molecules

* P. Nalbach, I. Pugliesi, H. Langhals, M. Thorwart*

### Quantum noise properties of multiphoton transitions in driven nonlinear resonators

*V. Leyton, V. Peano, M. Thorwart*

We investigate the quantum noise properties of a weakly nonlinear Duffing resonator in the deep quantum regime, where only a few quanta are excited. This regime is dominated by the appearance of coherent multiphoton resonances in the nonlinear response of the resonator to the modulation. We determine simple expressions for the photon noise spectrum and find that the multiphoton resonances also induce a multiple peak structure in that spectrum. When the corresponding multiphoton Rabi oscillations are underdamped, zero-temperature quantum fluctuations determine comparable populations of all quasienergy states which belong to a resonant multiphoton doublet. Most interestingly, the quantum fluctuations probe the multiphoton transitions by inducing several peaks in the noise spectrum of the resonator observables. In particular, the noise of the photon number contains complete information about the multiphoton states and their stationary populations via pairs of nearly symmetric peaks at opposite frequencies. Their widths are determined by the damping of the Rabi oscillations and their heights are proportional to the stationary nonequilibrium populations. A finite detuning from a multiphoton resonance generates a quasielastic noise peak at zero frequency. In addition, we relate the stationary populations of the quasienergy states with an effective quantum temperature and discuss the role of a finite temperature.

## Project C9

### Formation of Spontaneous Density-Wave Patterns in dc Driven Lattices

*H. P. Zahn, V. P. Singh, M. N. Kosch, L. Asteria, L. Freystatzky, K. Sengstock, L. Mathey, and C. Weitenberg*

Driving a many-body system out of equilibrium induces phenomena such as the emergence and decay of transient states, which can manifest itself as pattern and domain formation. The understanding of these phenomena expands the scope of established thermodynamics into the out-of-equilibrium domain. Here, we experimentally and theoretically study the out-of-equilibrium dynamics of a bosonic lattice model subjected to a strong dc field, realized as ultracold atoms in a strongly tilted triangular optical lattice. We observe the emergence of pronounced density-wave patterns—which spontaneously break the underlying lattice symmetry—using a novel single-shot imaging technique with two-dimensional single-site resolution in three-dimensional systems, which also resolves the domain structure. Our study suggests that the short-time dynamics arises from resonant pair tunneling processes within an effective description of the tilted Hubbard model. More broadly, we establish the far out-of-equilibrium regime of lattice models subjected to a strong dc field, as an exemplary and paradigmatic scenario for transient pattern formation.

### Squeezed-field path-integral description of second sound in Bose-Einstein condensates

*Ilias M. H. Seifie, Vijay Pal Singh, and L. Mathey*

We propose a generalization of the Feynman path integral using squeezed coherent states. We apply thisapproach to the dynamics of Bose-Einstein condensates, which gives an effective low-energy descriptionthat contains both a coherent field and a squeezing field. We derive the classical trajectory of this action,which constitutes a generalization of the Gross-Pitaevskii equation, at linear order. We derive the low-energyexcitations, which provides a description of second sound in weakly interacting condensates as a squeezingoscillation of the order parameter. This interpretation is also supported by a comparison to a numerical c-fieldmethod.

### Critical behavior of a chiral superfluid in a bipartite square lattice

*J. Okamoto, W.-M. Huang, R. Höppner and L. Mathey*

We study the critical behavior of Bose–Einstein condensation in the second band of a bipartite optical square lattice in a renormalization group framework at one-loop order. Within our field theoretical representation of the system, we approximate the system as a two-component Bose gas in three dimensions. We demonstrate that the system is in a different universality class than the previously studied condensation in a frustrated triangular lattice due to an additional Umklapp scattering term, which stabilizes the chiral superfluid order at low temperatures. We derive the renormalization group flow of the system and show that this order persists in the low energy limit. Furthermore, the renormalization flow suggests that the phase transition from the thermal phase to the chiral superfluid state is first order.

### Influence of electron-phonon coupling on the low-temperature phases of metallic single-wall carbon nanotubes

*J. Okamoto, L. Mathey and W.-M. Huang*

We investigate the effect of electron-phonon coupling on low-temperature phases in metallic single-wall carbon nanotubes. We obtain low-temperature phase diagrams of armchair and zigzag type nanotubes with screened interactions with a weak-coupling renormalization group approach. In the absence of electron-phonon coupling, two types of nanotubes have similar phase diagrams. A D-Mott phase or d-wave superconductivity appears when the on-site interaction is dominant, while a charge-density wave or an excitonic insulator phase emerges when the nearest neighbor interaction becomes comparable to the on-site interaction. The electron-phonon coupling, treated by a two-cutoff scaling scheme, leads to different behavior in two types of nanotubes. For strong electron-phonon interactions, phonon softening is induced and a Peierls insulator phase appears in armchair nanotubes. We find that this softening of phonons may occur for any intraband scattering phonon mode. On the other hand, the effect of electron-phonon coupling is negligible for zigzag nanotubes. The distinct behavior of armchair and zigzag nanotubes against lattice distortion is explained by analysis of the renormalization group equations.

### Light-induced coherence in an atom-cavity system

*C. Georges, J. G. Cosme, L. Mathey, A. Hemmerich*

We demonstrate a light-induced formation of coherence in a cold atomic gas system that utilizes the suppression of a competing density wave (DW) order. The condensed atoms are placed in an optical cavity and pumped by an external optical standing wave, which induces a long-range interaction mediated by photon scattering and a resulting DW order above a critical pump strength. We show that the light-induced temporal modulation of the pump wave can suppress this DW order and restore coherence. This establishes a foundational principle of dynamical control of competing orders analogous to a hypothesized mechanism for light-induced superconductivity in high*-T*_{c} cuprates.

### Dynamics of Ultracold Quantum Gases in the Dissipative Fermi-Hubbard Model

* K. Sponselee, L. Freystatzky, B. Abeln, M. Diem, B. Hundt, A. Kochanke, T. Ponath, B. Santra, L. Mathey, K. Sengstock and C. Becker*

Abstract. We employ metastable ultracold 173-Yb atoms to study dynamics in the 1D dissipative Fermi-Hubbard model experimentally and theoretically, and observe a complete inhibition of two-body losses after initial fast transient dynamics. We attribute the suppression of particle loss to the dynamical generation of a highly entangled Dicke state. For several lattice depths and for two- and six-spin component mixtures we find very similar dynamics, showing that the creation of strongly correlated states is a robust and universal phenomenon. This offers interesting opportunities for precision measurements.

arXiv:1805.11853 (2018)

https://arxiv.org/abs/1805.11853

https://iopscience.iop.org/article/10.1088/2058-9565/aadccd/meta

### Fermion pairing in mixed-dimensional atomic mixtures

*Junichi Okamoto,1,2 Ludwig Mathey,1,2 and Wen-Min Huang*

We investigate the quantum phases of mixed-dimensional cold atom mixtures. In particular, we consider a mixture of a Fermi gas in a two-dimensional lattice, interacting with a bulk Fermi gas or a Bose-Einstein condensate in a three-dimensional lattice. The effective interaction of the two-dimensional system mediated by the bulk system is determined. We perform a functional renormalization group analysis, and demonstrate that by tuning the properties of the bulk system, a subtle competition of several superconducting orders can be controlled among *s*−wave, *p*−wave, *d _{x2−y2}*−wave, and

*g*−wave pairing symmetries. Other instabilities such as a charge-density-wave order are also demonstrated to occur. In particular, we find that the critical temperature of the

_{xy(x2−y2)}*d*−wave pairing induced by the next-nearest-neighbor interactions can be an order of magnitude larger than that of the same pairing induced by doping in the simple Hubbard model. We expect that by combining the nearest-neighbor interaction with the next-nearest-neighbor hopping (known to enhance

*d*−wave pairing), an even higher critical temperature may be achieved

### Dynamic Kosterlitz-Thouless transition in two-dimensional Bose mixtures of ultracold atoms

*L. Mathey, Kenneth J. Günter, Jean Dalibard, and A. Polkovnikov*

We propose a realistic experiment to demonstrate a dynamic Kosterlitz-Thouless transition in ultracold atomic gases in two dimensions. With a numerical implementation of the truncated Wigner approximation we simulate the time evolution of several correlation functions, which can be measured via matter wave interference. We demonstrate that the relaxational dynamics is well described by a real-time renormalization-group approach and argue that these experiments can guide the development of a theoretical framework for the understanding of critical dynamics.

### Observation of Topological Bloch-State Defects and Their Merging Transition

*Matthias Tarnowski, Marlon Nuske, Nick Fläschner, Benno Rem, Dominik Vogel, Lukas Freystatzky, Klaus Sengstock, Ludwig Mathey, and Christof Weitenberg*

Topological defects in Bloch bands, such as Dirac points in graphene, and their resulting Berry phases play an important role for the electronic dynamics in solid state crystals. Such defects can arise in systems with a two-atomic basis due to the momentum-dependent coupling of the two sublattice states, which gives rise to a pseudospin texture. The topological defects appear as vortices in the azimuthal phase of this pseudospin texture. Here, we demonstrate a complete measurement of the azimuthal phase in a hexagonal optical lattice employing a versatile method based on time-of-flight imaging after off-resonant lattice modulation. Furthermore, we map out the merging transition of the two Dirac points induced by beam imbalance. Our work paves the way to accessing geometric properties in optical lattices also with spin-orbit coupling and interactions.

### Implementing supersymmetric dynamics in ultracold-atom systems

*M. Lahrz, C. Weitenberg, L. Mathey*

Supersymmetric systems derive their properties from conserved supercharges which form a supersymmetric algebra. These systems naturally factorize into two subsystems, which, when considered as individual systems, have essentially the same eigenenergies, and their eigenstates can be mapped onto each other. We propose a Mach-Zehnder interference experiment to detect supersymmetry in quantum-mechanical systems, which can be realized with current technology. To demonstrate this interferometric technique, we first propose a one-dimensional ultracold-atom setup to realize a pair of supersymmetric systems. Here, a single-atom wave packet evolves in a superposition of the subsystems and gives an interference contrast that is sharply peaked if the subsystems form a supersymmetric pair. Second, we propose a two-dimensional setup that implements supersymmetric dynamics in a synthetic gauge field.

### Emergence in Driven Solid-State and Cold-Atom Systems

*L. Mathey, J. Okamoto*

While phase transitions in equilibrium systems have been studied extensively, the emergence of order in non-equilibrium systems, such as periodically driven systems, continues to pose conceptual questions as well as offer intriguing possibilities. In particular, two recent experimental developments emphasise the urgency of further exploration. A new direction of research is taking place in the field of ultra-cold atoms. In a recent set of experiments, atoms in an optical lattice were subjected to a lattice shaking protocol. With this new feature of ultra-cold atom systems, it has been possible to create effective, renormalised Hamiltonians. In particular, the lattice shaking could be performed in such a manner to create frustrated systems and synthetic gauge fields, see e.g. [1]. A parallel development is taking place in solid state physics. A new research direction was established by using ultrafast light pulses to stabilise superconducting order. Here, the periodic driving was used to achieve a remarkable, counterintuitive result: to enhance superconductivity with high frequency and high intensity driving, see e.g

### Sudden and slow quenches into the antiferromagnetic phase of ultracold fermions

*M. Ojekhile, R. Höppner, H. Moritz, L. Mathey*

We propose a method to reach the antiferromagnetic state of two-dimensional Fermi gases trapped in optical lattices: Independent subsystems are prepared in suitable initial states and then connected by a sudden or slow quench of the tunneling between the subsystems. Examples of suitable low-entropy subsystems are double wells or plaquettes, which can be experimentally realised in Mott insulating shells using optical super-lattices. We estimate the effective temperature T* of the system after the quench by calculating the distribution of excitations created using the spin wave approximation in a Heisenberg model. We investigate the effect of an initial staggered magnetic field and find that for an optimal polarisation of the initial state the effective temperature can be significantly reduced from T*≈1.7 Tc at zero polarisation to T*<0.65Tc, where Tc is the crossover temperature to the antiferromagnetic state. The temperature can be further reduced using a finite quench time. We also show that T* decreases logarithmically with the linear size of the subsystem.

### Hierarchical equations of motion approach to transport through an Anderson impurity coupled to interacting Luttinger liquid leads

*Jun-ichi Okamoto, Ludwig Mathey, and Rainer Härtle*

We generalize the hierarchical equations of motion method to study electron transport through a quantum dot or molecule coupled to one-dimensional interacting leads that can be described as Luttinger liquids. Such leads can be realized, for example, by quantum wires or fractional quantum Hall edge states. In comparison to noninteracting metallic leads, Luttinger liquid leads involve many-body correlations and the single-particle tunneling density of states shows a power-law singularity at the chemical potential. Using the generalized hierarchical equations of motion method, we assess the importance of the singularity and the next-to-leading order many-body correlations. To this end, we compare numerically converged results with second- and first-order results of the hybridization expansion that is inherent to our method. As a test case, we study transport through a single-level quantum dot or molecule that can be described by an Anderson impurity model. Cotunneling effects turn out to be most pronounced for attractive interactions in the leads or repulsive ones if an excitonic coupling between the dot and the leads is realized. We also find that an interaction-induced negative differential conductance near the Coulomb blockade thresholds is slightly suppressed as compared to a first-order and/or rate equation result. Moreover, we find that the two-particle (*n*-particle) correlations enter as a second-order (*n*-order) effect and are, thus, not very pronounced at the high temperatures and parameters that we consider.

### Bose-Einstein condensation in a frustrated triangular optical lattice

*Peter Janzen, Wen-Min Huang, and L. Mathey*

The recent experimental condensation of ultracold atoms in a triangular optical lattice with a negative effective tunneling parameter paves the way for the study of frustrated systems in a controlled environment. Here, we explore the critical behavior of the chiral phase transition in such a frustrated lattice in three dimensions. We represent the low-energy action of the lattice system as a two-component Bose gas corresponding to the two minima of the dispersion. The contact repulsion between the bosons separates into intra- and intercomponent interactions, referred to as V0 and V12, respectively. We first employ a Huang-Yang-Luttinger approximation of the free energy. For V12/V0=2, which corresponds to the bare interaction, this approach suggests a first-order phase transition, at which both the U(1) symmetry of condensation and the Z2 symmetry of the emergent chiral order are broken simultaneously. Furthermore, we perform a renormalization-group calculation at one-loop order. We demonstrate that the coupling regime 0<V12/V0≤1 shares the critical behavior of the Heisenberg fixed point at V12/V0=1. For V12/V0>1 we show that V0 flows to a negative value, while V12 increases and remains positive. This results in a breakdown of the effective quartic-field theory due to a cubic anisotropy and, again, suggests a discontinuous phase transition.

### Realizing and optimizing an atomtronic SQUID

*A. C. Mathey, L. Mathey*

We demonstrate how a toroidal Bose–Einstein condensate with a movable barrier can be used to realize an atomtronic SQUID. The magnitude of the barrier height, which creates the analogue of an SNS junction, is of crucial importance, as well as its ramp-up and -down protocol. For too low of a barrier, the relaxation of the system is dynamically suppressed, due to the small rate of phase slips at the barrier. For a higher barrier, the phase coherence across the barrier is suppressed due to thermal fluctuations, which are included in our Truncated Wigner approach. Furthermore, we show that the ramp-up protocol of the barrier can be improved by ramping up its height first, and its velocity after that. This protocol can be further improved by optimizing the ramp-up and ramp-down time scales, which is of direct practical relevance for on-going experimental realizations.

### Probing superfluidity of Bose-Einstein condensates via laser stirring

*Vijay Pal Singh, Wolf Weimer, Kai Morgener, Jonas Siegl, Klaus Hueck, Niclas Luick, Henning Moritz, Ludwig Mathey*

We investigate the superfluid behavior of a Bose-Einstein condensate of 6Li molecules. In the experiment by Weimer et al., Phys. Rev. Lett. 114, 095301 (2015) a condensate is stirred by a weak, red-detuned laser beam along a circular path around the trap center. The rate of induced heating increases steeply above a velocity vc, which we define as the critical velocity. Below this velocity, the moving beam creates almost no heating. In this paper, we demonstrate a quantitative understanding of the critical velocity. Using both numerical and analytical methods, we identify the non-zero temperature, the circular motion of the stirrer, and the density profile of the cloud as key factors influencing the magnitude of vc. A direct comparison to the experimental data shows excellent agreement.

Phys. Rev. A 93, 023634 (2016)

http://lanl.arxiv.org/abs/1509.02168

### Theory of enhanced interlayer tunneling in optically driven high Tc superconductors

*J. Okamoto, A. Cavalleri, L. Mathey*

Motivated by recent pump-probe experiments indicating enhanced coherent c-axis transport in underdoped YBCO, we study Josephson junctions periodically driven by optical pulses. We propose a mechanism for this observation by demonstrating that a parametrically driven Josephson junction shows an enhanced imaginary part of the low-frequency conductivity when the driving frequency is above the plasma frequency, implying an effectively enhanced Josephson coupling. We generalize this analysis to a bilayer system of Josephson junctions modeling YBCO. Again, the Josephson coupling is enhanced when the pump frequency is blue-detuned to either of the two plasma frequencies of the material. We show that the emergent driven state is a genuine, non-equilibrium superconducting state, in which equilibrium relations between the Josephson coupling, current fluctuations, and the critical current no longer hold.

### Magnus expansion approach to parametric oscillator systems in a thermal bath

*B. Zhu, T. Rexin, L. Mathey*

We develop a Magnus formalism for periodically driven systems which provides an expansion both in the driving term and the inverse driving frequency, applicable to isolated and dissipative systems. We derive explicit formulas for a driving term with a cosine dependence on time, up to fourth order. We apply these to the steady state of a classical parametric oscillator coupled to a thermal bath, which we solve numerically for comparison. Beyond dynamical stabilization at second order, we find that the higher orders further renormalize the oscillator frequency, and additionally create a weakly renormalized effective temperature. The renormalized oscillator frequency is quantitatively accurate almost up to the parametric instability, as we confirm numerically. Additionally, a cut-off dependent term is generated, which indicates the break-down of the hierarchy of time scales of the system, as a precursor to the instability. Finally, we apply this formalism to a parametrically driven chain, as an example for the control of the dispersion of a many-body system.

arXiv:1604.01010

http://arxiv.org/abs/1604.01010

* *

### Sudden-quench dynamics of Bardeen-Cooper-Schrieffer states in deep optical lattices

*Marlon Nuske, L. Mathey, Eite Tiesinga*

We determine the exact dynamics of an initial Bardeen-Cooper-Schrieffer (BCS) state of ultracold atoms in a deep hexagonal optical lattice. The dynamical evolution is triggered by a quench of the lattice potential such that the interaction strength Uf is much larger than the hopping amplitude Jf. The quench initiates collective oscillations with frequency ∣∣Uf|/2π in the momentum occupation numbers and imprints an oscillating phase with the same frequency on the BCS order parameter Δ. The oscillation frequency of Δ is not reproduced by treating the time evolution in mean-field theory. In our theory, the momentum noise (i.e., density-density) correlation functions oscillate at frequency ∣∣Uf|/2π as well as at its second harmonic. For a very deep lattice, with zero tunneling energy, the oscillations of momentum occupation numbers are undamped. Nonzero tunneling after the quench leads to dephasing of the different momentum modes and a subsequent damping of the oscillations. The damping occurs even for a finite-temperature initial BCS state, but not for a noninteracting Fermi gas. Furthermore, damping is stronger for larger order parameter and may therefore be used as a signature of the BCS state. Finally, our theory shows that the noise correlation functions in a honeycomb lattice will develop strong anticorrelations near the Dirac point.

### Dynamical phase transition in the open Dicke model

*J. Klinder, H. Keßler, M. Wolke, L. Mathey, A. Hemmerich*

The Dicke model with a weak dissipation channel is realized by coupling a Bose–Einstein condensate to an optical cavity with ultranarrow bandwidth. We explore the dynamical critical properties of the Hepp–Lieb–Dicke phase transition by performing quenches across the phase boundary. We observe hysteresis in the transition between a homogeneous phase and a self-organized collective phase with an enclosed loop area showing power-law scaling with respect to the quench time, which suggests an interpretation within a general framework introduced by Kibble and Zurek. The observed hysteretic dynamics is well reproduced by numerically solving the mean-field equation derived from a generalized Dicke Hamiltonian. Our work promotes the understanding of nonequilibrium physics in open many-body systems with infinite range interactions.

### The critical velocity in the BEC-BCS crossover

* W. Weimer, K. Morgener, V. P. Singh, J. Siegl, K. Hueck, N. Luick, L. Mathey, H. Moritz*

We map out the critical velocity in the crossover from Bose-Einstein condensation (BEC) to Bardeen-Cooper-Schrieffer superfluidity with ultracold 6Li gases. A small attractive potential is dragged along lines of constant column density. The rate of the induced heating increases steeply above a critical velocity vc. In the same samples, we measure the speed of sound vs by exciting density waves and compare the results to the measured values of vc. We perform numerical simulations in the BEC regime and find very good agreement, validating the approach. In the strongly correlated regime, where theoretical predictions only exist for the speed of sound, our measurements of vc provide a testing ground for theoretical approaches.

Phys. Rev. Lett. 114, 095301 (2015)

http://arxiv.org/abs/1408.5239

### Observation of chiral superfluid order by matter wave interference

* T. Kock, M. Ölschläger, A. Ewerbeck, W.-M. Huang, L. Mathey, A. Hemmerich*

The breaking of time reversal symmetry via the spontaneous formation of chiral order is ubiquitous in nature. Here, we present an unambiguous demonstration of this phenomenon for atoms Bose-Einstein condensed in the second Bloch band of an optical lattice. As a key tool we use a matter wave interference technique, which lets us directly observe the phase properties of the superfluid order parameter and allows us to reconstruct the spatial geometry of certain low energy excitations, associated with the formation of domains of different chirality. Our work marks a new era of optical lattices where orbital degrees of freedom play an essential role for the formation of exotic quantum matter, similarly as in electronic systems.

Physical Review Letters 114, 115301 (2015)

http://arxiv.org/abs/1411.3483

### Redistribution of phase fluctuations in a periodically driven cuprate superconductor

*R. Höppner, B. Zhu, T. Rexin, A. Cavalleri, and L. Mathey*

We study the thermally fluctuating state of a bilayer cuprate superconductor under the periodic action of a staggered field oscillating at optical frequencies. This analysis distills essential elements of the recently discovered phenomenon of light-enhanced coherence in YBa_{2}Cu_{3}O_{6+x}, which was achieved by periodically driving infrared active apical oxygen distortions. The effect of a staggered periodic perturbation is studied using a Langevin and Fokker-Planck description of driven, coupled Josephson junctions, which represent two neighboring pairs of layers and their two plasmons. In a toy model including only two junctions, we demonstrate that the external driving leads to a suppression of phase fluctuations of the low-energy plasmon, an effect which is amplified via the resonance of the high-energy plasmon. When extending the modeling to the full layers, we find that this reduction becomes far more pronounced, with a striking suppression of the low-energy fluctuations, as visible in the power spectrum. We also find that this effect acts on the in-plane fluctuations, which are reduced on long length scales. All these findings provide a physical framework to describe light control in cuprates.

### Noise correlations of two-dimensional Bose gases

*V. P. Singh, L. Mathey*

We analyze density-density correlations of expanding clouds of weakly interacting two-dimensional Bose gases below and above the Berezinskii-Kosterlitz-Thouless transition, with particular focus on short-time expansions. During time-of-flight expansion, phase fluctuations of the trapped system translate into density fluctuations, in addition to the density fluctuations that exist in situ. We calculate the correlations of these fluctuations both in real space and in momentum space and derive analytic expressions in momentum space. Below the transition, the correlation functions show an oscillatory behavior, controlled by the scaling exponent of the quasicondensed phase, due to constructive interference. We argue that this can be used to extract the scaling exponent of the quasicondensate experimentally. Above the transition, the interference is rapidly suppressed when the atoms travel an average distance beyond the correlation length. This can be used to distinguish the two phases qualitatively.

### Quantum Phases of quadrupolar Fermi gases in optical lattices

*S. G. Bhongale, L. Mathey, E. Zhao, S. F. Yelin, M. Lemeshko*

We introduce a new platform for quantum simulation of many-body systems based on non-spherical particles with zero dipole moment but possessing a significant value of the electric quadrupole moment. Considering a quadrupolar quantum gas trapped in a 2D optical lattice, we show that the peculiar symmetry and broad tunability of the quadrupole-quadrupole interactions results in a rich phase diagram encompassing unconventional BCS and charge density wave phases, and paves the way to create topological superfluid ground states of px + i py symmetry. Quadrupolar species, such as metastable alkaline-earth atoms and homonuclear molecules, are stable against chemical reactions and collapse and are readily available in experiment at high densities.

Phys. Rev. Lett. 110, 155301 (2013)

http://arxiv.org/abs/1211.3317

### Unconventional Spin Density Waves in Dipolar Fermi Gases

*S. G. Bhongale, L. Mathey, S. Tsai, C. W. Clark, E. Zhao*

The conventional spin density wave (SDW) phase (Overhauser, 1962), as found in antiferromagnetic metal for example (Fawcett 1988), can be described as a condensate of particle-hole pairs with zero angular momentum, $\ell=0$, analogous to a condensate of particle-particle pairs in conventional superconductors. While many unconventional superconductors with Cooper pairs of finite $\ell$ have been discovered, their counterparts, density waves with non-zero angular momenta, have only been hypothesized in two-dimensional electron systems (Nayak, 2000). Using an unbiased functional renormalization group analysis, we here show that spin-triplet particle-hole condensates with $\ell=1$ emerge generically in dipolar Fermi gases of atoms (Lu, Burdick, and Lev, 2012) or molecules (Ospelkaus et al., 2008; Wu et al.) on optical lattice. The order parameter of these exotic SDWs is a vector quantity in spin space, and, moreover, is defined on lattice bonds rather than on lattice sites. We determine the rich quantum phase diagram of dipolar fermions at half-filling as a function of the dipolar orientation, and discuss how these SDWs arise amidst competition with superfluid and charge density wave phases.

Phys. Rev. A 87, 043604 (2012)

http://arxiv.org/abs/1209.2671

### Decay of a superfluid current of ultra-cold atoms in a toroidal trap

*Amy C. Mathey, Charles W. Clark, L. Mathey*

Using a numerical implementation of the truncated Wigner approximation, we simulate the experiment reported by Ramanathan et al. in Phys. Rev. Lett. 106, 130401 (2011), in which a Bose-Einstein condensate is created in a toroidal trap and set into rotation via a Gauss-Laguerre beam. A potential barrier is then placed in the trap to study the decay of the superflow. We find that the current decays via thermally activated phase slips, which can also be visualized as vortices crossing the barrier region in radial direction. Adopting the notion of critical velocity used in the experiment, we determine it to be lower than the local speed of sound at the barrier. This result is in agreement with the experimental findings, but in contradiction to the predictions of the Gross-Pitaevskii equation. This emphasizes the importance of thermal fluctuations in the experiment.