# Theory Group Ultra-cold atoms and solid state systems

## Prof. Dr. Ludwig Mathey

We investigate the theory of ultracold atom and solid state systems, with particular emphasis on many-body effects. One of our primary research interest is many-body dynamics, such as dynamic phase transitions and dynamic control of many-body systems, as well as properties of superfluidity and superconductivity. Furthermore, we study the equilibrium phase diagrams of ultra-cold atoms, at both finite and zero temperature. Another objective is the advancement of the technology of ultra-cold atom systems, for example by investigating new cooling and detection methods.

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**Recent**** activities:**

### Observing light-induced Floquet band gaps in the longitudinal conductivity of graphene

#### Lukas Broers, Ludwig Mathey

We propose optical longitudinal conductivity as a realistic observable to detect light-induced Floquet band gaps in graphene. These gaps manifest as resonant features in the conductivity, when resolved with respect to the probing frequency and the driving field strength. We demonstrate these features via a dissipative master equation approach which gives access to a frequency- and momentum-resolved electron distribution. This distribution follows the light-induced Floquet-Bloch bands, resulting in a natural interpretation as occupations of these bands. Furthermore, we show that there are population inversions of the Floquet-Bloch bands at the band gaps for sufficiently strong driving field strengths. This strongly reduces the conductivity at the corresponding frequencies. Therefore our proposal puts forth not only an unambiguous demonstration of light-induced Floquet-Bloch bands, which advances the field of Floquet engineering in solids, but also points out the control of transport properties via light, that derives from the electron distribution on these bands.

### Squeezed-field path-integral description of BCS superconductors

#### Kazuma Nagao, Dapeng Li, Ludwig Mathey

We develop a squeezed-field path-integral representation for BCS superconductors utilizing a generalized completeness relation of squeezed-fermionic coherent states. We derive a Grassmann path integral of fermionic quasiparticles that explicitly includes the collective degrees of freedom of the order-parameter dynamics governed by the classical Anderson pseudospin model. Based on this method, we analyze the spectral function of the single-particle excitations, and show that the squeezed-field path integral for the BCS Hamiltonian describes the dispersion relation and the mass gap of the Higgs amplitude mode of BCS superconductors, as well as the quasiparticle and quasihole excitation branches described by the BCS mean-field approximation.

### Observation of a dissipative time crystal

#### Hans Keßler, Phatthamon Kongkhambut, Christoph Georges, Ludwig Mathey, Jayson G. Cosme, Andreas Hemmerich

The formation of a phase of matter can be associated with the spontaneous breaking of a symmetry. For crystallization, this broken symmetry is the spatial translation symmetry, as the atoms spontaneously localize in a periodic fashion. In analogy to spatial crystals, the spontaneous breaking of temporal translation symmetry results in the formation of time crystals. While recent and on-going experiments on driven isolated systems aim to minimize dissipative processes, as it is an undesired source of decay, well-designed dissipation has been put forth as a constitutive ingredient in the formation of dissipative time crystals (DTCs). Here, we present the first experimental realisation of a DTC, implemented in an atom-cavity system. Its defining feature is a period doubled switching between distinct chequerboard density wave patterns, induced by controlled cavity-dissipation and cavity-mediated interactions. We demonstrate the robustness of this phase against system parameter changes and temporal perturbations of the driving. Our work provides a framework for realising phases of matter with spatiotemporal order in presence of dissipation. We note that this is the natural environment of matter, and therefore shapes its physical phenomena profoundly, making its study imperative.

### Fluctuations of squeezing fields beyond the Tomonaga--Luttinger liquid paradigm

#### Kazuma Nagao, Ludwig Mathey

The concept of Tomonaga--Luttinger liquids (TLL) on the basis of the free-boson models is ubiquitous in theoretical descriptions of low-energy properties in one-dimensional quantum systems. In this work, we develop a squeezed-field path-integral description for gapless one-dimensional systems beyond the free-boson picture of the TLL paradigm. In the squeezed-field description, the parameter of the Bogoliubov transformation for the TL Hamiltonian becomes a dynamical squeezing field, and its fluctuations give rise to corrections to the free-boson results. We derive an effective nonlinear Lagrangian describing the dispersion relation of the squeezing field, and interactions between the excitations of the TLL and the squeezing modes. Using the effective Lagrangian, we analyze the imaginary-time correlation function of a vertex operator in the non-interacting limit. We show that a side-band branch emerges due to the fluctuation of the squeezing field, in addition to the standard branch of the free-boson model of the TLL paradigm. Furthermore, we perturbatively analyze the spectral function of the density fluctuations for an ultracold Bose gas in one dimension. We evaluate the renormalized values of the phase velocities and spectral weights of the TLL and side-band branches due to the interaction between the TLL and the squeezing modes. At zero temperature, the renormalized dispersion relations are linear in the momentum, but at nonzero temperatures, these acquire a nonlinear dependence on the momentum due to the thermal population of the excitation branches.

### Higgs-mediated enhancement of interlayer transport in high-Tc superconductors

#### Guido Homann, Jayson G. Cosme, Junichi Okamoto, Ludwig Mathey

We put forth a mechanism for enhancing the interlayer transport in cuprate superconductors, by optically driving plasmonic excitations with a frequency that is blue-detuned from the Higgs frequency. The plasmonic excitations induce a collective oscillation of the Higgs field which induces parametric enhancement of the superconducting response, as we demonstrate with a minimal analytical model. Furthermore, we perform relativistic U(1) lattice gauge simulations and find good agreement with our analytical prediction. We map out the renormalization of the interlayer coupling as a function of the parameters of the optical field and demonstrate that the Higgs-mediated enhancement can be larger than 50%.

### Collective modes and superfluidity of a two-dimensional ultracold Bose gas

#### Vijay Pal Singh and Ludwig Mathey

The collective modes of a quantum liquid shape and impact its properties profoundly, including its emergent phenomena such as superfluidity. Here we present how a two-dimensional Bose gas responds to a moving lattice potential. In particular we discuss how the induced heating rate depends on the interaction strength and the temperature. This study is motivated by the recent measurements of Sobirey, et al., arXiv:2005.07607 (2020), for which we provide a quantitative understanding. Going beyond the existing measurements, we demonstrate that this probing method allows to identify first and second sound in quantum liquids. We show that the two sound modes undergo hybridization as a function of interaction strength, which we propose to detect experimentally. This gives a novel insight into the two regimes of Bose gases, defined via the hierarchy of sounds modes.

### Dynamical control of the conductivity of an atomic Josephson junction

#### Beilei Zhu, Vijay Pal Singh, Junichi Okamoto, Ludwig Mathey

We propose to dynamically control the conductivity of a Josephson junction composed of two weakly coupled one dimensional condensates of ultracold atoms. A current is induced by a periodically modulated potential difference between the condensates, giving access to the conductivity of the junction. By using parametric driving of the tunneling energy, we demonstrate that the low-frequency conductivity of the junction can be enhanced or suppressed, depending on the choice of the driving frequency. The experimental realization of this proposal provides a quantum simulation of optically enhanced superconductivity in pump-probe experiments of high temperature superconductors.

Phys. Rev. Research 3, 013111 (2021)

### Orbital many-body dynamics of bosons in the second Bloch band of an optical lattice

#### M. Nuske, R. Eichberger, C. Hippler, L. Mathey, A. Hemmerich

A Bose-Einstein condensate (BEC) of rubidium atoms is prepared in one of two degenerate energy minima in the second Bloch band of an optical square lattice. A subsequent oscillation of the BEC between the two energy minima is observed, which is driven by two distinct collision processes: the conventional Hubbard-type on-site collision and a collision process that changes the orbital flavor. The oscillation frequency scales with the relative strength of these collisional interactions, which can be readily tuned via an experimentally well controlled distortion of the unit cell. The observations are compared to a quantum model of two single-particle modes and to a semi-classical multi-band tight-binding simulation of 12x12 tubular sites of the lattice. Both models reproduce the observed oscillatory quantum many-body dynamics and show the correct dependence of the oscillation frequency on the ratio between the strengths of the on-site and flavor-changing collision processes.

### Three-body bound states of an atom in a Fermi mixture

#### Ali Sanayei, Ludwig Mathey

We determine the three-body bound states of an atom in a Fermi mixture. Compared to the Efimov spectrum of three atoms in vacuum, we show that the Fermi seas deform the Efimov spectrum systematically. We demonstrate that this effect is more pronounced near unitarity, for which we give an analytical estimate. We show that in the presence of Fermi seas, the three-body bound states obey a generalized discrete scaling law. For an experimental confirmation of our prediction, we propose three signatures of three-body bound states of an ultracold Fermi mixture of Yb isotopes, and provide an estimate for the onset of the bound state and the binding energy.

### Floquet dynamics in light-driven solids

#### M. Nuske, L. Broers, B. Schulte, G. Jotzu, S. A. Sato, A. Cavalleri, A. Rubio, J. W. McIver, L. Mathey

We demonstrate how the properties of light-induced electronic Floquet states in solids impact natural physical observables, such as transport properties, by capturing the environmental influence on the electrons. We include the environment as dissipative processes, such as inter-band decay and dephasing, often ignored in Floquet predictions. These dissipative processes determine the Floquet band occupations of the emergent steady state, by balancing out the optical driving force. In order to benchmark and illustrate our framework for Floquet physics in a realistic solid, we consider the light-induced Hall conductivity in graphene recently reported by J. W. McIver, et al., Nature Physics (2020). We show that the Hall conductivity is estimated by the Berry flux of the occupied states of the light-induced Floquet bands, in addition to the kinetic contribution given by the average band velocity. Hence, Floquet theory provides an interpretation of this Hall conductivity as a geometric-dissipative effect. We demonstrate this mechanism within a master equation formalism, and obtain good quantitative agreement with the experimentally measured Hall conductivity, underscoring the validity of this approach which establishes a broadly applicable framework for the understanding of ultrafast non-equilibrium dynamics in solids.

Phys. Rev. Research 2, 043408 (2020)

### From a continuous to a discrete time crystal

#### Hans Keßler, Jayson G. Cosme, Christoph Georges, Ludwig Mathey, Andreas Hemmerich

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.

### Higgs Time Crystal in a High-Tc Superconductor

#### Guido Homann, Jayson G. Cosme, Ludwig Mathey

We propose to induce a time crystalline state in a high-Tc superconductor, by optically driving a sum resonance of the Higgs mode and a Josephson plasma mode. The generic cubic process that couples these fundamental excitations converts driving of the sum resonance into simultaneous resonant driving of both modes, resulting in an incommensurate subharmonic motion. We use a numerical implementation of a semi-classical driven-dissipative lattice gauge theory on a 3D layered lattice, which models the geometry of cuprate superconductors, to demonstrate the robustness of this motion against thermal fluctuations. We demonstrate this light-induced time crystalline phase for mono- and bilayer systems, and show that this order can be detected for pulsed driving under realistic technological conditions.

Phys. Rev. Research 2, 043214 (2020)

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Time for a new state of matter in high-temperature superconductors SFB925

### Metastable order via 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.

Phys. Rev. Research 2, 043210 (2020)

### Nonequilibrium density wave order in an atom-cavity system

#### Christoph Georges, Jayson G. Cosme, Hans Keßler, Ludwig Mathey, Andreas Hemmerich

We demonstrate the emergence of a nonequilibrium density wave order in a driven-dissipative 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 superradiant checkerboard order of the atoms stabilized by a strong intracavity light field. When in addition the pump is modulated close to a resonance frequency, the atoms occupy higher momentum modes leading to a nonequilibrium subradiant density wave order as pump-induced light scattering into the cavity is suppressed. Our observations together with theoretical modelling reveal the distinct dynamical nature of this higher order density wave state.

### Second sound in the BEC-BCS crossover

#### Daniel Kai Hoffmann, Vijay Pal Singh, Thomas Paintner, Manuel Jäger, Wolfgang Limmer, Ludwig Mathey, Johannes Hecker Denschlag

Second sound is an entropy wave which propagates in the superfluid component of a quantum liquid. Because it is an entropy wave, it probes the thermodynamic properties of the quantum liquid which are determined, e.g., by the interaction strength between the particles of the quantum liquid and their temperature. Here, we study second sound propagation for a large range of interaction strengths within the crossover between a Bose-Einstein condensate (BEC) and the Bardeen-Cooper-Schrieffer (BCS) superfluid. In particular, we investigate the strongly-interacting regime where currently theoretical predictions only exist in terms of an interpolation between the BEC, BCS and unitary regimes. Working with a quantum gas of ultracold fermionic 6Li atoms with tunable interactions, we show that the second sound speed varies only slightly in the crossover regime. We gain deeper insights into sound propagation and excitation of second sound by varying the excitation procedure which ranges from a sudden force pulse to a gentle heating pulse at the cloud center. These measurements are accompanied by classical-field simulations which help with the interpretation of the experimental data. Furthermore, we determine the spatial extension of the superfluid phase and estimate the superfluid density. In the future, this may be used to construct the so far unknown equation of state throughout the crossover.

### Josephson junction dynamics in a two-dimensional ultracold Bose gas

#### Vijay Pal Singh, Niclas Luick, Lennart Sobirey, Ludwig Mathey

We investigate the Berezinskii-Kosterlitz-Thouless (BKT) scaling of the critical current of Josephson junction dynamics across a barrier potential in a two-dimensional (2D) Bose gas, motivated by recent experiments by Luick et al., arXiv:1908.09776. Using classical-field dynamics, we determine the dynamical regimes of this system, as a function of temperature and barrier height. As a central observable we determine the current-phase relation, as a defining property of these regimes. In addition to the ideal junction regime, we find a multimode regime, a second-harmonic regime, and an overdamped regime. For the ideal junction regime, we derive an analytical estimate for the critical current, which predicts the BKT scaling. We demonstrate this scaling behavior numerically for varying system sizes. The estimates of the critical current show excellent agreement with the numerical simulations and the experiments. Furthermore, we show the damping of the supercurrent due to phonon excitations in the bulk, and the nucleation of vortex-antivortex pairs in the junction.

Phys. Rev. Research 2, 033298 (2020)

### Sound propagation in a two-dimensional Bose gas across the superfluid transition

#### Vijay Pal Singh, Ludwig Mathey

Motivated by recent experiments in Phys. Rev. Lett. 121, 145301 (2018), we study sound propagation in a two-dimensional (2D) Bose gas across the superfluid-thermal transition using classical field dynamics. Below the transition temperature we find a Bogoliubov and a non-Bogoliubov mode, above it we find the normal sound mode and the diffusive mode, as we determine from the dynamical structure factor. Our simulations of the experimental procedure agree with the measured velocities, and show that below the transition temperature the measurements detect the Bogoliubov mode. Above the transition, they either detect the normal sound mode for low densities or weak interactions, or the diffusive mode for high densities or strong interactions. As a key observation, we discuss the weak coupling regime in which the non-Bogoliubov mode has a higher velocity than the Bogoliubov mode, in contrast to a hydrodynamic scenario. We propose to detect this regime via step-pulse density perturbation, which simultaneously detects both sound modes.

Phys. Rev. Research 2, 023336 (2020)

**Time crystals in a shaken atom-cavity system**

**Jayson G. Cosme, Jim Skulte, Ludwig Mathey**

We demonstrate the emergence of a time crystal of atoms in a high-finesse optical cavity driven by a phase-modulated transverse pump field, resulting in a shaken lattice. This shaken system exhibits macroscopic oscillations in the number of cavity photons and order parameters at noninteger multiples of the driving period, which signals the appearance of an incommensurate time crystal. The subharmonic oscillatory motion corresponds to dynamical switching between symmetry-broken states, which are nonequilibrium bond ordered density wave states. Employing a semiclassical phase-space representation for the driven-dissipative quantum dynamics, we confirm the rigidity and persistence of the time crystalline phase. We identify experimentally relevant parameter regimes for which the time crystal phase is long-lived, and map out the dynamical phase diagram. We compare and contrast the incommensurate time crystal with the commensurate Dicke time crystal in the amplitude-modulated case.

Phys. Rev. A 100, 053615 (2019)

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

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

Two-dimensional structures are present in almost all known superconductors with high critical temperatures, but the role of the reduced dimensionality is still under debate. Recently, ultracold atoms have emerged as an ideal model system to study such strongly correlated 2D systems. Here, we report on the realisation 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.

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

#### Hans Keßler, Jayson G. Cosme, Michal Hemmerling, Ludwig Mathey, 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.

Phys. Rev. A 99, 053605 (2019)

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

#### Junichi Okamoto, Ludwig Mathey, Wen-Min 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.

Phys. Rev. B 98, 205122 (2018)

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

#### Ilias. M. H. Seifie, Vijay Pal Singh, L. Mathey

We propose a generalization of the Feynman path integral using squeezed coherent states. We apply this approach to the dynamics of Bose-Einstein condensates, which gives an effective low energy description that 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 energy excitations, which provides a description of second sound in weakly interacting condensates as a squeezing oscillation of the order parameter. This interpretation is also supported by a comparison to a numerical c-field method.

Phys. Rev. A 100, 013602 (2019)

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A new path to understanding second sound in Bose-Einstein condensates CUI UHH

### Electron trimer states in conventional superconductors

#### Ali Sanayei, Pascal Naidon, Ludwig Mathey

We expand the Cooper problem by including a third electron in an otherwise empty band. We demonstrate the formation of a trimer state of two electrons above the Fermi sea and the third electron, for sufficiently strong inter-band attractive interaction. We show that the critical interaction strength is the lowest for small Fermi velocities, large masses of the additional electron and large Debye energy. This trimer state competes with the formation of the two-electron Cooper pair, and can be created transiently via optical pumping.

Phys. Rev. Research 2, 013341 (2020)

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

#### Christoph Georges, Jayson G. Cosme, Ludwig Mathey, Andreas Hemmerich

We demonstrate 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 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-Tc cuprates.

Phys. Rev. Lett. 121, 220405 (2018)

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

#### Jayson G. Cosme, Christoph Georges, Andreas Hemmerich, 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, non-equilibrium density wave orders, which do not exist in equilibrium, can be stabilized dynamically. Finally, for strong driving, chaotic dynamics emerges.

Phys. Rev. Lett. 121, 153001 (2018)

Disrupting crystalline order to restore superfluidity CUI UHH

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

#### Koen Sponselee, Lukas Freystatzky, Benjamin Abeln, Marcel Diem, Bastian Hundt, André Kochanke, Thomas Ponath, Bodhaditya Santra, Ludwig Mathey, Klaus Sengstock, Christoph Becker

We employ metastable ultracold 173Yb 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.

Quantum Sci. Technol. 4, 014002 (2018)

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

#### Junichi Okamoto, Wen-Min Huang, Robert Höppner, Ludwig 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.

### Transiently enhanced interlayer tunneling in optically driven high Tc superconductors

#### Jun-ichi Okamoto, Wanzheng Hu, Andrea Cavalleri, Ludwig Mathey

Recent pump-probe experiments reported an enhancement of superconducting transport along the c-axis of underdoped YBa2Cu3O6+δ (YBCO), induced by a mid-infrared optical pump pulse tuned to a specific lattice vibration. To understand this transient non-equilibrium state, we develop a pump-probe formalism for a stack of Josephson junctions, and we consider the tunneling strengths in presence of modulation with an ultrashort optical pulse. We demonstrate that a transient enhancement of the Josephson coupling can be obtained for pulsed excitation and that this can be even larger than in a continuously driven steady-state. Especially interesting is the conclusion that the effect is largest when the material is parametrically driven at a frequency immediately above the plasma frequency, in agreement with what is found experimentally. For bilayer Josephson junctions, an enhancement similar to that experimentally is predicted below the critical temperature Tc. This model reproduces the essential features of the enhancement measured below Tc. To reproduce the experimental results above Tc, we will explore extensions of this model, such as in-plane and amplitude fluctuations, elsewhere.

Phys. Rev. B 96, 144505 (2017)

### Probing optically silent superfluid stripes in cuprates

#### Srivats Rajasekaran, Jun-ichi Okamoto, Ludwig Mathey, Michael Fechner, Vivek Thampy, Genda D. Gu, Andrea Cavalleri

Unconventional superconductivity in the cuprates emerges from, or coexists with, other types of electronic order. However, these orders are sometimes invisible because of their symmetry. For example, the possible existence of superfluid charge stripes in the normal state of single layer cuprates cannot be validated with infrared optics, because interlayer tunneling fluctuations vanish on average. Similarly, it is not easy to establish if charge orders are responsible for dynamical decoupling of the superconducting layers over broad ranges of doping and temperatures. Here, we show that TeraHertz pulses can excite nonlinear tunneling currents between linearly de-coupled charge-ordered planes. A giant TeraHertz third harmonic signal is observed in La1.885Ba0.115CuO4 far above Tc=13 K and up to the charge ordering temperature TCO = 55 K. We model these results by considering large order-parameter-phase oscillations in a pair density wave condensate, and show how nonlinear mixing of optically silent tunneling modes can drive large dipole-carrying super-current oscillations. Our results provide compelling experimental support for the presence of hidden superfluid order in the normal state of cuprates. These experiments also underscore the power of nonlinear TeraHertz optics as a sensitive probe of frustrated excitations in quantum solids.

### 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 first propose a one-dimensional ultracold atom setup to realize such a pair of supersymmetric systems. We propose a Mach-Zehnder interference experiment which we demonstrate for this system, and which can be realized with current technology. In this interferometer, a single atom wave packet that evolves in a superposition of the subsystems, gives an interference contrast that is sharply peaked if the subsystems form a supersymmetric pair. Secondly, we propose a two-dimensional setup that implements supersymmetric dynamics in a synthetic gauge field.

Phys. Rev. A 96, 043624 (2017)

### 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, 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 pseudo-spin texture. The topological defects appear as vortices in the azimuthal phase of this pseudo-spin 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.

Phys. Rev. Lett. 118, 240403 (2017)

### Superfluidity and relaxation dynamics of a laser-stirred 2D Bose gas

#### Vijay Pal Singh, Christof Weitenberg, Jean Dalibard, Ludwig Mathey

We investigate the superfluid behavior of a two-dimensional (2D) Bose gas of 87Rb atoms using classical field dynamics. In the experiment by R. Desbuquois \textit{et al.}, Nat. Phys. \textbf{8}, 645 (2012), a 2D quasicondensate in a trap is stirred by a blue-detuned laser beam along a circular path around the trap center. Here, we study this experiment from a theoretical perspective. The heating induced by stirring increases rapidly above a velocity vc, which we define as the critical velocity. We identify the superfluid, the crossover, and the thermal regime by a finite, a sharply decreasing, and a vanishing critical velocity, respectively. We demonstrate that the onset of heating occurs due to the creation of vortex-antivortex pairs. A direct comparison of our numerical results to the experimental ones shows good agreement, if a systematic shift of the critical phase-space density is included. We relate this shift to the absence of thermal equilibrium between the condensate and the thermal wings, which were used in the experiment to extract the temperature. We expand on this observation by studying the full relaxation dynamics between the condensate and the thermal cloud.

Phys. Rev. A 95, 043631 (2017)

### Fermion pairing in mixed-dimensional atomic mixtures

#### Junichi Okamoto, Ludwig Mathey, 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, dx2−y2-wave, and gxy(x2−y2)-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 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.

Phys. Rev. A 95, 053633 (2017)

### Editorial: Emergence in driven solid-state and cold-atom systems

#### Ludwig Mathey, Junichi Okamoto

We briefly summarize the current status of driven solid-state and cold-atom systems, and introduce articles compiled in the Focus Section in Zeitschrift für Naturforschung A, Volume 71, Issue 10 (2016)

### Observation of a dynamical topological phase transition

#### Nick Fläschner, Dominik Vogel, Matthias Tarnowski, Benno S. Rem, Dirk-Sören Lühmann, Markus Heyl, Jan Carl Budich, Ludwig Mathey, Klaus Sengstock, Christof Weitenberg

Phase transitions are a fundamental concept in science describing diverse phenomena ranging from, e.g., the freezing of water to Bose-Einstein condensation. While the concept is well-established in equilibrium, similarly fundamental concepts for systems far from equilibrium are just being explored, such as the recently introduced dynamical phase transition (DPT). Here we report on the first observation of a DPT in the dynamics of a fermionic many-body state after a quench between two lattice Hamiltonians. With time-resolved state tomography in a system of ultracold atoms in optical lattices, we obtain full access to the evolution of the wave function. We observe the appearance, movement, and annihilation of vortices in reciprocal space. We identify their number as a dynamical topological order parameter, which suddenly changes its value at the critical times of the DPT. Our observation of a DPT is an important step towards a more comprehensive understanding of non-equilibrium dynamics in general.

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

#### Jun-ichi Okamoto, Ludwig Mathey, 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.

Phys. Rev. B 94, 235411 (2016)

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

#### Jun-ichi Okamoto, Andrea Cavalleri, Ludwig 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.

Phys. Rev. Lett. 117, 227001 (2016)

### Designing exotic many-body states of atomic spin and motion in photonic crystals

#### Marco T. Manzoni, Ludwig Mathey, Darrick E. Chang

Cold atoms coupled to photonic crystals constitute an exciting platform for exploring quantum many-body physics. Here we investigate the strong coupling between atomic internal ("spin") degrees of freedom and motion, which arises from spin-dependent forces associated with the exchange of guided photons. We show that this system can realize a remarkable and extreme limit of quantum spin-orbital systems, where both the direct spin exchange between neighboring sites and the kinetic energy of the orbital motion vanish. We find that this previously unexplored system has a rich phase diagram of emergent orders, including spatially dimerized spin-entangled pairs, a fluid of composite particles comprised of joint spin-phonon excitations, phonon-induced Néel ordering, and a fractional magnetization plateau associated with trimer formation.

### 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.

### 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 ultra-cold 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. Non-zero 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 non-interacting 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 anti-correlations near the Dirac point.

Phys. Rev. A 94, 023607 (2016)

### Realizing and optimizing an atomtronic SQUID

#### Amy 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.

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

#### Peter Janzen, Wen-Min Huang, Ludwig Mathey

The recent experimental condensation of ultracold atoms in a triangular optical lattice with negative effective tunneling energies paves the way to study 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 inter-component 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.

Phys. Rev. A 94, 063614 (2016)

### 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)

### Optimization of collisional Feshbach cooling of an ultracold nondegenerate gas

#### Marlon Nuske, Eite Tiesinga, L. Mathey

We optimize a collision-induced cooling process for ultracold atoms in the nondegenerate regime. It makes use of a Feshbach resonance, instead of rf radiation in evaporative cooling, to selectively expel hot atoms from a trap. Using functional minimization we analytically show that for the optimal cooling process the resonance energy must be tuned such that it linearly follows the temperature. Here, optimal cooling is defined as maximizing the phase-space density after a fixed cooling duration. The analytical results are confirmed by numerical Monte-Carlo simulations. In order to simulate more realistic experimental conditions, we show that background losses do not change our conclusions, while additional non-resonant two-body losses make a lower initial resonance energy with non-linear dependence on temperature preferable.

Phys. Rev. A 91, 043626 (2015)

### Exotic roton excitations in quadrupolar Bose-Einstein condensates

#### M. Lahrz, Mikhail Lemeshko, L. Mathey

We investigate the occurrence of rotons in a quadrupolar Bose-Einstein condensate confined to two dimensions. Depending on the particle density, the ratio of the contact and quadrupole-quadrupole interactions, and the alignment of the quadrupole moments with respect to the confinement plane, the dispersion relation features two or four point-like roton minima, or one ring-shaped minimum. We map out the entire parameter space of the roton behavior and identify the instability regions. We propose to observe the exotic rotons by monitoring the characteristic density wave dynamics resulting from a short local perturbation, and discuss the possibilities to detect the predicted effects in state-of-the-art experiments with ultracold homonuclear molecules.

### Observing 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.

Phys. Rev. Lett. 114, 115301 (2015)

### 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 ultra-narrow 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

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

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

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

We study the thermally fluctuating state of a bi-layer 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 YBa2Cu3O6+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 onto the in-plane fluctuations, which are reduced on long length scales. All these findings provide a physical framework to describe light control in cuprates.

Phys. Rev. B 91, 104507 (2015)

### Noise correlations of two-dimensional Bose gases

#### Vijay Pal Singh, Ludwig 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 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 quasi-condensed phase, due to constructive interference. We argue that this can be used to extract the scaling exponent of the quasi-condensate 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.

Phys. Rev. A 89, 053612 (2014)

### Detecting quadrupole interactions in ultracold Fermi gases

#### M. Lahrz, Mikhail Lemeshko, Klaus Sengstock, Christoph Becker, L. Mathey

We propose to detect quadrupole interactions of neutral ultra-cold atoms via their induced mean-field shift. We consider a Mott insulator state of spin-polarized atoms in a two-dimensional optical square lattice. The quadrupole moments of the atoms are aligned by an external magnetic field. As the alignment angle is varied, the mean-field shift shows a characteristic angular dependence, which constitutes the defining signature of the quadrupole interaction. For the 3P2 states of Yb and Sr atoms, we find a frequency shift of the order of tens of Hertz, which can be realistically detected in experiment with current technology. We compare our results to the mean-field shift of a spin-polarized quasi-2D Fermi gas in continuum.

Phys. Rev. A 89, 043616 (2014)

### Quantum phases of quadrupolar Fermi gases in coupled one-dimensional systems

#### Wen-Min Huang, M. Lahrz, L. Mathey

Following the recent proposal to create quadrupolar gases [S.G. Bhongale et al., Phys. Rev. Lett. 110, 155301 (2013)], we investigate what quantum phases can be created in these systems in one dimension. We consider a geometry of two coupled one-dimensional systems, and derive the quantum phase diagram of ultra-cold fermionic atoms interacting via quadrupole-quadrupole interaction within a Tomonaga-Luttinger-liquid framework. We map out the phase diagram as a function of the distance between the two tubes and the angle between the direction of the tubes and the quadrupolar moments. The latter can be controlled by an external field. We show that there are two magic angles θcB,1 and θcB,2 between 0 to π/2, where the intratube quadrupolar interactions vanish and change signs. Adopting a pseudo-spin language with regards to the two 1D systems, the system undergoes a spin-gap transition and displays a zig-zag density pattern, above θcB,2 and below θcB,1. Between the two magic angles, we show that polarized triplet superfluidity and a planar spin-density wave order compete with each other. The latter corresponds to a bond order solid in higher dimensions. We demonstrate that this order can be further stabilized by applying a commensurate periodic potential along the tubes.

Phys. Rev. A 89, 013604 (2014)

### 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.

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

#### Julian Struck, Malte Weinberg, Christoph Ölschläger, Patrick Windpassinger, Juliette Simonet, Klaus Sengstock, Robert Höppner, Philipp Hauke, André Eckardt, Maciej Lewenstein, Ludwig Mathey

Emulation of gauge fields for ultracold atoms provides access to a class of exotic states arising in strong magnetic fields. Here we report on the experimental realisation of tunable staggered gauge fields in a periodically driven triangular lattice. For maximal staggered magnetic fluxes, the doubly degenerate superfluid ground state breaks both a discrete Z2 (Ising) symmetry and a continuous U(1) symmetry. By measuring an Ising order parameter, we observe a thermally driven phase transition from an ordered antiferromagnetic to an unordered paramagnetic state and textbook-like magnetisation curves. Both the experimental and theoretical analysis of the coherence properties of the ultracold gas demonstrate the strong influence of the Z2 symmetry onto the condensed phase.

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

#### Satyan G. Bhongale, Ludwig Mathey, Erhai Zhao, Susanne F. Yelin, Mikhail Lemeshko

We introduce a new platform for quantum simulation of many-body systems based on nonspherical atoms or molecules with zero dipole moment but possessing a significant value of electric quadrupole moment. We consider a quadrupolar Fermi gas trapped in a 2D square optical lattice, and show that the peculiar symmetry and broad tunability of the quadrupole-quadrupole interaction results in a rich phase diagram encompassing unconventional BCS and charge density wave phases, and opens up a perspective to create topological superfluid. 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)

### Unconventional Spin Density Waves in Dipolar Fermi Gases

#### S. G. Bhongale, L. Mathey, Shan-Wen Tsai, Charles W. Clark, Erhai 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, ℓ=0, analogous to a condensate of particle-particle pairs in conventional superconductors. While many unconventional superconductors with Cooper pairs of finite ℓ 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 ℓ=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 (2013)

### Intrinsic Photoconductivity of Ultracold Fermions in Optical Lattices

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

We report on the experimental observation of an analog to 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. While particles in the excited band exhibit long-lived oscillations with a momentum dependent frequency a strikingly different behavior is observed for holes in the lowest band. An initial fast collapse is followed by subsequent periodic revivals. Both observations are fully explained by mapping the system onto a nonlinear pendulum.

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

### 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 phase imprinting technique. 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 the 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, in contradiction to the predictions of the zero-temperature Gross-Pitaevskii equation. We map out the superfluid decay rate and critical velocity as a function of temperature and observe a strong dependence. Thermal fluctuations offer a partial explanation of the experimentally observed reduction of the critical velocity from the phonon velocity.

Phys. Rev. A 90, 023604 (2014)

### Counterflow superfluid of polaron pairs in Bose-Fermi mixtures in optical lattices

#### Ippei Danshita, L. Mathey

We study the quantum phases of one-dimensional Bose-Fermi mixtures in optical lattices. Assuming repulsive interparticle interactions, equal mass, and unit total filling, we calculate the ground-state phase diagram by means of both Tomonaga-Luttinger liquid theory and time-evolving block decimation method. We demonstrate the existence of a counterflow superfluid (CFSF) phase of polaron pairs, which are composite particles consisting of two fermions and two bosonic holes, in a broad range of the parameter space. We find that this phase naturally emerges in 174Yb-173Yb mixtures, realized in recent experiments, at low temperatures.

Phys. Rev. A 87, 021603(R), (2013)

### Dynamic Kosterlitz-Thouless transition in 2D Bose mixtures of ultra-cold atoms

#### L. Mathey, Kenneth J. Günter, Jean Dalibard, A. Polkovnikov

We propose a realistic experiment to demonstrate a dynamic Kosterlitz-Thouless transition in ultra-cold 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.

Phys. Rev. A 95, 053630 (2017)

### Bond order solid of two-dimensional dipolar fermions

#### S. G. Bhongale, L. Mathey, Shan-Wen Tsai, Charles W. Clark, Erhai Zhao

Recent experimental realization of dipolar Fermi gases near or below quantum degeneracy provides opportunity to engineer Hubbard-like models with long range interactions. Motivated by these experiments, we chart out the theoretical phase diagram of interacting dipolar fermions on the square lattice at zero temperature and half filling. We show that in addition to p-wave superfluid and charge density wave order, two new and exotic types of bond order emerge generically in dipolar fermion systems. These phases feature homogeneous density but periodic modulations of the kinetic hopping energy between nearest or next-nearest neighbors. Similar, but manifestly different, phases of two-dimensional correlated electrons have previously only been hypothesized and termed "density waves of nonzero angular momentum". Our results suggest that these phases can be constructed flexibly with dipolar fermions, using currently available experimental techniques.

Phys. Rev. Lett. 108, 145301 (2012)

### Detecting paired and counterflow superfluidity via dipole oscillations

#### Anzi Hu, L. Mathey, Eite Tiesinga, Ippei Danshita, Carl J. Williams, Charles W. Clark

We suggest an experimentally feasible procedure to observe paired and counterflow superfluidity in ultra-cold atom systems. We study the time evolution of one-dimensional mixtures of bosonic atoms in an optical lattice following an abrupt displacement of an additional weak confining potential. We find that the dynamic responses of the paired superfluid phase for attractive inter-species interactions and the counterflow superfluid phase for repulsive interactions are qualitatively distinct and reflect the quasi long-range order that characterizes these states. These findings suggest a clear experimental procedure to detect these phases, and give an intuitive insight into their dynamics.

Phys. Rev. A 84, 041609(R), (2011)

### Phase fluctuations in anisotropic Bose condensates: from cigars to rings

#### L. Mathey, A. Ramanathan, K. C. Wright, S. R. Muniz, W. D. Phillips, Charles W. Clark

We study the phase-fluctuating condensate regime of ultra-cold atoms trapped in a ring-shaped trap geometry, which has been realized in recent experiments. We first consider a simplified box geometry, in which we identify the conditions to create a state that is dominated by thermal phase-fluctuations, and then explore the experimental ring geometry. In both cases we demonstrate that the requirement for strong phase fluctuations can be expressed in terms of the total number of atoms and the geometric length scales of the trap only. For the ring-shaped trap we discuss the zero temperature limit in which a condensate is realized where the phase is fluctuating due to interactions and quantum fluctuations. We also address possible ways of detecting the phase fluctuating regime in ring condensates.

Phys. Rev. A 82, 033607 (2010)

### Noise correlations of one-dimensional Bose mixtures in optical lattices

#### Anzi Hu, L. Mathey, Carl J. Williams, Charles W. Clark

We study the noise correlations of one-dimensional binary Bose mixtures, as a probe of their quantum phases. In previous work, we found a rich structure of many-body phases in such mixtures, such as paired and counterflow superfluidity. Here we investigate the signature of these phases in the noise correlations of the atomic cloud after time-of-flight expansion, using both Luttinger liquid theory and the time-evolving block decimation (TEBD) method. We find that paired and counterflow superfluidity exhibit distinctive features in the noise spectra. We treat both extended and inhomogeneous systems, and our numerical work shows that the essential physics of the extended systems is present in the trapped-atom systems of current experimental interest. For paired and counterflow superfluid phases, we suggest methods for extracting Luttinger parameters from noise correlation spectroscopy.

Phys. Rev. A 81, 063602 (2010)

### Light cone dynamics and reverse Kibble-Zurek mechanism in two-dimensional superfluids following a quantum quench

#### L. Mathey, A. Polkovnikov

We study the dynamics of the relative phase of a bilayer of two-dimensional superfluids after the two superfluids have been decoupled. We find that on short time scales the relative phase shows "light cone" like dynamics and creates a metastable superfluid state, which can be supercritical. We also demonstrate similar light cone dynamics for the transverse field Ising model. On longer time scales the supercritical state relaxes to a disordered state due to dynamical vortex unbinding. This scenario of dynamically suppressed vortex proliferation constitutes a reverse-Kibble-Zurek effect. We study this effect both numerically using truncated Wigner approximation and analytically within a newly suggested time dependent renormalization group approach (RG). In particular, within RG we show that there are two possible fixed points for the real time evolution corresponding to the superfluid and normal steady states. So depending on the initial conditions and the microscopic parameters of the Hamiltonian the system undergoes a non-equilibrium phase transition of the Kosterlitz-Thouless type. The time scales for the vortex unbinding near the critical point are exponentially divergent, similar to the equilibrium case.

Phys. Rev. A 81, 033605 (2010)

### Counterflow and paired superfluidity in one-dimensional Bose mixtures in optical lattices

#### Anzi Hu, L. Mathey, Ippei Danshita, Eite Tiesinga, Carl J. Williams, Charles W. Clark

We study the quantum phases of mixtures of ultra-cold bosonic atoms held in an optical lattice that confines motion or hopping to one spatial dimension. The phases are found by using Tomonaga-Luttinger liquid theory as well as the numerical method of time evolving block decimation (TEBD). We consider a binary mixture with repulsive intra-species interactions, and either repulsive or attractive inter-species interaction. For a homogeneous system, we find paired- and counterflow-superfluid phases at different filling and hopping energies. We also predict parameter regions in which these types of superfluid order coexist with charge density wave order. We show that the Tomonaga-Luttinger liquid theory and TEBD qualitatively agree on the location of the phase boundary to superfluidity. We then describe how these phases are modified and can be detected when an additional harmonic trap is present. In particular, we show how experimentally measurable quantities, such as time-of-flight images and the structure factor, can be used to distinguish the quantum phases. Finally, we suggest applying a Feshbach ramp to detect the paired superfluid state, and a π/2 pulse followed by Bragg spectroscopy to detect the counterflow superfluid phase.

Phys. Rev. A 80, 023619 (2009)

### A supercritical superfluid and vortex unbinding following a quantum quench

#### L. Mathey, A. Polkovnikov

We study the dynamics of the relative phase of a bilayer of two-dimensional superfluids after the two superfluids have been decoupled, using truncated Wigner approximation. On short time scales the relative phase shows "light cone" like thermalization and creates a metastable superfluid state, which can be supercritical. On longer time scales this state relaxes to a disordered state due to dynamical vortex unbinding. This scenario of dynamically suppressed vortex proliferation constitutes a {\it reverse-Kibble-Zurek effect}. We observe dynamics of creation of vortex anti-vortex pairs and their consequent motion. Our predictions can be directly measured in interference experiments, see Ref 1.

Phys. Rev. A 80, 041601(R), (2009)

### Ultracold Atomic Gases: Novel States of Matter

#### L. Mathey, S.-W. Tsai, A. H. Castro Neto

Article to appear in the Encyclopedia of Complexity and Systems Science, Dr. R. A. Meyers (Ed.) (Springer Heidelberg, 2009).

### Collisional cooling of ultra-cold atom ensembles using Feshbach resonances

#### L. Mathey, Eite Tiesinga, Paul S. Julienne, Charles W. Clark

We propose a new type of cooling mechanism for ultra-cold fermionic atom ensembles, which capitalizes on the energy dependence of inelastic collisions in the presence of a Feshbach resonance. We first discuss the case of a single magnetic resonance, and find that the final temperature and the cooling rate is limited by the width of the resonance. A concrete example, based on a p-wave resonance of 40K, is given. We then improve upon this setup by using both a very sharp optical or radio-frequency induced resonance and a very broad magnetic resonance and show that one can improve upon temperatures reached with current technologies.

Phys. Rev. A 80, 030702(R), (2009)

### Noise Correlations in low-dimensional systems of ultracold atoms

#### L. Mathey, A. Vishwanath, E. Altman

We derive relations between standard order parameter correlations and the noise correlations in time of flight images, which are valid for systems with long range order as well as low dimensional systems with algebraic decay of correlations. Both Bosonic and Fermionic systems are considered. For one dimensional Fermi systems we show that the noise correlations are equally sensitive to spin, charge and pairing correlations and may be used to distinguish between fluctuations in the different channels. This is in contrast to linear response experiments, such as Bragg spectroscopy, which are only sensitive to fluctuations in the particle-hole channel (spin or charge). For Bosonic systems we find a sharp peak in the noise correlation at opposite momenta that signals pairing correlations in the depletion cloud. In a condensate with true long range order, this peak is a delta function and we can use Bogoliubov theory to study its temperature dependence. Interestingly we find that it is enhanced with temperature in the low temperature limit. In one dimensional condensates with only quasi-long range (i.e. power-law) order the peak in the noise correlations also broadens to a power-law singularity.

Phys. Rev. A 79, 013609 (2009)

### Creating a supersolid in one-dimensional Bose mixtures

#### L. Mathey, Ippei Danshita, Charles W. Clark

We identify a one-dimensional supersolid phase in a binary mixture of near-hardcore bosons with weak, local inter-species repulsion. We find realistic conditions under which such a phase, defined here as the coexistence of quasi-superfluidity and quasi-charge density wave order, can be produced and observed in finite ultra-cold atom systems in a harmonic trap. Our analysis is based on Luttinger liquid theory supported with numerical calculations using the time-evolving block decimation method. Clear experimental signatures of these two orders can be found, respectively, in time-of-flight interference patterns, and the structure factor S(k) derived from density correlations.

Phys. Rev. A 79, 011602(R), (2009)

### Phase-locking transition of coupled low-dimensional superfluids

#### L. Mathey, A. Polkovnikov, A.H. Castro Neto

We study the phase-locking transition of two coupled low-dimensional superfluids, either two-dimensional superfluids at finite temperature, or one-dimensional superfluids at zero temperature. We find that the superfluids have a strong tendency to phase-lock. The phase-locking is accompanied by a sizeable increase of the transition temperature Tc (in 2D systems) of the resulting double-layer superfluid to thermal Bose gas transition, compared to the Kosterlitz-Thouless temperature TKT of the uncoupled 2D systems, which suggests a plausible way of observing the Kibble-Zurek mechanism in two-dimensional cold atom systems by rapidly varying the tunneling rate between the superfluids. If there is also interaction between atoms in different layers present we find additional phases, while no sliding phase, characterized by order or quasi long range order (QLRO) either in the symmetric or the antisymmetric sector of the system.

### Exotic Superconducting Phases of Ultracold Atom Mixtures on Triangular Lattices

#### L. Mathey, S.-W. Tsai, A.H. Castro Neto

We study the phase diagram of two-dimensional Bose-Fermi mixtures of ultracold atoms on a triangular optical lattice, in the limit when the velocity of bosonic condensate fluctuations is much larger than the Fermi velocity.

We contrast this work with our previous results for a square lattice system in Phys. Rev. Lett. {\bf 97}, 030601 (2006).

Using functional renormalization group techniques we show that the phase diagrams for a triangular lattice contain exotic superconducting phases. For spin-1/2 fermions on an isotropic lattice we find a competition of s-, p-, extended d-, and f-wave symmetry, as well as antiferromagnetic order. For an anisotropic lattice, we further find an extended p-wave phase. A Bose-Fermi mixture with spinless fermions on an isotropic lattice shows a competition between p- and f-wave symmetry.

These phases can be traced back to the geometric shapes of the Fermi surfaces in various regimes, as well as the intrinsic frustration of a triangular lattice.

Phys. Rev. B 75, 174516 (2007)

### Commensurate mixtures of ultra-cold atoms in one dimension

#### L. Mathey

We study binary mixtures of ultra-cold atoms, confined to one dimension in an optical lattice, with commensurate densities. Within a Luttinger liquid description, which treats various mixtures on equal footing, we derive a system of renormalization group equations at second order, from which we determine the rich phase diagrams of these mixtures. These phases include charge/spin density wave order, singlet and triplet pairing, polaron pairing, and a supersolid phase. Various methods to detect our results experimentally are discussed.

Phys. Rev. B 75, 144510 (2007)

### Phase diagrams of one-dimensional Bose-Fermi mixtures of ultra-cold atoms

#### L. Mathey, D.-W. Wang

We study the quantum phase diagrams of Bose-Fermi mixtures of ultracold atoms confined to one dimension in an optical lattice. For systems with incommensurate densities, various quantum phases, e.g. charge/spin density waves, pairing, phase separation, and the Wigner crystal, are found to be dominant in different parameter regimes within a bosonization approach. The structure of the phase diagram leads us to propose that the system is best understood as a Luttinger liquid of polarons (i.e. atoms of one species surrounded by screening clouds of the other species). Special fillings, half-filling for fermions and unit filling for bosons, and the resulting gapped phases are also discussed, as well as the properties of the polarons and the experimental realization of these phases.

Phys. Rev. A 75, 013612 (2007)

### Competing Orders in two-dimensional Bose-Fermi mixtures

#### L. Mathey, S-W. Tsai, A.H. Castro Neto

Using a functional renormalization group approach we study the zero temperature phase diagram of two-dimensional Bose-Fermi mixtures of ultra-cold atoms in optical lattices, in the limit when the velocity of bosonic condensate fluctuations are much larger than the Fermi velocity.

For spin-1/2 fermions we obtain a phase diagram, which shows a competition of pairing phases of various orbital symmetry (s, p, and d) and antiferromagnetic order. We determine the value of the gaps of various phases close to half-filling, and identify subdominant orders as well as short-range fluctuations from the RG flow. For spinless fermions we find that p-wave pairing dominates the phase diagram.

Phys. Rev. Lett 97, 030601 (2006)

### Noise Correlations in one-dimensional systems of ultra-cold fermions

#### L.Mathey, E.Altman, A.Vishwanath

Time of flight images reflect the momentum distribution of the atoms in the trap, but the spatial noise in the image holds information on more subtle correlations. Using Bosonization, we study such noise correlations in generic one dimensional systems of ultra cold fermions. Specifically, we show how pairing as well as spin and charge density wave correlations may be identified and extracted from the time of flight images. These incipient orders manifest themselves as power law singularities in the noise correlations, that depend on the Luttinger parameters, which suggests a general experimental technique to obtain them.

Phys. Rev. Lett 100, 240401 (2008)

### Luttinger liquid of polarons in one-dimensional boson-fermion mixtures

#### L. Mathey, D.-W. Wang, W. Hofstetter, M. D. Lukin, Eugene Demler

We use bosonization approach to investigate quantum phases in mixtures of bosonic and fermionic atoms confined in one dimensional optical lattices. The phase diagrams can be well understood in terms of polarons, which correspond to atoms that are "dressed" by screening clouds of the other atom species. For a mixture of single species of fermionic and bosonic atoms we find a charge density wave phase, a phase with fermion pairing, and a regime of phase separation. For a mixture of two species of fermionic atoms and one species of bosonic atoms we obtain spin and charge density wave phases, a Wigner crystal phase, singlet and triplet paired states of fermions, and a phase separation regime. Equivalence between the Luttinger liquid description of polarons and the canonical polaron transformation is established and the techniques to detect the resulting quantum phases are discussed.

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

We demonstrate 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...

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

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...

### Electron trimer states in conventional superconductors

We expand the Cooper problem by including a third electron in an otherwise empty band. We demonstrate the formation of a trimer state of two electrons above the Fermi sea and the third electron, for sufficiently strong inter-band attractive...