Novel Paradigms in Many-Body Physics from Open Quantum Systems

Superradiant phases occur when matter-light coupling is strong enough to sustain a steady state with non-zero photon numbers. For the open Dicke model, this requires counter-rotating matter-light coupling to create photons at a rate to compensate cavity losses. This mechanism differs from that of a laser, where counter-rotating terms are not important, but where inversion due to pumping is instead required. We show how a driven dissipative Dicke model can support both "lasing" and "superradiant" phases, and how these can be distinguished. We find that the competition between pumping and counter-rotating terms can produce chaotic phases [1,2]. The need for counter-rotating terms to balance cavity loss means that for such an open single model problem, continuous symmetry breaking is not possible. However, as recently realized experimentally [3], one can engineer models with continuous symmetries by coupling to two cavity modes. Considering specifically internal states of atoms, we show an extended range of parameters for which continuous symmetry breaking can occur, and classify the distinct steady and time-dependent states that arise in this extended parameter regime [4]. [1] P. Kirton and J. Keeling, New J. Phys, Accepted online [2] P. Kirton and J. Keeling, Phys. Rev. Lett. 118 123602 (2017) [3] J. Leonard et al., Nature 543 87 (2017) [4] R. I. Moodie, K. E. Ballantine and J. Keeling, arXiv:1711.03915

Nonequilibrium phase transitions exist in damped-driven open quantum systems when the continuous tuning of an external parameter leads to a transition between two robust steady states. In second-order transitions this change is abrupt at a critical point, whereas in first-order transitions the two phases can coexist in a critical hysteresis domain. We report the observation of a first-order dissipative quantum phase transition in a driven circuit quantum electrodynamics system. It takes place when the photon blockade of the driven cavity-atom system is broken by increasing the drive power. The observed experimental signature is a bimodal phase space distribution with varying weights controlled by the drive strength. Our measurements show an improved stabilization of the classical attractors up to the millisecond range when the size of the quantum system is increased from one to three artificial atoms.

Bose-Einstein condensation has been observed with cold atomic gases, exciton-polaritons, and more recently with photons in a dye-solution filled optical microcavity. I will here describe measurements of my Bonn group observing the transition between usual lasing dynamics and photon Bose-Einstein condensation. The photon Bose-Einstein condensate is generated in a wavelength-sized optical cavity, where the small mirror spacing imprints a low-frequency cutoff and photons confined in the resonator thermalize to room temperature by absorption re-emission processes on the dye molecules. This allows for a particle-number conserving thermalization, with photons showing a thermodynamic phase transition to a macroscopically occupied ground state, the Bose-Einstein condensate. When the thermalization by absorption and re-emission is faster than the photon loss rate in the cavity, the photons accumulate at lower energy states above the cavity cutoff, and the system finally thermalizes to a Bose-Einstein condensate of photons. On the other hand, for a small reabsorption with respect to the photon loss, the state remains laser-like. I will also report recent measurements of the heat capacity of the photon gas, which were performed under the conditions of the thermalization being much faster than both photon loss and pumping. At the Bose-Einstein phase transition, the observed specific heat shows a cusp-like singularity, as in the λ-transition of liquid helium, illustrating critical behavior of the photon gas. In my talk, I will begin with a general introduction and give an account of current work and future plans of the Bonn photon gas experiment.

The equilibrium configuration of a chain of trapped ions subjected to an optical potential depends, in general, on the competition between the forces resulting from the trap, the Coulomb repulsion, and the optical field. We theoretically explore new possibilities that come into play when the optical potential is provided by a laser-pumped high-quality resonator: photons leaking out of the cavity can be used to non-destructively monitor the system, and the optomechanical coupling of fluctuations allows one to implement cavity cooling of the chain motion, to the ground state in appropriate regimes. Furthermore, when the back-action of the particles on the field intensity is relevant, it leads to an infinitely ranged effective interaction between ions, introducing drastic changes in the critical properties of the chain across structural transitions. In particular, we study the transitions between linear and zigzag configurations, and between sliding and pinned phases, and observe the appearance of hysteresis and bistability in otherwise continuous transitions. Finally, we show that this kind of setup can also be used to study the transition between commensurate and incommensurate configurations, namely, a regime where the interparticle distance is pinned at a multiple of the potential periodicity, and another one in which particle density can vary due to the formation of defects.

The coupling of a quantum gas to the field of an optical high-finesse cavity can be employed to induce global-range atomic interactions. If these are sufficiently strong, such a driven-dissipative many-body system undergoes a structural phase transition. Introducing a 3D optical lattice to this system, the collisional short-range interactions can be brought to competition with these global-range interactions and – at the same time – with the zero-point motion of the particles. The resulting phase diagram hosts four distinct phases – a superfluid, a lattice supersolid, a Mott insulator and a charge density wave. We study the metastable dynamics when driving the system from the Mott insulator to the charge density wave phase. In a different set of experiments, we couple a superfluid cloud of atoms simultaneously to two intersecting optical cavities. This arrangement leads to a symmetry enhancement and the resulting system exhibits a continuous spatial U(1)-symmetry. The combination of two continuous symmetries – the gauge symmetry of the superfluid and the spatial symmetry – is a prerequisite for a supersolid state of matter, which we investigate in our experiments.

I will speak about Bose condensation in non-equilibrium steady states of driven-dissipative Bose gases, considering three basic scenarios: periodically forced (Floquet) systems coupled to a heat bath [1], systems coupled to two heat baths of different temperature [2], and pumped lossy photonic systems interacting with a heat bath [3]. Unlike equilibrium states, which are determined completely by a few thermodynamic variables like the bath temperature only, non-equilibrium steady states obey less restrictions and depend sensitively on the very details of the environment. This offers great freedom to tailor the properties of a quantum system by engineering its environment. Among others, we show that this freedom can be used for the robust preparation of excited-state and fragmented Bose condensates. We also demonstrate that Bose condensation can be induced by coupling a system to two competing baths, both of which have temperatures well above the equilibrium critical temperature [2]. Moreover, for a broad class of models describing a variety of complex photonic systems, we predict a cascade of transitions when the pump power is ramped up [3]: First, above a threshold, the mode with the largest effective gain becomes macroscopically occupied (corresponding to simple lasing). Ramping up the pump further, further transitions can occur where single modes acquire or loose macroscopic occupation, eventually leading to a macroscopic occupation of the ground state alone (resembling equilibrium Bose condensation). Our theory describes experimental data for a two-mode microcavity and exciton-polaritons in a double well [Galbiati et al. PRL 108, 126403 (2012)]. Based on references: [1] D. Vorberg et al., PRL 111, 240405 (2013) & PRE 92, 062119 (2015) [2] A. Schnell et al., PRL 119, 140602 (2017) [3] A. Leymann et al., PRX 7, 021045 (2017); D. Vorberg, R. Ketzmerick, A. Eckardt, in preparation

I report on recent work on periodically-driven (Floquet) many-body systems, focussing on the case where nontrivial phases (including a pi-spin glass or "discrete time crystal") are found.

In this presentation I introduce {\it boundary time-crystals}. Here {\it continuous} time-translation symmetry breaking occurs at the boundary (or generically in a macroscopic portion) of a many-body quantum system. I describe definition and properties, and then analyse in details a solvable model. I provide examples of other systems where boundary time crystalline phases can occur and underline the intimate connection to the emergence of a time-periodic steady state in the thermodynamic limit.

Open dissipative systems have provided a mean to study many-body phenomena well beyond isolated systems obeying unitary evolution. The next step enriching physical picture is to exploit the quantum nature of the measurement process for preparing nontrivial many-body states, which can be then controlled using feedback. Such a setting includes the description of dissipation as a special case (by the ignorance of measurement results), but enables to achieve many-body states and phenomena, which are typical to neither closed systems described by Hamiltonians, nor open systems described by e.g. Lindblad master equation [1,2]. Considering ultracold bosons and fermions in optical lattices, we show that it is not only density variables that are accessible by quantum measurements of scattered light, but the matter-phase variables (intersite bonds or links) as well [1]. We prove that the quantum backaction of weak global measurement constitutes a novel source of competitions in many-body systems [3], leading to novel effects: multimode oscillations of macroscopic superposition states, protection and break-up of fermion pairs [3], as well as generation of antiferromagnetic states [4]. Novel processes beyond the standard Hubbard models can be designed by the measurement, entering the field of non-Hermitian (while being non-dissipative) many-body physics: long-range correlated pair tunnelling and Raman-like second-order transitions beyond the typical quantum Zeno dynamics [1,5]. We show that the feedback control can drive the system to stable many-body strongly correlated phases [2]. [1] W. Kozlowski, S. F. Caballero-Benitez, and I. B. Mekhov, Scientific Rep. 7, 42597 (2017); [2] G. Mazzucchi, S. F. Caballero-Benitez, D. A. Ivanov, and I. B. Mekhov, Optica (OSA) 3, 1213 (2016); [3] G. Mazzucchi, W. Kozlowski, S. F. Caballero-Benitez, T. J. Elliott, and I. B. Mekhov, Phys. Rev. A 93, 023632 (2016); [4] G. Mazzucchi, S. F. Caballero-Benitez, and I. B. Mekhov, Scientific Rep. 6, 31196 (2016); [5] W. Kozlowski and I. B. Mekhov, Phys. Rev. A 94, 012123 (2016).

There have been concerted efforts in recent years to realize the next generation of clocks using alkaline earth atoms in an optical lattice. Assuming that the atoms are independent, such a clock would benefit from a sqrt(N) enhancement in its stability, associated with the improved signal-to-noise ratio of a large atom number. Any kind of atomic interactions, however, could result in a clock inaccuracy. Here, we investigate the effect of one type of interaction, dipole-dipole, in which atoms excited during the clock protocol emit and re-absorb photons. Taking a simple system consisting of a 1D atomic array, we find that dipole-dipole interactions in fact result in open critical dynamics, as a set of collective excitations acquire a decay rate approaching zero in the thermodynamic limit due to subradiance. As one particular consequence, the decay of atomic excited population at long times exhibits power-law behavior, instead of the exponential expected for non-interacting atoms. The population distribution also exhibits fermionic spatial correlations at long times, due to the microscopic properties of the subradiant states. These results suggest that lattice clocks and collectively emitting atomic ensembles in general constitute rich many-body open systems.

We discuss ideas for developing an understanding of how quantum effects may impact on the dynamics of neural networks. We implement the dynamics of neural networks in terms of Markovian open quantum systems, which allows us to treat thermal and quantum coherent effects on the same footing. This leads quite naturally to an open quantum generalisation of the Hopfield neural network, the simplest toy model of associative memory. We determine its phase diagram and show that quantum fluctuations may give rise to a qualitatively new non-equilibrium phase. This phase is characterised by limit cycles corresponding to high-dimensional stationary manifolds that may be regarded as a generalisation of storage patterns to the quantum domain.

Algebraic order in two-dimensional systems with U(1) symmetry is destroyed by even the slightest deviation from detailed-balance conditions which are characteristic of thermal equilibrium. Such a violation of equilibrium conditions alters the interaction of topological defects in a drastic way: instead of being long-ranged and thus capable of holding together pairs of vortices and anti-vortices in the ordered phase, it becomes screened, and defects proliferate. Here, we show that the structure of defects and their interaction can equally dramatically be modified by breaking rotational symmetry. For sufficiently strong spatial anisotropy, the force that binds pairs of defects can even be enhanced up to parametrically large scales. As a consequence, the vortex-unbinding crossover in such finite-size systems exhibits peculiar universal behavior. In the thermodynamic limit, we argue that the modified structure of defects allows for a stable ordered phase. These results, which we obtain by analyzing the compact anisotropic Kardar-Parisi-Zhang (caKPZ) equation, are relevant for a wide variety of physical systems, ranging from strongly coupled light-matter quantum systems to recently proposed classical time crystals.

Dark states are stationary states of a dissipative, Lindblad-type time evolution with zero von Neumann entropy, therefore representing examples of pure, steady quantum states. Quantum dynamics featuring a dark state recently gained a lot of attraction since their implementation in the context of driven-open quantum systems represents a viable possibility to engineer unique, pure quantum states without the problem of extensive cooling. In this work, we analyze a driven many-body spin system, which undergoes a transition from a dark steady state to a mixed steady state as a function of the driving strength. Since this transition connects a zero entropy (dark) state with a finite entropy (mixed) state, it goes beyond the realm of equilibrium statistical mechanics and becomes a genuine non-equilibrium transition. We focus on a parameter regime, where the transition is predicted to be discontinuous and analyze the relevant long-wavelength fluctuations driving the transition in the framework of the renormalization group. This allows us to approach the non-equilibrium dark state transition and identify similarities and clear differences to common, equilibrium phase transitions and to derive the phenomenology for a first order dark state phase transition.

Understanding and probing phase transitions in non-equilibrium systems is an ongoing challenge in physics. A particular instance are phase transitions that occur between a non-fluctuating absorbing phase, e.g., an extinct population, and an active phase in which the relevant order parameter, such as the population density, assumes a finite value. I will present recent results on the observation of such a non-equilibrium phase transition in an open driven quantum system. In our experiment, rubidium atoms in a cold disordered gas are laser-excited to Rydberg states under so-called facilitation conditions. This conditional excitation process competes with spontaneous decay and leads to a crossover between a stationary state with no excitations (absorbing state) and one with a finite number of Rydberg excitations (active state). I will discuss the implications of our findings for future investigations into non-equilibrium phase transitions in open many-body quantum systems.

A long-lived prethermal state may emerge upon a sudden quench of a quantum system. In this talk, I consider a quantum quench of an initial critical state, and show that the resulting prethermal state exhibits a genuinely quantum and dynamical universal behavior. Specifically, I consider a scenario where the “speed of light” characterizing the propagation of local perturbations is suddenly quenched at criticality. I also briefly discuss how the system approaches the prethermal state in a universal way described by a new exponent that characterizes a kind of quantum aging.

Ultracold quantum gases are usually well isolated from the environment. This allows for the study of ground state properties and non-equilibrium dynamics of many-body quantum systems under almost ideal conditions. Introducing a controlled coupling to the environment “opens” the quantum system and non-unitary dynamics can be investigated. Such an approach provides new opportunities to study fundamental quantum phenomena and to engineer robust many-body quantum states. I will present an experimental platform [1,2] that allows for the controlled engineering of dissipation in ultracold quantum gases by means of localized particle losses. This is exploited to study quantum Zeno dynamics in a Bose-Einstein condensate [3], where we find that the particle losses are well described by an imaginary potential in the system’s Hamiltonian. We also investigate the steady-states in a driven-dissipative Josephson array [4]. We find that for small dissipation strength, the steady-states are a direct manifestation of coherent perfect absorption. This phenomenon is known from linear optics and is usually challenging to observe. Surprisingly, due to the nonlinearity of the condensate, coherent perfect absorption can be observed much even easier and the phenomenon is very robust. References [1] T. Gericke et al., Nature Physics 4, 949 (2008). [2] P. Würtz et al., Phys. Rev. Lett. 103, 080404 (2009). [3] G. Barontini et al., Phys. Rev. Lett. 110, 035302 (2013). [4] R. Labouvie et al. Phys. Rev. Lett. 116, 235302 (2016).

Ultracold Rydberg atoms provide an ideal testbed for study the interplay between strong coherent interactions and dissipative processes, a subject that has recently seen great attention following the discovery of dissipative state engineering for tailored many-body quantum states. However, in contrast to their fully coherent counterparts, our insights into dissipative many-body dynamics are still in its infancy. I will present the first steps towards a deeper understanding of driven-dissipative Rydberg gases based on a recently developed variational principle [1] as well as numerical simulations based on tensor network operators [2]. Specifically, I will investigate phase transitions of the steady state, including the presence of a multicritical point that is triggered by the dissipation within the system [3]. References [1]H. Weimer, Phys. Rev. Lett. 114, 040402 (2015) [2]A. Kshetrimayum, H. Weimer, R. Orus, Nature Commun. 8, 1291 (2017). [3]V. R. Overbeck, M. F. Maghrebi, A. V. Gorshkov, H. Weimer, Phys. Rev. A 95, 042133 (2017)

Critical phenomena in the nonequilibrium steady state of driven-dissipative quantum systems are emerging as a novel class of phase transitions that may be studied in modern hybrid quantum platforms like ultracold atoms or superconducting circuits. Arrays of coupled, nonlinear optical resonators can in particular simulate the physics of the driven-dissipative Bose-Hubbard model. The closed Bose-Hubbard lattice at zero temperature presents a quantum phase transition with a spontaneous breaking of the $U(1)$ symmetry of the Bose field. This has no analogue in the case of one-photon coherent driving, as the $U(1)$ symmetry is lifted by the driving field. Here, we show [1] that in presence of two-photon driving, the $U(1)$ symmetry is only partially lifted and a discrete $\mathbb{Z}_2$ symmetry is left. This symmetry can be spontaneously broken giving rise to a second order phase transition in analogy with a lattice spin model. We present the results of modeling this phase transition in the framework of the Gutzwiller mean-field ansatz and study its dependence on the system parameters. We then present an analysis of the phase transition beyond mean field, obtained using the corner-space renormalization method [2], and discuss the dependence on the system dimensionality. [1] V. Savona, Physical Review A 96, 033826 (2017). [2] S. Finazzi, et al., Phys. Rev. Lett. 115, 080604 (2015).

Driven dissipative systems opened up a novel research area in the field of quantum criticality. Phase transitions in these systems lie beyond the standard classification of classical dynamical or equilibrium phase transitions and define completely new universality classes. In an open quantum system, the critical behavior appears in the state formed by the dynamical equilibrium of the external driving and dissipation processes. The correlation functions at the critical point are determined by nonequilibrium noise rather than thermal or ground-state quantum fluctuations. We study the open-system realization of the Dicke model, where a bosonic cavity mode couples to a large spin formed by two motional modes of an atomic Bose-Einstein condensate. The cavity mode is driven by a high-frequency laser and it decays to a Markovian bath, while the atomic mode interacts with a colored bath. We demonstrate that the critical exponent of the superradiant phase transition is determined by the low-frequency behavior of the spectral density function of the colored bath. We show that a finite temperature in the colored bath leads to qualitatively similar dependence on the spectral density function.