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arXiv:2403.15237 · PRX Quantum

Driven-Dissipative Phase Separation in Free-Space Atomic Ensembles

D. Goncalves, Lisa Bombieri, Giovanni Ferioli, Sara Pancaldi, Igor Ferrier-Barbut, Antoine Browaeys, Ephraim Shahmoon, Darrick E. Chang

Published Apr 3, 2025• 17 citations
arXivHTMLPDF

Abstract

The driven Dicke model, wherein an ensemble of atoms is driven by an external field and undergoes collective spontaneous emission due to coupling to a leaky cavity mode, is a paradigmatic example of a system exhibiting a driven-dissipative phase transition as a function of driving strength. Recently, a similar phenomenon was experimentally observed, not in a cavity setting, but rather in a free-space atomic ensemble. The reason why similar behavior should emerge in free space is not obvious, as the system interacts with a continuum of optical modes, which encodes light-propagation effects. Here, we present and solve a simple model to explain the behavior of the free-space system, based on the one-dimensional Maxwell-Bloch equations. On one hand, we show that a free-space ensemble at a low optical depth can exhibit similar behavior as the cavity system, as spatial propagation effects are negligible. On the other hand, in the thermodynamic limit of large atom number, we show that certain observables such as the transmittance or the atomic excited population exhibit nonanalytic behavior as a function of the driving intensity, reminiscent of a phase transition. However, a closer analysis reveals that the atomic properties are highly inhomogeneous in space, and based on this we argue that the free-space system does not undergo a phase transition but rather a “phase separation,” roughly speaking, between saturated and unsaturated regions.

DOI
10.1103/prxquantum.6.020303

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The driven Dicke model, wherein an ensemble of atoms is driven by an external field and undergoes collective spontaneous emission due to coupling to a leaky cavity mode, is a paradigmatic example of a system exhibiting a driven-dissipative phase transition as a function of driving strength. Recently, a similar phenomenon was experimentally observed, not in a cavity setting, but rather in a free-space atomic ensemble. The reason why similar behavior should emerge in free space is not obvious, as the system interacts with a continuum of optical modes, which encodes light-propagation effects. Here, we present and solve a simple model to explain the behavior of the free-space system, based on the one-dimensional Maxwell-Bloch equations. On one hand, we show that a free-space ensemble at a low optical depth can exhibit similar behavior as the cavity system, as spatial propagation effects are negligible. On the other hand, in the thermodynamic limit of large atom number, we show that certain observables such as the transmittance or the atomic excited population exhibit nonanalytic behavior as a function of the driving intensity, reminiscent of a phase transition. However, a closer analysis reveals that the atomic properties are highly inhomogeneous in space, and based on this we argue that the free-space system does not undergo a phase transition but rather a “phase separation,” roughly speaking, between saturated and unsaturated regions.
Present-day experiments have started to couple traditional simulations of ultracold atom experiments with quantum light-fields in cavities. This has provided a wealth of op- portunities to enlarge the number of interaction potentials: cavity mediated long-range interactions compete with kinetic energies, longitudinal fields, short-ranged collisional or magnetic spin-spin interactions. The intracavity many-body lattice models often have to be maintained far from equilibrium through the presence of external driving lasers that help to boost and engineer the various interaction potentials. The steady influx of energy is compensated by a steady stream of energy out of the atom-cavity system for example by photon losses or atomic spontaneous emission. Several experiments have demonstrated that such environments can give rise to new competing quantum phases. \nBut this long-standing ambition to push for models with tailorable interaction potentials can bring with it also considerable challenges in their theoretical description, since sponta- neous symmetry breaking transitions in many body lattice systems coupled to dynamical light-fields with single-photon character occur in the presence of drive and dissipation for the photonic force carriers. This clearly calls for model systems where the above men- tioned interplay of interactions, drive, dissipation and cooperative many-body behaviour can be theoretically studied to provide simple, experimentally verifiable predictions. \nThe Dicke model is, through its simplicity (an exactly solved ferromagnet with infinite range atom-atom interactions mediated by a single cavity mode), an exceptionally well- suited candidate. As the generic model for atom-light interactions, it has been experimen- tally realized in a variety of modern quantum optical systems, highlighting its relevance for present-day research. The Dicke model is also highly versatile itself. It has been extended into the dissipative realm, was promoted to account for multiple optical light modes and was used to describe multiple, coupled single-mode cavity structures. It was adapted to treat spin-selective coupling to a cavity to describe superradiance phase tran- sitions in multi-level atomic systems. Moreover, it was realised in electronic circuits where the dipole coupling of real atoms to single mode fields is replaced by a capacitive cou- pling of artificial atoms to a resonator mode. This illustrates that the Dicke model and its extended variants are ’future-proof’ and continue to be of relevance for fundamental light-matter interactions and for driven-dissipative phase transitions. \nIn this thesis, we investigate magnetic phase transitions in driven-dissipative atomic en- sembles interacting with quantum light. We present three research projects on variants of cooperative radiation of an ensemble of laser driven two-level atoms in a single mode optical cavity, as described by the Dicke model. \nThroughout the chapters 2,3 and 4 that contain the main body of research of this thesis, we investigate phase transitions between non-equilibrium stationary states in engineered quantum-optical systems each of which extends the conventional Dicke model physics. As a starting point, we map the quantum equations of motion onto a set of semiclassical nonlinear stochastic equations and analyse their stationary states and instabilities with master equations for atomic spin and photon mean-field amplitudes. These are used to obtain experimentally relevant parameters such as critical atom-light couplings for phase transitions, phase diagrams and properties of stationary non-equilibrium states in addition to cavity output spectra that identify the imprint of magnetic correlations in the light-field. \nIn chapter 2, we help resolve a discrepancy between earlier experimental investigations of the critical atom-light coupling strength for the superradiance transition in the Dicke model: higher external pumping strengths than theoretically predicted were needed to observe a coherent, superradiant state of the light field in an optical cavity. By including incoherent spontaneous emission of atomic excitations, we extend the dissipative Dicke model to a two loss channel variant containing both photon leakage and atomic decay that reproduces the experimentally observed critical atom-light coupling. \nRecent experiments have started to interface quantum many body lattice models with coherent cavity fields, thereby allowing to realize new quantum phases through competing atom-cavity and atom-atom interactions. In chapter 3, we consider a simplified model for such a set-up where a single quantized mode of the light-field interacts with an ensemble of Rydberg-dressed atoms inside a high finesse optical cavity. This model provides a base case for further studies of quantum magnets in optical cavities. At the heart of this model is a competition of short- (dipolar atom-atom) and long-range (atom-light) interactions at the Hamiltonian level in the presence of both spontaneous emission and photon leak- age through the cavity mirrors. We show that different magnetic phases can coexist with coherent atomic radiation and provide clear experimental signatures to identify the mag- netic structure and intra-cavity dynamics. We suggest an experimental level-scheme for a quantum optical implementation of our model. \nIn chapter 4 we consider a generic, collective decay for many-body excitations in the paradigmatic Dicke model. This extension drastically enriches the dynamics as it induces a bicritical point and a bistable regime dominated by true non-equilibrium fluctuations that induce a dissipative first-order phase transition that can only be resolved by including finite fluctuation corrections with the help of stochastic Langevin equations. We investigate the hysteretic response to time-dependent ramps of the atom-light coupling. Discontinuous first-order phase transitions where metastable states coexist in a hysteresis domain have been investigated in recent dissipative quantum-optical experiments. We review noise- activation far from thermal equilibrium in chapter 5.
The Dicke model describes the coherent interaction of a laser-driven ensemble of two-level atoms with a quantized light field. It is realized within cavity QED experiments, which in addition to the coherent Dicke dynamics feature dissipation due to, e.g., atomic spontaneous emission and cavity photon loss. Spontaneous emission supports the uncorrelated decay of individual atomic excitations as well as the enhanced collective decay of an excitation that is shared by NNN atoms and whose strength is determined by the cavity geometry. We derive a many-body master equation for the dissipative Dicke model including both spontaneous emission channels and analyze its dynamics on the basis of Heisenberg-Langevin and stochastic Bloch equations. We find that the collective loss channel leads to a region of bistability between the empty and the superradiant state. Transitions between these states are driven by nonthermal Markovian noise. The interplay between dissipative and coherent elements leads to a genuine nonequilibrium dynamics in the bistable regime, which is expressed via a nonconservative force and a multiplicative noise kernel appearing in the stochastic Bloch equations. We present a semiclassical approach, based on stochastic nonlinear optical Bloch equations, which for the infinite-range Dicke model become exact in the large-NNN-limit. The absence of an effective free-energy functional, however, necessitates the inclusion of fluctuation corrections with O(1/N)O(1/N)O(1/N) for finite $N<\ensuremath{\infty}$ to locate the nonthermal first-order phase transition between the superradiant and the empty cavity.
A class of systems exists in which dissipation, external drive and interactions compete and give rise to non equilibrium phases that would not exist without the drive. There, phase transitions could occur without the breaking of any symmetry, yet with a local order parameter, in contrast with the Landau theory of phase transitions at equilibrium. One of the simplest driven dissipative quantum systems consists of two-level atoms enclosed in a volume smaller than the wavelength of the atomic transition cubed, driven by a light field. The competition between collective coupling of the atoms to the driving field and their cooperative decay should lead to a transition between a phase where all the atomic dipoles are phaselocked and a phase governed by superradiant spontaneous emission. Here, we realize this model using a pencil-shaped cloud of laser cooled atoms in free space, optically excited along its main axis, and observe the predicted phases. Our demonstration is promising in view of obtaining free-space superradiant lasers or to observe new types of time crystals.
The transmission of light through an ensemble of two-level emitters in a one-dimensional geometry is commonly described by one of two emblematic models of quantum electrodynamics (QED): the driven-dissipative Dicke model or the Maxwell-Bloch equations. Both exhibit distinct features of phase transitions and phase separations, depending on system parameters such as optical depth and external drive strength. Here, we explore the crossover between these models via a parent spin model from bidirectional waveguide QED, by varying positional disorder among emitters. Solving mean-field equations and employing a second-order cumulant expansion for the unidirectional model—equivalent to the Maxwell-Bloch equations—we study phase diagrams, the emitter's inversion, and transmission depending on optical depth, drive strength, and spatial disorder. We find in the thermodynamic limit the emergence of phase separation with a critical value that depends on the degree of spatial order but is independent of Doppler broadening effects. Even far from the thermodynamic limit, this critical value marks a special point in the emitter's correlation landscape of the unidirectional model and is also observed as a maximum in the magnitude of inelastically transmitted photons. We conclude that a large class of effective one-dimensional systems without tight control of the emitters spatial ordering can be effectively modeled using a unidirectional waveguide approach.
We investigate the near-critical dynamics of atomic density fluctuations in the nonequilibrium self-organization transition of an optically driven quantum gas coupled to a single mode of a cavity. In this system cavity-mediated long-range interactions between atoms, tunable by the drive strength, lead to softening of an excitation mode recently observed in experiments. This phenomenon has previously been studied within a two-mode approximation for the collective motional degrees of freedom of the atomic condensate, which results in an effective open-system Dicke model. Here, including the full spectrum of atomic modes we find a finite lifetime for a rotonlike mode in the Bogoliubov excitation spectrum that is strongly pump dependent. The corresponding decay rate and critical exponents for the phase transition are calculated explaining the nonmonotonic pump-dependent atomic damping rate observed in recent experiments. We compute the near-critical behavior of the intracavity field fluctuations that has been previously shown to be enhanced with respect to the equilibrium Dicke model in a two-mode approximation. We highlight the role of the finite size of the system in the suppression of it below the expectations of the open Dicke model.
We observe a non-equilibrium phase transition in a driven dissipative quantum system consisting of an pencil-shape cloud of up to N≈2000N\approx 2000N≈2000 laser-cooled atoms in free space, optically excited along its main axis. We find that our data are well reproduced by the Driven Dicke model, which assumes a sub-wavelength sample volume, by simply using an effective atom number. By measuring the excited state population of the atoms and the light emitted in the superradiant mode, we characterize the dynamics of the system and its steady-state properties. In particular, we observe the characteristic N2N^2N2 scaling of the photon emission rate in the superradiant phase, thus demonstrating steady-state superradiance in free space. Finally, we observe a modification of the statistics of the superradiant light as we cross the phase transition.