Keldysh Field Theory for Driven Open Quantum Systems L. M. Sieberer,1 M. Buchhold,2 and S. Diehl2, 3, 4 1Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel 2Institute of Theoretical Physics, TU Dresden, D-01062 Dresden, Germany 3Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA 4Institut f¨urTheoretische Physik, Universit¨atzu K¨oln,D-50937 Cologne, Germany Recent experimental developments in diverse areas - ranging from cold atomic gases over light-driven semi- conductors to microcavity arrays - move systems into the focus, which are located on the interface of quantum optics, many-body physics and statistical mechanics. They share in common that coherent and driven-dissipative quantum dynamics occur on an equal footing, creating genuine non-equilibrium scenarios without immediate counterpart in condensed matter. This concerns both their non-thermal flux equilibrium states, as well as their many-body time evolution. It is a challenge to theory to identify novel instances of universal emergent macro- scopic phenomena, which are tied unambiguously and in an observable way to the microscopic drive conditions. In this review, we discuss some recent results in this direction. Moreover, we provide a systematic introduction to the open system Keldysh functional integral approach, which is the proper technical tool to accomplish a merger of quantum optics and many-body physics, and leverages the power of modern quantum field theory to driven open quantum systems. CONTENTS II Applications 32 I. Introduction2 III. Non-equilibrium stationary states: spin models 32 A. New phenomena2 A. Ising spins in a single-mode cavity 32 B. Theoretical concepts and techniques3 1. Large-spin Holstein-Primakoff representation 33 C. Experimental platforms5 2. Effective Ising spin action for the single-mode cavity 35 1. Cold atoms in an optical cavity, and microcavity arrays: driven-dissipative spin-boson models5 B. Ising spins in a random multi-mode cavity 36 1. Keldysh action and saddle point equations 37 2. Exciton-polariton systems: driven open interacting bosons7 2. Non-equilibrium glass transition 38 3. Thermalization of photons and atoms in the glass 3. Cold atoms in optical lattices: heating phase 39 dynamics9 4. Quenched and Markovian bath coupling in the D. Outline and scope of this review 10 Keldysh formalism 40 IV. Non-equilibrium stationary states: bosonic models 40 I Theoretical background 11 A. Density-phase representation of the Keldysh action 42 II. Keldysh functional integral for driven open systems 11 B. Critical dynamics in 3D 44 A. From the quantum master equation to the Keldysh 1. Effective action for driven-dissipative functional integral 11 condensation 45 B. Examples 16 2. Non-equilibrium FRG flow equations 46 1. Single-mode cavity, and some exact properties 16 3. Scaling solutions and critical behavior 48 2. Driven-dissipative condensate 20 C. Absence of algebraic order in 2D 49 C. Semiclassical limit of the Keldysh action 21 D. KPZ scaling in 1D 51 D. Symmetries of the Keldysh action 23 1. Thermodynamic equilibrium as a symmetry of V. Universal heating dynamics in 1D 52 the Keldysh action 24 A. Heating an interacting Luttinger liquid 53 2. Semiclassical limit of the thermal symmetry 26 B. Kinetic equation 54 3. The Noether theorem in the Keldysh formalism 27 C. Self-consistent Born approximation 56 4. Closed systems: energy, momentum and particle 1. Kinetics for small momenta 58 number conservation 28 2. Scaling of the self-energy 58 5. Extended continuity equation in open systems 29 D. Heating and universality 58 6. Spontaneous symmetry breaking and the 1. Phonon densities 58 Goldstone theorem 30 2. Non-equilibrium scaling 59 E. Open system functional renormalization group 31 3. Observability 59 2 VI. Conclusions and outlook 60 techniques to be used, and calls for the development of new theo- retical tools. The physical framework sparks broader theoretical VII. Acknowledgements 60 questions on the existence of new phases of bosonic [18, 19] and fermionic [20–22] matter, the nature of phase transitions in such A. Functional differentiation 61 driven systems [10, 23–26], and the observable consequences of quantum mechanics at the largest scales [27]. Beyond stationary B. Gaussian functional integration 61 states, a fundamental challenge is set by the time evolution of interacting quantum systems, which is currently explored theo- References 62 retically [28–33] and experimentally in cold atomic [34–38] and photonic systems [39]. A key goal is to identify universal dy- namical regimes that hold beyond specific realizations or precise I. INTRODUCTION initial conditions. Combining idealized closed system evolution with the intrinsic open system character of any real world ex- periment takes this setting to the next stage, and exhibits emer- Understanding the quantum many-particle problem is one of gent dynamics markedly different from closed systems both for the grand challenges of modern physics. In thermodynamic short [40, 41] and long evolution times [42–47]. equilibrium, the combined effort of experimental and theoretical The interplay of coherent and driven-dissipative dynamics can research has made tremendous progress over the last decades, be a natural consequence of the driving necessary to maintain revealing the key concepts of emergent phenomena and univer- a certain many-body state. Going one step further, it is possi- sality. This refers to the observation that the relevant degrees ble to exploit and further develop the toolbox of quantum optics of freedom governing the macrophysics may be vastly differ- for the driven-dissipative manipulation of many-body systems. ent from those of the microscopic physics, but on the other Recently, it has been recognized that the concept of dissipative hand are constrained by basic symmetries on the short distance state preparation in quantum optics [48, 49] can be developed scale, restoring predictive power. Regarding the role and power into a many-body context, both theoretically [50–53] and exper- of these concepts in out-of-equilibrium situations, there is a imentally [54–56]. Suitably tailored dissipation then does not large body of work in the context of classical near equilibrium necessarily act as an adversary to subtle quantum mechanical and non-equilibrium many-body physics and statistical mechan- correlations such as phase coherence or entanglement [57–62]. ics [1–5]. However, analogous scenarios and theoretical tools In contrast, it can even create these correlations, and dissipa- for non-equilibrium quantum systems are much less developed. tion then represents the dominant resource of many-body dy- This review addresses recent theoretical progress in an impor- namics. In particular, even topologically ordered states in spin tant and uprising class of dynamical non-equilibrium phases of systems [63] or of fermionic matter [64, 65] can be induced dis- quantum matter, which emerge in driven open quantum systems sipatively ([52] and [66–68], respectively). These developments where a Hamiltonian is not the only resource of dynamics. This open up a new arena for many-body physics, where the quan- concerns both non-equilibrium stationary states, but also the dy- tum mechanical microscopic origin is of key importance despite namics of such ensembles. Strong motivation for the exploration a dominantly dissipative dynamics. of non-thermal stationary states comes from a recent surge of Summarizing, this is a fledging topical area, where first results experiments in diverse areas: in cold atomic gases [6–8], hy- underpin the promise of these systems to exhibit genuinely new brid light-matter systems of Bose-Einstein condensates placed physics. In the remainder of the introduction, we will discuss in optical cavities are created [9, 10], or driven Rydberg ensem- in some more detail the three major challenges which emerge in bles are prepared [11]; light-driven semiconductor heterostruc- these systems. The first one concerns the identification of novel tures realize Bose condensation of exciton-polaritons [12, 13]; macroscopic many-body phenomena, which witness the micro- coupled microcavity arrays are pushed to a regime demonstrat- scopic driven open nature of such quantum systems. Second, we ing both strong coupling of light and matter [14] and scalabil- anticipate the theoretical machinery which allows us to perform ity [15]; large ensembles of trapped ions implement varieties of the transition from micro- to macrophysics in a non-equilibrium driven spin models [16, 17]. All those systems are genuinely context in practice. Third, we describe some representative ex- made of quantum ingredients, and share in common that coher- perimental platforms, which motivate the theoretical efforts, and ent and driven-dissipative dynamics occur on equal footing. This in which the predictions can be further explored. creates close ties to typical setups in quantum optics. But on the other hand, they exhibit a continuum of spatial degrees of free- dom, characteristic for many-body physics. Systems located at this new interface are not guaranteed to thermalize, due to the A. New phenomena absence of energy conservation and the resulting breaking of detailed balance by the external drive at the microscale. They As pointed out above, one of the key goals of the research re- rather converge to flux-equilibrium states of matter, creating viewed here is the identification of new macroscopic many-body scenarios without counterpart in condensed, equilibrium matter. phenomena, which can be uniquely traced back to the micro- This rules out conventional theoretical equilibrium concepts and scopic driven open nature of such systems, and do not have an 3 immediate counterpart in equilibrium systems. cal behavior is characterized by a fine structure in a new and in- The driven nature common to the systems considered here can dependent critical exponent, which measures decoherence, and always be associated to the fact that the underlying microscopic whose value distinguishes equilibrium and non-equilibrium dy- Hamiltonian is time dependent, with a time dependence relating namics, see Sec.IVB.