Keldysh Field Theory for Driven Open Quantum Systems

TL;DR

This work develops and applies a comprehensive Keldysh functional integral framework for driven open quantum systems, unifying quantum optics and many-body physics to address genuine nonequilibrium phenomena. It maps Markovian quantum master equations to a two-branch Keldysh action, clarifies symmetry properties (including a thermal equilibrium symmetry) and Noether currents, and employs open-system FRG to access universal critical behavior. The approach is then used to analyze spin models in cavities (including Dicke-like transitions and glassy phases) and bosonic condensates, revealing emergent thermal behavior in 3D and KPZ universality in low dimensions, as well as rich nonequilibrium critical dynamics. In 1D heating problems, a self-consistent kinetic theory and FRG reveal universal scaling laws for phonon populations and self-energies, offering experimentally accessible signatures in Bragg spectroscopy and related measurements. Overall, the framework provides a powerful, broadly applicable toolkit to characterize universal nonequilibrium phenomena in driven open quantum matter and to guide future experimental explorations.

Abstract

Recent experimental developments in diverse areas - ranging from cold atomic gases over light-driven semiconductors 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 macroscopic 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.

Figures (13)

• Figure 1: (a) Schematic of two Bragg mirrors forming a microcavity, in which a quantum well (QW) is embedded. In the regime of strong light-matter interaction, the cavity photon and the exciton hybridize and form new eigenmodes, which are called exciton-polaritons. (b) Energy dispersion of the upper and lower polariton branches as a function of in-plane momentum $q$. In the experimental scheme illustrated in this figure (cf. Ref. Kasprzak2006), the incident laser is tuned to highly excited states of the quantum well. These undergo relaxation via emission of phonons and scattering from polaritons, and accumulate at the bottom of the lower polariton branch. In the course of the relaxation process, coherence is quickly lost, and the effective pumping of lower polaritons is incoherent.
• Figure 2: Illustration of the coherent (a) and the incoherent (b) contribution to the dynamics stemming from the coupling of the atoms to a laser with amplitude $|\Omega|$ and large detuning $\Delta$. (a) Via the AC Stark effect, stimulated absorption and emission lead to a coherent periodic potential with amplitude $\frac{|\Omega|^2}{4\Delta}$. (b) Stimulated absorption and subsequent spontaneous emission lead to effective decoherence of the atomic state with rate $\Gamma\frac{|\Omega|^2}{4\Delta}$, where $\Gamma$ is the microscopic spontaneous emission rate.
• Figure 3: Idea of the Keldysh functional integral. a) According to the Schrödinger equation, the time evolution of a pure state vector is described by the unitary operator $U(t,t_0) = e^{- i H \left( t-t_0 \right)}$. In the Feynman functional integral construction, the time evolution is chopped into infinitesimal steps of length $\delta_t$, and completeness relations in terms of coherent states are inserted in between consecutive time steps. This insertion is signalled by the red arrows. b) In contrast, if the state is mixed, a density matrix must be evolved, and thus two time branches are needed. As explained in the text, the dynamics need not necessarily be restricted to unitary evolution. The most general time-local (Markovian) dynamics is generated by a Liouville operator in Lindblad form. c) For the analysis of the stationary state, we are interested in the real time analog of a partition function $Z = \mathop{\mathrm{tr}} \rho(t_f)$, starting from $t_0 =-\infty$ and running until $t_f=+\infty$. The trace operation connects the two time branches, giving rise to the closed Keldysh contour.
• Figure 4: Illustration of a homodyne detection measurement which determines the response function $G^R(t,t')$ of the cavity photons, see Ref. Buchhold2013. The system of atoms and photons inside the cavity is perturbed by the laser field $\eta(t)$, entering the cavity through the left mirror. The response of the system is encoded in the light field which is leaking out of the cavity on the right mirror with a rate $\kappa$. It can be measured by a standard homodyne detection measurement in which the reference laser $\beta(t)$ and a beam splitter are used in order to obtain information on the system's coherence $\langle a_c(t)\rangle$. Figure copied from Ref. Buchhold2013. (Copyright (2013) by The American Physical Society.)
• Figure 5: Equivalent descriptions on varying length scales. The quantum and classical Langevin equations are stochastic equations of motions for the field operators, or for classical field variables. In contrast, the descriptions in the middle column are deterministic equations of motion, where either a density operator or a probability distribution (diagonals of a density matrix) are evolved. In the functional integral formulation, the basic object in both quantum and classical cases is an action, which is averaged over all possible realizations of field configurations. The semiclassical limit is valid at mesoscopic scales as dicsussed in Sec. \ref{['sec:semicl-limit-keldysh']}. An effective description at macroscopic scales can be obtained by means of renormalization group methods (see Sec. \ref{['sec:open-sys-FRG']}). Generically in Markovian systems, in order to reduce the complexity of the problem it is useful to first perform the semiclassical limit before doing a renormalization group computation. However, in driven open quantum systems there are also circumstances where this is inappropriate, and the full quantum problem has to be analyzed, cf. Marino2015.