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arXiv:2007.14590 · Physical Review A

Method for driven-dissipative problems: Keldysh-Heisenberg equations

Yuanwei Zhang, Gang Chen

Published Dec 8, 2020• 3 citations
arXivHTMLPDF

Abstract

Driven-dissipative systems have recently attracted great attention due to the existence of novel physical phenomena with no analog in the equilibrium case. The Keldysh path-integral theory is a powerful tool to investigate these systems. However, it has still been a challenge to study strong nonlinear effects implemented by recent experiments, since in this case the photon number is few and quantum fluctuations play a crucial role in the dynamics of the system. Here we develop an approach for deriving exact steady states of driven-dissipative systems by introducing the Keldysh partition function in the Fock-state basis and then mapping the standard saddle-point equations into Keldysh-Heisenberg equations. We take the strong Kerr nonlinear resonators with and without the nonlinear driving as two examples to illustrate our method. It is found that, in the absence of the nonlinear driving, the exact steady state obtained does not exhibit bistability and agrees well with the complex $P$-representation solution. While in the presence of the nonlinear driving, the multiphoton resonance effects are revealed and are consistent with the qualitative analysis. Our method provides an intuitive way to explore a variety of driven-dissipative systems especially with strong correlations.

DOI
10.1103/physreva.102.062205

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Driven-dissipative systems have recently attracted great attention due to the existence of novel physical phenomena with no analog in the equilibrium case. The Keldysh path-integral theory is a powerful tool to investigate these systems. However, it has still been a challenge to study strong nonlinear effects implemented by recent experiments, since in this case the photon number is few and quantum fluctuations play a crucial role in the dynamics of the system. Here we develop an approach for deriving exact steady states of driven-dissipative systems by introducing the Keldysh partition function in the Fock-state basis and then mapping the standard saddle-point equations into Keldysh-Heisenberg equations. We take the strong Kerr nonlinear resonators with and without the nonlinear driving as two examples to illustrate our method. It is found that, in the absence of the nonlinear driving, the exact steady state obtained does not exhibit bistability and agrees well with the complex PPP-representation solution. While in the presence of the nonlinear driving, the multiphoton resonance effects are revealed and are consistent with the qualitative analysis. Our method provides an intuitive way to explore a variety of driven-dissipative systems especially with strong correlations.
We present a new approach for deriving exact closed-form solutions for the steady state of a wide class of driven-dissipative nonlinear resonators that is distinct from more common complex-P-function methods. Our method generalizes the coherent quantum-absorber approach of Stannigel et al. [New J. Phys. 14, 063014 (2012)] to include nonlinear driving and dissipation and relies crucially on exploiting the Segal-Bargmann representation of Fock space. Our solutions and method reveal a wealth of previously unexplored observable phenomena in these systems, including new generalized photon-blockade and antiblockade effects and an infinite number of new parameter choices that yield quantum bistability.
Recent experimental developments in diverse areas-ranging from cold atomic gases to 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 equilibrium condensed matter physics. This concerns both their non-thermal stationary states and 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.
Recent experimental developments in diverse areas-ranging from cold atomic gases to 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 equilibrium condensed matter physics. This concerns both their non-thermal stationary states and 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.
Pure quantum states, as described in quantum mechanics textbooks, are ideal representations inevitably deteriorated in real systems by any dissipative connection to the environment. In this work we derive a unified theoretical frame to study the non-Hermitian dynamics of a driven-dissipative quantum system and validate it by comparison to experimental data of matter-wave diffraction. To highlight the role of dissipation, we perform the comparison in a near-resonant regime of light-matter interaction, where perturbative approaches fail. Here, we show that the developed formalism enables both exact simulations and an intuitive interpretation based on modal analysis. The theoretical analysis is based on the general master equation description of driven-dissipative systems, from which we derive an effective complex potential Veff(Ω,Δ)V_\mathrm{eff}(Ω,Δ)Veff​(Ω,Δ) that depends on control parameters such as the amplitude and detuning of the light matter interaction. Experimentally, we drive a 87^{87}87Rb BEC near resonance using periodic excited-state engineering, mapping the spatially modulated Veff[Ω,Δ(x)]V_{\rm eff}[Ω,Δ(x)]Veff​[Ω,Δ(x)] into momentum space for precise quantification. The experimental results agree with numerical simulations of the master equation and demonstrate the interplay between coherent drive and dissipation, with an exceptional signature: a reduced decay rate with increasing drive. Such dynamics is then interpreted by a modal analysis of the non-Hermitian Hamiltonian Heff=p2/2m+VeffH_\mathrm{eff}=p^2/2m+V_\mathrm{eff}Heff​=p2/2m+Veff​ which provide an intuitive and qualitative explanation for such driven-dissipative system.
In this paper, we investigate periodically driven open quantum systems within the framework of Floquet-Lindblad master equations. Specifically, we discuss Lindblad master equations in the presence of a coherent, time-periodic driving and establish their general spectral features. We also clarify the notions of transient and nondecaying solutions from this spectral perspective, and then prove that any physical system described by a Floquet-Lindblad equation must have at least one physical nonequilibrium steady state (NESS), corresponding to an eigenoperator of the Floquet-Lindblad evolution superoperator UF{\mathcal{U}}_{F}UF​ with unit eigenvalue. Since the Floquet-Lindblad formalism encapsulates the entire information regarding the NESS, it in principle enables us to obtain nonlinear effects to all orders at once. The Floquet-Lindblad formalism thus provides a powerful tool for studying driven-dissipative solid-state systems, which we illustrate by deriving the nonlinear optical response of a simple two-band model of an insulating solid and comparing it with prior results established through Keldysh techniques.
Driven nonlinear quantum oscillators are a central platform for quantum technologies, yet their dissipative dynamics are typically described using Lindblad or Caldeira-Leggett master equations derived under assumptions that exclude nonlinearities and driving. Here, we derive a generalized Caldeira-Leggett master equation for driven nonlinear oscillators by retaining the full nonlinear and time-dependent system dynamics in the construction of the dissipator. For position- and momentum-dependent system-bath coupling, the dissipator itself becomes dynamically dressed, generating nonlinear and drive-dependent dissipative channels beyond conventional fixed-dissipator approaches. This produces nonlinear damping together with dissipation-induced corrections to the effective drive. The resulting dissipative dynamics suppress large-amplitude excitations and reduce phase-space fluctuations. For a driven Kerr oscillator, this leads to the suppression of bistability, asymmetric resonance responses, and strongly modified fluctuation distributions. More broadly, our results establish a microscopic framework in which nonlinear dynamics and driving directly reshape the dissipative sector of driven open quantum systems.