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Driven-Dissipative Quantum Kerr Resonators: New Exact Solutions, Photon Blockade and Quantum Bistability

David C K Roberts, Aashish A. Clerk

Published Apr 29, 2020• 67 citations
Publisher

Abstract

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.

DOI
10.6082/1k3w4-kvf25

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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.
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.
An arbitrary initial state of an optical or microwave field in a lossy driven nonlinear cavity can be changed into a partially incoherent superposition of only the vacuum and the single-photon states. This effect is known as single-photon blockade, which is usually analyzed for a Kerr-type nonlinear cavity parametrically driven by a single-photon process assuming single-photon loss mechanisms. We study photon blockade engineering via a nonlinear reservoir, i.e., a quantum reservoir, where only two-photon absorption is allowed. Namely, we analyze a lossy nonlinear cavity parametrically driven by a two-photon process and allowing two-photon loss mechanisms, as described by the master equation derived for a two-photon absorbing reservoir. The nonlinear cavity engineering can be realized by a linear cavity with a tunable two-level system via the Jaynes-Cummings interaction in the dispersive limit. We show that by tuning properly the frequencies of the driving field and the two-level system, the steady state of the cavity field can be the single-photon Fock state or a partially incoherent superposition of several Fock states with photon numbers, e.g., (0,2), (1,3), (0,1,2), or (0,2,4). At the right (now fixed) frequencies, we observe that an arbitrary initial coherent or incoherent superposition of Fock states with an even (odd) number of photons is changed into a partially incoherent superposition of a few Fock states of the same photon-number parity. We find analytically approximate formulas for these two kinds of solutions for several differently tuned systems. A general solution for an arbitrary initial state is a weighted mixture of the above two solutions with even and odd photon numbers, where the weights are given by the probabilities of measuring the even and odd numbers of photons of the initial cavity field, respectively. This can be interpreted as two separate evolution-dissipation channels for even and odd-number states. Thus, in contrast to the standard predictions of photon blockade, we prove that the steady state of the cavity field, in the engineered photon blockade, can depend on its initial state. To make our results more explicit, we analyze photon blockades for some initial infinite-dimensional quantum and classical states via the Wigner and photon-number distributions.
We present an exact solution in arbitrary dimensions for the steady states of a class of quantum driven-dissipative bosonic models, where a set of modes is subject to arbitrary two-photon driving, single-photon loss, and a global Hubbard (or Kerr)-like interaction. Our solutions reveal a wealth of striking phenomena, including the emergence of dissipative phase transitions, nontrivial mode competition physics and symmetry breaking, and the stabilization of many-body SU(1,1) pair-coherent states. Our exact solutions enable the description of spatial correlations, and are fully valid in regimes where traditional mean-field and semiclassical approaches break down.
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.