Quantum master equations from classical Lagrangians with two stochastic forces

Abstract

We show how a large family of master equations, describing quantum Brownian motion of a harmonic oscillator with translationally invariant damping, can be derived within a phenomenological approach, based on the assumption that an environment can be simulated by two classical stochastic forces. This family is determined by three time-dependent correlation functions (besides the frequency and damping coefficients), and it includes as special cases the known master equations, whose dissipative part is bilinear with respect to the operators of coordinate and momentum.

How ‘cold’ can a Markovian dissipative cavity QED system be?

“…We can arrive at the same conclusion via the simplified phenomenological approach. The most general master equation for a single cavity field mode [51,52], preserving the normalization and hermiticity of the statistical operator ρ and containing only bilinear forms of operators x and p, is given by equation ( 2) with the effective Hamiltonian…”

“…Therefore, if the anti-resonant cavity-reservoir interactions are taken into account, under the Markovian approximation the asymptotic mean photon number in the dissipative cavity is larger than zero. For example, taking µ = κ, which can be deduced from a phenomenological classical Lagrangian [52], and using inequality (11) one obtains…”

Quantum integrals of motion for variable quadratic Hamiltonians

“…The damped oscillations have been studied to a great extent in classical mechanics [4], [5] and [79]. Their quantum analogs are introduced and analyzed from different viewpoints by many authors; see, for example, [13], [19], [23], [24], [25], [27], [32], [33], [35], [37], [40], [82], [83], [69], [104], [110], [139], [140], [143], [146], and references therein. The quantum parametric oscillator with variable frequency is also largely studied in view of its physical importance; see, for example, [22], [39], [62], [63], [81], [95], [97], [117], [118], [121], [122], [130], and [133]; a detailed bibliography is given in [14].…”

Errors in zero-excitation state preparation due to anti-rotating terms in two-atom Markovian cavity QED

“…where the superoperators L κ (L λk ) and L γ k describe cavity (atom) damping by a thermal reservoir and pure atomic dephasing, respectively (k = 1, 2). There are several analytical expressions for the Liouvillian [14], deduced under different approximations and assumptions, but one of the most used in the cavity QED scenario corresponds to the 'standard master equation' (SME), which can be deduced microscopically by Markovian approximation on the system-reservoir interaction [13][14][15]. Since in this paper we investigate the bounds for errors in the zero-excitation state preparation, we consider the zero-temperature limit of the SME:…”

“…Since the ARTs in Hamiltonian (1) appear naturally throughout the microscopic deduction of light-matter interaction [12], while the master equation is deduced under several ad hoc assumptions [13][14][15]19], this result gave rise to some debates concerning the applicability of the Markovian master equation in this case [20,21]. Here, we do not enter this nontrivial discussion, noting that the applicability of the master equation to the cavity QED setup depends on concrete implementation (see [14,15,[17][18][19]21] for more discussion). Instead we respond to a question of practical interest: if one uses the master equation ( 2) to describe the two-atom cavity QED system, how large is the error committed by assuming that the asymptotic state is the zero-excitation state?…”