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Journal of Physics A Mathematical and General

Quantum friction with a stochastic force

Robert Alicki, Joachim A. Messer

Published Nov 1, 1982• 10 citations
Publisher

Abstract

The Hamiltonian quantum theory of the damped harmonic oscillator is investigated by using an explicitly time-dependent Hamiltonian including a commutative stochastic force. The propagator is computed and shown to describe an inhomogeneous quantum dynamical semigroup evolution on the Banach cone of density matrices. The moments tend to the thermal equilibrium values independently of the initial conditions, when time tends to infinity and some rescaling procedure is performed. With respect to the rescaling neither the Heisenberg nor the Schrodinger picture exists. Furthermore the energy dissipates below the zero-point energy in violation of the Heisenberg uncertainty relation.

DOI
10.1088/0305-4470/15/11/032

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The Hamiltonian quantum theory of the damped harmonic oscillator is investigated by using an explicitly time-dependent Hamiltonian including a commutative stochastic force. The propagator is computed and shown to describe an inhomogeneous quantum dynamical semigroup evolution on the Banach cone of density matrices. The moments tend to the thermal equilibrium values independently of the initial conditions, when time tends to infinity and some rescaling procedure is performed. With respect to the rescaling neither the Heisenberg nor the Schrodinger picture exists. Furthermore the energy dissipates below the zero-point energy in violation of the Heisenberg uncertainty relation.
We begin by revisiting the so-called Caldirola - Kanai Hamiltonian and discuss its inherent ambiguity: does it represent a dissipative harmonic oscillator (HO) subject to a friction force, or does it describe an HO with a time-dependent mass (TDM)? Although classically both descriptions do coexist, in the quantum domain the solution of the Schrödinger equation (or Heisenberg equations of motion) with a TDM does not present inconsistencies, however, the dissipative Hamiltonian shows violation of the Heisenberg uncertainty principle. This violation is avoided by introducing a stochastic force in the equations of motion, which will take care of the fluctuations due to the environment. Once the distinction between the dissipative and amplifying Hamiltonian is made clear, we consider the problem of the quantum TDM HO subject to dissipation, showing that both phenomena may be merged and described by a single Hamiltonian, the amplifying - dissipative Hamiltonian . We obtain the solutions of the Heisenberg equations of motion for the canonical momentum and position; next, we specialize on the weak damping limit and analyse the effects of the amplifying - dissipative process on the mean values of the physical variables.
Considering the Lagrangian proposed by Havas, that describes the classical damped motion of a particle, new momentum and position are defined in order to write a Hamiltonian that is subsequently quantized and expressed in terms of non-Hermitian operators. Using the c-number formalism proposed by Lax and Yuen, we associate to the quantum Liouville equation a Fokker–Planck one in terms of c-numbers. From the properties of this equation we obtain the mean values of the position, momentum, and energy of a brownian particle and we also verify the uncertainty principle. We observe that when the system is considered under the Markov hypothesis, the stochastic force is intimately related to the uncertainty principle and to the zero point energy.
The damped harmonic oscillator is used to represent the damping processes in nature. Therefore it is necessary to find the Hamiltonian for the damped oscillator. The Hamiltonian is developed for the harmonic oscillator with frictional force proportional to velocity. The solution of the Schr\"odinger equation gives nonstationary states. The quantum friction coefficient is derived to be zero for massless particles. It is thought that velocity-dependent forces exist near the nucleus and therefore, the scattering amplitude for the deep inelastic process is derived.
This paper investigates the evolution of a quantum particle in a harmonic oscillator driven by time-dependent forces. The perturbations are small, but they act long enough so that we can solve the problem in the asymptotic framework corresponding to a perturbation amplitude that tends to zero and a perturbation duration that tends to infinity. We describe the effective evolution equation of the state vector, which reads as a stochastic partial differential equation. We exhibit a closed-form equation for the transition probabilities, which can be interpreted in terms of a jump process. Using standard probability tools, we are then able to compute explicitly the probabilities for observing the different energy eigenstates and give the exact statistical distribution of the energy of the particle.
A previously developed novel theory for the formal canonical quantization of classically dissipating systems will be the starting point for a detailed discussion of the quantum statistical aspects of the simple linearly damped harmonic oscillator. The formalism essentially involves complex classical canonical coordinates and momenta, and thus quite naturally leads to the possibility of creation and annihilation operators. Furthermore, the occurrence of quantal noise operators appears to be of principal importance for the conservation of the fundamental commutator in the course of time, as will be expressed in a simple fluctuation-dissipation relation. Making a canonical transformation back to the real, Cartesian Hermitian position and momentum an "effective" Hermitian Hamiltonian will be derived, with which a transformation is made from the Heisenberg frame to the Schr\"odinger frame where the density operator equation will be computed. This will make it obvious that no proper Schr\"odinger equation exists for the dissipative subsystem on its own, thus reflecting an incomplete knowledge. The master equation will then be translated into its Wigner representation. The intimate connection between the diffusion coefficients in the resulting Fokker-Planck equation and the uncertainty relation will be demonstrated in a clear fashion.
The in-in formalism and its influence functional generalization are widely used to describe the out-of-equilibrium dynamics of unitary and open quantum systems, respectively. In this paper, we build on these techniques to develop an effective theory of a quantum damped harmonic oscillator and use it to study initial state-dependence, decoherence, and thermalization. We first consider a Gaussian initial state and quadratic influence functional and obtain general equations for the Green's functions of the oscillator. We solve the equations in the specific case of time-local dissipation and use the resulting Green's functions to obtain the purity and unequal-time two-point correlations of the oscillator. We find that the dynamics must include a nonvanishing noise term to yield physical results for the purity and that the oscillator decoheres in time such that the late-time density operator is thermal. We show that the frequency spectrum or unequal-time correlations can, however, distinguish between the damped oscillator and an isolated oscillator in thermal equilibrium and obtain a generalized fluctuation-dissipation relation for the damped oscillator. We briefly consider time-nonlocal dissipation as well, to show that the fluctuation-dissipation is satisfied for a specific choice of dissipation kernels. Lastly, we develop a double in-out path integral approach to go beyond Gaussian initial states and show that our equal-time results for time-local dissipation are in fact nonperturbative in the initial state.