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Europhysics Letters (EPL)

Quantum Langevin Equation of Parametrically Driven Linear Oscillators

R. Graham

Published Jul 15, 1988• 6 citations
Publisher

Abstract

The quantum Langevin force of a parametrically driven damped harmonic oscillator is determined from the condition of the preservation of the canonical commutation relations. The arguments are independent of the microscopic modelling of the heat reservoir. For damping with and without retardation the Langevin force is found to be independent of the time-dependent parametric driving force and determined by the damping rate alone.

DOI
10.1209/0295-5075/6/6/001

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The quantum Langevin force of a parametrically driven damped harmonic oscillator is determined from the condition of the preservation of the canonical commutation relations. The arguments are independent of the microscopic modelling of the heat reservoir. For damping with and without retardation the Langevin force is found to be independent of the time-dependent parametric driving force and determined by the damping rate alone.
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The heat-bath frequency distribution, necessary to maintain the canonical commutation relations for all time for a quantum mechanical oscillator with non-Markovian features like memory-dependent damping, is shown to satisfy certain constraint relations. An algorithm is given to explicitly find the heat-bath frequency distribution in terms of a series expansion for all processes where the timescale for the non-Markovian memory kernel is much smaller than the inverse of the strength of damping term. In the non-Markovian case, the heat-bath distribution function exhibits dependence on the system characteristics. The KS periodicity conditions on the system Green functions as the system approaches equilibrium are established for the present case.
The macroscopic description of a quantum particle with passive dissipation and moving in an arbitrary external potential is formulated in terms of the generalized Langevin equation. The coupling with the heat bath corresponds to two terms: a mean force characterized by a memory function \ensuremath{\mu}(t) and an operator-valued random force. Explicit expressions are given for the correlation and commutator of the random force. The random force is never Markovian. It is shown that \ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(z), the Fourier transform of the memory function, must be a positive real function, analytic in the upper half-plane and with Re[\ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{}(\ensuremath{\omega}+i0+{0}^{+}0+)] a positive distribution on the real axis. This form is then derived for the independent-oscillator model of a heat bath. It is shown that the most general quantum Langevin equation can be realized by this simple model. A critical comparison is made with a number of other models that have appeared in the literature.
Abstract. The quantum theory of the damped harmonic oscillator has been a subject of continual investigation since the 1930s. The obstacle to quantization created by the dissipation of energy is usually dealt with by including a discrete set of additional harmonic oscillators as a reservoir. But a discrete reservoir cannot directly yield dynamics such as Ohmic damping (proportional to velocity) of the oscillator of interest. By using a continuum of oscillators as a reservoir, we canonically quantize the harmonic oscillator with Ohmic damping and also with general damping behaviour. The dynamics of a damped oscillator is determined by an arbitrary effective susceptibility that obeys Kramers-Kronig relations. This approach offers an alternative description of nano-mechanical oscillators and opto-mechanical systems. PACS numbers: 03.65.-w, 03.65.Yz, 03.70.+k 1.