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Journal of the Optical Society of America B

Rigged-reservoir response I General theory

M.‐A. Dupertuis, Stig Stenholm

Published Jul 1, 1987• 65 citations
Publisher

Abstract

We present the general theory of the linear response of a single boson mode embedded in an arbitrary background treated as a reservoir. This gives the master equation for linear amplifiers and attenuators. A conservation law similar to the Manley–Rowe relation aids the derivations. For a rigged reservoir the master equation is found to contain anomalous terms. We divide the reservoir correlation functions into linear-response and fluctuation contributions. The former give gain or loss; the latter determine the noise. The equation is written for the Wigner function, because this is suitable to describe squeezed states. The anomalous terms make the noise properties anisotropic, and the output may be squeezed, in contrast to that emerging from a device interacting with a thermal bath. For both components the output satisfies the added-noise inequality of Caves. We derive expressions for the excess noise in terms of reservoir correlation functions. In a thermal bath a fluctuation-dissipation theorem guarantees that the amplifier output cannot be squeezed when the gain exceeds the cloning value 2.

DOI
10.1364/josab.4.001094

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We present the general theory of the linear response of a single boson mode embedded in an arbitrary background treated as a reservoir. This gives the master equation for linear amplifiers and attenuators. A conservation law similar to the Manley–Rowe relation aids the derivations. For a rigged reservoir the master equation is found to contain anomalous terms. We divide the reservoir correlation functions into linear-response and fluctuation contributions. The former give gain or loss; the latter determine the noise. The equation is written for the Wigner function, because this is suitable to describe squeezed states. The anomalous terms make the noise properties anisotropic, and the output may be squeezed, in contrast to that emerging from a device interacting with a thermal bath. For both components the output satisfies the added-noise inequality of Caves. We derive expressions for the excess noise in terms of reservoir correlation functions. In a thermal bath a fluctuation-dissipation theorem guarantees that the amplifier output cannot be squeezed when the gain exceeds the cloning value 2.
In this paper we utilize the idea of multiatom squeezed states to build a rigged reservoir for linear amplification or attenuation. The reservoir state is constructed such that the processed boson mode is driven by a collective reservoir operator that offers squeezed stationary noise. Then the amplifier can retain squeezing for arbitrary gain, and the amplifier added noise can be made widely different in the two quadrature components. The attenuator drives the signal toward a squeezed noise state. In contrast to the case of a multimode squeezed reservoir, we can in this case make the noise exactly zero for one component, but only at the expense of a vanishing linear gain (or loss) coefficient. The noise-over-gain ratio can, however, become arbitrarily small. The results are discussed and related to earlier work.
As a specific example of the linear response induced by a rigged reservoir, we construct one from an assembly of harmonic oscillators in a multimode squeezed state. In the rotating frame of the boson mode the squeezed reservoir offers noise properties corresponding to those characterizing a stationary random process. With a squeezed reservoir an attenuator can produce a squeezed output, irrespective of its input. For an arbitrarily squeezed reservoir the amplifier can provide a squeezed output for a gain even larger than 2, thus beating the cloning limit.
Conventional amplifiers add quantum noise to a signal. Both amplifiers and attenuators lead to a reduction of the signal-to-noise ratio. We find, however, that devices exhibiting squeezed fluctuations in the gain or loss medium lead to an improved signal-to-noise ratio, in one quadrature of the output, compared to the limits associated with conventional devices. We investigate amplifier and attenuator noise properties using a simple model in which the gain or loss medium is a broad-band reservoir in a specially prepared, or ‘rigged’, state.
The theory of linear quantum amplifiers is reviewed. A simplified method using a "faked vacuum" is introduced to derive amplifier master equations. Properties of various amplifiers are summarized. The solution is derived and used as the amplifier transfer function. The Caves' theory for amplifier added noise is discussed. The properties of squeezed states are reviewed, the use of the Wigner function to describe them is advocated. Finally the amplification of squeezed states is discussed in the Wigner representation.
A linear response theory for open quantum systems is formulated by means of the time-local and time-nonlocal quantum master equations, where a relevant quantum system interacts with a thermal reservoir as well as with an external classical field. A linear response function that characterizes how a relaxation process deviates from its intrinsic process by a weak external field is obtained by extracting the linear terms with respect to the external field from the quantum master equation. It consists of four parts. One represents the linear response of a quantum system when system-reservoir correlation at an initial time and correlation between reservoir states at different times are neglected. The others are correction terms due to these effects. The linear response function is compared with the Kubo formula in the usual linear response theory. To investigate the properties of the linear response of an open quantum system, an exactly solvable model for a stochastic dephasing of a two-level system is examined. Furthermore, the method for deriving the linear response function is applied for calculating two-time correlation functions of open quantum systems. It is shown that the quantum regression theorem is not valid for open quantum systems unless their reduced time evolution is Markovian.
Scattering of quantum fluctuations by a linear system is entirely characterized by impedance functions that also allow the determination of the system fluctuations. We present a generalization of this linear input-output theory to the case of a nonlinear scattering system. We define response functions that are similar to the susceptibility functions of linear response theory, but behave as noncommuting quantities. These functions contain a ``dynamical'' part determined by the relaxation of the system and a ``structural'' one related to the commutators between the system observables. The generalized linear input-output theory provides us with a complete description of the system fluctuations as well as of the output reservoir fields. The consistency of the results is ensured by general relations existing between the response functions and the correlation functions.