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Fluctuation and Noise Letters

DAMPING AND NOISE IN QUANTUM MECHANICS

N. G. van Kampen

Published Jun 1, 2001• 5 citations
Publisher

Abstract

In a quantum system damping and noise enter through the coupling with an external large bath. The usual treatments computes the density matrix of the system by expanding in the coupling strength to the bath and averaging over the bath. The result, however, is known to be unacceptable as it leads to negative probabilities. It is here proposed to study correlation functions rather than the density matrix itself. Equations for them are derived and compared with the result obtained by the usual method.

DOI
10.1142/s0219477501000299

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In a quantum system damping and noise enter through the coupling with an external large bath. The usual treatments computes the density matrix of the system by expanding in the coupling strength to the bath and averaging over the bath. The result, however, is known to be unacceptable as it leads to negative probabilities. It is here proposed to study correlation functions rather than the density matrix itself. Equations for them are derived and compared with the result obtained by the usual method.
Dissipative effects on a microscopic level are included in the Schrödinger equation. When the decay between different local levels as a result of the coupling to a bath, the Schrödinger equation no longer conserves energy, but the probability of the states is conserved. The procedure is illustrated with several examples that include direct electronic decay and damping of local phonons (vibrational levels). This method significantly reduces the calculational effort compared to conventional density matrix techniques.
In the modeling of damped quantum systems by coupling to a heat bath, it has been usual to assume the initial factorization of the density matrix into system and bath components. For realistic couplings, this does not represent a physically reasonable state. One of the simplest such models is that of phase damping, and an alternative characterization of its initial states can be obtained exactly and without ambiguity. The consequences of these new states are particularly interesting when the system is considered as a primitive model of number measurement: The local system cannot, with a finite excitation energy, be put into a superposition of number states. This is a stronger result than from previous treatment of this system, which only showed that such superpositions would decay to mixtures. Similar conclusions can be drawn for other quantum nondemolition measurements.
Dissipative effects on a microscopic level are included in the Schr\odinger equation. When the decay between different local levels as a result of the coupling to a bath, the Schr\odinger equation no longer conserves energy, but the probability of the states is conserved. The procedure is illustrated with several examples that include direct electronic decay and damping of local phonons (vibrational levels). This method significantly reduces the calculational effort compared to conventional density matrix techniques.
In the spirit of the linear response theory, a simple derivation is given of a Langevin equation for collective modes of a macroscopic system. Unlike in the usual treatment, we separate a weak interaction of individual particles with collective modes which is allowed to depend on time. In nonequilibrium cases, we have shown that a damping constant of the collective modes can be expressed in terms of a time correlation function, in which its average is taken with respect to a nonequilibrium density matrix rather than a usual canonical density matrix. A microscopic expression for a Langevin random force is obtained in terms of fluctuations of a density matrix.
In many problems in quantum optics, damping plays an important role. These include, for example, the decay of an atom in an excited state to a lower state and the decay of the radiation field inside a cavity with partially transparent mirrors. In general, damping of a system is described by its interaction with a reservoir with a large number of degrees of freedom. We are interested, however, in the evolution of the variables associated with the system only. This requires us to obtain the equations of motion for the system of interest only after tracing over the reservoir variables. There are several different approaches to deal with this problem. In this chapter, we present a theory of damping based on the density operator in which the reservoir variables are eliminated by using the reduced density operator for the system in the Schrödinger (or interaction) picture. We also present a ‘quantum jump’ approach to damping. In the next chapter, the damping of the system will be considered using the noise operator method in the Heisenberg picture. An insight into the damping mechanism is obtained by considering the decay of an atom in an excited state inside a cavity. The atom may be considered as a single system coupled to the radiation field inside the cavity. Even in the absence of photons in the cavity, there are quantum fluctuations associated with the vacuum state. As discussed in Chapter 1, the field may be visualized as a large number of harmonic oscillators, one for each mode of the cavity.
In the previous chapter, we developed the equation of motion for a system as it evolved under the influence of an unobserved (reservoir) system. We used the density matrix approach and worked in the interaction picture. In this chapter, we consider the same problem of the system-reservoir interaction using a quantum operator approach. We again eliminate the reservoir variables. The resulting equations for the system operators include, in addition to the damping terms, the noise operators which produce fluctuations. These equations have the form of classical Langevin equations, which describe, for example, the Brownian motion of a particle suspended in a liquid. The Heisenberg–Langevin approach discussed in this chapter is particularly suitable for the calculation of two-time correlation functions of the system operator as is, for example, required for the determination of the natural linewidth of a laser. We first consider the damping of the harmonic oscillator by an interaction with a reservoir consisting of many other simple harmonic oscillators. This system describes, for example, the damping of a single-mode field inside a cavity with lossy mirrors. The reservoir, in this case, consists of a large number of phonon-like modes in the mirrors. We also consider the decay of the field due to its interaction with an atomic reservoir. An interesting application of the theory of the system–reservoir interaction is the evolution of an atom inside a damped cavity. It is shown that the spontaneous transition rate of the atom can be substantially enhanced if it is placed in a resonant cavity.