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Journal of Modern Optics

Decoherence as the process of generation of coherent states

S. M. Dutra

Published Apr 1, 1998• 11 citations
Publisher

Abstract

Coherent states are the only quantum states that recover the harmonic oscillator classical motion. However, rather than being deduced as the classical limit of the theory, coherent states were introduced into physics in a different context and then found to have properties that we can identify with our familiar classical experience. If classicality is really brought about by decoherence, coherent states should emerge naturally as the unique privileged states selected by the interaction with the environment. This is shown explicitly here, for the field in a single-mode cavity with ordinary one-photon losses, by calculating them directly as the only pure state solutions of the master equation at zero temperature.

DOI
10.1080/09500349808230619

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Coherent states are the only quantum states that recover the harmonic oscillator classical motion. However, rather than being deduced as the classical limit of the theory, coherent states were introduced into physics in a different context and then found to have properties that we can identify with our familiar classical experience. If classicality is really brought about by decoherence, coherent states should emerge naturally as the unique privileged states selected by the interaction with the environment. This is shown explicitly here, for the field in a single-mode cavity with ordinary one-photon losses, by calculating them directly as the only pure state solutions of the master equation at zero temperature.
The rate at which pure initial states deteriorate into mixtures is computed for a harmonic oscillator interacting with an environment in thermal equilibrium. The decoherence process resulting from this interaction selects a set of states characterized by maximal stability (or minimal loss of predictive power) which can be quantified by the rate of increase in either linear or statistical entropy. In the weak coupling limit, coherent states are shown to produce the least entropy, thus becoming the natural counterparts of classical points in phase space.
Quantum decoherence is a proposed mechanism for the emergence of classical physics from the quantum mechanics. It has been developed extensively in recent years, but is sufficiently technically complicated to discourage widespread understanding. In this paper we provide a gentle introduction to quantum decoherence. We introduce state operators and their density matrix representations to describe composite systems, such as an experiment and its environment. We illustrate how the loss of information about a subsystem can cause a quantum system to appear classical. We first analyze a discrete example of phase randomization, then a Bell state, and finally a continuous system. In the final case we provide an accessible derivation of a major early result of decoherence theory, the master equation of quantum Brownian motion. We conclude by applying the master equation to the decoherence of a simple harmonic oscillator, with results reminiscent of our earlier discrete examples.
A thoroughgoing and consistent application of quantum theory shows that the connection between classical and quantum physics is very different from what can be found in standard textbooks. In the last two decades it has been shown that the elements of classical physics emerge through an irreversible quantum process called ‘decoherence’. According to decoherence theories, quantum objects acquire classical properties only through interactions with their natural environment as a consequence of the holistic features of quantum theory. In view of the interpretation problem of quantum theory, only two directions out of the host of proposals suggested in the course of time appear to remain consistent solutions. Whatever solution is preferred, the message of decoherence remains the same: classical notions are not needed as the starting point for physics. Instead, these emerge through the dynamical process of decoherence from the quantum substrate.
We present a correspondence principle framework that maps a classical field-amplitude decay law Γ(t) onto a vacuum-loss quantum master equation for a single bosonic mode. Under the strict assumption that an initially coherent state remains pure, the ansatz selects a vacuum reservoir loss model rather than fixing general open-system quantum dynamics. While classical attenuation alone does not fix the noise structure accordingly, the derived generator is a compatible quantum model under explicit zero-temperature assumptions, not a consequence of the classical envelope alone-as thermal noise or environmental fluctuations can produce different quantum states with identical mean amplitudes. After reviewing representative classical regimes we derive a generalized master equation for a single bosonic mode whose complex amplitude is αΓ(t), where Γ(t) is a real, positive, strictly monotonic envelope. The resulting dynamics reduces to the standard amplitude-damping Lindblad equation for Γ(t)=exp(-γt/2) and preserves the coherent-state form |αΓ(t)⟩⟨αΓ(t)|. We clarify the exact physical assumptions required for this construction and distinguish CP-divisibility from non-Markovian memory effects. We prove uniqueness of this generator only within the stated pure-loss, single-mode, real-envelope assumptions, clarify its exact equivalence to a time-reparametrized pure-loss channel and discuss the limits of validity in dispersive absorbing media within macroscopic QED.
The elementary quantum theory of quasi-classical states for mechanical harmonic oscillators is applied to the radiation oscillators describing light, and thus used to define quasi-classical states of light. These states are called “coherent”, because the expectation value of any normally ordered product of creation and annihilation operators factorizes for them. A theoretical discussion of possible sources for coherent states is followed by experimental evidence for their existence. The mathematical properties of coherent states, e.g., nonorthogonality and overcompleteness, are explained in order to use them to represent states and operators. These representations provide a suitable replacement for the classical phase space description of light, which is forbidden by the uncertainty principle. The quantum phase space approach leads to the definition of the Wigner distribution, the Q-function, the P-function, and Gaussian mixed states.
In recent years decoherence has been widely discussed and accepted as a mechanism responsible for the emergence of classicality in quantum measurements and the absence in the real world of Schrödinger-cat-like states [ 2 , 6 , 7 , 11 , 12 ]. Let me start with recalling the Schrödinger cat paradox. It was a hypothetical experiment proposed by Schrödinger in 1935 which can be briefly described as follows. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.