Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Appearance of classical coherent oscillator states under environment-induced decoherence

Kanwarpreet S. Grewal

Published Apr 15, 2002• 4 citations
Publisher

Abstract

The existence of well-defined quasiprobabilities has long been known to be the signature of a classical state. Recent studies on environment induced decoherence have suggested the use of the Wigner function to illustrate the destruction of coherence on interaction with the environment. Due to the overcompleteness of coherent states, any state can be represented as a mixture of coherent states with a relative weight $P(v).$ This is known as the P representation of the state. In this paper we use the P representation of a state to see how its classical form emerges on interaction with the environment. We also use the concept of ``depth of classicality'' introduced recently to see how classicality is increased on interaction with the environment. We use the example of superposition of Gaussian wave packets for all the calculations. We also attempt to implement the predictability sieve using the P representation of a state. We use this definition of the predictability sieve to show that coherent states are more robust to environment-induced decoherence than squeezed states.

DOI
10.1103/physreva.65.052103

Related papers

Reduction of the wave packet: Preferred observable and decoherence time scale

Juan Pablo Paz, Salman Habib, Wojciech Hubert Zurek

Environment-induced destruction of quantum coherence is investigated in a simple model where the system is a harmonic oscillator, the environment is a collection of harmonic oscillators, and the interaction between them is linear in the position coordinate of the system. We study decoherence for initial states consisting of coherent superpositions of two Gaussian wave packets in either position or momentum. A new measure of the effectiveness of decoherence appropriate to the model and choice of initial conditions is proposed. By studying the dependence of the decoherence rate on the location of the initial peaks of the Wigner function, we clarify the sense in which position is a preferred observable even though position eigenstates are not the pointer states of this model. We analyze decoherence in the low-temperature regime and show that the usual "high-temperature" approximation is remarkably accurate in its domain of applicability. We also examine the relationship between the decoherence process and the frequency distribution of the environment oscillators (in particular, we focus attention on a specific "supra-Ohmic" environment). Implications of our results for the quantum to classical transition in various contexts are briefly explored.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Jan 15, 1993•283 citations

Predictability sieve, pointer states, and the classicality of quantum trajectories

Diego A. R. Dalvit, Jacek Dziarmaga, Wojciech Hubert Zurek

We study various measures of classicality of the states of open quantum systems subject to decoherence. Classical states are expected to be stable in spite of decoherence, and are thought to leave conspicuous imprints on the environment. Here these expected features of environment-induced superselection are quantified using four different criteria: predictability sieve (which selects states that produce least entropy), purification time (which looks for states that are the easiest to find out from the imprint they leave on the environment), efficiency threshold (which finds states that can be deduced from measurements on a smallest fraction of the environment), and purity loss time (that looks for states for which it takes the longest to lose a set fraction of their initial purity). We show that when pointer states---the most predictable states of an open quantum system selected by the predictability sieve---are well defined, all four criteria agree that they are indeed the most classical states. We illustrate this with two examples: an underdamped harmonic oscillator, for which coherent states are unanimously chosen by all criteria, and a free particle undergoing quantum Brownian motion, for which most criteria select almost identical Gaussian states (although, in this case, the predictability sieve does not select well defined pointer states).

arXiv:quant-ph/0509174•Physical Review A•Dec 1, 2005•33 citations

Stability of states in a squeezed environment

Kanwarpreet S. Grewal

Several recent studies on the subject of environment induced decoherence have suggested that the predictability sieve selects coherent states over squeezed states. Most of such studies have considered a system consisting of an oscillator linearly coupled to a bath of harmonic oscillators. In this paper, we consider an environment consisting of squeezed oscillators. We derive the master equation for an oscillator coupled to such an environment. We solve this master equation for the case of the oscillator being in a coherent state and for the oscillator in a squeezed state. We then compare the stability of coherent states and squeezed states in the general squeezed environment. Our aim is to find if coherent states are more stable than squeezed states in a squeezed environment as well or is this fact true only in an environment of coherent oscillator states? We argue that unless we show that coherent states are preferred in a nonclassical environment as well, we cannot assume that coherent states form most environments and therefore the study of the stability of states only in an environment of coherent oscillator states is not a general case.

Physical Review A•Feb 13, 2003•11 citations

Preferred States, Predictability, Classicality and the Environment-Induced Decoherence

Wojciech Hubert Zurek

Selection of the preferred classical set of states in the process of decoherence is discussed with an emphasis on the role of correlations, information loss and entropy. Persistence of correlations between the observables of two systems (for instance, a record and a state of a system evolved from the initial conditions described by that record) in the presence of the environment is used to define classical behavior. From the viewpoint of an observer (or any system capable of maintaining records and utilising information) predictability is a measure of such persistence. \texitPredictability sieve – a procedure which employs both the statistical and algorithmic entropies to systematically explore all of the Hilbert space of an open system in order to eliminate the majority of the unpredictable and non-classical states, and to locate the islands of predictability including the preferred pointer basis, is proposed. The relation between the consistent histories approach and the preferred sets of states is considered. It is demonstrated that histories of sequences of events corresponding to projections onto the states of the pointer basis are consistent. The correspondence between the perceived reality and the evolving state of the quantum Universe based on a prolonged existence of states of individual systems and correlations between them is proposed. This emphasis on predictable existence of states of classical systems and records of classical observers suggests existential interpretation of quantum theory. Existential interpretation addresses the long-standing problem of the “collapse of the wave function” in a manner which goes beyond Everett's relative state and Bohr's Copenhagen interpretations, by identifying the collapse with the decay of non-classical superpositions of states (and of the observers records) into mixtures of preferred states chosen by the environment-induced superselection.

Progress of Theoretical Physics•Feb 1, 1993•525 citations

Coherent states

J. C. Garrison, R. Y. Chiao

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.

Oxford University Press eBooks•Jun 5, 2008•0 citations

Predictability sieve and quantum trajectories: For pointer states quantum unravellings are classically robust

Diego A. R. Dalvit, Jacek Dziarmaga, Wojciech Hubert Zurek

We study various measures of classicality of the states of open quantum systems subject to decoherence and continuous quantum measurement. Classical states are expected to be stable in spite of decoherence, and are thought to leave conspicuous imprints on the environment. Here these expected features of environment-induced superselection (einselection) are quantified using four different criteria: predictability sieve (which selects states that produce least entropy), purification time (which looks for states that are the easiest to find out from the imprint they leave on the environment), efficiency threshold (which finds states that can be deduced from measurements on a smallest fraction of the environment), and purity loss time (that looks for states for which it takes the longest to lose a set fraction of their initial purity). We show that when pointer states, the most predictable states of an open quantum system selected by the predictability sieve, are well defined, all four criteria agree that they are indeed the most classical states. We illustrate this with two examples: an underdamped harmonic oscillator, for which coherent states are unanimously chosen by all criteria, and a free particle undergoing quantum Brownian motion, for which most criteria select almost identical Gaussian states (although, in this case, predictability sieve does not select well defined pointer states.)

•Sep 23, 2005•0 citations
The existence of well-defined quasiprobabilities has long been known to be the signature of a classical state. Recent studies on environment induced decoherence have suggested the use of the Wigner function to illustrate the destruction of coherence on interaction with the environment. Due to the overcompleteness of coherent states, any state can be represented as a mixture of coherent states with a relative weight P(v).P(v).P(v). This is known as the P representation of the state. In this paper we use the P representation of a state to see how its classical form emerges on interaction with the environment. We also use the concept of ``depth of classicality'' introduced recently to see how classicality is increased on interaction with the environment. We use the example of superposition of Gaussian wave packets for all the calculations. We also attempt to implement the predictability sieve using the P representation of a state. We use this definition of the predictability sieve to show that coherent states are more robust to environment-induced decoherence than squeezed states.
Environment-induced destruction of quantum coherence is investigated in a simple model where the system is a harmonic oscillator, the environment is a collection of harmonic oscillators, and the interaction between them is linear in the position coordinate of the system. We study decoherence for initial states consisting of coherent superpositions of two Gaussian wave packets in either position or momentum. A new measure of the effectiveness of decoherence appropriate to the model and choice of initial conditions is proposed. By studying the dependence of the decoherence rate on the location of the initial peaks of the Wigner function, we clarify the sense in which position is a preferred observable even though position eigenstates are not the pointer states of this model. We analyze decoherence in the low-temperature regime and show that the usual "high-temperature" approximation is remarkably accurate in its domain of applicability. We also examine the relationship between the decoherence process and the frequency distribution of the environment oscillators (in particular, we focus attention on a specific "supra-Ohmic" environment). Implications of our results for the quantum to classical transition in various contexts are briefly explored.
We study various measures of classicality of the states of open quantum systems subject to decoherence. Classical states are expected to be stable in spite of decoherence, and are thought to leave conspicuous imprints on the environment. Here these expected features of environment-induced superselection are quantified using four different criteria: predictability sieve (which selects states that produce least entropy), purification time (which looks for states that are the easiest to find out from the imprint they leave on the environment), efficiency threshold (which finds states that can be deduced from measurements on a smallest fraction of the environment), and purity loss time (that looks for states for which it takes the longest to lose a set fraction of their initial purity). We show that when pointer states---the most predictable states of an open quantum system selected by the predictability sieve---are well defined, all four criteria agree that they are indeed the most classical states. We illustrate this with two examples: an underdamped harmonic oscillator, for which coherent states are unanimously chosen by all criteria, and a free particle undergoing quantum Brownian motion, for which most criteria select almost identical Gaussian states (although, in this case, the predictability sieve does not select well defined pointer states).
Several recent studies on the subject of environment induced decoherence have suggested that the predictability sieve selects coherent states over squeezed states. Most of such studies have considered a system consisting of an oscillator linearly coupled to a bath of harmonic oscillators. In this paper, we consider an environment consisting of squeezed oscillators. We derive the master equation for an oscillator coupled to such an environment. We solve this master equation for the case of the oscillator being in a coherent state and for the oscillator in a squeezed state. We then compare the stability of coherent states and squeezed states in the general squeezed environment. Our aim is to find if coherent states are more stable than squeezed states in a squeezed environment as well or is this fact true only in an environment of coherent oscillator states? We argue that unless we show that coherent states are preferred in a nonclassical environment as well, we cannot assume that coherent states form most environments and therefore the study of the stability of states only in an environment of coherent oscillator states is not a general case.
Selection of the preferred classical set of states in the process of decoherence is discussed with an emphasis on the role of correlations, information loss and entropy. Persistence of correlations between the observables of two systems (for instance, a record and a state of a system evolved from the initial conditions described by that record) in the presence of the environment is used to define classical behavior. From the viewpoint of an observer (or any system capable of maintaining records and utilising information) predictability is a measure of such persistence. \texitPredictability sieve – a procedure which employs both the statistical and algorithmic entropies to systematically explore all of the Hilbert space of an open system in order to eliminate the majority of the unpredictable and non-classical states, and to locate the islands of predictability including the preferred pointer basis, is proposed. The relation between the consistent histories approach and the preferred sets of states is considered. It is demonstrated that histories of sequences of events corresponding to projections onto the states of the pointer basis are consistent. The correspondence between the perceived reality and the evolving state of the quantum Universe based on a prolonged existence of states of individual systems and correlations between them is proposed. This emphasis on predictable existence of states of classical systems and records of classical observers suggests existential interpretation of quantum theory. Existential interpretation addresses the long-standing problem of the “collapse of the wave function” in a manner which goes beyond Everett's relative state and Bohr's Copenhagen interpretations, by identifying the collapse with the decay of non-classical superpositions of states (and of the observers records) into mixtures of preferred states chosen by the environment-induced superselection.
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.
We study various measures of classicality of the states of open quantum systems subject to decoherence and continuous quantum measurement. Classical states are expected to be stable in spite of decoherence, and are thought to leave conspicuous imprints on the environment. Here these expected features of environment-induced superselection (einselection) are quantified using four different criteria: predictability sieve (which selects states that produce least entropy), purification time (which looks for states that are the easiest to find out from the imprint they leave on the environment), efficiency threshold (which finds states that can be deduced from measurements on a smallest fraction of the environment), and purity loss time (that looks for states for which it takes the longest to lose a set fraction of their initial purity). We show that when pointer states, the most predictable states of an open quantum system selected by the predictability sieve, are well defined, all four criteria agree that they are indeed the most classical states. We illustrate this with two examples: an underdamped harmonic oscillator, for which coherent states are unanimously chosen by all criteria, and a free particle undergoing quantum Brownian motion, for which most criteria select almost identical Gaussian states (although, in this case, predictability sieve does not select well defined pointer states.)