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

States of the phase-damped oscillator

Matthew J. Collett

Published Nov 1, 1988• 0 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.38.4907

Related papers

Damping of quantum coherence: The master-equation approach

Craig M. Savage, Daniel F. Walls

We solve the master equation for the coordinate-coordinate damped harmonic oscillator for initial superpositions of coherent states. In the zero-temperature case the solution remains a simple superposition of coherent states. While the underdamped oscillator evolves all initial superpositions into mixtures of coherent states the overdamped oscillator does so selectively. For finite temperatures coherent states are no longer preserved, and we find a decrease in the variance of the off-diagonal coordinate-basis density-matrix elements below the coherent-state value. This variance decreases with increasing bath temperature. In the overdamped case there is negligible associated spreading of the diagonal coordinate-basis density-matrix elements. Thus the coordinate basis is an example of Zurek's pointer basis and the coordinate damped oscillator models the coordinate-basis density-matrix diagonalization which occurs in a coordinate measurement.

Physical Review A•Oct 1, 1985•101 citations

Damped harmonic oscillator: Pure states of the bath and exact master equations

Andrey Pereverzev

Time evolution of a harmonic oscillator linearly coupled to a heat bath is compared for three classes of initial states for the bath modes-grand canonical ensemble, number states, and coherent states. It is shown that for a wide class of number states the behavior of the oscillator is similar to the case of the equilibrium bath. If the bath modes are initially in coherent states, then the variances of the oscillator coordinate and momentum, as well as its entanglement to the bath, asymptotically approach the same values as for the oscillator at zero temperature and the average coordinate and momentum show a Brownian-like behavior. We derive an exact master equation for the characteristic function of the oscillator valid for arbitrary factorized initial conditions. In the case of the equilibrium bath this equation reduces to an equation of the Hu-Paz-Zhang type, while for the coherent states bath it leads to an exact stochastic master equation with a multiplicative noise.

arXiv:quant-ph/0306061•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 14, 2003•15 citations

Phase-diffusion pattern in quantum-nondemolition systems

Subhashish Banerjee, Joyee Ghosh, Rupamanjari Ghosh

We quantitatively analyze the dynamics of the quantum phase distribution associated with the reduced density matrix of a system, as the system evolves under the influence of its environment with an energy-preserving quantum nondemolition (QND) type of coupling. We take the system to be either an oscillator (harmonic or anharmonic) or a two-level atom (or equivalently, a spin-1/2 system), and model the environment as a bath of harmonic oscillators, initially in a general squeezed thermal state. The impact of the different environmental parameters is explicitly brought out as the system starts out in various initial states. The results are applicable to a variety of physical systems now studied experimentally with QND measurements.

arXiv:quant-ph/0703055•Physical Review A•Jun 14, 2007•19 citations

Phase diffusion in quantum dissipative systems

Subhashish Banerjee, R. Srikanth

We study the dynamics of the quantum phase distribution associated with the reduced density matrix of a system for a number of situations of practical importance, as the system evolves under the influence of its environment, interacting via a quantum nondemolition type of coupling, such that there is decoherence without dissipation, as well as when it interacts via a dissipative interaction, resulting in decoherence as well as dissipation. The system is taken to be either a two-level atom (or, equivalently, a spin-$1∕2$ system) or a harmonic oscillator, and the environment is modeled as a bath of harmonic oscillators, starting out in a squeezed thermal state. The impact of the different environmental parameters on the dynamics of the quantum phase distribution for the system starting out in various initial states is explicitly brought out. An interesting feature that emerges from our work is that the relationship between squeezing and temperature effects depends on the type of system-bath interaction. In the case of a quantum nondemolition type of interaction, squeezing and temperature work in tandem, producing a diffusive effect on the phase distribution. In contrast, in the case of a dissipative interaction, the influence of temperature can be counteracted by squeezing, which manifests as a resistance to randomization of phase. We make use of the phase distributions to bring out a notion of complementarity in atomic systems. We also study the variance of the phase using phase distributions conditioned on particular initial states of the system.

arXiv:0706.3633•Physical Review A•Dec 13, 2007•23 citations

Quantum theory of the damped harmonic oscillator

Edward G. Harris

The quantum theory of a damped harmonic oscillator is developed by considering an exactly solvable model, previously discussed by Unruh and Zurek [Phys. Rev. D 40, 1071 (1989)]. A one-dimensional harmonic oscillator is coupled to a scalar field. The field is assumed to be initially in thermal equilibrium. The Heisenberg equations of motion are solved without approximation, and the first and second moments, 〈q(t)〉, 〈p(t)〉, 〈${\mathit{q}}^{2}$(t)〉, 〈${\mathit{p}}^{2}$(t)〉, and 〈q(t)p(t)+p(t)q(t)〉 of the coordinate and momentum of the oscillator are calculated. The high-temperature, zero-temperature, free-particle, and weak-damping limits are discussed. A Wigner function of Gaussian form is constructed. The density-matrix equation derived by Louisell $[---Quantum Statistical Properties of Radiation (Wiley, New York, 1973)] is examined and shown to have a solution in agreement with our weak-damping limit.

Physical Review A•Oct 1, 1990•26 citations

Quantum coherence and interference of damped free particles

Craig M. Savage, Daniel F. Walls

We use a Markovian master equation to solve for the dynamics of a free particle damped by the coupling of its coordinate to a bath of harmonic oscillators. In the coordinate basis the reduced density matrix for the free particle tends to be diagonalized. Considering the evolution of an initial superposition of two plane waves, we find that the interference-pattern fringe contrast is decreased by the damping. However, for fixed times and sufficiently large damping, the fringe contrast increases with increasing damping. Considering an initial coordinate-space Gaussian wave function, we find that for sufficiently large damping the coupling to the environment models a quantum measurement of the particle's coordinate.

Physical Review A•Dec 1, 1985•20 citations
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.
We solve the master equation for the coordinate-coordinate damped harmonic oscillator for initial superpositions of coherent states. In the zero-temperature case the solution remains a simple superposition of coherent states. While the underdamped oscillator evolves all initial superpositions into mixtures of coherent states the overdamped oscillator does so selectively. For finite temperatures coherent states are no longer preserved, and we find a decrease in the variance of the off-diagonal coordinate-basis density-matrix elements below the coherent-state value. This variance decreases with increasing bath temperature. In the overdamped case there is negligible associated spreading of the diagonal coordinate-basis density-matrix elements. Thus the coordinate basis is an example of Zurek's pointer basis and the coordinate damped oscillator models the coordinate-basis density-matrix diagonalization which occurs in a coordinate measurement.
Time evolution of a harmonic oscillator linearly coupled to a heat bath is compared for three classes of initial states for the bath modes-grand canonical ensemble, number states, and coherent states. It is shown that for a wide class of number states the behavior of the oscillator is similar to the case of the equilibrium bath. If the bath modes are initially in coherent states, then the variances of the oscillator coordinate and momentum, as well as its entanglement to the bath, asymptotically approach the same values as for the oscillator at zero temperature and the average coordinate and momentum show a Brownian-like behavior. We derive an exact master equation for the characteristic function of the oscillator valid for arbitrary factorized initial conditions. In the case of the equilibrium bath this equation reduces to an equation of the Hu-Paz-Zhang type, while for the coherent states bath it leads to an exact stochastic master equation with a multiplicative noise.
We quantitatively analyze the dynamics of the quantum phase distribution associated with the reduced density matrix of a system, as the system evolves under the influence of its environment with an energy-preserving quantum nondemolition (QND) type of coupling. We take the system to be either an oscillator (harmonic or anharmonic) or a two-level atom (or equivalently, a spin-1/2 system), and model the environment as a bath of harmonic oscillators, initially in a general squeezed thermal state. The impact of the different environmental parameters is explicitly brought out as the system starts out in various initial states. The results are applicable to a variety of physical systems now studied experimentally with QND measurements.
We study the dynamics of the quantum phase distribution associated with the reduced density matrix of a system for a number of situations of practical importance, as the system evolves under the influence of its environment, interacting via a quantum nondemolition type of coupling, such that there is decoherence without dissipation, as well as when it interacts via a dissipative interaction, resulting in decoherence as well as dissipation. The system is taken to be either a two-level atom (or, equivalently, a spin-1∕21∕21∕2 system) or a harmonic oscillator, and the environment is modeled as a bath of harmonic oscillators, starting out in a squeezed thermal state. The impact of the different environmental parameters on the dynamics of the quantum phase distribution for the system starting out in various initial states is explicitly brought out. An interesting feature that emerges from our work is that the relationship between squeezing and temperature effects depends on the type of system-bath interaction. In the case of a quantum nondemolition type of interaction, squeezing and temperature work in tandem, producing a diffusive effect on the phase distribution. In contrast, in the case of a dissipative interaction, the influence of temperature can be counteracted by squeezing, which manifests as a resistance to randomization of phase. We make use of the phase distributions to bring out a notion of complementarity in atomic systems. We also study the variance of the phase using phase distributions conditioned on particular initial states of the system.
The quantum theory of a damped harmonic oscillator is developed by considering an exactly solvable model, previously discussed by Unruh and Zurek [Phys. Rev. D 40, 1071 (1989)]. A one-dimensional harmonic oscillator is coupled to a scalar field. The field is assumed to be initially in thermal equilibrium. The Heisenberg equations of motion are solved without approximation, and the first and second moments, 〈q(t)〉, 〈p(t)〉, 〈q2{\mathit{q}}^{2}q2(t)〉, 〈p2{\mathit{p}}^{2}p2(t)〉, and 〈q(t)p(t)+p(t)q(t)〉 of the coordinate and momentum of the oscillator are calculated. The high-temperature, zero-temperature, free-particle, and weak-damping limits are discussed. A Wigner function of Gaussian form is constructed. The density-matrix equation derived by Louisell $[---Quantum Statistical Properties of Radiation (Wiley, New York, 1973)] is examined and shown to have a solution in agreement with our weak-damping limit.
We use a Markovian master equation to solve for the dynamics of a free particle damped by the coupling of its coordinate to a bath of harmonic oscillators. In the coordinate basis the reduced density matrix for the free particle tends to be diagonalized. Considering the evolution of an initial superposition of two plane waves, we find that the interference-pattern fringe contrast is decreased by the damping. However, for fixed times and sufficiently large damping, the fringe contrast increases with increasing damping. Considering an initial coordinate-space Gaussian wave function, we find that for sufficiently large damping the coupling to the environment models a quantum measurement of the particle's coordinate.