Interpretation of the Quantum Measurement: Example of a Linearly Coupled Harmonic Oscillator

We argue that it may be possible to consistently explain the quantum measurement by assuming that the wave function is in one-to-one correspondence with objective physical reality and has no probabilistic interpretation. In the context of such approach we consider the model of a harmonic oscillator linearly coupled to a heat bath and treat the oscillator as the system being measured. Three classes of initial pure states for the bath are considered. Exact expressions for the average values and variances of the oscillator coordinate and momentum as functions of time are considered for each class of pure states. It is shown that these quantities exhibit different asymptotic behavior for different classes of initial states of the bath. In particular, if each mode of the bath is initially in a coherent state, then for an arbitrary initial state of the oscillator the variances of the oscillator coordinate and momentum asymptotically approach the same values as for a coherent state of the free oscillator, while the averages of coordinate and momentum show a Brownian-like behavior. We argue that such behavior shows several features of the quantum measurement and supports our interpretation of the wave function.     

Thermodynamic anomalies in the presence of general linear dissipation: from the free particle to the harmonic oscillator

A free particle coupled to a heat bath can exhibit a number of thermodynamic anomalies like a negative specific heat, reentrant classicality or a nonmonotonic entropy. These low-temperature phenomena are expected to be modified at very low temperatures where finite-size effects associated with the discreteness of the energy spectrum become relevant. In this paper, we explore in which form the thermodynamic anomalies visible in the specific heat and the entropy of the free damped particle appear for a damped harmonic oscillator. Since the discreteness of the oscillator’s energy spectrum is fully accounted for, the results are valid for arbitrary temperatures. As expected, they are in agreement with the third law of thermodynamics and indicate how the thermodynamic anomalies of the free damped particle can be reconciled with the third law. Particular attention is paid to the transition from the harmonic oscillator to the free particle when the limit of the oscillator frequency to zero is taken.     

Quantum fidelity for Gaussian states describing the evolution of open systems

Using the expression of the fidelity for the most general Gaussian quantum states, the quantum fidelity is studied for the states of a harmonic oscillator interacting with an environment, in particular with a thermal bath. The time evolution of the considered system is described in the framework of the theory of open systems based on quantum dynamical semigroups. By taking a correlated squeezed Gaussian state as initial state, we calculate the quantum fidelity for both undisplaced and displaced states. The time evolution of the quantum fidelity is analyzed depending on the squeezing and correlation parameters characterizing the initial Gaussian state and on the dissipation constant and temperature of the thermal bath.     

Once upon a time,in the Soviet Union …

Radiation reaction (but, more generally, fluctuations and dissipation) occurs when a system interacts with a heat bath, a particular case being the interaction of an electron with the radiation field. We have developed a general theory for the case of a quantum particle in a general potential (but, in more detail, an oscillator potential) coupled to an arbitrary heat bath at arbitrary temperature, and in an external time-dependent c-number field. The results may be applied to a large variety of problems in physics but we concentrate by showing in detail the application to the blackbody radiation heat bath, giving an exact result for the radiation reaction problem which has no unsatisfactory features such as the runaway solutions associated with the Abraham–Lorentz theory. In addition, we show how atomic energy and free energy shifts due to temperature may be calculated. Finally, we give a brief review of applications to Josephson junctions, quantum statistical mechanics, mesoscopic physics, quantum information, noise in gravitational wave detectors, Unruh radiation and the violation of the quantum regression theorem.     

Three-step approach to Fokker-Planck equation for an interacting open waveguide system and localizable entanglement dynamics

We consider a system of three coupled single-mode waveguides each locally interacting with its own Gaussian environment and present a general solution for this coupled system initially in any Gaussian state using the symplectic operations. We investigate the dynamics of two-mode localizable entanglement contained in the evolved state when the system is initially in three-mode bisymmetric Gaussian state in contact with the independent decoherence. We show that such an entanglement exhibits a damped oscillation in a regime of weak waveguide-waveguide coupling and small mean photon numbers of the bath. Remarkably, we find that the entanglement can reappear after the long-time death and arrives at a steady-state oscillation, whose maximum depends strongly on both the squeezing of the bath and the coupling strength between these waveguides. Finally, we generalize the approach to a common squeezed environment case.     

Admissible states in quantum phase space

We address the question of which phase space functionals might represent a quantum state. We derive necessary and sufficient conditions for both pure and mixed phase space quantum states. From the pure state quantum condition we obtain a formula for the momentum correlations of arbitrary order and derive explicit expressions for the wave functions in terms of time-dependent and independent Wigner functions. We show that the pure state quantum condition is preserved by the Moyal (but not by the classical Liouville) time evolution and is consistent with a generic stargenvalue equation. As a by-product Baker's converse construction is generalized both to an arbitrary stargenvalue equation, associated to a generic phase space symbol, as well as to the time-dependent case. These results are properly extended to the mixed state quantum condition, which is proved to imply the Heisenberg uncertainty relations. Globally, this formalism yields the complete characterization of the kinematical structure of Wigner quantum mechanics. The previous results are then succinctly generalized for various quasi-distributions. Finally, the formalism is illustrated through the simple examples of the harmonic oscillator and the free Gaussian wave packet. As a by-product, we obtain in the former example an integral representation of the Hermite polynomials.     

Quantum fidelity of Gaussian states in open systems

Using the expression of the fidelity for the most general Gaussian quantum states, the behaviour of the quantum fidelity is described for the states of a harmonic oscillator interacting with an environment, in particular with a thermal bath. By taking a correlated squeezed Gaussian state as initial state, we calculate the quantum fidelity for both kinds of undisplaced and displaced states, and for different values of the squeezing and correlation parameters and of the environment temperature.     

Uncertainty inequalities as entanglement criteria for negative partial-transpose States

In this Letter, we show that the fulfillment of uncertainty relations is a sufficient criterion for a quantum-mechanically permissible state. We specifically construct two pseudospin observables for an arbitrary nonpositive Hermitian matrix whose uncertainty relation is violated. This method enables us to systematically derive separability conditions for all negative partial-transpose states in experimentally accessible forms. In particular, generalized entanglement criteria are derived from the Schr?dinger-Robertson inequalities for bipartite continuous-variable states.     

瓮模型中多粒子动力学的研究

研究了处于热库中的多颗粒在两个和多个瓮中的运动.通过求解含噪声项的一维朗之万方程,获得颗粒的位置和速度,并分析了其运动状态.研究发现，在高温下系统处于对称态的时间较长,反之系统将会出现多个定态.所有运动颗粒的速率分布都满足Gauss分布,非对称态的有效温度T₂与弹性恢复系数r有良好的指数关系. 关键词： 多颗粒 多瓮 速率分布 有效温度     

Universal quantum uncertainty relations: Minimum-uncertainty wave packet depends on measure of spread

Based on the statistical concept of the median, we propose a quantum uncertainty relation between semi-interquartile ranges of the position and momentum distributions of arbitrary quantum states. The relation is universal, unlike that based on the mean and standard deviation, as the latter may become non-existent or ineffective in certain cases. We show that the median-based one is not saturated for Gaussian distributions in position. Instead, the Cauchy-Lorentz distributions in position turn out to be the one with the minimal uncertainty, among the states inspected, implying that the minimum-uncertainty state is not unique but depends on the measure of spread used. Even the ordering of the states with respect to the distance from the minimum uncertainty state is altered by a change in the measure. We invoke the completeness of Hermite polynomials in the space of all quantum states to probe the median-based relation. The results have potential applications in a variety of studies including those on the quantum-to-classical boundary and on quantum cryptography.     

Gaussian transformations and distillation of entangled Gaussian states

We prove that it is impossible to distill more entanglement from a single copy of a two-mode bipartite entangled Gaussian state via local Gaussian operations and classical communication. More generally, we show that any hypothetical distillation protocol for Gaussian states involving only Gaussian operations would be a deterministic protocol. Finally, we argue that the protocol considered by Eisert et al. [preceding Letter, Phys. Rev. Lett. 89, 137903 ()]] is the optimum Gaussian distillation protocol for two copies of entangled Gaussian states.     

Nonlinear conductance fluctuations in a mesoscopic ring

We study the conductance of a single particle on a ring subject to an arbitrary dc electric field, which is generated by a linearly in time increasing magnetic flux. The full quantum mechanical time development is calculated numerically by splitting the dynamics into independent consecutive Zener tunneling transitions and free motion on the ring. The Zener transitions occur near the avoided crossings of the bandstructure which arises from the adiabatic eigenstates as a function of flux in the presence of a static scattering potential. To account for the necessary dissipation the particle is coupled to an appropriate oscillator bath which is adjusted to give a strictly linear current-voltage characteristic for arbitrary voltage and temperature in the absence of scattering. Taking a single δ-function scatterer we find that the dissipative coupling eliminates the localization in energy space found previously and leads to a well defined resistive steady state. The scattering introduces reproducible fluctuations around the average Ohmic behavior which are caused by coherent backscattering. Their magnitude depends on the strength of the scattering potential and decays slowly for large voltages. The associated correlation energy is determined by the uncertainty of the eigenstates due to the dissipative bath coupling. Thermal averaging leads to a decrease of the conductance fluctuations proportional to T^?1.     

The harmonic oscillator in a heat bath

We study the time evolution of a quantum-mechanical harmonic oscillator in interaction with an infinite heat bath, which is supposed to be initially in the canonical equilibrium at some temperature. We show that the oscillator relaxes from an arbitrary initial state to its canonical state at the same temperature, and that in the weak coupling limit the relaxation is Markovian, that is exponential. In contrast to earlier treatments of the problem [4, 5], the results are obtained without assuming any particular special form for the self-interaction of the heat bath. No use is made of coarse graining, finite memory assumptions or randomly varying Hamiltonians.     

A higher order GUP with minimal length uncertainty and maximal momentum II: Applications

In a recent paper, we presented a nonperturbative higher order Generalized Uncertainty Principle (GUP) that is consistent with various proposals of quantum gravity such as string theory, loop quantum gravity, doubly special relativity, and predicts both a minimal length uncertainty and a maximal observable momentum. In this Letter, we find exact maximally localized states and present a formally self-adjoint and naturally perturbative representation of this modified algebra. Then we extend this GUP to D dimensions that will be shown it is noncommutative and find invariant density of states. We show that the presence of the maximal momentum results in upper bounds on the energy spectrum of the free particle and the particle in box. Moreover, this form of GUP modifies blackbody radiation spectrum at high frequencies and predicts a finite cosmological constant. Although it does not solve the cosmological constant problem, it gives a better estimation with respect to the presence of just the minimal length.     

Exact c-number representation of non-Markovian quantum dissipation

The reduced dynamics of a quantum system interacting with a linear heat bath finds an exact representation in terms of a stochastic Schr?dinger equation. All memory effects of the reservoir are transformed into noise correlations and mean-field friction. The classical limit of the resulting stochastic dynamics is shown to be a generalized Langevin equation, and conventional quantum state diffusion is recovered in the Born-Markov approximation. The non-Markovian exact dynamics, valid at arbitrary temperature and damping strength, is exemplified by an application to the dissipative two-state system.     

Minimal set of local measurements and classical communication for two-mode Gaussian state entanglement quantification

We develop the minimal requirements for the complete entanglement quantification of an arbitrary two-mode bipartite Gaussian state via local measurements and a classical communication channel. The minimal set of measurements is presented as a reconstruction protocol of local covariance matrices and no previous knowledge of the state is required but its Gaussian character. The protocol becomes very simple mostly when dealing with Gaussian states transformed to its standard form, since photocounting or intensity measurements define the whole set of entangled states. In addition, conditional on some prior information, the protocol is also useful for a complete global state reconstruction.     

Quantum Speed Limit for Dynamics of Central Spin Model with Initial System-Bath Correlations

We investigate the quantum speed limit (QSL) time of an electronic spin coupled to a bath of nuclear spins. We consider three types of initial states with different correlations between the system and bath, i.e., quantum correlation, classical correlation, and no any correlation. Interestingly, we show that the QSL times of the central spin for these three types of initial correlations are identical when the couplings are homogeneous. However, it is remarkable different for inhomogenous couplings. The QSL time of the central spin is sensitive to the initial states, the average coupling strength, the distribution of the couplings between the system and bath and the number of the nuclear spins in the bath. Furthermore, we find that the coherence in the initial state has significant influences on the QSL time of the system, and can lead to the increase of QSL time for homogeneous couplings.     

Pair Production and Solutions of Wave Equation for Arbitrary-Spin Particles

We investigate the theory of particles with arbitrary spin and magnetic moment in the Lorentz representation (0, s) (s, 0) in an external constant and uniform electromagnetic field. We obtain the density matrix of free particles in pure spin states. The differential probability of pair producing particles with arbitrary spin by an external constant and uniform electromagnetic field is found using the exact solutions. We calculate the imaginary and real parts of the Lagrangian in an electromagnetic field that takes into account the vacuum polarization.     

Phase behavior and collective excitations of the Morse ring chain

Using primarily numerical methods we study clustering processes and collective excitations in a one-dimensional ring chain. The ring chain is constituted by N identical point particles with next neighbors interacting via nonlinear Morse springs. If the system is coupled to a heat bath (Gaussian white noise and viscous friction), then depending on the particle density and the bath temperature different phase-like states can be distinguished. This will be illustrated by means of numerically calculated phase diagrams. In order to identify collective excitations activated by the heat bath we calculate the spectrum of the normalized dynamical structure factor (SDF). Our numerical results show that the transition regions between different phase-like states are typically characterized by a 1/f-type SDF spectrum, reflecting the fact that near critical points correlations on all length and time scales become important. In the last part of the paper we also discuss a non-equilibrium effect, which occurs if an additional nonlinearly velocity-dependent force is included in the equations of motions. In particular it will be shown that such additional dissipative effects may stabilize cluster configurations.Received: 27 June 2003, Published online: 2 October 2003PACS: 05.70.Fh Phase transitions: general studies - 05.70.Ln Non-equilibrium and irreversible processes - 05.40.-a Fluctuation phenomena, random processes, noise and Brownian motion     

Calculation of Covariance Matrix for Multi-mode Gaussian States in Decoherence Processes

We investigate the dynamics of n single-mode continuous variable systems in a generic Gaussian state under the influence of the independent and correlated noises making use of the characteristic function method. In two models the bath is assumed to be a squeezed thermal one. We derive an explicit input-output expression between the initial and final covariance matrices. As an example, we study the evolution of entanglement of three-mode Gaussian state embedded in two noisy models.