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arXiv:quant-ph/9909005 · Physical Review A

Pointer states via decoherence in a quantum measurement

Anu Venugopalan

Published Dec 8, 1999• 34 citations
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Abstract

We consider the interaction of a quantum system (spin-$\frac{1}{2})$ with a macroscopic quantum apparatus (harmonic oscillator) which in turn is coupled to a bath of harmonic oscillators. Exact solutions of the Markovian master equation show that the reduced density matrix of the system-apparatus combination decoheres to a statistical mixture where up and down spins eventually correlate with pointer states of the apparatus (harmonic oscillator), with associated probabilities in accordance with quantum principles. For the zero-temperature bath these pointer states turn out to be coherent states of the harmonic oscillator (apparatus) for arbitrary initial states of the apparatus. Further, we see that the decoherence time is inversely proportional to the square of the separation between the two coherent states with which the spins correlate. For a high-temperature bath, pointer states no longer remain coherent states but are Gaussian distributions (generalized coherent states). Spin up and down states of the system now correlate with nearly diagonal distributions in position of these generalized coherent states. The diagonalization in position increases with the temperature of the bath. The off-diagonal elements in spin space decohere over a time scale which goes inversely as the square of the separation between the peaks of the two position distributions that correlate with the spin states. Zurek's earlier approximate result for the decoherence time is consistent with our exact results. Our analysis brings out the importance of looking at a measurementlike scenario where definite correlations are established between the system and apparatus to determine the nature of the pointer basis of the apparatus. Further, our exact results demonstrate in an unambiguous way that the pointer states in this measurement model emerge independent of the initial state of the apparatus.

DOI
10.1103/physreva.61.012102

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We consider the interaction of a quantum system (spin-12)\frac{1}{2})21​) with a macroscopic quantum apparatus (harmonic oscillator) which in turn is coupled to a bath of harmonic oscillators. Exact solutions of the Markovian master equation show that the reduced density matrix of the system-apparatus combination decoheres to a statistical mixture where up and down spins eventually correlate with pointer states of the apparatus (harmonic oscillator), with associated probabilities in accordance with quantum principles. For the zero-temperature bath these pointer states turn out to be coherent states of the harmonic oscillator (apparatus) for arbitrary initial states of the apparatus. Further, we see that the decoherence time is inversely proportional to the square of the separation between the two coherent states with which the spins correlate. For a high-temperature bath, pointer states no longer remain coherent states but are Gaussian distributions (generalized coherent states). Spin up and down states of the system now correlate with nearly diagonal distributions in position of these generalized coherent states. The diagonalization in position increases with the temperature of the bath. The off-diagonal elements in spin space decohere over a time scale which goes inversely as the square of the separation between the peaks of the two position distributions that correlate with the spin states. Zurek's earlier approximate result for the decoherence time is consistent with our exact results. Our analysis brings out the importance of looking at a measurementlike scenario where definite correlations are established between the system and apparatus to determine the nature of the pointer basis of the apparatus. Further, our exact results demonstrate in an unambiguous way that the pointer states in this measurement model emerge independent of the initial state of the apparatus.
We study a class of quantum measurement models. A microscopic object is entangled with a macroscopic pointer such that each eigenvalue of the measured object observable is tied up with a specific pointer deflection. Different pointer positions mutually decohere under the influence of a bath. Object-pointer entanglement and decoherence of distinct pointer readouts proceed simultaneously. Mixtures of macroscopically distinct object-pointer states may then arise without intervening macroscopic superpositions. Initially, object and apparatus are statistically independent while the latter has pointer and bath correlated according to a metastable local thermal equilibrium. We obtain explicit results for the object-pointer dynamics with temporal coherence decay in general neither exponential nor Gaussian. The decoherence time does not depend on details of the pointer-bath coupling if it is smaller than the bath correlation time, whereas in the opposite Markov regime the decay depends strongly on whether that coupling is Ohmic or super-Ohmic.
We report and give an alternative derivation of some results on a model for a quantum measurement studied in [1]. The measured microscopic system is coupled to the position of a macroscopic pointer, which itself interacts with its environment via its momentum. The entanglement between the system and the pointer produced by their mutual interaction is simultaneous with the decoherence of distinct pointer readings resulting from leakage of information to the environment. After a discussion on the various time scales in the model we calculate the matrix elements of the system-pointer density operator between eigenstates of the measured observable with distinct eigenvalues. In general, the decay with time of these coherences is neither exponential nor gaussian. We determine the decoherence (decay) time in terms of the strength of the system-pointer and pointer-environment couplings. This decoherence time does not depend upon the details of the pointer-bath coupling as soon as it is smaller than the bath correlation time (non-Markov regime). In contrast, in the Markov regime it depends strongly on whether this coupling is Ohmic or super-Ohmic.
We study a model of dephasing (decoherence) in a two-state quantum system (qubit) coupled to a bath of harmonic oscillators. An exact analytic solution for the reduced dynamics of a two-state system in this model has been obtained previously for factorizing initial states of the combined system. We show that the model admits exact solutions for a large class of correlated initial states which are typical in the theory of quantum measurements. We derive exact expressions for the off-diagonal elements of the qubit density matrix, which hold for an arbitrary strength of coupling between the qubit and the bath. The influence of initial correlations on decoherence is considered for different bath spectral densities. Time behavior of the qubit entropy in the decoherence process is discussed.
We present a general analytic method for evaluating the generally time-dependent pointer states of a subsystem, which are defined by their capability not to entangle with the states of another subsystem. We explore the conditions under which the pointer states of the system become independent of time; so that a preferred basis of measurement can be realized. We relate the mathematical conditions for having time-independent pointer states to some classes of possible symmetries in the Hamiltonian of the total composite system. Indeed, our theory would serve as a generalization of the existing theory for determination of the preferred basis of measurement. By exploiting this new theory we can obtain those regimes of the parameter space for a given total Hamiltonian defining our system-environment model for which a preferred basis of measurement can be realized. Moreover, we can predict the corresponding preferred basis of measurement for each regime. We can also obtain the time-dependent pointer states of the system and the environment in most of the other regimes where the pointer states of the system are time-dependent and a preferred basis of measurement cannot be realized at all. This ability to obtain time-dependent pointer states is specifically important in decoherence studies; as these pointer states, although they evolve with time and cannot represent the preferred basis of measurement, they correspond to those initial conditions for the state of the system and the environment for which we can have longer decoherence times. In the next step, we consider a spin-boson Hamiltonian which is generalized such that the Hamiltonians for the system and the interaction with the environment do not commute with each other. Considering a single-mode quantized field in exact resonance with the tunneling matrix element of the system, we calculate the time-dependent pointer states of the system and the environment for the case that the environment initially is prepared in the coherent state. We also obtain a closed form for the offdiagonal element of the reduced density matrix of the system and study the decoherence of the central system in our model. We will show that for the case that the system initially is prepared in one of its pointer states, the offdiagonal element of the reduced density matrix of the system will be a sinusoidal function with a slow decaying envelop which is characterized by a decay time proportional to n-bar.
We study the effect of the environment on the process of the measurement of a state of a microscopic spin-half system. The measuring apparatus is a heavy particle, whose center-of-mass coordinates can be considered at the end of the measurement as approximately classical, and thus can be used as a pointer. The state of the pointer, which is the result of its interaction with the spin, is transformed into a mixed state by the coupling of the pointer to the environment. The environment is considered to be a gas reservoir, whose particles interact with the pointer. This results in a Fokker-Planck equation for the reduced density matrix of the pointer. The solution of the equation shows that the quantum coherences, which are characteristic to the entangled state between the probabilities to find the pointer in one of two positions, decays exponentially fast in time. We calculate the exponential decay function of this decoherence effect, and express it in terms of the parameters of the model.
It is not always justified to model the environment of a realistic quantum system as a collection of harmonic oscillators or two-level systems. To explore interesting physics associated with a composite environment, here we study the dynamics of a qubit coupled to a second two-level system, which is in turn coupled to a harmonic-oscillator bath. That is, the qubit of interest is in the presence of a composite environment consisting of the second two-level system and a conventional harmonic-oscillator bath. We investigate the issue of pointer states of decoherence for the qubit for different coupling strengths between the qubit and such a composite environment. It is shown that for weak-, intermediate-, or strong-coupling strengths, the qubit's reduced density matrix has a coupling-strength-dependent diagonal representation, thus yielding coupling-strength-dependent computational pointer states. Interestingly, the associated decoherence rate can decrease as the coupling strength increases.