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arXiv:1104.4405

Time-dependent pointer states and determination of the preferred basis of measurement

Hoofar Daneshvar, G. W. F. Drake

Published Apr 22, 2011• 2 citations
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Abstract

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. In this way, we show how in practice the global state of the system and the environment may evolve into a diagonal state as a result of the natural evolution of the total composite system. 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. As we show, these conditions include the so-called quantum limit of decoherence and the so-called quantum measurement limit; as well as some other specific conditions which are discussed in the paper. 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 pointer states correspond to those initial conditions for the state of the system and the environment for which we can have longer decoherence times.

DOI
10.48550/arxiv.1104.4405

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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. In this way, we show how in practice the global state of the system and the environment may evolve into a diagonal state as a result of the natural evolution of the total composite system. 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. As we show, these conditions include the so-called quantum limit of decoherence and the so-called quantum measurement limit; as well as some other specific conditions which are discussed in the paper. 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 pointer states correspond to those initial conditions for the state of the system and the environment for which we can have longer decoherence times.
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 consider the quantized atom-field model and for the regime ; where , and respectively represent the self-Hamiltonians of the environment, the system, and the interaction between the system and the environment. Considering a single-mode quantized field we obtain the time evolution operator for the model. Using our time evolution operator we calculate the time-dependent pointer states of the system and the environment, which are characterized by their property of not entangling with states of another subsystem. We obtain a closed form expression for the off-diagonal element of the reduced density matrix of the system and study the decoherence of the central system in our model. We show that for the case where the system initially is not prepared in one of its pointer states, the off-diagonal element of the reduced density matrix of the system decays with a decoherence time that is inversely proportional to the square root of the mass of the bosonic field particles.
It is well-known that the pointer basis of a quantum system satisfies the condition to diagonalize the interaction Hamiltonian between the subsystems. We show that this condition can be translated into the form δΛ=0,δΛ=0,δΛ=0, where ΛΛΛ, so-called the action, is the time integrated interaction energy: it is found out naturally in the phase of state vectors due to diagonal interaction. The careful treatment of a two states system demonstrates that the states of the total system branch into the states with different values of the action. Mathematically the pointer states are selected out by the saddle point condition on the phase ΛΛΛ. This study helps us to understand the precise mechanism and the general dynamics of decoherence.
The consensus regarding quantum measurements rests on two statements: (i) von Neumann’s standard quantum measurement theory leaves undetermined the basis in which observables are measured, and (ii) the environmental decoherence of the measuring device (the “meter”) unambiguously determines the measuring (“pointer”) basis. The latter statement means that the environment monitors (measures) selected observables of the meter and (indirectly) of the system. Equivalently, a measured quantum state must end up in one of the “pointer states” that persist in the presence of the environment. We find that, unless we restrict ourselves to projective measurements, decoherence does not necessarily determine the pointer basis of the meter. Namely, generalized measurements commonly allow the observer to choose from a multitude of alternative pointer bases that provide the same information on the observables, regardless of decoherence. By contrast, the measured observable does not depend on the pointer basis, whether in the presence or in the absence of decoherence. These results grant further support to our notion of Quantum Lamarckism, whereby the observer’s choices play an indispensable role in quantum mechanics.
Selection of the preferred classical set of states in the process of decoherence -- so important for cosmological considerations -- is discussed with an emphasis on the role of information loss and entropy. {\it 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) {\it predictability} is a measure of such persistence. {\it Predictability 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 {\it pointer basis} is proposed. Predictably evolving states of decohering systems along with the time-ordered sequences of records of their evolution define the effectively classical branches of the universal wavefunction in the context of the ``Many Worlds Interpretation". The relation between the consistent histories approach and the preferred basis is considered. It is demonstrated that histories of sequences of events corresponding to projections onto the states of the pointer basis are consistent.
Quantum decoherence is a physical process resulting from a system’s entanglement with an environment. This chapter examines the explanatory wealth of decoherence when brought to bear on foundational issues such as the question of why certain measurement bases seem to be preferred by nature – including the pointer basis of measuring apparatuses – the old familiar measurement problem, and the emergence of classicality. Decoherence satisfactorily explains the behavior of the ammonia molecule. Decoherence introduces no additional nonlinearity but “simply takes real, inevitable, interaction with external degrees of freedom to bear in the Hamiltonian”. The basis-specific nature of decoherence will allow for a satisfying explanation to one of quantum mechanics’ oldest mysteries: that of the pointer basis. The basis-specific nature of decoherence will allow for a satisfying explanation to one of quantum mechanics’ oldest mysteries: that of the pointer basis.