Subsystem dynamics under random Hamiltonian evolution

Vinayak,; Žnidarič, Marko

doi:10.1088/1751-8113/45/12/125204

Cited by 69 publications

(88 citation statements)

References 46 publications

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“…[340] for an early review and Refs. [341,342,343,344] for further results. Note that, in each subensemble E sub , the expectation values of the observables of S + M may evolve according to (11.11) on the time lapse τ sub ; however, they remain constant for the full ensemble sinceD(t) has already reached its stationary value: When the subensembles E k of some decomposition (11.4) of E are put back together, the time dependences issued from (11.11) compensate one another.…”

Section: Subensemble Relaxation Of the Pointer Alonementioning

confidence: 95%

Understanding quantum measurement from the solution of dynamical models

Allahverdyan¹,

Balian²,

2013

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The quantum measurement problem, to wit, understanding why a unique outcome is obtained in each individual experiment, is currently tackled by solving models. After an introduction we review the many dynamical models proposed over the years for elucidating quantum measurements. The approaches range from standard quantum theory, relying for instance on quantum statistical mechanics or on decoherence, to quantum-classical methods, to consistent histories and to modifications of the theory. Next, a flexible and rather realistic quantum model is introduced, describing the measurement of the z-component of a spin through interaction with a magnetic memory simulated by a Curie-Weiss magnet, including N 1 spins weakly coupled to a phonon bath. Initially prepared in a metastable paramagnetic state, it may transit to its up or down ferromagnetic state, triggered by its coupling with the tested spin, so that its magnetization acts as a pointer. A detailed solution of the dynamical equations is worked out, exhibiting several time scales. Conditions on the parameters of the model are found, which ensure that the process satisfies all the features of ideal measurements. Various imperfections of the measurement are discussed, as well as attempts of incompatible measurements. The first steps consist in the solution of the Hamiltonian dynamics for the spin-apparatus density matrixD(t). Its off-diagonal blocks in a basis selected by the spin-pointer coupling, rapidly decay owing to the many degrees of freedom of the pointer. Recurrences are ruled out either by some randomness of that coupling, or by the interaction with the bath. On a longer time scale, the trend towards equilibrium of the magnet produces a final stateD(t f ) that involves correlations between the system and the indications of the pointer, thus ensuring registration. AlthoughD(t f ) has the form expected for ideal measurements, it only describes a large set of runs. Individual runs are approached by analyzing the final states associated with all possible subensembles of runs, within a specified version of the statistical interpretation. There the difficulty lies in a quantum ambiguity: There exist many incompatible decompositions of the density matrixD(t f ) into a sum of sub-matrices, so that one cannot infer from its sole determination the states that would describe small subsets of runs. This difficulty is overcome by dynamics due to suitable interactions within the apparatus, which produce a special combination of relaxation and decoherence associated with the broken invariance of the pointer. Any subset of runs thus reaches over a brief delay a stable state which satisfies the same hierarchic property as in classical probability theory; the reduction of the state for each individual run follows. Standard quantum statistical mechanics alone appears sufficient to explain the occurrence of a unique answer in each run and the emergence of classicality in a measurement process. Finally, pedagogical exercises are proposed and lessons for future works on m...

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Section: Subensemble Relaxation Of the Pointer Alonementioning

confidence: 95%

Understanding quantum measurement from the solution of dynamical models

Allahverdyan¹,

Balian²,

2013

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“…When the initial state is spread over many energy levels, and we consider the set of observables for which this state is an eigenstate, most observables are initially out of equilibrium yet equilibrate rapidly. Moreover, all two-outcome measurements, where one of the projectors is of low rank, equilibrate rapidly.The topic of equilibration time scales has been of much interest lately [1][2][3][4][5][6][7][8][9]. Given that it has been shown that quantum systems equilibrate under rather general conditions [10][11][12], it is important to understand the time scale for the process.…”

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confidence: 99%

Quantum systems equilibrate rapidly for most observables

et al. 2014

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Considering any Hamiltonian, any initial state, and measurements with a small number of possible outcomes compared to the dimension, we show that most measurements are already equilibrated. To investigate non-trivial equilibration we therefore consider a restricted set of measurements. When the initial state is spread over many energy levels, and we consider the set of observables for which this state is an eigenstate, most observables are initially out of equilibrium yet equilibrate rapidly. Moreover, all two-outcome measurements, where one of the projectors is of low rank, equilibrate rapidly.The topic of equilibration time scales has been of much interest lately [1][2][3][4][5][6][7][8][9]. Given that it has been shown that quantum systems equilibrate under rather general conditions [10][11][12], it is important to understand the time scale for the process. However, attempts to derive an upper bound on equilibration time have resulted in very large time scales. Short and Farrelly [1], for instance, obtain a very general bound, of which we give an improved derivation in appendix A, which scales with the dimension of the system, typically exponentially in the number of particles.A quantum system is said to undergo equilibration when its quantum state spends most of its time almost indistinguishable from a fixed (time-invariant) steadystate. This is not the same as thermalization, in which the steady-state is a Gibbs state. Thus, thermalization is a special case of equilibration, and understanding equilibration times is a key step in understanding thermalization times.When one discusses quantum equilibration, it is common to refer to either subsystem equilibration [10,13], in which a small system equilibrates due to contact with a bath, or observable equilibration, in which a fully closed system appears to equilibrate due to the limited information offered by outcomes of a particular set of observables. The latter was initially shown by Reimann [11,14], as a statement that the expectation values of quantum observables stay predominantly close to a static value, and was later built-on by Short [15], who showed that these results apply even if one considers all the information that can be gathered from the observable, instead of just the expectation value.In this paper we consider any finite-dimensional system and any Hamiltonian, and show that most N -outcome observables are initially in equilibrium (for N small compared to the dimension). To investigate timescales we therefore turn to a natural class of observables which are initially typically out of equilibrium -those with a definite initial value (i.e. observables for which the initial state is an eigenstate). We show that, for pure initial states spread over many energy levels, most of these observables equilibrate in very short times -in fact, most equilibrate essentially as fast as possible. Moreover, in the case of two-outcome observables where one of the projectors is of low rank we show that all observables equilibrate fast (for any initial state spread...

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“…[20,21,48] They later found their way into mathematics [22,24,49,50,51,52] and back into other areas of physics [53,54,55,56].…”

Section: Wgmentioning

confidence: 99%

Energy-dependent correlations in the S-matrix of chaotic systems

Novaes

2016

Journal of Mathematical Physics

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The M -dimensional unitary matrix S(E), which describes scattering of waves, is a strongly fluctuating function of the energy for complex systems such as ballistic cavities, whose geometry induces chaotic ray dynamics. Its statistical behaviour can be expressed by means of correlation functions of the kind Sij(E + ǫ)S † pq (E − ǫ) , which have been much studied within the random matrix approach. In this work, we consider correlations involving an arbitrary number of matrix elements and express them as infinite series in 1/M , whose coefficients are rational functions of ǫ. From a mathematical point of view, this may be seen as a generalization of the Weingarten functions of circular ensembles.

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Subsystem dynamics under random Hamiltonian evolution

Cited by 69 publications

References 46 publications

Understanding quantum measurement from the solution of dynamical models

Understanding quantum measurement from the solution of dynamical models

Quantum systems equilibrate rapidly for most observables

Energy-dependent correlations in the S-matrix of chaotic systems

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