6,211 research outputs found
Elementary solution to the time-independent quantum navigation problem
A quantum navigation problem concerns the identification of a time-optimal Hamiltonian that realizes a required quantum process or task, under the influence of a prevailing ‘background’ Hamiltonian that cannot be manipulated. When the task is to transform one quantum state into another, finding the solution in closed form to the problem is nontrivial even in the case of timeindependent Hamiltonians. An elementary solution, based on trigonometric analysis, is found here when the Hilbert space dimension is two. Difficulties arising from generalizations to higher-dimensional systems are discussed
Note on exponential families of distributions
We show that an arbitrary probability distribution can be represented in
exponential form. In physical contexts, this implies that the equilibrium
distribution of any classical or quantum dynamical system is expressible in
grand canonical form.Comment: 5 page
Entropy and Temperature of a Quantum Carnot Engine
It is possible to extract work from a quantum-mechanical system whose
dynamics is governed by a time-dependent cyclic Hamiltonian. An energy bath is
required to operate such a quantum engine in place of the heat bath used to run
a conventional classical thermodynamic heat engine. The effect of the energy
bath is to maintain the expectation value of the system Hamiltonian during an
isoenergetic expansion. It is shown that the existence of such a bath leads to
equilibrium quantum states that maximise the von Neumann entropy. Quantum
analogues of certain thermodynamic relations are obtained that allow one to
define the temperature of the energy bath.Comment: 4 pages, 1 figur
Random Hamiltonian in thermal equilibrium
A framework for the investigation of disordered quantum systems in thermal
equilibrium is proposed. The approach is based on a dynamical model--which
consists of a combination of a double-bracket gradient flow and a uniform
Brownian fluctuation--that `equilibrates' the Hamiltonian into a canonical
distribution. The resulting equilibrium state is used to calculate quenched and
annealed averages of quantum observables.Comment: 8 pages, 4 figures. To appear in DICE 2008 conference proceeding
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Causal contribution and dynamical encoding in the striatum during evidence accumulation.
A broad range of decision-making processes involve gradual accumulation of evidence over time, but the neural circuits responsible for this computation are not yet established. Recent data indicate that cortical regions that are prominently associated with accumulating evidence, such as the posterior parietal cortex and the frontal orienting fields, may not be directly involved in this computation. Which, then, are the regions involved? Regions that are directly involved in evidence accumulation should directly influence the accumulation-based decision-making behavior, have a graded neural encoding of accumulated evidence and contribute throughout the accumulation process. Here, we investigated the role of the anterior dorsal striatum (ADS) in a rodent auditory evidence accumulation task using a combination of behavioral, pharmacological, optogenetic, electrophysiological and computational approaches. We find that the ADS is the first brain region known to satisfy the three criteria. Thus, the ADS may be the first identified node in the network responsible for evidence accumulation
Geometric Phase and Modulo Relations for Probability Amplitudes as Functions on Complex Parameter Spaces
We investigate general differential relations connecting the respective
behavior s of the phase and modulo of probability amplitudes of the form
\amp{\psi_f}{\psi}, where is a fixed state in Hilbert space
and is a section of a holomorphic line bundle over some complex
parameter space. Amplitude functions on such bundles, while not strictly
holomorphic, nevertheless satisfy generalized Cauchy-Riemann conditions
involving the U(1) Berry-Simon connection on the parameter space. These
conditions entail invertible relations between the gradients of the phase and
modulo, therefore allowing for the reconstruction of the phase from the modulo
(or vice-versa) and other conditions on the behavior of either polar component
of the amplitude. As a special case, we consider amplitude functions valued on
the space of pure states, the ray space , where
transition probabilities have a geometric interpretation in terms of geodesic
distances as measured with the Fubini-Study metric. In conjunction with the
generalized Cauchy-Riemann conditions, this geodesic interpretation leads to
additional relations, in particular a novel connection between the modulus of
the amplitude and the phase gradient, somewhat reminiscent of the WKB formula.
Finally, a connection with geometric phases is established.Comment: 11 pages, 1 figure, revtex
Complex Extension of Quantum Mechanics
It is shown that the standard formulation of quantum mechanics in terms of
Hermitian Hamiltonians is overly restrictive. A consistent physical theory of
quantum mechanics can be built on a complex Hamiltonian that is not Hermitian
but satisfies the less restrictive and more physical condition of space-time
reflection symmetry (PT symmetry). Thus, there are infinitely many new
Hamiltonians that one can construct to explain experimental data. One might
expect that a quantum theory based on a non-Hermitian Hamiltonian would violate
unitarity. However, if PT symmetry is not spontaneously broken, it is possible
to construct a previously unnoticed physical symmetry C of the Hamiltonian.
Using C, an inner product is constructed whose associated norm is positive
definite. This construction is completely general and works for any
PT-symmetric Hamiltonian. Observables exhibit CPT symmetry, and the dynamics is
governed by unitary time evolution. This work is not in conflict with
conventional quantum mechanics but is rather a complex generalisation of it.Comment: 4 Pages, Version to appear in PR
Hidden variable interpretation of spontaneous localization theory
The spontaneous localization theory of Ghirardi, Rimini, and Weber (GRW) is a
theory in which wavepacket reduction is treated as a genuine physical process.
Here it is shown that the mathematical formalism of GRW can be given an
interpretation in terms of an evolving distribution of particles on
configuration space similar to Bohmian mechanics (BM). The GRW wavefunction
acts as a pilot wave for the set of particles. In addition, a continuous stream
of noisy information concerning the precise whereabouts of the particles must
be specified. Nonlinear filtering techniques are used to determine the dynamics
of the distribution of particles conditional on this noisy information and
consistency with the GRW wavefunction dynamics is demonstrated. Viewing this
development as a hybrid BM-GRW theory, it is argued that, besides helping to
clarify the relationship between the GRW theory and BM, its merits make it
worth considering in its own right.Comment: 13 page
An alternative to the conventional micro-canonical ensemble
Usual approach to the foundations of quantum statistical physics is based on
conventional micro-canonical ensemble as a starting point for deriving
Boltzmann-Gibbs (BG) equilibrium. It leaves, however, a number of conceptual
and practical questions unanswered. Here we discuss these questions, thereby
motivating the study of a natural alternative known as Quantum Micro-Canonical
(QMC) ensemble. We present a detailed numerical study of the properties of the
QMC ensemble for finite quantum systems revealing a good agreement with the
existing analytical results for large quantum systems. We also propose the way
to introduce analytical corrections accounting for finite-size effects. With
the above corrections, the agreement between the analytical and the numerical
results becomes very accurate. The QMC ensemble leads to an unconventional kind
of equilibrium, which may be realizable after strong perturbations in small
isolated quantum systems having large number of levels. We demonstrate that the
variance of energy fluctuations can be used to discriminate the QMC equilibrium
from the BG equilibrium. We further suggest that the reason, why BG equilibrium
commonly occurs in nature rather than the QMC-type equilibrium, has something
to do with the notion of quantum collapse.Comment: 25 pages, 6 figure
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