2,410 research outputs found
Hamiltonian formalism of fractional systems
In this paper we consider a generalized classical mechanics with fractional
derivatives. The generalization is based on the time-clock randomization of
momenta and coordinates taken from the conventional phase space. The fractional
equations of motion are derived using the Hamiltonian formalism. The approach
is illustrated with a simple-fractional oscillator in a free state and under an
external force. Besides the behavior of the coupled fractional oscillators is
analyzed. The natural extension of this approach to continuous systems is
stated. The interpretation of the mechanics is discussed.Comment: 16 pages, 5 figure
Fingerprints of Classical Instability in Open Quantum Dynamics
The dynamics near a hyperbolic point in phase space is modelled by an
inverted harmonic oscillator. We investigate the effect of the classical
instability on the open quantum dynamics of the oscillator, introduced through
the interaction with a thermal bath, using both the survival probability
function and the rate of von Neumann entropy increase, for large times. In this
parameter range we prove, using influence functional techniques, that the
survival probability function decreases exponentially at a rate, K', depending
not only on the measure of instability in the model but also on the strength of
interaction with the environment. We also show that K' determines the rate of
von Neumann entropy increase and that this result is independent of the
temperature of the environment. This generalises earlier results which are
valid in the limit of vanishing dissipation. The validity of inferring similar
rates of survival probability decrease and entropy increase for quantum chaotic
systems is also discussed.Comment: 13 pages, to be published in Physical Review
Scale invariant distribution functions in quantum systems with few degrees of freedom
Scale invariance usually occurs in extended systems where correlation
functions decay algebraically in space and/or time. Here we introduce a new
type of scale invariance, occurring in the distribution functions of physical
observables. At equilibrium these functions decay over a typical scale set by
the temperature, but they can become scale invariant in a sudden quantum
quench. We exemplify this effect through the analysis of linear and non-linear
quantum oscillators. We find that their distribution functions generically
diverge logarithmically close to the stable points of the classical dynamics.
Our study opens the possibility to address integrability and its breaking in
distribution functions, with immediate applications to matter-wave
interferometers.Comment: 8+10 pages. Scipost Submissio
Scale invariant distribution functions in quantum systems with few degrees of freedom
Scale invariance usually occurs in extended systems where correlation
functions decay algebraically in space and/or time. Here we introduce a new
type of scale invariance, occurring in the distribution functions of physical
observables. At equilibrium these functions decay over a typical scale set by
the temperature, but they can become scale invariant in a sudden quantum
quench. We exemplify this effect through the analysis of linear and non-linear
quantum oscillators. We find that their distribution functions generically
diverge logarithmically close to the stable points of the classical dynamics.
Our study opens the possibility to address integrability and its breaking in
distribution functions, with immediate applications to matter-wave
interferometers.Comment: 8+10 pages. Scipost Submissio
Parametric Competition in non-autonomous Hamiltonian Systems
In this work we use the formalism of chord functions (\emph{i.e.}
characteristic functions) to analytically solve quadratic non-autonomous
Hamiltonians coupled to a reservoir composed by an infinity set of oscillators,
with Gaussian initial state. We analytically obtain a solution for the
characteristic function under dissipation, and therefore for the determinant of
the covariance matrix and the von Neumann entropy, where the latter is the
physical quantity of interest. We study in details two examples that are known
to show dynamical squeezing and instability effects: the inverted harmonic
oscillator and an oscillator with time dependent frequency. We show that it
will appear in both cases a clear competition between instability and
dissipation. If the dissipation is small when compared to the instability, the
squeezing generation is dominant and one can see an increasing in the von
Neumann entropy. When the dissipation is large enough, the dynamical squeezing
generation in one of the quadratures is retained, thence the growth in the von
Neumann entropy is contained
Energy Dissipation Via Coupling With a Finite Chaotic Environment
We study the flow of energy between a harmonic oscillator (HO) and an
external environment consisting of N two-degrees of freedom non-linear
oscillators, ranging from integrable to chaotic according to a control
parameter. The coupling between the HO and the environment is bilinear in the
coordinates and scales with system size with the inverse square root of N. We
study the conditions for energy dissipation and thermalization as a function of
N and of the dynamical regime of the non-linear oscillators. The study is
classical and based on single realization of the dynamics, as opposed to
ensemble averages over many realizations. We find that dissipation occurs in
the chaotic regime for a fairly small N, leading to the thermalization of the
HO and environment a Boltzmann distribution of energies for a well defined
temperature. We develop a simple analytical treatment, based on the linear
response theory, that justifies the coupling scaling and reproduces the
numerical simulations when the environment is in the chaotic regime.Comment: 7 pages, 10 figure
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