212 research outputs found
OTOC, complexity and entropy in bi-partite systems
There is a remarkable interest in the study of Out-of-time ordered
correlators (OTOCs) that goes from many body theory and high energy physics to
quantum chaos. In this latter case there is a special focus on the comparison
with the traditional measures of quantum complexity such as the spectral
statistics, for example. The exponential growth has been verified for many
paradigmatic maps and systems. But less is known for multi-partite cases. On
the other hand the recently introduced Wigner separability entropy (WSE) and
its classical counterpart (CSE) provide with a complexity measure that treats
equally quantum and classical distributions in phase space. We have compared
the behavior of these measures in a system consisting of two coupled and
perturbed cat maps with different dynamics: double hyperbolic (HH), double
elliptic (EE) and mixed (HE). In all cases, we have found that the OTOCs and
the WSE have essentially the same behavior, providing with a complete
characterization in generic bi-partite systems and at the same time revealing
them as very good measures of quantum complexity for phase space distributions.
Moreover, we establish a relation between both quantities by means of a
recently proven theorem linking the second Renyi entropy and OTOCs.Comment: 6 pages, 5 figure
Classical to quantum correspondence in dissipative directed transport
We compare the quantum and classical properties of the (Quantum) Isoperiodic
Stable Structures -- (Q)ISSs -- which organize the parameter space of a
paradigmatic dissipative ratchet model, i.e. the dissipative modified kicked
rotator. We study the spectral behavior of the corresponding classical
Perron-Frobenius operators with thermal noise and the quantum superoperators
without it for small values. We find a remarkable similarity
between the classical and quantum spectra. This finding significantly extends
previous results -- obtained for the mean currents and asymptotic distributions
only -- and on the other hand unveils a classical to quantum correspondence
mechanism where the classical noise is qualitatively different from the quantum
one. This is crucial not only for simple attractors but also for chaotic ones,
where just analyzing the asymptotic distribution reveals insufficient.
Moreover, we provide with a detailed characterization of relevant eigenvectors
by means of the corresponding Weyl-Wigner distributions, in order to better
identify similarities and differences. Finally, this model being generic, it
allows us to conjecture that this classical to quantum correspondence mechanism
is a universal feature of dissipative systems.Comment: 7 pages, 6 figure
Semiclassical Coherent States propagator
In this work, we derived a semiclassical approximation for the matrix
elements of a quantum propagator in coherent states (CS) basis that avoids
complex trajectories, it only involves real ones. For that propose, we used
the, symplectically invariant, semiclassical Weyl propagator obtained by
performing a stationary phase approximation (SPA) for the path integral in the
Weyl representation. After what, for the transformation to CS representation
SPA is avoided, instead a quadratic expansion of the complex exponent is used.
This procedure also allows to express the semiclassical CS propagator uniquely
in terms of the classical evolution of the initial point, without the need of
any root search typical of Van Vleck Gutzwiller based propagators. For the case
of chaotic Hamiltonian systems, the explicit time dependence of the CS
propagator has been obtained. The comparison with a
\textquotedbl{}realistic\textquotedbl{} chaotic system that derives from a
quadratic Hamiltonian, the cat map, reveals that the expression here derived is
exact up to quadratic Hamiltonian systems.Comment: 13 pages, 2 figure. Accepted for publication in PR
Stable classical structures in dissipative quantum chaotic systems
We study the stability of classical structures in chaotic systems when a
dissipative quantum evolution takes place. We consider a paradigmatic model,
the quantum baker map in contact with a heat bath at finite temperature. We
analyze the behavior of the purity, fidelity and Husimi distributions
corresponding to initial states localized on short periodic orbits (scar
functions) and map eigenstates. Scar functions, that have a fundamental role in
the semiclassical description of chaotic systems, emerge as very robust against
environmental perturbations. This is confirmed by the study of other states
localized on classical structures. Also, purity and fidelity show a
complementary behavior as decoherence measures.Comment: 4 pages, 3 figure
Relevant OTOC operators: footprints of the classical dynamics
The out-of-time order correlator (OTOC) has recently become relevant in
different areas where it has been linked to scrambling of quantum information
and entanglement. It has also been proposed as a good indicator of quantum
complexity. In this sense, the OTOC-RE theorem relates the OTOCs summed over a
complete base of operators to the second Renyi entropy. Here we have studied
the OTOC-RE correspondence on physically meaningful bases like the ones
constructed with the Pauli, reflection, and translation operators. The
evolution is given by a paradigmatic bi-partite system consisting of two
perturbed and coupled Arnold cat maps with different dynamics. We show that the
sum over a small set of relevant operators, is enough in order to obtain a very
good approximation for the entropy and hence to reveal the character of the
dynamics, up to a time t 0 . In turn, this provides with an alternative natural
indicator of complexity, i.e. the scaling of the number of relevant operators
with time. When represented in phase space, each one of these sets reveals the
classical dynamical footprints with different depth according to the chosen
base.Comment: 8 pages, 10 figure
Quantum Lyapunov exponent in dissipative systems
The out-of-time order correlator (OTOC) has been widely studied in closed
quantum systems. However, there are very few studies for open systems and they
are mainly focused on isolating the effects of scrambling from those of
decoherence. Adopting a different point of view, we study the interplay between
these two processes. This proves crucial in order to explain the OTOC behavior
when a phase space contracting dissipation is present, ubiquitous not only in
real life quantum devices but in the dynamical systems area. The OTOC decay
rate is closely related to the classical Lyapunov exponent -- with some
differences -- and more sensitive in order to distinguish the chaotic from the
regular behavior than other measures. On the other hand, it reveals as a
generally simple function of the longest lived eigenvalues of the quantum
evolution operator. We find no simple connection with the Ruelle-Pollicott
resonances, but by adding Gaussian noise of size to the
classical system we recover the OTOC decay rate, being this a consequence of
the correspondence principle put forward in [Physical Review Letters 108 210605
(2012) and Physical Review E 99 042214 (2019)]Comment: 5 pages, 7 figure
Hyperbolic Scar Patterns in Phase Space
We develop a semiclassical approximation for the spectral Wigner and Husimi
functions in the neighbourhood of a classically unstable periodic orbit of
chaotic two dimensional maps. The prediction of hyperbolic fringes for the
Wigner function, asymptotic to the stable and unstable manifolds, is verified
computationally for a (linear) cat map, after the theory is adapted to a
discrete phase space appropriate to a quantized torus. The characteristic
fringe patterns can be distinguished even for quasi-energies where the fixed
point is not Bohr-quantized. The corresponding Husimi function dampens these
fringes with a Gaussian envelope centered on the periodic point. Even though
the hyperbolic structure is then barely perceptible, more periodic points stand
out due to the weakened interference.Comment: 12 pages, 10 figures, Submited to Phys. Rev.
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