4 research outputs found
Particle Trajectories for Quantum Maps
We study the trajectories of a semiclassical quantum particle under repeated
indirect measurement by Kraus operators, in the setting of the quantized torus.
In between measurements, the system evolves via either Hamiltonian propagators
or metaplectic operators. We show in both cases the convergence in total
variation of the quantum trajectory to its corresponding classical trajectory,
as defined by propagation of a semiclassical defect measure. This convergence
holds up to the Ehrenfest time of the classical system, which is larger when
the system is less chaotic. In addition, we present numerical simulations of
these effects.Comment: 35 pages, 7 figure
The Spectrum of an Almost Maximally Open Quantized Cat Map
We consider eigenvalues of a quantized cat map (i.e. hyperbolic symplectic
integer matrix), cut off in phase space to include a fixed point as its only
periodic orbit on the torus. We prove a simple formula for the eigenvalues on
both the quantized real line and the quantized torus in the semiclassical limit
as . We then consider the case with no fixed points, and prove a
superpolynomial decay bound on the eigenvalues. The results are illustrated
with numerical calculations.Comment: 35 pages, 3 figure
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Applications of Semiclassical Analysis on the Quantized Torus
The quantized torus is a finite-dimensional Hilbert space that represents quantum mechanicswith periodic phase space. The space can act as a toy model for many quantum effects, and
it has the benefit of admitting numerical illustrations. Taking the dimension to infinity
corresponds to taking a semiclassical limit, and allows us to visualize the quantum-classical
correspondence in a simple setting.
In this thesis, we examine several instances of such a semiclassical limit. We begin in
the setting of quantum dynamics, and consider eigenvalues of a quantized cat map (i.e.
hyperbolic symplectic integer matrix), cut off in phase space to include a fixed point as its
only periodic orbit on the torus. We prove a simple formula for the eigenvalues on both the
quantized real line and the quantized torus in the semiclassical limit as h → 0. We then
consider the case with no fixed points, and prove a superpolynomial decay bound on the
eigenvalues.
We then study the trajectories of a semiclassical quantum particle under repeated indirect
measurement by Kraus operators, in the setting of the quantized torus. In between mea-
surements, the system evolves via either Hamiltonian propagators or metaplectic operators.
We show in both cases the convergence in total variation of the quantum trajectory to its
corresponding classical trajectory, as defined by propagation of a semiclassical defect mea-
sure. This convergence holds up to the Ehrenfest time of the classical system, which is larger
when the system is “less chaotic”.
Finally, we apply semiclassical analysis to the field of quantum simulation, and improve
bounds on the basic Trotterization algorithm in the setting of the semiclassical Schr ̈odinger
equation. We show that the number of Trotter steps used for the observable evolution can
be O(1). We then apply the theory of the quantized torus to extend our results to the
discretized case, which is amenable to quantum computation models