Time-dependence in non-Hermitian quantum systems
- URL: http://arxiv.org/abs/2002.01977v1
- Date: Wed, 5 Feb 2020 20:19:03 GMT
- Title: Time-dependence in non-Hermitian quantum systems
- Authors: Thomas Frith
- Abstract summary: We present a coherent and consistent framework for explicit time-dependence in non-Hermitian quantum mechanics.
We create an elegant framework for Darboux and Darboux/Crum for time-dependent non-Hermitian Hamiltonians.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: We present a coherent and consistent framework for explicit time-dependence
in non-Hermitian quantum mechanics. The area of non-Hermitian quantum mechanics
has been growing rapidly over the past twenty years. This has been driven by
the fact that $\mathcal{PT}$-symmetric non-Hermitian systems exhibit real
energy eigenvalues and unitary time evolution. Historically, the introduction
of time into the world of non-Hermitian quantum mechanics has been a
conceptually difficult problem to address, as it requires the Hamiltonian to
become unobservable. We solve this issue with the introduction of a new
observable energy operator and explain why its instigation is a necessary and
natural progression in this setting. For the first time, the introduction of
time has allowed us to make sense of the parameter regime in which the
$\mathcal{PT}$-symmetry is spontaneously broken. Ordinarily, in the
time-independent setting, the energy eigenvalues become complex and the wave
functions are asymptotically unbounded. We demonstrate that in the
time-dependent setting this broken symmetry can be mended and analysis on the
spontaneously broken $\mathcal{PT}$ regime is indeed possible. We provide many
examples of this mending on a wide range of different systems, beginning with a
$2\times2$ matrix model and extending to higher dimensional matrix models and
coupled harmonic oscillator systems with infinite Hilbert space. Furthermore,
we use the framework to perform analysis on time-dependent quasi-exactly
solvable models. We present the "eternal life" of entropy in this thesis.
Ordinarily, for entangled quantum systems coupled to the environments, the
entropy decays rapidly to zero. However, in the spontaneously broken regime, we
find the entropy decays asymptotically to a non-zero value. We create an
elegant framework for Darboux and Darboux/Crum transformations for
time-dependent non-Hermitian Hamiltonians.
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