387 research outputs found
On the Integrability of the Rabi Model
The exact spectrum of the Rabi hamiltonian is analytically found for
arbitrary coupling strength and detuning. I present a criterion for
integrability of quantum systems containing discrete degrees of freedom which
shows that in this case a finite symmetry group may entail integrability, even
without the presence of conserved charges beyond the hamiltonian itself.
Moreover, I introduce and solve a natural generalization of the Rabi model
which has no symmetries and is therefore probably the smallest non-integrable
physical system.Comment: 7 pages, 4 figures, figure captions correcte
Solution of the Dicke model for N=3
The N=3 Dicke model couples three qubits to a single radiation mode via
dipole interaction and constitutes the simplest quantum-optical system allowing
for Greenberger-Horne-Zeilinger states. In contrast to the case N=1 (the Rabi
model), it is non-integrable if the counter-rotating terms are included. The
spectrum is determined analytically, employing the singularity structure of an
associated differential equation. While quasi-exact eigenstates known from the
Rabi model do not exist, a novel type of spectral degeneracy becomes possible
which is not associated with a symmetry of the system.Comment: 10 pages, 4 figure
Note on the Analytical Solution of the Rabi Model
It is shown that a recent critique (arXiv:1210.1130 and arXiv:1211.4639)
concerning the analytical solution of the Rabi model is unfounded.Comment: This version contains the reply to a comment on the previous versio
Fermi's golden rule and the second law of thermodynamics
We present a Gedankenexperiment that leads to a violation of detailed balance
if quantum mechanical transition probabilities are treated in the usual way by
applying Fermi's "golden rule". This Gedankenexperiment introduces a collection
of two-level systems that absorb and emit radiation randomly through
non-reciprocal coupling to a waveguide, as realized in specific chiral quantum
optical systems. The non-reciprocal coupling is modeled by a hermitean
Hamiltonian and is compatible with the time-reversal invariance of unitary
quantum dynamics. Surprisingly, the combination of non-reciprocity with
probabilistic radiation processes entails negative entropy production. Although
the considered system appears to fulfill all conditions for Markovian
stochastic dynamics, such a dynamics violates the Clausius inequality, a
formulation of the second law of thermodynamics. Several implications
concerning the interpretation of the quantum mechanical formalism are
discussed.Comment: thoroughly revised, 30.5 pages, 9 figures, published online in
Foundations of Physic
Integrability and weak diffraction in a two-particle Bose-Hubbard model
A recently introduced one-dimensional two-particle Bose-Hubbard model with a
single impurity is studied on finite lattices. The model possesses a discrete
reflection symmetry and we demonstrate that all eigenstates odd under this
symmetry can be obtained with a generalized Bethe ansatz if periodic boundary
conditions are imposed. Furthermore, we provide numerical evidence that this
holds true for open boundary conditions as well. The model exhibits
backscattering at the impurity site -- which usually destroys integrability --
yet there exists an integrable subspace. We investigate the non-integrable even
sector numerically and find a class of states which have almost the Bethe
ansatz form. These weakly diffractive states correspond to a weak violation of
the non-local Yang-Baxter relation which is satisfied in the odd sector. We
bring up a method based on the Prony algorithm to check whether a numerically
obtained wave function is in the Bethe form or not, and if so, to extract
parameters from it. This technique is applicable to a wide variety of other
lattice models.Comment: 13.5 pages, 11 figure
Bound states in the one-dimensional two-particle Hubbard model with an impurity
We investigate bound states in the one-dimensional two-particle Bose-Hubbard
model with an attractive () impurity potential. This is a
one-dimensional, discrete analogy of the hydrogen negative ion H problem.
There are several different types of bound states in this system, each of which
appears in a specific region. For given , there exists a (positive) critical
value of , below which the ground state is a bound state.
Interestingly, close to the critical value (), the ground
state can be described by the Chandrasekhar-type variational wave function,
which was initially proposed for H. For , the ground state is no
longer a bound state. However, there exists a second (larger) critical value
of , above which a molecule-type bound state is established and
stabilized by the repulsion. We have also tried to solve for the eigenstates of
the model using the Bethe ansatz. The model possesses a global \Zz_2-symmetry
(parity) which allows classification of all eigenstates into even and odd ones.
It is found that all states with odd-parity have the Bethe form, but none of
the states in the even-parity sector. This allows us to identify analytically
two odd-parity bound states, which appear in the parameter regions
and , respectively. Remarkably, the latter one can be \textit{embedded}
in the continuum spectrum with appropriate parameters. Moreover, in part of
these regions, there exists an even-parity bound state accompanying the
corresponding odd-parity bound state with almost the same energy.Comment: 18 pages, 18 figure
Symmetries in the quantum Rabi model
The quantum Rabi model is the simplest and most important theoretical description of light–matter interaction for all experimentally accessible coupling regimes. It can be solved exactly and is even integrable due to a discrete symmetry, the Z 2 or parity symmetry. All qualitative properties of its spectrum, especially the differences to the Jaynes–Cummings model, which possesses a larger, continuous symmetry, can be understood in terms of the so-called “G-functions” whose zeroes yield the exact eigenvalues of the Rabi Hamiltonian. The special type of integrability appearing in systems with discrete degrees of freedom is responsible for the absence of Poissonian level statistics in the spectrum while its well-known “Juddian” solutions are a natural consequence of the structure of the G-functions. The poles of these functions are known in closed form, which allows drawing conclusions about the global spectrum
Continued Fractions and the Rabi Model
Techniques based on continued fractions to compute numerically the spectrum
of the quantum Rabi model are reviewed. They are of two essentially different
types. In the first case, the spectral condition is implemented using a
representation in the infinite-dimensional Bargmann space of analytic
functions. This approach is shown to approximate the correct spectrum of the
full model if the continued fraction is truncated at sufficiently high order.
In the second case, one considers the limit of a sequence of models defined in
finite-dimensional state spaces. Contrary to the first, the second approach is
ambiguous and can be justified only through recourse to the analyticity
argument from the first method.Comment: published versio
Spectral determinant of the two‐photon quantum Rabi model
The various generalized spectral determinants (G-functions) of the two-photon quantum Rabi model are analyzed with emphasis on the qualitative aspects of the regular spectrum. Whereas all of them yield at least a subset of the exact regular eigenvalues, only the G-function proposed by Chen et al. in 2012 exhibits an explicitly known pole structure which dictates the approach to the collapse point. This function is derived rigorously employing the Z4-symmetry of the model and shown that its zeros correspond to the complete regular spectrum
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