477 research outputs found
Concise analytic solutions to the quantum Rabi model with two arbitrary qubits
Using extended coherent states, an analytical exact study has been carried
out for the quantum Rabi model (QRM) with two arbitrary qubits in a very
concise way. The -functions with determinants are generally
derived. For the same coupling constants, the simplest -function, resembling
that in the one-qubit QRM, can be obtained. Zeros of the -function yield the
whole regular spectrum. The exceptional eigenvalues, which do not belong to the
zeros of the function, are obtained in the closed form. The Dark states in
the case of the same coupling can be detected clearly in a continued-fraction
technique. The present concise solution is conceptually clear and practically
feasible to the general two-qubit QRM and therefore has many applications.Comment: 13 pages, 3 figure
Quantum criticality of the sub-Ohmic spin-boson model within displaced Fock states
The spin-boson model is analytically studied using displaced Fock states
(DFS) without discretization of the continuum bath. In the orthogonal displaced
Fock basis, the ground-state wavefunction can be systematically improved in a
controllable way. Interestingly, the zeroth-order DFS reproduces exactly the
well known Silbey-Harris results. In the framework of the second-order DFS, the
magnetization and the entanglement entropy are exactly calculated. It is found
that the magnetic critical exponent is converged to in the whole
sub-Ohmic bath regime , compared with that by the exactly solvable
generalized Silbey-Harris ansatz. It is strongly suggested that the system with
sub-Ohmic bath is always above its upper critical dimension, in sharp contrast
with the previous findings. This is the first evidence of the violation of the
quantum-classical Mapping for .Comment: 8 pages, 4 figure
Improved Silbey-Harris polaron ansatz for the spin-boson model
In this paper, the well-known Silbey-Harris (SH) polaron ansatz for the
spin-boson model is improved by adding orthogonal displaced Fock states. The
obtained results for the ground state in all baths converge very quickly within
finite displaced Fock states and corresponding SH results are corrected
considerably. Especially for the sub-Ohmic spin-boson model, the converging
results are obtained for 0 < s < 1/2 in the fourth-order correction and very
accurate critical coupling strengths of the quantum phase transition are
achieved. Converging magnetization in the biased spin-boson model is also
arrived at. Since the present improved SH ansatz can yield very accurate, even
almost exact results, it should have wide applications and extensions in
various spin-boson model and related fields.Comment: 9 pages, 6 figures. arXiv admin note: substantial text overlap with
arXiv:1410.099
Exact solvability, non-integrability, and genuine multipartite entanglement dynamics of the Dicke model
In this paper, the finite size Dicke model of arbitrary number of qubits is
solved analytically in an unified way within extended coherent states. For the
or Dicke models ( is an integer), the -function, which is
only an energy dependent determinant, is derived in a transparent
manner. The regular spectrum is completely and uniquely given by stable zeros
of the -function. The closed-form exceptional eigenvalues are also derived.
The level distribution controlled by the pole structure of the -functions
suggests non-integrability for model at any finite coupling in the sense
of recent criterion in literature. A preliminary application to the exact
dynamics of genuine multipartite entanglement in the finite Dicke model is
presented using the obtained exact solutions.Comment: 18 pages, 5 figure
The mixed quantum Rabi model
The analytical exact solutions to the mixed quantum Rabi model (QRM)
including both one- and two-photon terms are found by using Bogoliubov
operators. Transcendental functions in terms of determinants
responsible for the exact solutions are derived. These so-called -functions
with pole structures can be reduced to the previous ones in the unmixed QRMs.
The zeros of -functions reproduce completely the regular spectra. The
exceptional eigenvalues can also be obtained by another transcendental
function. From the pole structure, we can derive two energy limits when the
two-photon coupling strength tends to the collapse point. All energy levels
only collapse to the lower one, which diverges negatively. The level crossings
in the unmixed QRMs are relaxed to avoided crossings in the present mixed QRM
due to absence of parity symmetry. In the weak two-photon coupling regime, the
mixed QRM is equivalent to an one-photon QRM with an effective positive bias,
suppressed photon frequency and enhanced one-photon coupling, which may pave a
highly efficient and economic way to access the deep-strong one-photon coupling
regime.Comment: 11 pages, 8 figure
Distributed Bandit Learning: Near-Optimal Regret with Efficient Communication
We study the problem of regret minimization for distributed bandits learning,
in which agents work collaboratively to minimize their total regret under
the coordination of a central server. Our goal is to design communication
protocols with near-optimal regret and little communication cost, which is
measured by the total amount of transmitted data. For distributed multi-armed
bandits, we propose a protocol with near-optimal regret and only
communication cost, where is the number of arms. The communication cost is
independent of the time horizon , has only logarithmic dependence on the
number of arms, and matches the lower bound except for a logarithmic factor.
For distributed -dimensional linear bandits, we propose a protocol that
achieves near-optimal regret and has communication cost of order
, which has only logarithmic dependence on
Quantum Rabi-Stark model: Solutions and exotic energy spectra
The quantum Rabi-Stark model, where the linear dipole coupling and the
nonlinear Stark-like coupling are present on an equal footing, are studied
within the Bogoliubov operators approach. Transcendental functions responsible
for the exact solutions are derived in a compact way, much simpler than
previous ones obtained in the Bargmann representation. The zeros of
transcendental functions reproduce completely the regular spectra. In terms of
the explicit pole structure of these functions, two kinds of exceptional
eigenvalues are obtained and distinguished in a transparent manner. Very
interestingly, a first-order quantum phase transition indicated by level
crossing of the ground state and the first excited state is induced by the
positive nonlinear Stark-like coupling, which is however absent in any previous
isotropic quantum Rabi models. When the absolute value of the nonlinear
coupling strength is equal to twice the cavity frequency, this model can be
reduced to an effective quantum harmonic oscillator, and solutions are then
obtained analytically. The spectra collapse phenomenon is observed at a
critical coupling, while below this critical coupling, infinite discrete
spectra accumulate into a finite energy from below.Comment: 16 pages, 4 figure
Quantum phase transitions in the spin-boson model without the counterrotating terms
We study the spin-boson model without the counterrotating terms by a
numerically exact method based on variational matrix product states.
Surprisingly, the second-order quantum phase transition (QPT) is observed for
the sub-Ohmic bath in the rotating-wave approximations. Moreover, first-order
QPTs can also appear before the critical points. With the decrease of the bath
exponents, these first-order QPTs disappear successively, while the
second-order QPT remains robust. The second-order QPT is further confirmed by
multi-coherent-states variational studies, while the first-order QPT is
corroborated with the exact diagonalization in the truncated Hilbert space.
Extension to the Ohmic bath is also performed, and many first-order QPTs appear
successively in a wide coupling regime, in contrast to previous findings. The
previous pictures for many physical phenomena for the spin-boson model in the
rotating-wave approximation have to be modified at least at the strong
coupling.Comment: 10 pages, 10 figure
Towards Understanding Learning Representations: To What Extent Do Different Neural Networks Learn the Same Representation
It is widely believed that learning good representations is one of the main
reasons for the success of deep neural networks. Although highly intuitive,
there is a lack of theory and systematic approach quantitatively characterizing
what representations do deep neural networks learn. In this work, we move a
tiny step towards a theory and better understanding of the representations.
Specifically, we study a simpler problem: How similar are the representations
learned by two networks with identical architecture but trained from different
initializations. We develop a rigorous theory based on the neuron activation
subspace match model. The theory gives a complete characterization of the
structure of neuron activation subspace matches, where the core concepts are
maximum match and simple match which describe the overall and the finest
similarity between sets of neurons in two networks respectively. We also
propose efficient algorithms to find the maximum match and simple matches.
Finally, we conduct extensive experiments using our algorithms. Experimental
results suggest that, surprisingly, representations learned by the same
convolutional layers of networks trained from different initializations are not
as similar as prevalently expected, at least in terms of subspace match.Comment: 17 pages, 6 figure
Rich phase diagram of quantum phases in the anisotropic subohmic spin-boson model
We study the anisotropic spin-boson model (SBM) with the subohmic bath by a
numerically exact method based on variational matrix product states. A rich
phase diagram is found in the anisotropy-coupling strength plane by calculating
several observables. There are three distinct quantum phases: a delocalized
phase with even parity (phase I), a delocalized phase with odd parity (phase
II), and a localized phase with broken symmetry (phase III), which
intersect at a quantum tricritical point. The competition between those phases
would give overall picture of the phase diagram. For small power of the
spectral function of the bosonic bath, the quantum phase transition (QPT) from
phase I to III with mean-field critical behavior is present, similar to the
isotropic SBM. The novel phase diagram full with three different phases can be
found at large power of the spectral function: For highly anisotropic case, the
system experiences the QPTs from phase I to II via 1st-order, and then to the
phase III via 2nd-order with the increase of the coupling strength. For low
anisotropic case, the system only experiences the continuous QPT from phase I
to phase III with the non-mean-field critical exponents. Very interestingly, at
the moderate anisotropy, the system would display the continuous QPTs for
several times but with the same critical exponents. This unusual reentrance to
the same localized phase is discovered in the light-matter interacting systems.
The present study on the anisotropic SBM could open an avenue to the rich
quantum criticality.Comment: 9 pages, 6 figure
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