Daniel Braak

Integrability of the Rabi model.

The Rabi model is a paradigm for interacting quantum systems. It couples a bosonic mode to the smallest possible quantum model, a two-level system. I present the analytical solution which allows us to consider the question of integrability for quantum systems that do not possess a classical limit. A criterion for quantum integrability is proposed which shows that the Rabi model is integrable due to the presence of a discrete symmetry. Moreover, I introduce a generalization with no symmetries; the generalized Rabi model is the first example of a nonintegrable but exactly solvable system.

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.

Continued Fractions and the Rabi Model

Techniques based on continued fractions to numerically compute 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. In contrast to the first, the second approach is ambiguous and can be justified only through recourse to the analyticity argument from the first method.

Bound states in the continuum realized in the one-dimensional two-particle Hubbard model with an impurity.

We report a bound state of the one-dimensional two-particle (bosonic or fermionic) Hubbard model with an impurity potential. This state has the Bethe-ansatz form, although the model is nonintegrable. Moreover, for a wide region in parameter space, its energy is located in the continuum band. A remarkable advantage of this state with respect to similar states in other systems is the simple analytical form of the wave function and eigenvalue. This state can be tuned in and out of the continuum continuously.

Solution of the two-qubit quantum Rabi model and its exceptional eigenstates

We have studied the two-qubit quantum Rabi model in the asymmetric case and its generalizations with dipole and Heisenberg-type qubit-qubit interactions. The solutions are obtained analytically with eigenstates given in terms of the extended coherent states or photon number states. For identical qubit-photon couplings, a novel type of quasi-exact solution which exists for all coupling values with constant eigenenergy is found, leading to level crossings within the same parity subspace even for non-identical qubits. In contrast to the quasi-exact eigenstates of the single-qubit model, these exceptional eigenstates are formed by just a few Fock states (photon number bounded from above at one or three), and the condition for them depends only on a fine-tuning of the qubit level splittings with respect to the photon energy, but not on the coupling to the photon field if the qubit-qubit interaction is not taken into consideration. This makes them excellent candidates for direct experimental observation within circuit quantum electrodynamics (QED) and application in single photon experiments. Besides, exceptional solutions with finite photon numbers N are also found.

Exact real-time dynamics of the quantum Rabi model

We use the analytical solution of the quantum Rabi model to obtain absolutely convergent series expressions of the exact eigenstates and their scalar products with Fock states. This enables us to calculate the numerically exact time evolution of � σx(t)� and � σz(t)� for all regimes of the coupling strength, without truncation of the Hilbert space. We find a qualitatively different behavior of both observables which can be related to their representations in the invariant parity subspaces.

Two-photon Rabi model: analytic solutions and spectral collapse

The two-photon quantum Rabi model with quadratic coupling is studied using extended squeezed states and we derive

A generalized G‐function for the Quantum Rabi Model

G

Analytical Solutions of Basic Models in Quantum Optics

-functions for Bargmann index

Bound states in the one-dimensional two-particle Hubbard model with an impurity

q=1/4