Quantum Phase Model Research Articles

Linear response theory of Josephson junction arrays in a microwave cavity

Recent experiments on Josephson junction arrays (JJAs) in microwave cavities have opened up a new avenue for investigating the properties of these devices while minimising the amount of external noise coming from the measurement apparatus itself. These experiments have already shown promise for probing many-body quantum effects in JJAs. In this work, we develop a general theoretical description of such experiments by deriving a quantum phase model for planar JJAs containing quantized vortices. The dynamical susceptibility of this model is calculated for some simple circuits, and signatures of the injection of additional vortices are identified. The effects of decoherence are considered via a Lindblad master equation.

Fast and optimal generation of entanglement in bosonic Josephson junctions

We use an exact quantum phase model to study the dynamical generation of particle-entanglement in a bosonic Josephson junction composed by two coupled and interacting Bose-Einstein condensates. Using analytical arguments, we show that linear coupling can accelerate the creation of entanglement with respect to the well known one-axis twisting model where coupling is absent. Furthermore, with a numerical analysis of optimal control schemes, we identify the optimal parameters for the fast generation of metrologically useful entanglement on short time scales: the optimal evolution is generated for a specific ratio between the coupling and the interaction strength.

The phase model and the norm-trace generating function of plane partitions

A quantum phase model is considered and the temporal evolution of the exponential of the first moment of particles distribution is studied. The calculation of the temporal evolution is based on the properties of Schur functions. The form-factor of the considered distribution function is expressed in the determinant form and it is proved that in the q-parametrization it is equal to the generating function of the norm-trace generating function of the boxed plane partitions. The asymptotics of the temporal evolution in the double scaling limit and decreasing temperature is obtained. It is shown that the amplitudes of the leading asymptotics depend on the generating function of plane partitions with the fixed sum of diagonal elements.

Combinatorics of a strongly coupled boson system

We introduce a quantum phase model as a limit for very strong interactions of a strongly correlated q-boson hopping model. We describe the general solution of the phase model and express scalar products of state vectors in determinant form. The representation of state vectors in terms of Schur functions allows obtaining a combinatorial interpretation of the scalar products in terms of nests of self-avoiding lattice paths. We show that under a special parameterization, the scalar products are equal to the generating functions of plane partitions confined in a finite box. We consider the two-dimensional vertex model related to the phase model and express the vertex model partition function with special boundary conditions in terms of the scalar product of the phase model state vectors.

Dynamic phase diagram for the quantum phase model

We address the stability of superfluid currents in a system of interacting lattice bosons. We consider various Gutzwiller trial states for the quantum phase model which provides a good approximation for the Bose-Hubbard model in the limit of large interactions and boson populations. We thoroughly analyze the current-carrying stationary states of the dynamics ensuing from a Gaussian ansatz, and derive analytical results for the critical lines signaling their modulational and energetic instability, as well as the maximum of the carried current. We show that these analytical results are in good qualitative agreement with those obtained numerically in previous works on the Bose-Hubbard model, and in the present work for the quantum phase model.

Second Josephson excitations beyond mean field as a toy model for thermal pressure: exact quantum dynamics and the quantum phase model

A simple four-mode Bose–Hubbard model with intrinsic time-scale separation can be considered as a paradigm for mesoscopic quantum systems in thermal contact. In our previous work, we showed that in addition to coherent particle exchange, a novel slow collective excitation can be identified by a series of Holstein–Primakoff transformations. This resonant energy exchange mode is not predicted by the linear Bogoliubov theory, and its frequency is sensitive to interactions among Bogoliubov quasi-particles; it may be referred to as a second Josephson oscillation, in analogy to the second sound mode of liquid helium II. In this paper, we will explore this system beyond the Gross–Pitaevskii mean-field regime. We directly compare the classical mean-field dynamics to the exact full quantum many-particle dynamics and show good agreement over a wide range of system parameters. The second Josephson frequency becomes imaginary for stronger interactions however, indicating dynamical instability of the symmetric state. By means of a generalized quantum phase model for the full four-mode system, we then show that, in this regime, high-energy Bogoliubov quasiparticles tend to accumulate in one pair of sites, while the actual particles preferentially occupy the opposite pair. We interpret this as a simple model of thermal pressure.

Correlation functions for a strongly coupled boson system and plane partitions

A quantum phase model is introduced as a limit for very strong interactions of a strongly correlated q-boson hopping model. The exact solution of the phase model is reviewed, and solutions are also provided for two correlation functions of the model. Explicit expressions, including both amplitude and scaling exponent, are derived for these correlation functions in the low temperature limit. The amplitudes were found to be related to the number of plane partitions contained in boxes of finite size.

Nonlinear phenomena, optical and quantum solitons

Research in the theory of solitons and its applications has undergone impressive development in recent years and has affected a broad range of fields, including theory of special functions, quantum integrable systems, numerical analysis, cellular automata, representations of quantum groups and

The phase diagram of quantum disordered superconducting networks

I review the low energy dynamics and the phase diagram of two parallel coupled Josephson junction arrays (JJA) in the presence of charge and magnetic frustration. Starting from the quantum phase model known to capture the relevant physics of JJA, the system is mapped in the self-dual limit into an Abelian gauge theory with a mixed Chern-Simons term. The low energy dynamics is shown to be governed by Higgs fields associated with disordering effects due to electric charges and magnetic excitations minimally coupled to gauge fields related to the currents of Cooper pairs and vortices. A rich phase diagram not attainable by standard mean field theories emerges as a result of the condensation of disorder fields. The interplay between Hall quantization and interlayer coherence displays appealing features resulting into quantized Hall states coexisting with interlayer coherence, quantized Hall states without interlayer coherence, and interlayer coherent states without Hall quantization.

Amplitude Mode in the Quantum Phase Model

We derive the collective low-energy excitations of the quantum phase model of interacting lattice bosons within the superfluid state using a dynamical variational approach. We recover the well-known sound (or Goldstone) mode and derive a gapped (Higgs-type) mode that was overlooked in previous studies of the quantum phase model. This mode is relevant to ultracold atoms in a strong optical lattice potential. We predict the signature of the gapped mode in lattice modulation experiments and show how it evolves with increasing interaction strength.

Dissipation, topology, and quantum phase transition in a one-dimensional Josephson junction array

We study the phase diagram and quantum critical properties of a resistively shunted Josephson junction array in one dimension from a strong coupling analysis. After mapping the dissipative quantum phase model to an effective sine-Gordon model we study the renormalization group flow and the phase diagram. We try to bridge the phase diagrams obtained from the weak and the strong coupling renormalization group calculations to extract a more comprehensive picture of the complete phase diagram. The relevance of our theory to experiments in nanowires is discussed.

Superconductor-insulator transition driven by local dephasing

We consider a system where localized bound electron pairs form an array of "Andreev"-like scattering centers and are coupled to a fermionic subsystem of uncorrelated electrons. By means of a path-integral approach, which describes the bound electron pairs within a coherent pseudospin representation, we derive and analyze the effective action for the collective phase modes which arise from the coupling between the two subsystems once the fermionic degrees of freedom are integrated out. This effective action has features of a quantum phase model in the presence of a Berry phase term and exhibits a coupling to a field which describes at the same time the fluctuations of density of the bound pairs and those of the amplitude of the fermion pairs. Due to the competition between the local and the hopping induced non-local phase dynamics it is possible, by tuning the exchange coupling or the density of the bound pairs, to trigger a transition from a phase ordered superconducting to a phase disordered insulating state. We discuss the different mechanisms which control this occurrence and the eventual destruction of phase coherence both in the weak and strong coupling limit.

Hermitian-phase-operator treatment of the superfluid states in a quantum two-well problem

We apply the Hermitian phase operator method (HPOM), as discussed by Pegg and Barnett [D. T. Pegg and S. M. Barnett, Phys. Rev. A 39, 1665 (1989)] to a two-well Bose-Hubbard model. Application of the HPOM formalism yields an approximate quantum phase model, or a Schr\"odinger-like differential equation for the phase variable $\ensuremath{\theta}$, which is a conjugate variable to the half of the number difference between the wells, $m=({N}_{1}\ensuremath{-}{N}_{2})∕2$. In the construction of the model we take care so that the Hermiticity of the original Bose-Hubbard model in number representation is inherited. We demonstrate that the quantum phase model supersedes two theoretical models suggested for the group of so-called ``normal'' states, for which $⟨\ensuremath{\theta}⟩=⟨m⟩=0$: a non-Hermitian ``exact quantum phase model'' by Anglin, Drummond, and Smerzi [J. R. Anglin, P. Drummond, and A. Smerzi, Phys. Rev. A 64, 063605 (2001)], and the ``phonon model'' by Javanainen and Ivanov [J. Javanainen and M. Yu. Ivanov, Phys. Rev. A 60, 2351 (1999)]. We apply the HPOM to the so called ``superfluid'' states, for which $⟨m⟩=0$ and $⟨\ensuremath{\theta}⟩=\ensuremath{\pi}$, and find their superfluid phonon model. We find that the superfluid phonon model has a threshold point $\mathcal{M}=\ensuremath{\delta}∕(4gN)=1$, at which the model diverges. Here, $\ensuremath{\delta}$ is the tunneling rate between the wells and $g$ is the self-interaction, while $N$ is the total number of particles. The above threshold phonon model is quite accurate in the description of the number and phase statistics of the superfluid states. We show that at the threshold point HPOM itself can be exactly solved. However, the agreement between the exact solution and the HPOM solution is poor: $⟨{m}^{2}⟩$ close to the threshold assumes values $\ensuremath{\sim}{N}^{2}$, thus voiding the second order expansion the HPOM is based on, $⟨{m}^{2}⟩⪡{N}^{2}$.

QUANTUM TUNNELING OF MESOSCOPIC SPIN IN BOSE–EINSTEIN CONDENSATES

In this paper the exact quantum phase model is derived with the help of a mesoscopic spin model from weakly linked two-component Bose–Einstein condensates. It is shown that the π-phase state which is seen to be the degenerate ground state exists only in an extreme limit of parameter, Δ≪1. π-phase state becomes metastable in the parameter region 0<Δ<1 and unstable when Δ≥1. The low-lying energy band resulted from the quantum tunneling is obtained explicitly by means of instanton method. We also analyze the phase-dissipation due to quantum tunneling. Moreover, the decay of the metastable π-phase state via quantum tunneling is studied and the life time is found analytically.

Quantum dynamics of strongly interacting boson systems: atomic beam splitters and coupled Bose-Einstein condensates.

An effective boson Hamiltonian applicable to atomic beam splitters, coupled Bose-Einstein condensates, and optical lattices can be made exactly solvable by including all n-body interactions. The model can include an arbitrary number of boson components. In the strong interaction limit the model becomes a quantum phase model, which also describes a tight-binding lattice particle. Through exact results for dynamic correlation functions, it is shown how the previous weak interaction dynamics of these systems are extended to strong interactions, now becoming relevant in the experiments. The effect of the number of boson components is also analyzed.

ALGEBRAIC EQUIVALENCE BETWEEN CERTAIN MODELS FOR SUPERFLUID–INSULATOR TRANSITION

Algebraic contraction is proposed to realize mappings between Hamiltonian models. This transformation contracts the algebra of the degrees of freedom underlying the Hamiltonian. The rigorous mapping between the anisotropic XXZ Heisenberg model, the quantum phase model and the Bose Hubbard model is established as the contractions of the algebra u(2) underlying the dynamics of the XXZ Heisenberg model.

Time-dependent mean-field theory of the superfluid-insulator phase transition

We develop a time-dependent mean field approach, within the time-dependent variational principle, to describe the Superfluid-Insulator quantum phase transition. We construct the zero temperature phase diagram both of the Bose-Hubbard model (BHM), and of a spin-S Heisenberg model (SHM) with the XXZ anisotropy. The phase diagram of the BHM indicates a phase transition from a Mott insulator to a compressibile superfluid phase, and shows the expected lobe-like structure. The SHM phase diagram displays a quantum phase transition between a paramagnetic and a canted phases showing as well a lobe-like structure. We show how the BHM and Quantum Phase model (QPM) can be rigorously derived from the SHM. Based on such results, the phase boundaries of the SHM are mapped to the BHM ones, while the phase diagram of the QPM is related to that of the SHM. The QPM's phase diagram obtained through the application of our approach to the SHM, describes the known onset of the macroscopic phase coherence from the Coulomb blockade regime for increasing Josephson coupling constant. The BHM and the QPM phase diagrams are in good agreement with Quantum Monte Carlo results, and with the third order strong coupling perturbative expansion.

Charge frustration effects in capacitively coupled two-dimensional Josephson-junction arrays

We investigate the quantum phase transitions in two capacitively coupled two-dimensional Josephson-junction arrays with charge frustration. The system is mapped onto the S=1 and $S=1/2$ anisotropic Heisenberg antiferromagnets near the particle-hole symmetry line and near the maximal-frustration line, respectively, which are in turn argued to be effectively described by a single quantum phase model. Based on the resulting model, it is suggested that near the maximal frustration line the system may undergo a quantum phase transition from the charge-density wave to the super-solid phase, which displays both diagonal and off- diagonal long-range order.