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arXiv:1311.2348 · Physical Review A

Fermionized photons in the ground state of one-dimensional coupled cavities

Adam G. D’Souza, Barry C. Sanders, David L. Feder

Published Dec 2, 2013• 9 citations
arXivHTMLPDF

Abstract

The density matrix renormalization group algorithm is used to characterize the ground states of one-dimensional coupled cavities in the regime of low photon densities. Numerical results for photon and spin excitation densities, one- and two-body correlation functions, superfluid and condensate fractions, as well as the entanglement entropy and localizable entanglement are obtained for the Jaynes-Cummings-Hubbard (JCH) model and are compared with those for the Bose-Hubbard (BH) model where applicable. The results indicate that a Tonks-Girardeau phase, in which the photons are strongly fermionized, appears between the Mott-insulating and superfluid phases as a function of the intercavity coupling. In fact, the superfluid density is found to be zero in a wide region outside the Mott-insulator phase boundary. The presence of two different species of excitation (spin and photon) in the JCH model gives rise to properties with no analog in the BH model, such as the (quasi)condensation of spin excitations and the spontaneous generation of entanglement between the atoms confined to each cavity.

DOI
10.1103/physreva.88.063801

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The density matrix renormalization group algorithm is used to characterize the ground states of one-dimensional coupled cavities in the regime of low photon densities. Numerical results for photon and spin excitation densities, one- and two-body correlation functions, superfluid and condensate fractions, as well as the entanglement entropy and localizable entanglement are obtained for the Jaynes-Cummings-Hubbard (JCH) model and are compared with those for the Bose-Hubbard (BH) model where applicable. The results indicate that a Tonks-Girardeau phase, in which the photons are strongly fermionized, appears between the Mott-insulating and superfluid phases as a function of the intercavity coupling. In fact, the superfluid density is found to be zero in a wide region outside the Mott-insulator phase boundary. The presence of two different species of excitation (spin and photon) in the JCH model gives rise to properties with no analog in the BH model, such as the (quasi)condensation of spin excitations and the spontaneous generation of entanglement between the atoms confined to each cavity.
The rapid progress in quantum technology enables the implementation of artificial many-body systems with correlated photons and polaritons. A multiconnected Jaynes-Cummings (MCJC) lattice can be constructed by connecting qubits and cavities alternately. Such models can be realized with superconducting qubits coupled to superconducting microwave resonators or with quantum dots coupled to optical nanocavities. We study the physical properties of a one-dimensional MCJC lattice using the density-matrix renormalization-group method. This model has an intrinsic symmetry between the left and right qubit-cavity couplings. The competition between these couplings may drive the ground state either to a Mott-insulating phase or to a superfluid phase at integer fillings. We calculate the single-particle and density-density correlation functions, the correlation lengths in the Mott-insulating phase, and the Luttinger parameters in the superfluid phase and determine accurately the critical points that separate these two phases.
We study the ground-state phase diagrams of two-photon Dicke, one-dimensional Jaynes-Cummings-Hubbard (JCH), and Rabi-Hubbard (RH) models using mean-field, perturbation, quantum Monte Carlo (QMC), and density-matrix renormalization-group methods. We first compare mean-field predictions for the phase diagram of the Dicke model with exact QMC results and find excellent agreement. The phase diagram of the JCH model is then shown to exhibit a single Mott insulator lobe with two excitons per site, a superfluid (SF, superradiant) phase and a large region of instability where the Hamiltonian becomes unbounded. Unlike the one-photon model, there are no higher Mott lobes. Also, unlike the one-photon case, the SF phases above and below the Mott lobe are surprisingly different: Below the Mott lobe, the SF consists of photon pairs as opposed to above the Mott lobe, where it is SF of simple photons. The mean-field phase diagram of the RH model predicts a transition from a normal to a superradiant phase but none is found with QMC.
We use the density-matrix renormalization group method to investigate ground-state and dynamic properties of the one-dimensional Bose-Hubbard model, the effective model of ultracold bosonic atoms in an optical lattice. For fixed maximum site occupancy n b = 5, we calculate the phase boundaries between the Mott insulator and the "superfluid" phase for the lowest two Mott lobes. We extract the Tomonaga-Luttinger parameter from the density-density correlation function and determine accurately the critical interaction strength for the Mott transition. For both phases, we study the momentum distribution function in the homogeneous system, and the particle distribution and quasi-momentum distribution functions in a parabolic trap. With our zero-temperature method we determine the photoemission spectra in the Mott insulator and in the "superfluid" phase of the one-dimensional Bose-Hubbard model. In the insulator, the Mott gap separates the quasi-particle and quasi-hole dispersions. In the "superfluid" phase the spectral weight is concentrated around zero momentum.
Bipartite entanglement entropies are calculated for the ground state of the two-excitation subspace in a two-site coupled cavity model. Each region in the phase diagram (atomic insulator, polaritonic insulator, photonic superfluid, and polaritonic superfluid) is found to be characterized by unique entanglement properties. In particular, the polaritonic superfluid region exhibits genuine 4-partite entanglement among the two atoms and two-cavity fields. The difference in entanglement properties between the small- and large-hopping limits provides further evidence that the transitions in the two limits are qualitatively different and that the large-hopping limit cannot be described by a simple Bose-Hubbard-type model.
We use strong-coupling perturbation theory, the variational cluster approach (VCA), and the dynamical density-matrix renormalization group (DDMRG) method to investigate static and dynamical properties of the one-dimensional Bose--Hubbard model in both the Mott-insulating and superfluid phases. From the von Neumann entanglement entropy we determine the central charge and the transition points for the first two Mott lobes. Our DMRG results for the ground-state energy, momentum distribution function, boson correlation function decay, Mott gap, and single-particle spectral function are reproduced very well by the strong-coupling expansion to fifth order, and by VCA with clusters up to 12 sites as long as the ratio between the hopping amplitude and onsite repulsion, t/Ut/Ut/U, is smaller than 0.15 and 0.25, respectively. In addition, in the superfluid phase VCA captures well the ground-state energy and the sound velocity of the linear phonon modes. This comparison provides an authoritative estimate for the range of applicability of these methods. In strong-coupling theory for the Mott phase, the dynamical structure factor is obtained from the solution of an effective single-particle problem with an attractive potential. The resulting resonances show up as double-peak structures close to the Brillouin zone boundary. These high-energy features also appear in the superfluid phase which is characterized by a pronounced phonon mode at small momenta and energies, as predicted by Bogoliubov and field theory. In one dimension, there are no traces of an amplitude mode in the dynamical single-particle and two-particle correlation functions.
We study the superfluid-Mott-insulator transition of ultracold bosonic atoms in a one-dimensional optical lattice with a double-well confining trap using the density-matrix renormalization group method. At low density, the system behaves similarly as two separated ones inside each well. At high density, however, interesting features appear resulting from the quantum tunnelling between the two wells and the competition between the ``superfluid'' and Mott regions. These interesting features are characterized by a rich step-plateau structure in the visibility and the satellite peaks in the momentum distribution function, with an oscillating pattern as a function of the on-site repulsion. These properties shed light on the understanding of the quantum phase coherence between two coupled condensates.