We compare and contrast the mean-field and many-body properties of a Bose-Einstein condensate trapped in a double well potential with a single impurity atom. The mean-field solutions display a rich structure of bifurcations as parameters such as the boson-impurity interaction strength and the tilt between the two wells are varied. In particular, we study a pitchfork bifurcation in the lowest mean-field stationary solution which occurs when the boson-impurity interaction exceeds a critical magnitude. This bifurcation, which is present for both repulsive and attractive boson-impurity interactions, corresponds to the spontaneous formation of an imbalance in the number of particles between the two wells. If the boson-impurity interaction is large, the bifurcation is associated with the onset of a Schroedinger cat state in the many-body ground state. We calculate the coherence and number fluctuations between the two wells, and also the entanglement entropy between the bosons and the impurity. We find that the coherence can be greatly enhanced at the bifurcation.Comment: 19 pages, 17 figures. The second version contains minor corrections and some better figures (thicker lines

We study a model describing N identical bosonic atoms trapped in a double-well potential together with a single impurity atom, comparing and contrasting it throughout with the Dicke model. As the boson-impurity coupling strength is varied, there is a symmetry-breaking pitchfork bifurcation which is analogous to the quantum phase transition occurring in the Dicke model. Through stability analysis around the bifurcation point, we show that the critical value of the coupling strength has the same dependence on the parameters as the critical coupling value in the Dicke model. We also show that, like the Dicke model, the mean-field dynamics go from being regular to chaotic above the bifurcation and macroscopic excitations of the bosons are observed. Overall, the boson-impurity system behaves like a poor man's version of the Dicke model.

No abstract

Abstract. As part of the quest to uncover universal features of quantum dynamics, we study catastrophes that form in simple many-particle wave functions following a quench, focusing on two-mode systems that include the two-site Bose Hubbard model, and under some circumstances optomechanical systems and the Dicke model. When the wave function is plotted in Fock space certain characteristic shapes, that we identify as cusp catastrophes, appear under generic conditions. In the vicinity of a cusp the wave function takes on a universal structure described by the Pearcey function and obeys scaling relations which depend on the total number of particles N . In the thermodynamic limit (N → ∞) the cusp becomes singular, but at finite N it is decorated by an interference pattern. This pattern contains an intricate network of vortex-antivortex pairs, initiating a theory of topological structures in Fock space. In the case where the quench is a δ-kick the problem can be solved analytically and we obtain scaling exponents for the size and position of the cusp, as well as those for the amplitude and characteristic length scales of its interference pattern. Finally, we use these scalings to describe the wave function in the critical regime of a Z 2 symmetrybreaking dynamical phase transition.

We consider the "membrane in the middle" optomechanical model consisting of a laser pumped cavity which is divided in two by a flexible membrane that is partially transmissive to light and subject to radiation pressure. Steady state solutions at the mean-field level reveal that there is a critical strength of the light-membrane coupling above which there is a symmetry breaking bifurcation where the membrane spontaneously acquires a displacement either to the left or the right. This bifurcation bears many of the signatures of a second order phase transition and we compare and contrast it with that found in the Dicke model. In particular, by studying limiting cases and deriving dynamical critical exponents using the fidelity susceptibility method, we argue that the two models share very similar critical behaviour. For example, the obtained critical exponents indicate that they fall within the same universality class. Away from the critical regime we identify, however, some discrepancies between the two models. Our results are discussed in terms of experimentally relevant parameters and we evaluate the prospects for realizing Dicke-type physics in these systems.Comment: 14 pages, 6 figure

When two Bose-Einstein condensates are suddenly coupled by a tunneling junction, the Gross-Pitaevskii mean-field theory predicts that caustics will form in the number-difference probability distribution. The caustics are singular but are regularized by going to the many-body theory where atom number is quantized. However, if the system is subject to a weak continuous measurement the quantum state decoheres and classicality is restored. We investigate the emergence of singularities during the quantum-to-classical transition paying attention to the interplay between particle number N and the quantum noise introduced by the measurement.

We show that the light conelike structures that form in spin chains after a quench are quantum caustics. Their natural description is in terms of catastrophe theory and this implies (1) a hierarchy of light cone structures corresponding to the different catastrophes, (2) dressing by characteristic wave functions that obey scaling laws determined by the Arnol'd and Berry indices, and (3) a network of vortex-antivortex pairs in space-time inside the cone. We illustrate the theory by giving explicit calculations for the transverse field Ising model and the XY model, finding fold catastrophes dressed by the Airy functions and cusp catastrophes dressed by the Pearcey functions; multisite correlation functions are described by higher catastrophes such as the hyperbolic umbilic. Furthermore, we find that the vortex pairs created inside the cone are sensitive to phase transitions in these spin models with their rate of production being determined by the dynamical critical exponent. More broadly, this work illustrates how catastrophe theory can be applied to singularities in quantum fields.

We use fidelity susceptibility to calculate quantum critical scaling exponents for a system consisting of N identical bosons interacting with a single impurity atom in a double well potential (bosonic Josephson junction). Above a critical value of the boson-impurity interaction energy there is a spontaneous breaking of Z2 symmetry corresponding to a second order quantum phase transition from a balanced to an imbalanced number of particles in either the left or right hand well. We show that the exponents match those in the Lipkin-Meshkov-Glick and Dicke models suggesting that the impurity model is in the same universality class. The phase transition can be interpreted as a measurement of the position of the impurity by the bosons.