Phase transitions are commonly held to occur only in the thermodynamical limit of a large number of system components. Here, we exemplify at the hand of the exactly solvable Jaynes-Cummings (JC) model and its generalization to finite JC lattices that finite component systems of coupled spins and bosons may exhibit quantum phase transitions (QPTs). For the JC model we find a continuous symmetry-breaking QPT, a photonic condensate with a macroscopic occupation as the ground state, and a Goldstone mode as a low-energy excitation. For the two site JC lattice we show analytically that it undergoes a Mott-insulator to superfluid QPT. We identify as the underlying principle of the emergence of finite system QPTs the combination of increasing atomic energy and increasing interaction strength between the atom and the bosonic mode, which allows for the exploration of an increasingly large portion of the infinite dimensional Hilbert space of the bosonic mode. This suggests that finite system phase transitions will be present in a broad range of physical systems.
Phase transitions are commonly held to occur only in the thermodynamical limit of a large number of system components. Here, we exemplify at the hand of the exactly solvable Jaynes-Cummings (JC) model and its generalization to finite JC lattices that finite component systems of coupled spins and bosons may exhibit quantum phase transitions (QPTs). For the JC model we find a continuous symmetry-breaking QPT, a photonic condensate with a macroscopic occupation as the ground state, and a Goldstone mode as a low-energy excitation. For the two site JC lattice we show analytically that it undergoes a Mott-insulator to superfluid QPT. We identify as the underlying principle of the emergence of finite system QPTs the combination of increasing atomic energy and increasing interaction strength between the atom and the bosonic mode, which allows for the exploration of an increasingly large portion of the infinite dimensional Hilbert space of the bosonic mode. This suggests that finite system phase transitions will be present in a broad range of physical systems.
Quantum phase transitions (QPTs) describe when a many-body quantum system displays non-analytic behavior associated with a discontinuous change in a property of the ground state as a parameter is varied. The QPT in prototypical Dicke model is difficult to reach experimentally as the spin–field coupling strength must be quite large. In this work we describe a new model—the off-resonant Tavis–Cummings model where we drive the common mode, and discover a new type of QPT at quite low coupling strengths which are comparable with the geometric mean of the atomic and field detunings . Through analytic methods we demonstrate this QPT for both finite and infinite numbers of spins and show that |〈 J x ( J z )〉|/( N /2) ∼ | λ / λ c − 1| γ x ( γ z ) and 〈 a † a 〉/ N ∼ | λ / λ c − 1| γ a for λ ⩾ λ c , with critical exponents γ x ≈ 1/2, γ z ≈ 1 and γ a ≈ 1. We show that this QPT can be immediately observed by laboratory cavity-QED setups such as Bose–Einstein condensate in optical cavity and superconducting circuit-QED as well as a line of trapped ultracold ions.
We use the mean field approximation method to study the quantum phase transitions of the Jaynes-Cummings lattice model and the Rabi lattice model. The effective Hamiltonians are obtained for the JC and Rabi model including the Kerr nonlinear term. Numerically we diagonalized the Hamiltonian matrix and calculated the superfluidity order parameter and the two-photon correlation function by solving the iteration equations. We have explored the Mott insulating-superfluid quantum phase transition, the bunching-antibunching behavior of light, and the effect of Kerr nonlinear term on the quantum phase transition and photon statistical characteristics. Our results show that in the JC lattice model, by increasing J, a quantum phase transition takes place and the system is driven to a superfluid phase. The phase boundaries of the Mott lobes are N-dependent. However the photon will always be in a bunching statistical behavior irrelevant of the coupling strength between the two-level atom and the phonton and the nonlinear Kerr effect. In the Rabi lattice model, the anti-rotating wave term breaks Mott-lobe structure of the phase diagram and the increase of the two-level atom and photon interaction strength g and the photon transition strength J between the lattices drive the system from the Mott insulating phase to the superfluid phase. The photon statistical behavior changes from the bunching to the antibunching one when considering the anti-rotating wave term, which is important in the strongly coupled systems. Most interestingly, the increase of the Kerr nonlinear coefficient will inhibit the Mott insulating phase-superfluid phase transition, but favor the superfluid phase and the transition from the bunching to anti-bunching statistics.
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.
The connectivity and tunability of superconducting qubits and resonators provide us with an appealing platform to study the many-body physics of microwave excitations. Here we present a multiconnected Jaynes-Cummings lattice model which is symmetric with respect to the nonlocal qubit-resonator couplings. Our calculation shows that this model exhibits a Mott insulator-superfluid-Mott insulator phase transition at commensurate fillings, featured by symmetric quantum critical points. Phase diagrams in the grand canonical ensemble are also derived, which confirm the incompressibility of the Mott insulator phase. Different from a general-purposed quantum computer, it only requires two operations to demonstrate this phase transition: the preparation and the detection of commensurate many-body ground state. We discuss the realization of these operations in a superconducting circuit.
Aim and Scope of this Book A plethora of systems exhibit phase transitions as the temperature or some other parameter is changed. Examples range from the ice-water phase transition observed in our daily life to the loss of ferromagnetism in iron or to the more sophisticated Mott insulator-superfluid phase transition observed in optical lattices [343]. The last five decades have witnessed a tremendous upsurge in the studies of phase transitions at finite temperature [727, 149, 333, 136, 494, 541, 556]. The success of Landau-Ginzburg theories and the concepts of spontaneous symmetry breaking and the renormalization group [27, 410, 821, 578] in explaining many of the finite temperature phase transitions occurring in nature has been spectacular. In this book, we will consider only a subclass of phase transitions called quantum phase transitions (QPTs) [154, 658, 725, 799, 185, 63, 62, 66, 141, 744] and we will discuss these mainly from the view point of recent studies of information and dynamics. QPTs are zero temperature phase transitions which are driven by quantum fluctuations and are usually associated with a non-analyticity in the ground state energy density of a quantum many-body Hamiltonian. We will focus on continuous QPTs where the order parameter vanishes continuously at the quantum critical point (QCP) at some value of the parameters which characterize the Hamiltonian. We will not discuss first order quantum phase transitions associated with an abrupt change in the order parameter. Usually, a first order phase transition is characterized by a finite discontinuity in the first derivative of the ground state energy density. A continuous QPT is similarly characterized by a finite discontinuity, or divergence, in the second derivative of the ground state energy density, assuming that the first derivative is continuous. This is of course the classical definition; we will later mention some QPTs where the ground state energy density is not necessarily singular.
The nature of the ground states for a system composed of two coupled cavities with each containing a pair of dipole-coupled two-level atoms are studied over a wide range of detunings and dipole coupling strengths. The cases for three limits of exact resonance, large positive and negative detunings are discussed, and four types of the ground states are revealed. Then the phase diagrams of the ground state are plotted by choosing three different "order parameters". We find that the phase space, determined by the combinative action of detuning and the dipole coupling strength, is divided into four regions. This is different from the general Bose-Hubbard model and more richer physics are presented in the two-site coupled cavities system. That is, the insulator region may be polaritonic or atomic and the superfluid region may be polaritonic or photonic in nature.