Quantum Phase Transition in the Finite Jaynes-Cummings Lattice 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.

Figure 1

Analytic solution of the JC model. (a) Level crossings for the ground state for a frequency ratio $\eta =10$. (b) An effective potential $V_{\mathrm{eff}}^{\eta ,g}$ for $g=0.8$ (dashed) and $g=1.2$ (solid) for different values of $\eta =10$ and 100. (c) The total number of excitation of the ground state. As $\eta$ increases, the change near $g=1$ becomes progressively sharper, which is well described by $n_{\mathrm{sp}}(g)=\eta (g^2-g^{-2})/4$ (solid).

Figure 2

QPT of the JC model. The excitation energy $\epsilon (g)$ (left, blue solid) and the ground state coherence $⟨a⟩/\eta$ of the cavity field (right, red dashed) in the $\eta \to \infty$ limit. For $g>1$, the $U(1)$ symmetry of the JC model is broken, leading to a Goldstone mode and a nonzero coherence.

Figure 3

JC lattice model. (a) Phase diagram in the $(g,J)$ plane. (b) Excitation energy of the antisymmetric ($b_1$) and symmetric ($b_2$) normal mode as a function of $g$ for $J/{\omega }_0=0.1$. At the critical point, where the $b_1$ mode becomes the Goldstone mode, the first derivative of the excitation energy of the $b_2$ mode becomes discontinuous.