Finite-Component Multicriticality at the Superradiant Quantum Phase Transition

We demonstrate the existence of finite-component multicriticality in a qubit-boson model where biased qubits collectively coupled to a single-mode bosonic field. The interplay between biases and boson-qubit coupling produces a rich phase diagram which shows multiple superradiant phases and phase boundaries of different orders. In particular, multiple phases become indistinguishable in appropriate bias configurations, which is the signature of multicriticality. A series of universality classes characterizing these multicritical points are identified. Moreover, we present a trapped-ion realization with the potential to explore multicritical phenomena experimentally using a small number of ions. The results open a novel way to probe multicritical universality classes in experiments.

Figure 1

A schematic illustration of the qubit-boson model with staggered bias configuration [Eq. (1)]. Here a bosonic field with frequency $\omega$ is collectively coupled to $2N$ qubits (blue spheres) which are separated into $M$ subsets. Within each subset, qubits are divided into two halves and each half experiences a transverse field with the same magnitude and opposite direction to the other.

Figure 2

(a) Phase diagram in the tricritical case ($M=1$). The solid lines and dashed line represent the second-order critical lines and triple line, respectively. The first-order coexistence surfaces $S_0$ separate two superradiant phases with different order parameter sign, while $S_{\pm }$ separate a normal phase and two superradiant phases, respectively. (b) The ground state order parameter $z_G$ (left panels) and the excitation energy $ϵ$ (right panels) as functions of $\tilde{g}$ and $\tilde{h}$. (c) Phase diagram in the tetracritical case ($M=2$ and $N_1=3N_2$), where ${\mathrm{\Delta }}_g=\tilde{g}-{\tilde{g}}_r({\tilde{\epsilon }}_1,{\tilde{\epsilon }}_2)$ is the distance to $L_{\lambda }$ for given ${\tilde{\epsilon }}_{1,2}$. ${\tilde{g}}_r({\tilde{\epsilon }}_1,{\tilde{\epsilon }}_2)$ is the coupling value on $L_{\lambda }$ when ${\tilde{\epsilon }}_{1,2}$ are fixed, and is the solution to $r({\tilde{g}}_r,{\tilde{\epsilon }}_1,{\tilde{\epsilon }}_2)=0$ since $L_{\lambda }$ is determined by $r=0$. (d) The ground state order parameter $|z_G|$ for fixed ${\tilde{\epsilon }}_2$. (e),(f) Schematic illustrations of the energy functional $E(z)$: (e) The system moves across $L_{\chi }$ from the low- (dashed line) to the high-bosonic-coherence (solid line) superradiant phase. (f) The system moves along $L_{\chi }$ (dashed line) then into the line of end points (dotted line) and finally reaches the tetracritical point (solid line).

Figure 3

(a) Fits of the critical exponents ${\gamma }_{ϵ,w_1}$ and ${\gamma }_{ϵ,r}$ [inset of (a)] at critical points of different orders. (b) Fits of the critical exponent ${\delta }_{ϵ}$. (c) Numerical results of the rescaled residual qubit population $S={\eta }^{-1+{\gamma }_{ϵ,r}/{\xi }_r}{⟨J_z⟩}_r=\sqrt{2}(\omega /\mathrm{\Omega }{)}^{-1/2}{⟨J_z⟩}_r$ after quench as a function of the rescaled quench time $T=\omega \tau {\eta }^{(1+{\gamma }_{ϵ,r})/{\xi }_r}=2^{-3/2}(\omega /\mathrm{\Omega }{)}^{3/2}\omega \tau$ for different frequency ratio with noise effects included. The black dashed line is the nonequilibrium scaling function ${\mathcal{S}}_{J_z}$ obtained in the $\eta \to 0$ limit numerically.