Unified superradiant phase transitions

We prove that superradiant phase transitions (SPTs) of the Dicke model and its generalizations in the thermodynamic and classical oscillator limit are indeed of the same type. In this sense, we unify SPTs under both limits at zero and finite temperature. We show that the mean-field approximation for bosons is exact in both cases, and compute the structure and location of the phase transitions in parameter space using a concise analytic method. Moreover, we illustrate how SPT properties (first order, second order, or none) are related to symmetry. Finally, we uncover general features of the phase structure in the space of parameters of these models with dipolar couplings. There will be a region of normal phase in the neighborhood of the origin of the space of dipolar couplings $\vec{\gamma }$, and that generally one flows radially in this space to a superradiant phase.

Figure 1

Phase diagram of the Dicke model in thermodynamic limit ($N\to \infty$).

Figure 2

(a) Phase diagram of the Dicke model with one- and two-photon terms in the thermodynamic limit. (b) The rescaled average photon number and reduced free energy in the classical oscillator limit for finite $N$. $g^{\prime }/\omega =0.25,g_c/\sqrt{\mathrm{\Delta }\omega /2}=0.866$, and ${\langle a^{†}a\rangle }_c/C=2/3$ at zero or finite temperature $\mathrm{\Delta }/k_{B}T\to \infty$, coinciding with the $T=0$ result in (a). (c) Energies of the ground state (GS) and first excited state (ES) obtained by direct diagonalization of the Hamiltonian Eq. (21) with $N=1$ and $N\mathrm{\Delta }/\omega =200$. (d) The rescaled average photon number of the GS and first ES in (c). The numerical results (sharp avoided crossing and steep photon number change) match the exact analysis for $N\mathrm{\Delta }/\omega \to \infty$.

Figure 3

Nonidentical two-qubit Rabi model with $XYZ$ spin-spin interactions in the classical oscillator limit. ${\mathrm{\Delta }}_1/\mathrm{\Delta }=3,{\mathrm{\Delta }}_2/\mathrm{\Delta }=2,J^x/\mathrm{\Delta }=3,J^y/\mathrm{\Delta }=2$, and $J^z/\mathrm{\Delta }=1$. (a) Phase diagram at zero or finite temperatures $\mathrm{\Delta }/k_{B}T\to \infty$. (b) Rescaled average photon number for $g_2/\sqrt{\mathrm{\Delta }\omega }=1.5$ at zero or finite temperature $\mathrm{\Delta }/k_{B}T\to \infty$. (c), (d) Lowest-energy levels and their rescaled average photon number obtained by direct diagonalizing the Hamiltonian at $\mathrm{\Delta }/\omega =100$ for $g_2/\sqrt{\mathrm{\Delta }\omega }=1.5$, coinciding well with our results. At $\mathrm{\Delta }/\omega \gg 1$, asymptotic QPT behavior of nonanalyticity of the ground-state energy presents as a very sharp avoided crossing, and the symmetry spontaneous breaking appears as a nearly degenerate ground state at SP. A steep ground-state photon number change in (d) will be a mark of first order SPT at $\mathrm{\Delta }/\omega \to \infty$.

Figure 4

Critical cases of a positive solution to $\mathrm{\Phi }\left(u^{\prime }\right)=0$. $u_{+}^{\prime }$ is the only local minimum. (a) Critical solution $u_c^{\prime }>0$, corresponding to first order SPT. (b), (c) Critical solution $u_c^{\prime }\to 0$, corresponding to second-order SPT. (d) Two kinds of critical solutions are both possible, connecting first- and second-order SPT.

Figure 5

Left panel: The rescaled average photon number and reduced free energy of the identical two-qubit Rabi model with qubit-qubit dipole interaction in the $\mathrm{\Delta }/\omega \to \infty$ limit at zero or finite temperature $\mathrm{\Delta }/k_{B}T\to \infty$, obtained by finding the global minimum of $\varphi \left(u\right)$ numerically. Right panel: The lowest eigenenergy levels in each parity subspace and their average photon numbers obtained by diagonalizing the Hamiltonian numerically at $\mathrm{\Delta }/\omega \gg 1$. $\delta ={\mathrm{\Delta }}_i/\mathrm{\Delta }=1$. (a) ${\epsilon }_x=1$. ${\epsilon }_y={\epsilon }_z=0$. $g_c/\sqrt{\mathrm{\Delta }\omega }=0.803$. $\omega \langle a^{†}a\rangle /{\mathrm{\Delta }}_c\sim 1.308$. (b) $\mathrm{\Delta }/\omega =200,{\epsilon }_x=1$, and ${\epsilon }_y={\epsilon }_z=0$. $\omega \langle a^{†}a\rangle /{\mathrm{\Delta }}_c\sim 1.308$. (c) ${\epsilon }_x=0.2$. ${\epsilon }_y={\epsilon }_z=0$. $g_c/\sqrt{\mathrm{\Delta }\omega }=0.55387$. (d) $\mathrm{\Delta }/\omega =800$. ${\epsilon }_x=0.2$. ${\epsilon }_y={\epsilon }_z=0$. The numerical results match the exact analysis for $N\mathrm{\Delta }/\omega \to \infty$, where the avoided crossings in (b) and (d) become infinitely sharp and bring the nonanalyticity of the ground-state energy [33].

Figure 6

Left panel: Identical two-qubit Rabi model with isotropic $XYZ$ Heisenberg interaction obtained by finding the global minimum of $\varphi \left(u\right)$ numerically. Right panel: The lowest eigenenergy levels in each parity subspace and their reduced average photon numbers obtained by diagonalizing the Hamiltonian numerically at $\mathrm{\Delta }/\omega \gg 1$. $\delta ={\mathrm{\Delta }}_i/\mathrm{\Delta }=1$. $\epsilon ={\epsilon }_x={\epsilon }_y={\epsilon }_z$. (a) $\epsilon =0.6$. $g_c/\sqrt{\mathrm{\Delta }\omega }=0.6825;{\langle \frac{\omega a^{†}a}{\mathrm{\Delta }}\rangle }_c\sim 1.327$. (b) $\mathrm{\Delta }/\omega =200;\epsilon =0.6$. The ground state at $g/\sqrt{\mathrm{\Delta }\omega }\to 0$ is the spin singlet (dark state) with constant eigenenergy. Real level crossing happens because of the existence of this invariant subspace. QPT is expected to exist even for finite $\mathrm{\Delta }/\omega$. (c) $\epsilon =0.4;g_c/{\sqrt{\mathrm{\Delta }\omega }}_c=0.5$. (d) $\mathrm{\Delta }/\omega =1200;\epsilon =0.4$. We find asymptotic QPT behavior of gap closing around $g_c/\sqrt{\mathrm{\Delta }\omega }=0.5$ and symmetry spontaneous breaking there.