Hidden order in quantum many-body dynamics of driven-dissipative nonlinear photonic lattices

We study the dynamics of nonlinear photonic lattices driven by two-photon parametric processes. By means of matrix-product-state–based calculations, we show that a quantum many-body state with long-range hidden order can be generated from the vacuum. Although this order resembles that characterizing the Haldane insulator, our system is far from equilibrium due to the drive and photon loss. A possible explanation highlighting the role of the symmetry of the drive and the effect of photon loss is discussed. An implementation based on superconducting circuits is proposed and analyzed.

Figure 1

(a) Sketch of the one-dimensional lossy photonic lattice described by the EBH model and driven by parametric drive. The lower panels show energy spectra of the undriven Hamiltonian in Eq. (1). (b) The laboratory frame and (c) the rotating frame. The red arrow indicates the energy shift $L{\omega }_d$ of the HI state due to the rotating frame. Since the HI state is a unit-filled state, its energy will be lowered by $L{\omega }_d$.

Figure 2

Mean-field energy landscape $E_{MF}$ plotted against $b/a$ and $c/a$ with $V=2.8J$ and $U=5J$. (a) In the absence of drive $\mathrm{\Omega }=0$, the variational ferromagnetic ground state has fourfold ground-state degeneracy, reflecting the global $D_2$ symmetry of the HI phase. (b) With a weak parametric drive $\mathrm{\Omega }=0.1J$, the ground states remain fourfold degenerate. Hence the symmetry is unbroken.

Figure 3

DMRG calculations of SO and DWO of the ground state of ${\widehat{H}}_{\mathrm{tot}}^{\mathrm{R}}$ with $\mu =7.5J$, $U=5J$, and $V=3.3J$. It shows that weak parametric drives preserve the HI phase, which is characterized by a nonvanishing SO and a vanishing DWO. Both SO and DWO are zero when on-site drives are used, indicating that the HI phase is destroyed. The DMRG calculations were performed with open boundary conditions and the bond dimension of 200. The system's size is $L=300$. The local Hilbert space in the numerics is truncated at the four-photon Fock state ($\mathrm{\Delta }{E}_{\text{HI}}^R=\langle \text{HI}|{\widehat{H}}_0^R|\text{HI}\rangle \approx -1.22LJ$ and $\mathrm{\Delta }{E}_{\text{HI}}^L=\langle \text{HI}|{\widehat{H}}_0|\text{HI}\rangle \approx 6.28LJ$).

Figure 4

Hidden order at a transient time during the driven-dissipative dynamics. Time evolution of the system ($L=100$) evolving under Eq. (6) with parametric drive, obtained by using the quantum trajectories with 200 trajectories and the TEBD algorithm with the bond dimension of 100. (a) SO and (b) DWO measured at the time $t^{\prime }=0.25/J$ as a function of $|i-j|$ for different driving amplitudes $\mathrm{\Omega }/J=0.5$, 1.0, 1.25, 1.5, 2.0.

Figure 5

Rise and fall of the hidden order in the driven-dissipative dynamics. Time evolution of the system ($L=50$) evolving under Eq. (6) with parametric drive, obtained by using the quantum trajectories with 100 trajectories and the TEBD algorithm with the bond dimension of 100. In panels (a)–(c), we vary the amplitude of the drive with $\mu =6.28J$, $U=5J$, $V=3.3J$, and $\gamma =0.05J$. In panels (d)–(f), we vary the photon loss rate with $\mu =6.28J$, $U=5J$, $V=3.3J$, and $\mathrm{\Omega }=1.25J$. In panels (g)–(i), we vary $V$ with $\mu =6.28J$, $U=5J$, $\gamma =0.05J$, and $\mathrm{\Omega }=1.25J$.

Figure 6

Proposed circuit diagram that implements the Bose–Hubbard model and parametric driving.