Engineering indefinitely long-lived localization in cavity-QED arrays

By exploiting the nonlinear nature of the Jaynes Cumming interaction, one can get photon population trapping in cavity-QED arrays. However, the unavoidable dissipative effects in a realistic system would destroy the self-trapped state by continuous photon leakage, and the self-trapping remains only as a temporary effect. To circumvent this issue, we aim to achieve an indefinitely long-lived self-trapped steady state rather than a localization with limited lifetime. We show that a careful engineering of drive, dissipation, and Hamiltonian results in achieving indefinitely sustained self-trapping. We show that the intricate interplay between drive, dissipation, and light-matter interaction results in requiring an optimal window of drive strengths in order to achieve such nontrivial steady states. We treat the two-cavity and four-cavity cases by using exact open quantum many-body calculations. Additionally, in the semiclassical limit we scale up the system to a long one-dimensional chain and demonstrate localized and delocalized steady-states in a driven-dissipative cavity-QED lattice. Although our analysis is performed by keeping cavity-QED systems in mind, our work is applicable to other driven-dissipative systems where nonlinearity plays a pivotal role.

Figure 1

Schematic depiction of 1-D lattice of identical cavity-QEDs, each containing a two-level system with levels marked 1 and 2. Each two-level system is coupled to the cavity photons with a strength $g_i$. Additionally, $J$, $\kappa$, $\gamma$ represent intercavity tunneling, photon decay, and spontaneous decay of the two-level system, respectively. $d_i$ is the coherent photon microwave drive applied at the $i\mathrm{th}$ cavity.

Figure 2

Quantum dynamics for two coupled cavities are presented for initial state $\{10,0,-1/2,-1/2\}$ and $\kappa =\gamma =0.04J$. Panels (a)–(c) are undriven cases when $g_1=g_2=17.7J$. Panels (d)–(f) are drawn for $d_1=0.04J$, $d_2=0$, and $g_1=g_2=17.7J$, i.e., light-matter coupling in both cavities are turned on. Panels (g)–(i) are cases when $g_1=0$, $g_2=17.7J$, and the driving intensities are the same as in panel (d). $N_i$ and $s_i^z$ are mentioned with similar colors as the respective lines. This demonstrates that indefinite self-trapping can be achieved when one of the cavities does not have light-matter coupling.

Figure 3

(a) Dynamics of quantum correlation function $g^{\left(2\right)}\left(0\right)$. The horizontal dotted black line in panel (a) marks the $g^{\left(2\right)}\left(0\right)=1$ value. (b) $N_2$ dynamics. (c) Short-time dynamics of $z$ is shown for the two cases. Vertical dotted lines of respective colors mark the times when $g^{\left(2\right)}\left(0\right)$ attains 1 in all figures. Parameter values are same as in Fig. 2 [solid orange (gray), $g_1=g_2]$ and Fig. 2 [solid blue (dark), $g_1=0$, $g_2\ne 0]$, respectively.

Figure 4

Full quantum treatment of a two-cavity case with uniform couplings. Attainment of self-trapped steady state is established when $g_1=g_2=10J$, $\kappa =\gamma =1.6J$, $d_1=4J$, and $d_2=0$. The $N_i$ are mentioned with similar colors as the respective lines.

Figure 5

Quantum dynamics for a 1-D chain of four cavities, when the initial conditions are $\{2,0,2,0\}$ for the cavity photon population and $\{-1/2,-1/2,-1/2,-1/2\}$ for the qubits. The results are nonetheless independent of initial conditions as shown in Fig. 6 (and in Appendix pp1). (a) Dynamics of $z$ is presented for $\gamma =\kappa =.02J$, $M=4$, $g_2=g_4=17.8J$, and $g_1=g_3=0$. The undriven case having $d_i=0$ (thin light blue) is contrasted with a case where $d_1=d_3=.025J$ and $d_2=d_4=0$ (thick black). Panels (b)–(e) describe the photon population of four cavities.

Figure 6

Quantum dynamics and demonstration of initial-condition independence. Here $M=2$, $\gamma =\kappa =0.04J$, and $g_1=0$, $g_2=10.62J$. Plots are shown for two distinct initial conditions: (i) $N_1=N_2=0$ (both cavities are initially vacuum presented by solid black and dotted purple) and (ii) $N_1=N_2=5$ [presented by dashed blue and solid yellow (thin gray)]. For each initial condition we show results for two $\{d_1,d_2\}$ compositions: (i) $d_1=0.04J$, $d_2=0$ (black and dashed blue) and (ii) $d_1=0.2J$, $d_2=0$ [dotted purple and solid yellow (thin gray)]. Panels (a)–(c) plot $N_1$, $N_2$, and $z$ dynamics, respectively. The initial-condition independence can be seen in our results. The initial conditions for qubits in all cases were $\{-1/2,-1/2\}$ but we find that the initial-condition independence also holds for different initial conditions of the qubits (see Fig. 10 in Appendix pp1).

Figure 7

Semiclassical dynamics for population imbalance in a 1-D lattice consisting 100 identical cavity-QEDs (each hosting a qubit) is presented with $g_i$ and $d_i$ values mentioned with similar colors as the respective lines. In all cases every odd cavity is initialized with 20 photons and even cavity is kept vacant. $\kappa$, $\gamma$, and $g_i=g$ values are kept identical for each cavity in the array. (a) Depiction of delocalization-localization transition for a closed-system case (no drive and dissipation). (b) Open-system case (i.e., with drive and dissipation) with $\kappa =\gamma =0.04J$ which reflects stabilization (indefinitely long lived) of self-trapped state with appropriate drive. Here, $g_c=2.8J\sqrt{20}$ is only defined for the closed-system case in panel (a).

Figure 8

(a) Semiclassically obtained phase-space description of long-lived self-trapped state in $d_1\text{−}\kappa$ plane, when $\gamma =0.04J$. The cavity system is initialized to a state $\{20,0,-1/2,-1/2\}$ and the coupling for both the cavities $g_1=g_2=g=25J$. The three phases in this figure are (i) under-driven non-self-trapped, (ii) indefinitely self-trapped, and (iii) over-driven delocalized. The limiting values $d_1^{\min }$ and $d_1^{\max }$ lie on the lower and upper boundaries of phase (ii), respectively. The magenta line gives an analytical prediction for the curve $\kappa \left(d_1\right)$ representing the boundary between phases (ii) and (iii). (b) For various values of $g$, $d_1=d_1^{\max }$ is obtained for $\kappa =0.1J$ and $\gamma =0.04J$. $N_1^{ss}$ is numerically found for these parameter values and plotted (by “$\times$” markers) with respect to $g^2$. A solid line is plotted for $N_1^{ss}=(1/8)g^2$ that nicely fits the data points.

Figure 9

Demonstration of driven-dissipative steady-state self-trapping for an interaction which is subcritical if considered for a closed-system counterpart. The two-cavity system is initialized at $\{20,0,-1/2,-1/2\}$. Panels (a) and (b) describe quantum cases when $g_1=0$, $g_2=5J$. Panels (c) and (d) describe semiclassical cases when $g_1=g_2=11.27J$. The critical interaction $g_c=2.8\sqrt{20}J$ can only be defined for fully-closed-system cases in panel (a) $g_1=0$, $g_2=0.4g_c$ (expressed in terms of $g_c$) and panel (c) $g_1=g_2=0.9g_c$. Drive and dissipation rates are mentioned for driven-dissipative cases in panels (b) and (d) with respective colors.

Figure 10

Exact quantum simulation. Photon and qubit dynamics for various initial photon number distribution and qubit states. Here $\kappa =\gamma =0.04J$, $d_1=0.04J$, $d_2=0$, and $g_1=0$, $g_2=10.63J$. Various preparations are as follows: (i) $\{0,0,-1/2,-1/2\}$ (thick solid black), (ii) $\{0,0,-1/2,1/2\}$ (thin solid purple), (iii) $\{0,0,-1/2,0\}$, i.e., the qubit state in cavity 2 is an equal superposition of the excited and ground states (dashed blue), (iv) $\{0,2,-1/2,-1/2\}$ (dashed dotted yellow), and (v) $\{0,2,-1/2,1/2\}$ (dotted red). At long time the uniqueness of the steady state is evident.

Figure 11

Semiclassical behavior of population dynamics of two coupled cavities with varying $\gamma$, $\kappa$, $g_1=g_2=g$, and drive. Here, $g_c=2.8J\sqrt{20}$ is defined for closed system. (a) $g=0.9g_c$, $\gamma =0$, $\kappa =0$, and $d_1=d_2=0$ (closed system with subcritical light-matter coupling). (b) $g=11.27J$, $\gamma =0.6J$, $\kappa =0.6J$, and $d_1=d_2=0$ (dissipative system with no drive). (d) $g=g_c$, $\gamma =0$, $\kappa =0$, and $d_1=d_2=0$ (closed system with critical light-matter coupling). (e) $g=12.52J$, $\gamma =0.04J$, $\kappa =0.04J$, and $d_1=d_2=0$ (dissipative system with no drive). (g) $g=2g_c$, $\gamma =0$, $\kappa =0$, and $d_1=d_2=0$ (closed system with light-matter coupling greater than $g_c$). (h) $g=25J$, $\gamma =0.04J$, $\kappa =0.04J$. $d_1=d_2=0$ (dissipative in solid magenta) and $d_1=0.04J,d_2=0$ (driven-dissipative in dashed green). Panels (c), (f), and (i) are the population dynamics for the left and right cavities for parameter specifications as in panels (b), (e), and (h), respectively. The dotted gray vertical lines in panels (b), (e), and (h) mark the break time $t_{\mathrm{break}}$ when the localization starts getting destroyed.

Figure 12

Semiclassical behavior. (a) Temporal behavior of $z$ for various $d_1$ values $[d_1=0$, $0.02J$, $0.03J$, $0.18J$, $0.2J$ presented as solid blue (dark), solid magenta (gray), dashed black, thick green (light gray), and dotted red (dark), respectively] when $g=25J$ and $\gamma =\kappa =0.04J$. (b) Long-time values of $z=z_{\mathrm{long}}$ and (c) $t_{\mathrm{break}}$ for varying $d_1$ are plotted. These results in panels (b) and (c) show that semiclassical method predicts an abrupt change between no self-trapping and an indefinitely-long-lived trapped state. Here (i)–(iii) are the ranges of $d_1$ corresponding to the three phases already presented in Fig. 8.

Figure 13

Semiclassical results for various drives when $g_1=g_2=25J$, $\kappa =\gamma =0.04J$, and $d_2=0$. The plots are presented column-wise: Column (a) is the under-driven case, column (b) is indefinite self-trapping, columns (c) and (d) are the over-driven delocalized cases. Blue (dark) and orange (gray) solid lines in the lower panels present the first and second cavity populations, respectively. Vertical dashed lines mark the break time, where the localization is just destroyed.

Figure 14

Semiclassically obtained phase-space description of long-lived trapped state. (a) Description in $d_1\text{−}\kappa$ plane when $\gamma =0.04J.$ (b) Description in $d_1\text{−}\gamma$ plane when $\kappa =0.04J$. In both figures, the cavity system is initialized to a state $\{20,0,-1/2,-1/2\}$ and fixed coupling for both cases, $g_1=g_2=25J$. The phases (i)–(iii) are the same as in Fig. 8.