Inverse Superconductor-Insulator Transition in Weakly Monitored Josephson Junction Arrays

Abstract

Control and manipulation of quantum states by measurements and bath engineering in open quantum systems have emerged as new paradigms in many-body physics. Here, taking a prototypical example of Josephson junction arrays (JJAs), we show how repetitive monitoring through continuous weak measurements and feedback can transform an insulating state in these systems to a superconductor and vice versa. We show that, even in the absence of any external thermal bath, the monitoring leads to a long-time steady state characterized by an effective `quantum' temperature in a suitably defined semiclassical limit. However, we show that the quantum dissipation due to monitoring has fundamental differences with equilibrium quantum and/or thermal dissipation in the well-studied case of JJAs in contact with an Ohmic bath. In particular, using a variational approximation, and by considering various limiting cases, we demonstrate that this difference can give rise to re-entrant steady-state phase transitions, resulting in unusual inverse transition from an effective low-temperature insulating normal state to superconducting state at intermediate temperature. Our work emphasizes the role of quantum feedback, that acts as an additional knob to control the effective temperature of non-equilibrium steady state leading to a phase diagram, not explored in earlier works on monitored and open quantum systems.

Figures (3)

• Figure 1: Monitored Josephson junction array (JJA) and phase diagrams: (a) The schematics of the continuous weak measurement setup for a one-dimensional (1d) JJA is shown. For the $n$-th measurement the detectors on the JJ links, e.g., $(i,i+1)$, prepared in a Gaussian state just before ($t = n\tau^-$) the measurement, are coupled to the JJ phase differences ($\hat{\theta}_i-\hat{\theta}_{i+1}$) at time $t = n\tau$. The detectors are subsequently decoupled and measured at time $t = n\tau^+$. Further, a feedback is applied using the measurement outcomes (see the main text). (b) The phase diagram within the self-consistent harmonic approximation (SCHA) as a function of JJ coupling $J$ and feedback $\gamma$ exhibits superconductor (SC)-insulator (INS) transition, as indicated by the superfluid stiffness $D$ (color) for a fixed measurement strength $\hbar\Delta^{-1}/E_c=0.5$. (c) The $J-\gamma$ phase diagram of the monitored JJA is contrasted with zero-temperature phase diagram in the $J$- dissipation ($\alpha$) plane for Ohmic JJA (see Supplementary Sec.\ref{['sec:OhmicJJA_S']}). (d) Phase diagram for monitored JJA as a function of effective temperature $T_{eff}$ and $J$ is compared with the $T$ vs. $J$ phase diagram of Ohmic JJA for $\alpha=0.5$ in (e).
• Figure 2: Phase diagram of monitored JJA and weak-coupling renormalization group: (a) Schwinger-Keldysh contour, with forward ($s=+1$) and backward ($s=-1$) branches, used to describe non-unitary dynamics in the long-time steady of monitored JJA. (b) Phase diagram as a function of JJ coupling $J$ and measurement strength $\Delta^{-1}$ for a fixed feedback $\hbar\gamma/E_c=2$. The color indicates the superfluid stiffness $D$ that demarcates between insulating (INS) and superconducting (SC) states. The re-entrant insulator-superconductor-insulator transitions are shown in $D$ vs. $\Delta^{-1}$ plot in (c) for $J/E_c=0.3$ and $\hbar\gamma/E_c=2$, and in $D$ vs. $\gamma$ plot in (d) for $J/E_c=0.3$ and $\hbar\Delta^{-1}/E_c=0.5$. (e) The weak-JJ coupling renormalization group qualitatively agrees with the re-entrant nature of the phase diagrams as a function of $\gamma$ and $\Delta^{-1}$. The sign of the $\beta$ function (color) indicates the flow towards weak coupling ($\beta<0$, insulator) or strong coupling ($\beta>0$, superconductor).
• Figure S1: Ohmic JJA:(a) The schematic diagram of a JJA coupled to an equilibrium Ohmic heat bath at constant temperatures $T$ is shown. Red circles denote the superconducting island and the black lines represent the Josephson junctions. Heat bath degrees of freedom are depicted as oscillators. (b) A horizontal cut through $J /E_c = 0.1$ is taken in Fig.\ref{['fig1']}(c) (main text) for $d=1$. We can see a jump in the value of $D$ at the first-order transition within SCHA. (c) A horizontal cut through $J /E_c = 0.035$ is taken in Fig.\ref{['fig1']}(d) (main text). We can see a continuous phase transition at $\alpha = 1/d$.