Replica Keldysh field theory of quantum-jump processes: General formalism and application to imbalanced and inefficient fermion counting

TL;DR

This work develops a comprehensive replica Keldysh field theory for general quantum-jump processes, unifying efficient-pure-state trajectories with mixed-state dynamics under imperfect detection and incorporating non-Hermitian jumps. The framework robustly handles state-dependent jump rates and averages over jump times and types, establishing a bridge between measurement-induced phase transitions and nonequilibrium steady states in driven open quantum systems. Applying it to imbalanced and inefficient fermion counting on a 1D lattice, the authors show that imbalance removes the phase transition and induces an intermediate quantum-critical regime, while inefficiency introduces a finite correlation length that confines genuine entanglement and yields volume-law subsystem entropy for mixed trajectories. Numerical simulations corroborate the analytical predictions, highlighting a smooth crossover controlled by scales l0, lc, and ξ, and revealing how non-Hermitian jump dynamics shape long-wavelength fluctuations via a nonlinear sigma model framework.

Abstract

Measurement-induced phase transitions have largely been explored for projective or continuous measurements of Hermitian observables, assuming perfect detection without information loss. Yet such transitions also arise in more general settings, including quantum-jump processes with non-Hermitian jump operators, and under inefficient detection. A theoretical framework for treating these broader scenarios has been missing. Here we develop a comprehensive replica Keldysh field theory for general quantum-jump processes in both bosonic and fermionic systems. Our formalism provides a unified description of pure-state quantum trajectories under efficient detection and mixed-state dynamics emerging from inefficient monitoring, with deterministic Lindbladian evolution appearing as a limiting case. It thus establishes a direct connection between phase transitions in nonequilibrium steady states of driven open quantum matter and in measurement-induced dynamics. As an application, we study imbalanced and inefficient fermion counting in a one-dimensional lattice system: monitored gain and loss of fermions occurring at different rates, with a fraction of gain and loss jumps undetected. For imbalanced but efficient counting, we recover the qualitative picture of the balanced case: entanglement obeys an area law for any nonzero jump rate, with an extended quantum-critical regime emerging between two parametrically separated length scales. Inefficient detection introduces a finite correlation length beyond which entanglement, as quantified by the fermionic logarithmic negativity, obeys an area law, while the subsystem entropy shows volume-law scaling. Numerical simulations support our analytical findings. Our results offer a general and versatile theoretical foundation for studying measurement-induced phenomena across a wide class of monitored and open quantum systems.

Figures (5)

• Figure 1: Rescaled density correlation function \ref{['eq:connected-density-correlation-function']} in (a) momentum space and (b) real space for $n = 0.4$ and $\eta = 1$. (a) The numerical data deviate significantly from the Gaussian correlation function \ref{['eq:C-q-Gaussian']} (black dashed line) for $\tilde{q} l_0 \to 0$: while the Gaussian result approaches a finite value, $C_q/(g_0 \tilde{q}) \to 1$, the numerical data exhibit a maximum at $q_c$ and then decrease toward zero. Inset: The position of the maximum scales as $q_c \sim \gamma^2$. (b) In real space, we observe good agreement with the Gaussian prediction on short scales, $\tilde{l} \lesssim l_0$. Significant deviations appear for $\tilde{l} \gtrsim l_c$, where the Gaussian result decays algebraically, $\left\lvert C_l \right\rvert \sim \tilde{l}^{-2}$, while the numerical data decay more rapidly. Inset: The crossover scale at which deviations from the Gaussian result become pronounced scales as $l_c \sim \gamma^{-2}$.
• Figure 2: Rescaled density correlation function \ref{['eq:connected-density-correlation-function']} in (a) momentum space and (b) real space for $n = 0.4$ and $\gamma/J = 0.1$. (a) Finite detection inefficiency leads to a nonzero axis intercept for $\tilde{q} l_0 \to 0$. Inset: The intercept follows the expected scaling $\sim \delta \eta^{1/2}$. (b) In real space, detection inefficiency induces an exponential decay of correlations on large scales. Inset: The fitted correlation length $\xi$ agrees well with the predicted behavior $\xi \sim \delta \eta^{-1/2}$. Thick lines in (a) and (b) show fits to Eqs. \ref{['eq:C-q-fit']} and \ref{['eq:C-l-fit']}, respectively.
• Figure 3: (a) Rescaled entanglement entropy and (b) rescaled scale-dependent effective central charge for $n = 0.4$ and $\eta = 1$. (a) For short subsystem sizes $\tilde{\ell} \lesssim l_0$, the entanglement entropy follows volume-law scaling (green dashed line), consistent with the infinite-temperature behavior in Eq. \ref{['eq:S-l-volume-law']}. At larger scales, the data show an apparent logarithmic increase for small $\gamma/J$, which transitions to area-law behavior as $\gamma/J$ is increased. (b) The effective central charge rises in the volume-law regime, reaches a maximum at a characteristic scale $l_m$, and then decreases for $\tilde{\ell}/l_0 \gg 1$. This decay demonstrates that the logarithmic growth of entanglement observed at small $\gamma/J$ does not persist asymptotically. By contrast, the Gaussian approximation (black dashed line) for $c_{\ell}$ approaches a constant value, indicating sustained logarithmic entanglement growth. Inset: The position of the numerically observed maximum of $c_{\ell}$ is consistent with the expected scaling $l_m \sim \gamma^{-3/2}$.
• Figure 4: Rescaled subsystem entropy for $\gamma/J = 0.1$ and $n = 0.4$. For inefficient detection with $\delta \eta = 1 - \eta > 0$, the subsystem entropy exhibits volume-law scaling. In the limit $\delta \eta = 1$, the numerical data are consistent with Eq. \ref{['eq:S-l-volume-law']}, as expected for a fully mixed infinite-temperature state with fixed fermion density $n$.
• Figure 5: Rescaled logarithmic negativity for $\gamma/J = 0.1$ and $n = 0.4$ at varying measurement inefficiencies $\delta \eta = 1 - \eta$. For $\delta \eta = 0$, the data exhibit approximately logarithmic growth with subsystem size. Any finite $\delta \eta > 0$ leads to area-law scaling on scales $\tilde{\ell} \gtrsim \xi$. This region, with $\xi$ obtained by fitting Eq. \ref{['eq:C-l-fit']} to the data in Fig. \ref{['fig:C-l-q-eta<1']}(b), is highlighted by thicker lines. For $\delta \eta = 1$, the steady state \ref{['eq:rho-ss']} is completely disentangled, yielding $\mathcal{E}_{\ell} = 0$.