Dynamical Phase Transitions in Open Quantum Walks

TL;DR

Open quantum systems subject to periodic dephasing can be effectively described by non-equilibrium Markov dynamics, revealing dynamical phase transitions via crossings of Floquet exponents $\lambda_2$ and $\lambda_3$ and, under broken detailed balance, exceptional points where $\lambda_2=\lambda_3$. The authors demonstrate first-order transitions when time-reversal symmetry is preserved and second-order transitions at exceptional points when it is broken, in two models: a gauge-influenced ring quantum walk and an internal-state line walk. These insights reveal how decoherence-induced classicalization enables access to non-Hermitian spectral phenomena with potential technological relevance for quantum simulation and state control. The results are experimentally accessible in photonic lattices, trapped ions, and ultracold atoms, offering pathways to observe and harness non-equilibrium quantum dynamics.

Abstract

Dynamical phase transitions in the relaxation behavior of stochastic quantum walks are investigated, focusing on systems where coherent unitary evolution is periodically interrupted by dephasing. This interplay leads to a classicalization of the dynamics, effectively described by non-equilibrium Markovian processes that can violate detailed balance. As a result, such systems exhibit a richer and more complex spectral structure than their equilibrium counterparts. Extending recent insights from classical Markov dynamics [G. Teza {\it et al.}, Phys. Rev. Lett. {\bf 130}, 207103 (2023)], we demonstrate that these quantum-classical hybrid systems can host not only first-order dynamical phase transitions -- characterized by eigenvalue crossings -- but also second-order transitions marked by the coalescence of eigenvalues and eigenvectors at exceptional points. We analyze two paradigmatic models: a quantum walk on a ring under gauge fields and a walk on a finite line with internal degrees of freedom, both exhibiting distinct mechanisms for breaking detailed balance. These findings reveal a novel class of critical behavior in open quantum systems, where decoherence-induced classicalization enables access to non-Hermitian spectral phenomena. Beyond their fundamental interest, our results offer promising implications for quantum technologies, including quantum simulation, error mitigation, and the engineering of controllable non-equilibrium quantum states.

Figures (6)

• Figure 1: Dynamical phase transitions in the decohered quantum walk on a network. The panels (a) and (b) show typical behaviors of the real (lower panels) and imaginary parts (upper panels) of the Floquet exponents $\lambda_2$ and $\lambda_3$ of the first two dominant decay modes versus the control parameter $\beta$. At the critical value $\beta=\beta_c$ the two eigenvalues coalesce, i.e. $\lambda_2=\lambda_3$. Under time reversal symmetry, detailed balance is satisfied and the eigenvalue crossing corresponds to a jump of the right eigenvector $r^{(2)}$ (first-order phase transition), as shown in (a). Detailed balance can be violated when time reversal symmetry is broken. In this case eigenvalue crossing can correspond to a coalescence of both eigenvalues and eigenvectors, i.e. to an exceptional point, as shown in (b). In this case the right eigenvector $r^{(2)}$, i.e. direction to equilibration, is continuous as the critical point $\beta=\beta_c$ is crossed, leading to a dynamical analogue of a second-order phase transition. (c) Minimal network for the observation of first- and second-order dynamical phase transitions. The network comprises three nodes $| 1 \rangle$, $|2 \rangle$ and $| 3 \rangle$, with coherent hopping rates $J_1$, $J_2$ and $J_3$. A gauge field (magnetic flux) $\phi$ can be applied to the closed triangular path. Time reversal symmetry, and thus detailed balance, is violated for $\phi \neq 0, \pi$.
• Figure 2: First-order dynamical phase transitions in the minimal network of Fig.1(c) with time-reversal symmetry ($\phi=0$) for parameter values $J_1=J_2=1$ and $J_3=0.5$. The control parameter is $\beta=J_1 \tau= \tau$. (a) and (b) show the behaviors of the real [panel (a)] and imaginary parts [panel (b)] of the Floquet exponents $\lambda_2$ and $\lambda_3$ of the first two dominant decay modes versus the control parameter $\beta$. At the critical value $\beta=\beta_c \simeq 0.8127$ the two eigenvalues coalesce, i.e. $\lambda_2=\lambda_3$, while the respective eigenvectors are distinct. Crossing the critical point yields a discontinuity in the direction to equilibration. (c) Behavior of the right eigenvectors $r^{(2)}$ and $r^{(3)}$ for a few increasing values of $\beta$, showing the flipping of the eigenvectors as $\beta$ crosses the critical value. From top to bottom: $\beta=0.7$, $\beta=0.8$, $\beta=0.83$, and $\beta=1$. (d,e) Relaxation dynamics, showing the behaviors of the node occupation probabilities $P_n$ at successive time steps, for the initial condition $P_n^{(t=0)}=\delta_{n,1}$, corresponding to initial particle occupying site $|1 \rangle$. In (c) $\beta=0.79< \beta_c$, whereas in (d) $\beta=0.83> \beta_c$. The insets in (d) and (e) display an enlargement of the long-time relaxation dynamics.
• Figure 3: Same as Fig.2, but for a non-vanishing gauge field $\phi= \pi/3$. Time-reversal symmetry breaking yields a Markov matrix which violates detailed balance. An exceptional point is observed at the critical value $\beta=\beta_c=0.7412$. Crossing the critical point corresponds to a second-order (smooth) phase transition in the approach to equilibrium. Panel (c) shows the behavior of the right eigenvectors $r^{(2)}$ and $r^{(3)}$ of $Q$ for a few increasing values of $\beta$ crossing the critical point (from top to bottom: $\beta=0.6$, $\beta=0.74$, $\beta=0.76$, and $\beta=1$). Note the smooth change of the dominant decay eigenmode and eigenvector coalescence as the critical point is crossed. Panels (d) and (e) show the relaxation dynamics for $\beta=0.7 <\beta_c$ and $\beta=0.8 >\beta_c$, respectively.
• Figure 4: Second-order dynamical phase transition in the discrete-time quantum walk on a line (size $L=3$) with reflective boundaries under dephasing. The walker can be found at the three spatial positions $n=1,2$ and 3 with either internal state $|H \rangle$ or $|V \rangle$. (a,b) Behavior of the Floquet exponents $\lambda_l$ ($l=3,4,5,6$) of the four decay modes (real and imaginary parts) versus coin angle $\beta$. An exceptional point occurs at the critical value $\beta=\beta_c \simeq 0.4771 \times \pi/2$. The critical coin angle $\beta_c$ depends on the system size $L$, as shown in the inset of (b). (c,d) Relaxation dynamics toward equilibrium distribution for a coin angle $\beta$ below [$\beta=0.3 \times \pi/2$, panel (c)] and above [$\beta=0.7 \times \pi/2$, panel (d)]] the critical value $\beta_c$. The initial probability distribution is $X_1=1$ and $X_l=Y_l=0$ otherwise, corresponding to the walker placed at site $n=1$ with internal state $| H \rangle$. The curves in the plots show the site occupation probabilities $p_n=X_n+Y_n$ of the walker at the three sites $n=1,2,3$ versus time step, regardless of its internal state.
• Figure A1: Effect of the dephasing probability $q$ on first-order phase transition for the minimal network of Fig.1(c) with the same parameter values as in Fig.2 and for decreasing values of $q$. The upper panels show the behavior of the decay rates of second and third eigenvectors of the Liouvillian superoperator $\mathcal{L}$ versus the control parameter $\beta$, whereas the lower panels depict the corresponding behavior of the eigenmode overlapping parameter $g=g(\beta)$. The first column, $q=1$, corresponds to the classical limit discussed in the main text (Fig.2). As $q$ is decreased, the critical value $\beta_c$ of eigenvalue crossing decreases. No eigenvalue crossing occurs for $q<q_c \simeq 0.23$.