Turning Down the Noise: Power-Law Decay and Temporal Phase Transitions

Abstract

We determine the late-time dynamics of a generic spin ensemble with inhomogeneous broadening - equivalently, qubits with arbitrary Zeeman splittings - coupled to a dissipative environment with strength decreasing as $1/t$. The approach to the steady state follows a power law, reflecting the interplay between Hamiltonian dynamics and vanishing dissipation. The decay exponents vary non-analytically with the ramp rate, exhibiting a cusp singularity, and $n$-point correlation functions factorize into one- and two-point contributions. Our exact solution anchors a universality class of open quantum systems with explicitly time-dependent dissipation.

Figures (6)

• Figure 1: Schematic depiction of the system and main results. (a) Generic spin ensemble with inhomogeneous broadening (qubits with arbitrary Zeeman splittings) coupled to a dissipative bath whose strength decreases in time, $g_{\pm}(t)=1/(\nu t)$. (b) Correlation functions decay as power laws, $t^{-\alpha(\nu)}$. At the critical ramp rate $\nu_{\text{critical}}$, the decay exponent switches from $\alpha_{\leq}(\nu)$ to $\alpha_{\geq}(\nu)$, signaling a temporal phase transition. Solid (colored) lines show the realized decay law in each regime, while dashed (gray) lines indicate the competing power law that is suppressed.
• Figure 2: Log-log plot of numerically computed three-point ($n=3$) correlation functions (solid lines) of a system of dissipative spins starting out in a spin-coherent state, defined by a tensor product of $\rho_{\text{sc}}(\theta,\phi) = \psi_{\text{sc}}(\theta,\phi) \psi_{\text{sc}}(\theta,\phi)^\dagger$ with $\psi_{\text{sc}}(\theta,\phi) = \text{exp}\left(\frac{1}{2} \theta e^{i\phi} \hat{s}^- - \frac{1}{2} \theta e^{-i\phi}\hat{s}^+\right)\left|-1/2\right>$, for $\nu=n/\eta=6.0$. (a) The full dynamics of select correlation functions starting at an initial time $t_\text{init} = 10^{-5}$. (b), (c) Zoomed in plots of the long-time regime indicated by the colored squares in (a). The scaling as predicted by \ref{['eq:spin_1_general_asymptotic_correlation_function']} is plotted (dashed-dotted) just above or below the correlation functions with the value for $\alpha$ indicated.
• Figure 3: Decay exponent $\alpha$ as a function of $\nu$. For the correlator $c_{zz}$, the long-time decay is governed by $\alpha=(\nu+2)/\nu$ or $\alpha=4/\nu$ (solid and dashed curves, colored as indicated in the legend). The smallest exponent sets the true asymptotic decay to the steady state and is indicated using solid curves. Dots denote exponents extracted from numerical fits of $c_{zz}$. The regime labeled 'Analytics' requires the exact result \ref{['eq:czz_corr_function_exact']}, while the 'Asymptotics' regime is captured by the general formula \ref{['eq:spin_1_general_asymptotic_correlation_function']}. The critical point $\nu=2$ marks the temporal phase transition.
• Figure 4: Decay exponents $\alpha$ as a function of $\nu$ for varying higher order correlation functions. The solid lines indicate the dominant decay rate whereas the dashed lines are plotted to indicate how the decay rate would have continued, had there not been a transition at $\nu = 2$. Panel (a) compares the analytical predictions (solid/dashed lines) to the numerical results (circles/$X$ markers) for the $c_{+zz}$ correlation function of a $n=3$ non-Hermitian RG Hamiltonian \ref{['eq_App:NHRGHamiltonian']}. (b), (c) and (d) depict the decay exponent of fourth, fifth and sixth order correlation functions, denoted by $c_{jz}$ where $j$ indicates the number of $z$-indices in the correlator $c_{z\dots z}$ and the number of sites in \ref{['eq_App:NHRGHamiltonian']}. The legends indicate the predicted curves of $\alpha$ as described in Eq. \ref{["eq:alpha'_for_general_nu"]} in the main text. For all panels, the initial state for the data depicted by the circles is defined by the (normalized) weights of $\left(\sum_{j=1}^n \hat{S}_j^+\right)^{N_+}\!\bigotimes^{N+}\!\left|-1\right>$ and $\varepsilon_i = 50\times i$. The results for random initial states are also depicted using the $X$-markers. An initial time-cutoff at $t = 10^{-2}$ is introduced to avoid singularities in the numerical simulations.
• Figure 5: (a) Numerically computed correlation functions on a log-log scale. The initial state is defined by the (normalized) weights of $\left(\sum_{j=1}^n \hat{S}_j^+\right)^{N_+}\left|\odot\right>$ with an additional factor of $\frac{1}{2}$. This factor ensures a valid (positive, semi-definite) state for $\rho$. The asymptotic solution is valid in the regime beyond $t = 10$barik_higher_2025. (b) Zoom-in of the marked region in (a). In this plot, the predicted decay of the correlation functions with exponent $\alpha$ is also plotted with the dashed-dotted line. In this figure, $\nu = n/\eta$ and $\varepsilon_i = i/n$. For these choices of system parameters, we have $N_+ = n$, $N_1 = n$ and $J^z = 0$. Thus, the predicted scaling $\alpha = 0.4$ matches the numerically simulated results. As in the main text, the numerical simulations start at an initial time $t_{\text{init}} = 10^{-5}$, to avoid singularities.