Power-law distributions in nonequilibrium open quantum systems

Abstract

Power-law probability distributions are widely used to model extreme statistical events in complex systems, with applications to a vast array of natural phenomena ranging from earthquakes to stock market crashes to pandemics. We show that analogous heavy tails arise naturally in open quantum systems with nonlinear dissipation. Introducing a prototypical family of quantum dynamical models, we analytically prove the emergence of power-law tails in the steady state energy distribution, originating from an amplification of quantum noise whose microscopic fluctuations grow with energy. Moreover, our analysis suggests a general mechanism for heavy-tail statistics in the nonequilibrium steady states of open quantum systems: Nonlinear dissipation generically induces multiplicative quantum noise, enforced by the constraints of quantum mechanics, which is responsible for the heavy-tail behavior. This is supported by numerical simulations of a general class of nonlinear dynamics known as quantum Liénard systems. Remarkably, even when the corresponding classical system is stable, we find power-law tails in both steady-state populations and coherences, which occur for typical parameters without fine-tuning. This phenomenon can potentially be harnessed to develop extreme photon sources for novel applications in light-matter interaction and sensing.

Figures (5)

• Figure 1: Power-law distribution in the $M$-boson model. (a) A quantum particle trapped in a one-dimensional harmonic potential interacts with $M$ independent baths ($M = 2$ depicted). The particle exchanges $m$ excitation quanta with the $m$-th bath ($m = 1,\ldots,M$), at a dissipation rate of $\gamma_m$ and pumping rate of $\kappa_m$. (b) Simulated measurement for the number of excitation quanta $n$ on the steady state of Eq. \ref{['eq:Mphoton_mastereq']} with $M = 2$ and $500$ measurement samples. The dashed line indicates the median count $n = 1$, and the dotted line indicates the $90^{\text{th}}$ percentile count $n \approx 41.1$. Extreme events reaching $n \approx 16000$ are observed. Parameters are $\{\gamma_1,\kappa_1,\gamma_2,\kappa_2\} = \{6,0,1,1\}$. (c) The probability distribution $\rho_{n,n}$ for $n$ exhibits a power-law tail $\sim n^{-\nu}$ at the critical point $\kappa_2 = \gamma_2$, plotted for $\gamma_1/\gamma_2 = 6$ and $\kappa_1/\gamma_2 = 0$. The inset shows the power-law exponent $\nu$ against $\gamma_1/4\gamma_2$. The data points are obtained by numerically fitting with a power law, which agrees excellently with the analytical result \ref{['eq:powerlawtail']}, $\nu = \gamma_1/4\gamma_2$. Away from the critical point, the probability distribution follows a power law up to an exponential cutoff at $n \sim M/\delta$.
• Figure 2: Physical signatures of the power-law distribution. Numerical results for the $M$-boson model with $M = 2$ and $\kappa_1 = 0$. (a) Mean number of excitation quanta $\braket{\hat{a}^{\dag }\hat{a}^{}}$, (b, c) Normalized second-order correlation function $g^{(2)}(0)$. $\delta = 1 - \kappa_2/\gamma_2$ is the deviation from the critical point. The power-law exponent is $\nu = \gamma_1/4\gamma_2$ [Eq. \ref{['eq:powerlawtail']}]. In all the plots, the solid curves are obtained numerically for $\delta \in [10^{-4},10^{-2}]$. The Hilbert space truncation is set at $N_T = 5 \times 10^5$ to ensure numerical convergence. The black dashed lines indicate the corresponding analytical results \ref{['eq:meanphoton']} and \ref{['eq:g20']}. Grey shaded regions denote the regime in which the physical quantity is divergent near the critical point, with the divergence derived from the scaling theory indicated. At large $\nu$, $g^{(2)}(0)$ scales linearly with $\nu$.
• Figure 3: Power-law tails in the quantum Liénard system. Magnitudes of steady-state density matrix element $|\rho_{n,n^\prime}|$ against the number of excitation quanta $n$ for the quantum Liénard system \ref{['eq:quantum_lienard']} in the limit-cycle regime. Curves plotted are for even $n^\prime - n \in [0,16]$. Matrix elements for odd $n^\prime - n$ are identically zero. The black dashed line shows the power-law scaling $|\rho_{n,n^\prime}| \propto n^{-1.6}$ obtained from fitting the number distribution $\rho_{n,n}$. (Inset) Wigner quasiprobability distribution $W(q,p)$ of the steady state. $W(q,p)$ is mostly supported near the phase space origin, with a heavy tail that decays slowly. Parameters: $(k_0,k_1,k_2,k_3) = (0.169,0.12,1,0.035)$, with a Hilbert space truncation of $N_T = 10^3$ in the simulation for numerical convergence. Similar behavior can be observed for other parameters.
• Figure 4: Steady state populations $\rho_{n,n}$ of the quantum Liénard system against the excitation number $n$. The Hilbert space truncation $N_T$ in the simulation ranges from $200$ to $1000$. The black dashed line indicates the power-law fit for $n \in [50,300]$ and $N_T = 1000$, with the fitted exponent indicated in the figures. The parameters $(k_0,k_1,k_2,k_3)$ used are: (a) (0.465, 0.353, 1, 0.042), (b) (0.168, 0.821, 1, 0.405), (c) (0.723, 0.372, 1, 0.788).
• Figure 5: Steady state populations $\rho_{n,n}$ against the excitation number $n$, in the even parity sector. For odd values of $n$, $\rho_{n,n} = 0$ which are omitted from the plots. The Hilbert space truncation $N_T$ in the simulation ranges from $200$ to $1000$. The black dashed line indicates the power-law fit for $n \in [50,300]$ and $N_T = 1000$, with the fitted exponent indicated in the figures. The systems studied are: (a) Quantum Liénard model with $k_0 = k_1 = k_3 = 0$ and $k_2 = 1$. (b) Same as (a), but with the Hamiltonian $H = 0$. (c) Same as (a), but with the Hamiltonian $H = -i(\hat{a}^{\dag }\hat{a}^{3} - \hat{a}^{\dag 3} \hat{a}^{})/4$. (d) Same as (a), but with the Hamiltonian $H = -i(\hat{a}^{4} - \hat{a}^{\dag 4})/8$.