Ehrenfest's theorem beyond the Ehrenfest time

TL;DR

The paper addresses how to extend quantum–classical correspondence beyond the Ehrenfest time in chaotic systems by accounting for environment-induced decoherence. It proves a general bound for open systems with Hamiltonians of the form $\hat{H}=\hat{p}^2/2m+V(\hat{x})$ and Hermitian linear Lindblad operators: if diffusion satisfies $D \gg (\hbar/s_H)^{4/3} D_H$, quantum and classical evolutions remain closely aligned for all observables up to times exponentially longer than the closed-system Ehrenfest time. The authors represent the quantum state as a mixture of not-too-squeezed Gaussian states, evolving under harmonic approximations, and compare to the classical Fokker–Planck evolution; they also provide a provably efficient classical FP-based simulation under the diffusion threshold. A heuristic argument supports the optimality of the $\hbar^{4/3}$ scaling, and the work is illustrated with a driven chaotic oscillator and a dust-particle decoherence scenario. The results offer both a rigorous pathway to quantum–classical convergence in open chaotic systems and a practical classical algorithm for simulating Lindblad dynamics.

Abstract

In closed quantum systems, wavepackets can spread exponentially in time due to chaos, forming long-range superpositions in just seconds for ordinary macroscopic systems. A weakly coupled environment is conjectured to decohere the system and restore the quantum-classical correspondence while necessarily introducing diffusive noise -- but at what coupling strength, and under which conditions? For Markovian open systems with Hamiltonians of the form $H = p^2/2m+V(x)$ and Hermitian linear Lindblad operators, we prove the quantum and classical evolutions are close whenever the strength of the environment-induced diffusion satisfies $D \gg (\hbar/s_H)^{4/3} D_H$, where $s_H$ and $D_H$ are characteristic action and diffusion scales that we define precisely using the classical Hamiltonian $H$. The bound applies for all observables and for times exponentially longer than the Ehrenfest timescale, which is when the correspondence can break down in closed systems. The strength of the diffusive noise can vanish in the classical limit to give the appearance of reversible dynamics. The $4/3$ exponent may be optimal, suggested by heuristic arguments and prior numerical evidence. Based on our bound, we give an efficient classical algorithm for simulating quantum Lindblad dynamics, which becomes provably accurate when the strength of environmental coupling exceeds the above threshold.

Figures (5)

• Figure 1: We compare quantum and classical evolutions, for both open and closed systems, using the example of a chaotic, sinusoidally-driven double-well quartic oscillator. We confirm that when the open system has sufficient diffusion (induced by the environmental coupling), the quantum and classical evolutions agree. While the figure illustrates a special case, Theorem \ref{['thm:mainResult']} identifies the sufficient diffusion strength $D$ at which this transition happens in general systems. The driven oscillator uses Hamiltonian $H(x,p,t) = p^2/2m + B x^4 - Ax^2 + C x \cos( \omega t)$ with $A=C=10$, $B=1/2$, $m=1$, $\omega = 6.07$, evolved from Gaussian initial condition $\alpha_0 = (x_0,p_0)=(-3,8)$, $\sigma = (\sigma^{\mathrm{x}\mathrm{x}},\sigma^{\mathrm{p}\mathrm{p}})=(1/10,1)$ for time $T=16\pi/\omega\approx 8.3$. (Cf. Ref. habib1998decoherence.) The quantum system uses $\hbar=1/5$ and is simulated with the QuTiP framework johansson2013qutip. The "fully open" versions additionally have homogeneous diffusion matrix $D \approx \mathop{\mathrm{diag}}\nolimits(0.0010, 0.13)$, and the "slightly open" versions have weaker diffusion $D/10$; in the quantum cases these correspond to Lindblad operators proportional to $x$ and $p$. (a) The probability distribution of the closed classical system is strictly positive (blue) and develops features on arbitrarily small scales for large times. (b) In contrast, the Wigner function for the closed quantum system exhibits dramatic non-classicality, as witnessed by strong rapidly oscillating negativity (red) that indicates long-range coherence over phase space. Over time, the smallest features approach minimum scale $(\delta x,\delta p) \sim (\hbar/R_\mathrm{p},\hbar/R_\mathrm{x}) \sim (0.007,0.02)$, where $(R_\mathrm{x},R_\mathrm{p})\sim(10,30)$ are the dimensions of the energetically accessible region of phase space zurek2001subplanck. The vertical bulge in the classically forbidden region near $(x,p)\sim (0,10)$ reflects long-range coherence between $(\pm 3,10)$; the chance of a Gaussian POVM actually measuring the particle near $(0,10)$ is very small. (c) The slightly open classical system is softened by noise, with less delicate structure than the closed case. (d) The slightly open quantum system has substantially suppressed long-range coherence, as seen from the elimination of the highest frequency modes (e.g., the vertical bulge at $(0,10)$ is gone), but still exhibits shorter-range coherence and Wigner negativity. Finally, the fully open classical (e) and quantum (f) systems are highly similar and nearly indistinguishable for any observable. In particular the state of the open quantum system is approximately a mixture of Gaussian wavepackets, necessarily producing an approximately positive Wigner function.
• Figure 2: (a) The Wigner function for an initially pure quantum Gaussian state $\rho(t \mathord{=} 0)$ evolves in phase space. (b) At short times the dynamics admit a local harmonic (quadratic) approximation, broadening the distribution via diffusion (purple arrows) and possibly squeezing it via classical flow (red arrows). For diffusion strength $D$ and local Lyapunov exponent $\lambda_{\mathrm{L}}$ of the flow, the Gaussian state (ellipse) has a minimum thickness: the diffusion broadens the ellipse at speed $\dot{w} \sim D/w$, while the the Hamiltonian flow can shrink the width by at most $\dot{w} \sim -w /\lambda_{\mathrm{L}}$, with the competing effects balanced at $w \sim (D/\lambda_{\mathrm{L}})^{1/2}$. (c) After $\rho(t)$ becomes mixed due to diffusive broadening, it can be approximated by a mixture $\tilde{\rho}(t)$ of pure Gaussian states (ellipses) that are individually less squeezed. Each evolves by its own local harmonic dynamics while continuously being further decomposed. (d) As $\rho(t)$ spreads in phase space, our approximation $\tilde{\rho}(t)$ uses ellipses of fixed area $\hbar$ but varying amounts of squeezing. See Eq. \ref{['eq:simple-trajectories']}.
• Figure 3: An illustration of the characteristic scales of a Hamiltonian $H = p^2/2m + V(x)$, which defines the opaque purple surface over phase space. The potential $V(x)$ is projected in red and the kinetic term $p^2/2m$ is in blue. For this potential, the second derivative $\nabla^2 V$ takes its largest absolute value near a local minimum. The transparent green paraboloid denotes the second-order approximation to the Hamiltonian (i.e., the conservative part of the local harmonic dynamics) near this point. The inverse harmonic time $t_H^{-1} = \sqrt{\|\nabla^{2} V\|/m}$ is the characteristic frequency, at any amplitude, of the harmonic motion within this local approximation to the potential (e.g., around the thick green ellipse). The small solid-black ellipse represents a classical (Gaussian) Gibbs state for this quadratic potential. Although the area of the small ellipse is temperature dependent, the harmonic aspect parameter $\eta_H = \sqrt{m \|\nabla^{2} V\|}$ of the ellipse --- the ratio of its axes with units kg/s --- is not. This determines a natural choice of (unsqueezed) quantum coherent states, corresponding to similarly oriented ellipses in phase space with area $\sim \hbar$. The harmonic approximation will be accurate within the anharmonic length $x_H = \sqrt{s_H /\eta_H} = \|\nabla^{2} V\|/\|\nabla^{3} V\|$ in the $x$ direction and the anharmonic momentum $p_H = \sqrt{s_H \newline\eta_H} = \sqrt{m \|\nabla^{2} V\|^3}/\|\nabla^{3} V\|$ in the $p$ direction, which are determined by the largest anharmonicity $\|\nabla^{3} V\|$ and which satisfy $\eta_H = p_H/x_H$. This characterizes the region of harmonic approximation accuracy, the dotted-black ellipse, which has phase-space area of order the anharmonic action $s_H = x_H p_H = m^{1/2}\|\nabla^{2} V\|^{5/2}/\|\nabla^{3} V\|^{2}$, the unique action scale constructible from $m$, $\|\nabla^{2} V\|$, and $\|\nabla^{3} V\|$. The anharmonic diffusion matrix $D_H = 2s_H\sigma_*/(\hbar t_H) =\mathrm{diag}(x_H^2,p_H^2)/t_H$ is a characteristic diffusion scale above which diffusion appreciably affects the system, relative to the largest anharmonicities, on the harmonic time scale. As quantified in Theorem \ref{['thm:mainResult']} and Corollary \ref{['cor:Main']}, quantum and classical dynamics will be essentially indistinguishable for all observables when $D \gg (\hbar/s_H)^{4/3} D_H$, where $D$ is the actual diffusive noise from the open-system dynamics, and where $\hbar/s_H \ll 1$ for macroscopic systems.
• Figure 4: Consider the error rate incurred by evolving a pure Gaussian state (ellipse) using the local Harmonic approximation to the dynamics. The discrepancy is dominated by the leading-order anharmonicity $\nabla^3 V$, which is strongest (relative to the center) at the tips of the ellipse lying on either end of the long axis of length $v \sim \hbar/w$. As the third-order term in the Taylor approximation, this changes the speed of the local flow by $s \lesssim v^2 \|\nabla^3 V\|$. So the discrepancy (red shaded area) between the distribution under the exact dynamics (curved boomerang) and the area of the ellipse grows at rate $\lesssim sv$. Compared to the ellipse's area $\hbar$, this gives an error rate $r \sim sv/\hbar \lesssim \hbar^{-1} v^3 \|\nabla^3 V\|$. Because the eigenvalues of the covariance matrix $\sigma \sim \textrm{diag}(w^2,v^2)$ are just the squares of the semimajor axes, the error rate is bounded as $r \lesssim \hbar^{-1} \|\sigma\|^{\frac{3}{2}} \|\nabla^3 V\|$, as proved in Lemma \ref{['lem:HarmErrorQV']}. As explained in Fig. \ref{['fig:ellipses']}, the width $w$ of the narrow axis is lower bounded by the diffusion strength: $w \gtrsim (D/\lambda_{\mathrm{L}})^{1/2}$. So we can also write $r \lesssim (\hbar^{4/3}/D)^{3/2} \lambda_{\mathrm{L}}^{3/2} \|\nabla^3 V\|$, which is small when $D \gg \hbar^{4/3}$.
• Figure 5: An illustration of the proposed approximation algorithm for sampling from $f(T)$. Starting at an initial point $\alpha'_0$ sampled from the density $f(0)$, and assuming $\sigma'_0 = 0$ we approximately solve the harmonic evolution equations \ref{['eq:reuse-adot']}, \ref{['eq:reuse-sdot']} for time $\tau$ to find a new centroid $\alpha_1$ and covariance matrix $\sigma_1$, defining the Gaussian measure $\mu_1 := \mathcal{N}(\alpha_1,\sigma_1)$. To perform this approximate evolution we propose to use a Runge-Kutta explicit integrator with time step $\zeta \leq \tau$. We then draw a new starting points $\alpha'_1$ from $\mu_1$, integrate it forward (with $\sigma'_1=0$) to get $\alpha_{2}$ and $\sigma_{2}$, and so on. This is repeated a total of $n=T/\tau$ times, yielding a sequence random Gaussian measures $\mu_k$. The final distribution $\mu_n$, averaged over random draws, approximates $f(T)$. The exact trajectory of the Gaussian at time $\tau$ is shown in black and the approximate discretized evolution is shown in blue. The exactly integrated Gaussian $\nu_k$ is represented by a gray ellipse, and the numerically integrated Gaussian measure $\mu_k$ by a blue ellipse. So long as $\mu_k$ and $\nu_k$ have sufficient overlap the black and blue trajectories remain together.

Theorems & Definitions (13)

• Theorem 1: Main result
• Corollary 1: Indistinguishable observables
• Theorem 2: Lindblad simulation (simplified)
• proof : Proof of Theorem \ref{['thm:mainResult']}
• Theorem 3: Main result for separable Hamiltonians
• Lemma 3: Harmonic approximation error
• proof
• Theorem 4: Lindblad Simulation
• Proposition 4: Fokker-Planck simulation
• Proposition 4: Fokker-Planck simulation