Quantum Birthmarks: Ergodicity Breaking Beyond Scarring

TL;DR

This work introduces quantum birthmarks (QBs) as permanent non-ergodic memory imprinted by a non-stationary quantum state and its early-time evolution. QBs decompose into a universal component (UQB) arising from global symmetries and a revival-enhanced component (RQB) from early recurrences, together biasing long-time occupation probabilities away from the ergodic baseline bar{P}_{ab}=1/N and yielding bar{P}_{aa}/ar{P}_{ab} \,\approx\, P^{\text{UQB}} P^{\text{RQB}} \ge 2. The authors demonstrate these effects in a chaotic stadium billiard, showing QB signatures in coordinate-space densities ar{Q}(r) and phase-space measures like the number of explored cells N_t, and they connect QBs to quantum scars while clarifying that QBs extend beyond scar physics. The framework reconciles universal random-matrix behavior with system-specific short-time dynamics, offering pathways to experimental observation in quantum dots, microcavities, and quantum simulators, and it introduces the notion of birthplaces where partial barriers preserve memory across subspaces. Overall, QBs provide a dynamical, time-domain perspective on ergodicity and thermalization in quantum chaotic systems, unifying scarring phenomena with generic non-stationary states.

Abstract

A hallmark of classical ergodicity is the complete loss of memory of the initial conditions due to eventual uniform covering of {\it a priori} available phase space. In quantum counterparts of such systems, however, this classical ergodic ideal is fundamentally limited: Here, we introduce the concept of a \emph{quantum birthmark}, a permanent signature left by the initial state and its early-time evolution in a general quantum system, which gives rise to non-ergodic behavior persisting even in the infinite-time limit. We present a birthmark framework outlining a ubiquitous memory effect for an arbitrary, non-stationary state composed of two factors conspiring together: the universal and the revival-enhancement. The former sets the minimal amplification carried by the time evolution of a quantum state based on global symmetries, whereas the latter incorporates the further enhancement stemming from the early dynamics, particularly prominent in the presence of recurrences that occur before the Heisenberg time. As a concrete example, we identify quantum birthmarks in the venerable stadium billiard, where they can be significantly enhanced by quantum scars. Finally, we discuss the broader implications of quantum birthmarks, including their role as a natural extension of all types of scarring theories to generic non-stationary quantum systems and prospects for experimental observation. Generally, our work opens an unexplored avenue for understanding the elusive quantum nature of ergodicity.

Figures (12)

• Figure 1: Quantum birthmark: Figure displays the long-time average of the probability density for a Gaussian wavepacket initialized along a "bowtie" quantum scar at the center of a stadium billiard (see Sec. III for details). Contrary to the classical expectation of ergodic uniformity, both the initial state and the early-time dynamics induced by the scar are indelibly burned in, manifesting as a quantum birthmark (as explained in Sec. II).
• Figure 2: Average occupation probability See equation \ref{['Eq:quantum_phase_space_exploration']}. The localized and "dynamically distinguishable" states $\vert a \rangle$ and $\vert b \rangle$ are depicted schematically as circles, alongside representative spatial profiles of typical eigenstates of the considered system. At the top, corresponding to an integrable system, each eigenstate overlaps with either $\vert a \rangle$, or $\vert b \rangle$, or neither - importantly not both, in the cases shown. If $\vert b \rangle$ had been in the zone reached by $\vert a \rangle$, the overlap would be large. At the bottom, representing an “ergodic” system, each eigenstate overlaps with both $\vert a \rangle$ and $\vert b \rangle$. This highlights a fundamental distinction between regular and chaotic quantum dynamics. In the long-time limit ($t \to \infty$), the average occupation probabilities behave differently in the two cases. For integrable systems, one finds $\bar{P}_{ab} \to 0$ (or more generally $\bar{P}_{ab} \sim m/N$, with $m$ of order $N$), provided that the probe state lies within the manifold accessed from $\lvert a\rangle$. By contrast, for chaotic systems, the long-time average approaches the ergodic value $\bar{P}_{ab} \to 1/N$.
• Figure 3: Spectral decomposition and overlap. In the left panel corresponding to integrable behavior, the states $\vert a \rangle$ and $\vert b \rangle$ have mutually exclusive occupation of the eigenstates, and vanishing dynamical access of one starting from the other. However, in the chaotic system shown in the middle, the spectral intensities are fluctuating according to a $\chi^2$ distribution of two degree of freedom (the square of Gaussian distributed amplitudes). Since here the large fluctuations are always squared, the sum $\sum_n (p_n^a)^2$ rises to $2/N$, as discussed in Sec. II (b). On the right, the states $\vert a \rangle$ and $\vert b \rangle$ share the same envelopes, but differ in their overlaps with the eigenstates. In this case, it is random as to whether large probabilities in $p_n^a$ correspond to large probabilities in $p_n^b$, yielding the sum $\sum_n p_n^a p_n^b$ matching with the ergodic expectation of $1/N$.
• Figure 4: Spectra and long-term occupation of two different initial states $\vert a\rangle$ and $\vert a' \rangle$. The same Hamiltonian $\mathcal{H}$ applies to both states, and therefore they share the same set of eigenvalues. Nonetheless, the individual intensities differ: $\vert a\rangle$ had no early revivals possessing the RMT average (green envelope), but $\vert a' \rangle$ instead underwent one or more revivals, causing the low resolution structure (pink envelope). The spectral lumps for the state $\vert b \rangle$ mean the intensities must necessarily be distributed more widely, yielding $\bar{P}_{a'a'} > \bar{P}_{aa}$.
• Figure 5: Birthmarks in a soft Bunimovich stadium. Each panel shows the scaled long-time averaged coordinate space probability density $\bar{Q}(\mathbf{r})$ of quantum wavepackets propagated for Cases (A)--(D) specified in the text (corresponding to the panels from top panel to the bottom, respectively). The strength of early-time recurrences and the associated QB enhancement decreases progressively from top to bottom, as moving from initial conditions along POs to more generic ones. None of the cases exhibits classical ergodic behavior in the sense of the probability density becoming uniform at infinite time.