Observation of the Quantum Boomerang Effect

Anderson localization is a quintessential effect of disorder in a variety of physical systems [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Disordered lattice models enjoy a robust theoretical proof of localization and remain a hotbed for discovery even today. While transport in disordered media has been studied for more than 50 years, a fundamental dynamical feature of Anderson-localized matter known as the “quantum boomerang” effect was only recently theoretically predicted [13, 14, 15]. The central prediction of the quantum boomerang is that, for an appropriate average over disorder realizations, a wavepacket launched with any nonzero momentum will return to and localize at its initial position [13]. This surprising phenomenon can be understood as a consequence of both Anderson localization and time-reversal symmetry: the long-time density distribution for a wavepacket with an initial momentum $k_{0}$ (i) is time-independent because of localization and (ii) does not depend on the sign of $k_{0}$ because of time-reversal symmetry. Together, these features imply that at long times the wavepacket must settle at its initial position. The quantum boomerang effect is predicted to occur in a broad class of disordered or pseudo-disordered tight-binding models [15] including the quantum kicked rotor (QKR), a momentum-space realization of a disordered quasi-1D wire [16, 17] which provides an ideal platform for the study of the interplay between localization, chaos, interactions, and transport [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. In any context, the quantum boomerang effect contravenes classical predictions and serves as a sensitive diagnostic of the quantum nature of localization, with the potential, for example, to map mobility edges [13, 31].

We report the experimental observation and characterization of the quantum boomerang effect. Using a degenerate gas and a pair of phase-shifted optical lattices to create a kicked rotor with tunable initial conditions (Fig. 1), we observe the boomerang’s distinctive return dynamics. We experimentally characterize the dependence of the dynamics on symmetry properties, Floquet gauge, and initial state, and confirm that results agree with an analytical theory we develop in the appendices. Finally, we measure the controlled destruction of the boomerang effect by stochastic kicking. The atom-optics kicked-rotor realization of the quantum boomerang offers three key advantages: (i) finite-width quasimomentum distributions naturally achieve single-shot disorder averaging, (ii) working in momentum space rather than position space enables simple observation of boomerang dynamics with single-site resolution, and (iii) symmetry properties can be straightforwardly tuned by adjusting properties of the quench [32, 33, 34].

We begin each experiment by preparing a Bose-Einstein condensate (BEC) of $10^{5}$ ⁷Li atoms and then tuning interatomic interactions to zero with a Feshbach resonance. A typical experimental sequence from this point on is depicted in Fig. 1a. The atoms are exposed in turn to two coaxial optical lattices with lattice constant $d=532\,\mathrm{nm}$, wave vector $k_{\mathrm{L}}=\pi/d$, recoil energy $E_{\mathrm{R}}=\hbar^{2}k_{\mathrm{L}}^{2}/2M$, and tunable relative offset $x_{0}$. $M$ is the atomic mass. To prepare an initial wavefunction which is well-localized in position space within each lattice site, we adiabatically load the BEC into the first optical lattice over $50\,\mathrm{ms}$, reaching a final depth of $7\,E_{\mathrm{R}}$. At $t=0$, we suddenly turn off the first lattice and quench into a Hamiltonian created by repeated kicking with the second, spatially-shifted lattice:

As described by Eq. 1, the second, phase-shifted lattice is pulsed $n$ times to depth $V_{0}$ with pulse shape $f_{\tau}$, effective pulse width $\tau=500\,\mathrm{ns}$, and period $T$; the first pulse peaks a delay time $t=\alpha T$ after the quench. Having measured the finite time of this quench to be approximately $190\,\mathrm{ns}$, we define $t=0$ halfway through the quench and recognize a systematic $\pm 0.01$ uncertainty in the reported values of $\alpha$. Because of the adiabatic loading into the first lattice and the relative phase shift, the atomic ensemble starts out centered at a nonzero position given by $x_{0}$. This is analogous to the situation in a position-space quantum boomerang, where the initial state is centered at nonzero momentum.

To make concrete the connection to the kicked rotor we introduce dimensionless variables $P\equiv 2k_{\mathrm{L}}Tp/M$, $Q\equiv 2k_{\mathrm{L}}x$, and $\mathcal{T}\equiv t/T$ as well as dimensionless parameters $\mathchar 22\relax\mkern-9.0muk\equiv 8E_{\mathrm{R}}T/\hbar$, an effective Planck’s constant, and $K\equiv\mathchar 22\relax\mkern-9.0mukV_{0}\tau/2\hbar$, encoding kicking strength. Substituting these into Eq. 1, rescaling the Hamiltonian $\mathcal{H}_{\alpha}\equiv(4k_{\mathrm{L}}^{2}T^{2}/M)H_{\alpha}$, and taking the $\delta$-function limit of $f_{\tau}/\tau$, we recover the canonical QKR Hamiltonian $\mathcal{H}_{\alpha}=\frac{P^{2}}{2}+{K}\cos Q\sum_{n}\delta(\mathcal{T}-n-\alpha)$. This gives rise to a single-particle 1-cycle Floquet operator $U_{\alpha}=e^{-i(1-\alpha)P^{2}/2\mathchar 22\relax\mkern-9.0muk}e^{-iK\cos Q/\mathchar 22\relax\mkern-9.0muk}e^{-i\alpha P^{2}/2\mathchar 22\relax\mkern-9.0muk}$.

Figure 1b illustrates the main result of this work: we experimentally observe the departure from and return to zero average momentum which is the key signature of the quantum boomerang effect [13]. Here $V_{0}=41\,E_{\mathrm{R}}$, $T=8\,\upmu$s, $\phi\equiv 2k_{\mathrm{L}}x_{0}=0.9\pi/2$, and $\alpha=0.48$, which correspond to $\mathchar 22\relax\mkern-9.0muk=10$, $K=9$. While the first kick imparts momentum asymmetry due to the lattice phase offset, the system returns to zero average momentum at long times. The experimental result agrees well with fit-parameter-free time-dependent Schrödinger equation (TDSE) simulations of the finite-pulse width Hamiltonian (1) using a split-step method described in appendix D.

This demonstration of the quantum boomerang effect enables an in-depth experimental characterization of the circumstances under which it can occur. Our measurements reveal a rich interplay among symmetries and quench dynamics which is well described by the analytic theory we present in the appendices. Specifically, it is the relation between the symmetry properties of the initial state and the Floquet gauge $\alpha$ defined by the $t=0$ quench which dictates the presence or absence of the quantum boomerang effect in the quantum kicked rotor. This interplay offers a contrast to static disordered systems, in which initial state symmetries are also important but which lack a Floquet gauge degree of freedom. We note for clarity that the QKR Hamiltonian itself is of course time-reversal and parity symmetric independent of Floquet gauge and spatial phase choices, so long as one references appropriate symmetry axes. For the remainder of the paper, we will solely refer to time-reversal and parity symmetry with respect to inversions about $t=0$ and $x=0$ unless otherwise stated. $\mathcal{H}_{\alpha}$ is then clearly parity symmetric, and only when $\alpha=1/2$ does the evolution also become time-reversal invariant. As we show in Appendix A, the initial Bloch state ensemble we experimentally prepare satisfies the exact properties necessary to observe a boomerang effect under these conditions: it is (i) parity asymmetric to allow for nontrivial early-time momentum dynamics and (ii) time-reversal symmetric ($\psi^{*}(x)=\psi(x)$) to ensure a complete return to 0 average momentum at late-times. This interplay of symmetries arising from the QKR quench protocol enables the quantum boomerang effect to be switched on or off by adjusting the Floquet gauge.

Fig. 2 demonstrates this dependence on Floquet gauge with a detailed characterization of the average momentum evolution for various $\alpha$. Measurements at $\alpha$ values of 0.29, 0.48, and 0.73 (Fig. 2a-c) exhibit fundamental differences in the dynamical behavior. As discussed above, only near $\alpha=1/2$ do we observe the quantum boomerang effect: a sharp initial increase in $\langle p\rangle$ followed by a decline back to the initial value of 0. Resolution of the precise boomerang condition is limited experimentally by timing noise on the location of the first pulse and the finite quench time of the acousto-optic modulators. In contrast, evolution under the other two Floquet gauges converges to nonzero momentum values at long times, indicating the absence of the full quantum boomerang effect. Each data set agrees well with fit-parameter-free numerical predictions, and all late-time momenta match the analytical predictions of the Floquet diagonal ensemble (Eq. 6).

Fig. 2d shows that the observed energy evolution is qualitatively consistent for all three values of $\alpha$, with each ensemble exhibiting dynamical localization as expected for a non-interacting quantum kicked rotor [35, 18, 24]. We note that the energy for $\alpha=0.48$ appears to saturate at a slightly higher value than the other gauges, a result consistent with the predictions of Ref. [15].

A key feature of quantum boomerang dynamics which can be probed with a kicked rotor is the contrast between short- and long-time evolution. Fig. 2e maps out the Floquet gauge dependence of momentum in these two regimes. Since the energy saturates after only five kicks, we take 25 kicks as a reasonable proxy for the infinite-time limit predicted by relaxation to the Floquet diagonal ensemble (see Eq. 6). The initial momentum asymmetry imparted to the rotor after one kick arises purely from the Talbot effect, a periodic re-imaging of the wavefunction in time under free-evolution as a result of the spatially periodic lattice exciting only quantized momentum modes; this Talbot effect is intimately related to the extensively explored quantum resonance phenomena in the QKR [36, 37, 38, 39]. The Talbot oscillation for a Bloch state is illustrated in Fig. 3d, where for our experiment, $T_{\mathrm{Talbot}}=\pi\hbar/2E_{\mathrm{R}}\approx 9.95\,\upmu$s.

In contrast, the long-time behavior is governed by localization physics and symmetries surrounding the quench. One feature of both experimental and theoretical results in Fig. 2e is that the long-time momentum should be an odd function about $\alpha=1/2$, a result we prove in appendix C. This result is connected to the resemblance between forward time-evolution in the gauge $\alpha$ (propagation with $U_{\alpha}$) and backward time-evolution in the gauge $1-\alpha$ (propagation with $U_{1-\alpha}^{\dagger}$). The anti-symmetry about $\alpha=1/2$ is dictated by the time-reversal invariance of the initial Bloch state ensemble. We note that because this property arises due to similarities between forward evolution in one gauge and backward evolution in the other, it only manifests as an (anti-)symmetry in the forward evolution of both gauges in the long-time limit (see Eq. 7); at short times, no symmetry is generically expected, as evidenced by the 1-kick data.

The odd parity of the late-time momentum about $\alpha=0.5$ is further reflected in Fig. 2f, which shows the late-time momentum as a function of initial wavepacket position for three different values of $\alpha$. Here a shift in the relative lattice phase is exactly analogous to variation in the initial launch velocity of a position space boomerang. We observe that regardless of initial position, $\alpha$ values near 0 and 1 result in opposite-sign, non-zero late-time momenta. The only exception occurs when the initial state recovers parity symmetry at $\phi=0$ or $\phi=\pi$ and the average momentum dynamics become trivial. Near $\alpha=1/2$, the entire curve collapses toward 0. This indicates that, as mentioned above, it is time-reversal symmetry which plays the fundamental role in determining the return of the boomerang. The initial position which controls the degree of parity asymmetry is instead more closely related to the amplitude of the early-time boomerang dynamics.

The demonstrated control of symmetry properties by varying $\alpha$ opens up the possibility of realizing quantum boomerang dynamics starting from more exotic initial states. Fig. 3 explores the quench dynamics of an initial state with an essentially homogeneous distribution in position space, achieved by replacing the adiabatic lattice load with a single kick. The initial wavefunction, approximately $\psi(x)=e^{-iV\sin(2k_{\mathrm{L}}x)}$ with $V$ the effective kick impulse, is neither time-reversal symmetric nor well-localized in position. Intuitively one would not expect such a state to demonstrate quantum boomerang dynamics, as our understanding of the position-space quantum boomerang effect is based on a well-defined initial velocity. Despite these expectations, the data in Fig. 3a reveal a clear quantum boomerang effect, though at a Floquet gauge $\alpha=0.86$, very different from the value of 0.5 expected for a time-reversal symmetric state. This can be understood by observing that free evolution maps this initial state into a time-reversal but not parity invariant wavefunction after a time $T_{\mathrm{Talbot}}/4$, as illustrated numerically in Fig. 3c and proven analytically in Appendix B. Thus, one expects to observe the boomerang effect at $\alpha=1/2+T_{\mathrm{Talbot}}/4T\simeq 0.855$ for the kicked initial state, in good agreement with numerical and experimental results in Fig. 3. This finding demonstrates that even for quenched states completely unlike traditional boomerang starting points, the Floquet gauge freedom can restore the boomerang effect in certain cases where the Talbot effect defines a new time-inversion symmetric point.

We note that while the kicked state is not initially time-reversal symmetric, it is invariant under simultaneous parity-time inversion ($\psi^{*}(-x)=\psi(x)$). As we show in Appendix C, this different symmetry condition manifests in the late-time momentum exhibiting even rather than odd symmetry about $\alpha=1/2$. These different Floquet gauge properties for the two initial state symmetries we consider are demonstrated and contrasted in Fig. 3b, which clearly distinguishes the different parity about $\alpha=0.5$ (see also data for the adiabatically loaded initial condition plotted for a broader range of $\alpha$ in Fig. 2e). We note that the data are not perfectly symmetric on the appropriate intervals, as demonstrated by comparison to the Floquet diagonal ensemble predictions. We attribute this to the fact that the symmetry is only established in the infinite-time limit, with deviations present at any finite time. Nevertheless, such deviations from the long-time limit are small, and our finite-time observations clearly reflect the idealized theoretical predictions. For the kicked state data shown, the symmetry is most apparent when extending to cases where $\alpha$ is slightly greater than 1 ¹¹1 The experimental delay-time definition and Floquet gauge definition of $\alpha$ is only consistent on $\left[0,1\right]$ . For $\alpha>1$ data and simulations shown, we define $\alpha$ solely by the delay-time definition. To make the Floquet gauge definition consistent with this, one should redefine the quenched initial state as $e^{-i\left\lfloor\alpha\right\rfloor P^{2}/2\mathchar 22\relax\mkern-9.0muk}\ket{\psi_{0}}$, with $\ket{\psi_{0}}$ the original quenched state for $\absolutevalue{\alpha}\leq 1$ cases, $\left\lfloor\alpha\right\rfloor$ denoting the integer part of $\alpha$, and taking the Floquet gauge to be $\alpha\bmod 1$. .

The final set of results we describe (Fig. 4) probes the question: what happens to quantum boomerang dynamics as localization is destroyed? In the QKR context, dynamical localization can be tunably disrupted by introducing temporal stochasticity into the kicking sequence [41, 24]. To implement this, we add a deviation $\delta_{n}$ to the delay time after the $n^{\mathrm{th}}$ kick, where $\delta_{n}$ is randomly selected from a uniform distribution in the range $\left[-W/2,W/2\right]$. The parameter $\Delta=W/T$ quantifies the degree of randomness and can range from 0 to 1. Fig. 4a shows the time evolution of average momentum for $\Delta=0$ (no randomness) and $\Delta=0.6$, with the first kick fixed so that $\alpha=0.52$, and with the same experimental parameters as in Fig. 2b. In contrast to the characteristic boomerang-like evolution when $\Delta=0$, each of the five experimental realizations of randomness for $\Delta=0.6$ exhibits a different behavior. The mean momentum over these disorder realizations demonstrates the destruction of the quantum boomerang effect, with a long-time average momentum significantly different from zero, and agrees reasonably well with the result of disorder-averaged simulations. Fig. 4b shows the evolution of the average energy for the same experiments: the destruction of dynamical localization by random kicking and the approach to classical linear-time energy growth are clearly visible in both experiment and theory, and are associated with the destruction of the quantum boomerang effect in Fig. 4a. To directly probe the dependence of the quantum boomerang on the disorder strength, we plot in Fig. 4c the disorder-averaged mean momentum after 25 kicks as a function of $\Delta$. Measured long-time $\langle p\rangle$ increases monotonically up to $\Delta\approx 0.5$ and agrees reasonably well with disorder-averaged numerical predictions. An interesting topic for further study is the relationship between the degree of destruction of the boomerang effect (measured, for example, by long-time average momentum) and the exponent characterizing energy growth [42, 43, 44].

In this work we have presented the first experimental evidence for the quantum boomerang effect in an Anderson-localized system. The measurements confirm theoretical predictions [13, 15] and demonstrate that boomerang dynamics can occur from a variety of initial conditions including even a homogeneous density distribution. These experimental results, along with the analytical theory we present in the appendices, elucidate the crucial role of parity and time-reversal symmetry in determining the presence or absence of the quantum boomerang effect. We have probed the dependence of boomerang dynamics upon Anderson localization itself by measuring the effect of tunably random kicking sequences: results confirm that classical diffusion induced by stochasticity destroys both dynamical localization and the quantum boomerang effect. These experiments validate a new, powerful probe of the uniquely quantum-mechanical nature of localization applicable to a general class of disordered systems and suggest a variety of intriguing topics for future exploration. These include boomerang phenomena in higher-dimensional systems, different symmetry classes [45], more exotic initial states, and the presence or absence of many-body boomerang effects in interacting systems [14] or even many-body localized states. Applications of our demonstration — that adjusting the Floquet gauge can switch on or off the boomerang effect — are an intriguing possibility. Beyond the context of the quantum kicked rotor, the investigation of boomerang-like phenomena in condensed matter systems and implications of these results for ultrafast electron dynamics in disordered solids represent unexplored frontiers.