Steering Zitterbewegung in driven Dirac systems: From persistent modes to echoes

Phillipp Reck, Cosimo Gorini, and Klaus Richter

Phys. Rev. B 101, 094306 – Published 13 March 2020

Abstract

Although Zitterbewegung—the jittery motion of relativistic particles—was known since 1930 and was predicted in solid-state systems long ago, it has been directly measured so far only in so-called quantum simulators, i.e., quantum systems under strong control, such as trapped ions and Bose-Einstein condensates. A reason for the lack of further experimental evidence is the transient nature of wave-packet Zitterbewegung. Here, we study how the jittery motion can be manipulated in Dirac systems via time-dependent potentials with the goal of slowing down/preventing its decay or of generating its revival. For the harmonic driving of a mass term, we find persistent Zitterbewegung modes in pristine, i.e., scattering free, systems. Furthermore, an effective time-reversal protocol—the “Dirac quantum time mirror”—is shown to retrieve Zitterbewegung through echoes.

Received 23 October 2019
Revised 20 February 2020
Accepted 21 February 2020

DOI:https://doi.org/10.1103/PhysRevB.101.094306

Physics Subject Headings (PhySH)

Carrier dynamics Quantum interference effects Quantum transport

Rotating wave approximation

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Phillipp Reck , Cosimo Gorini , and Klaus Richter^*

Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany

^*klaus.richter@physik.uni-regensburg.de

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Vol. 101, Iss. 9 — 1 March 2020

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Figure 1
Frequencies of a driven ZB. (a) The parallel ZB $〈 v_{∥} (t) 〉$ is shown both as a function of time (black) and as its Fourier transform (blue), i.e., as function of $ω$ for $ω_{D} = 2 M_{0} / ℏ$ and $\tilde{M} = 1.2 M_{0}$ . The inset shows a closeup for longer times. Peaks appear in the Fourier transform at multiple integers of $ω_{D}$ with satellite peaks at a distance of $ω_{R}$ (orange line at the first peak) away from the major peaks. (b) and (c) Fourier transform of $〈 v_{∥} 〉$ as a function of driving frequency $ω_{D}$ (for fixed $\tilde{M} = 0.5 M_{0}$ ) and amplitude $\tilde{M}$ (for fixed $ω_{D} = 2 M_{0} / ℏ$ ) of the time-dependent mass term, respectively. The blue vertical line in (c) marks the function shown in panel (a) (indicated by the blue arrow). The expected frequencies from the RWA $ω_{D}, ω_{D} \pm ω_{R}$ , and $ω_{R}$ are shown (red, solid line) as well as higher-order terms in $ω_{D}$ (red, dashed line). For smaller $\tilde{M}$ , up to $\tilde{M} \approx 1.3 M_{0}$ , the RWA results Eq. (20) are well recovered whereas for larger $\tilde{M}$ , the RWA is not justified anymore. In all plots, we choose $k_{0}$ such that $ɛ_{+} (k_{0}) = 0.4 M_{0}$ , and, thus, the ZB frequency corresponding to the static $H_{0}$ is $Ω_{k_{0}}^{st} \approx 2.15 M_{0} / ℏ$ .
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Figure 2
Searching for infinitely long surviving modes of the Zitterbewegung. The left panels (a) and (c) show plots of all appearing ZB modes whereas the right panels (b) and (d) show the modes which survive a long time (as defined in the text). The modes of the perpendicular and parallel ZBs $〈 v_{⊥} (ω) 〉$ (upper panels) and $〈 v_{∥} (ω) 〉$ (lower panels) are shown as a function either of the parameter $ω_{D}$ or of $\tilde{M}$ . Analytically expected modes from the RWA (red, dashed lines) are indicated in the left panels. In general, the surviving modes for long times are the ones not depending on $k$ , e.g., multiples of $ω_{D}$ , or weakly $k$ dependent where $\partial ω_{R} / \partial k = 0$ [indicated by the green dashed lines in (b)].
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Figure 3
Retrieving decaying Zitterbewegung through a spin-echo-type time-reversal mechanism. Left panel: An initial wave packet with components in both bands separates due to the different band velocities $\propto \nabla_{k} ɛ (k)$ . The decreasing overlap between the subwave packets causes the ZB to decay [(A) to (B)], see the time line of the velocity expectation value $〈 v_{y} 〉$ in the right panel. Via a short QTM pulse at $t_{0}$ , the subwave packets switch bands, thus, reversing their velocities and returning to their initial positions [see (C)]. The recovery of the overlap in (D) leads to a ZB revival. The black dots in the right panel belong to the time instants sketched in the left panel.
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Figure 4
Comparison between ZB echo simulations and analytical estimates for a gapless [panels (a), (b)] and a gapped [panel (c)] Dirac system. In panel (a), both the velocity expectation value and the correlation $C^{2} (t)$ , Eq. (28), are shown as functions of time, where $C^{2} (t)$ and the ZB amplitude coincide at $t_{echo} ≃ 2 t_{0}$ , in agreement with Eq. (38). Panel (b) compares the echo strength $C (t_{echo}) \approx | A (k_{0}, Δ t) |$ (see Ref. [29]), to the relative amplitude of the revived ZB both in the $x$ - and the $y$ -directions as a function of $Δ t$ for fixed $κ_{0} = ℏ v_{F} k_{0} / M = 0.4$ . The expected agreement can be seen. Panel (c) shows analogous data for a gapped Dirac system, where $Δ t = 1.4 ℏ / M$ whereas the mean wave vector of the wave packet, i.e., $κ_{0}$ , is varied.
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Figure 5
Analysis explaining why the modes at the crossing ( $ω \approx n ω_{D}$ ) survive in Fig. 2. Here, the parallel velocity $| 〈 v_{∥} (ω) 〉 |^{2}$ is shown for $M_{0} = 0$ as a function of $\tilde{M}$ for $ω_{D} = 1.5 \tilde{ω}$ with the normalization frequency $\tilde{ω} = 2.5 v_{F} k_{0}$ for two different energies with $k = 0.36 \tilde{ω} / v_{F}$ [in (a) and (c)] and $k = 0.44 \tilde{ω} / v_{F}$ [in (b) and (d)]. The frequency of the ZB is quite independent at the crossings, i.e., (a) and (b) are very similar, whereas away from the crossing, it is energy dependent, i.e., (c) and (d) differ. Since only energy-independent modes survive as discussed in the main text, only modes at the crossings can be seen after at late times [Fig. 2].
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Figure 6
ZB echoes in the presence of disorder. (a) The time-dependent revival strength ${\tilde{v}}_{x} = B_{x}^{revival} / B_{x}^{initial}$ is shown as a function of the pulse time $t_{0}$ for different disorder strengths ranging from $u_{0} = 0.002 M$ to $0.014 M$ . An exponential decay can be seen in the weak disorder regime. (b) The decay rate $1 / τ_{0}$ is extracted by a fit [red lines in (a)] and plotted as a function of $u_{0}$ and compared to the analytically expected scattering time (black dotted) from Eq. (E3). The quadratic fit in $u_{0}$ (blue line) to the data points is close to the expected scattering time. Saturation is obtained for larger $u_{0}$ . For more details, see the discussion of disorder in Ref. [29].
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