Three-Dimensional Localized-Delocalized Anderson Transition in the Time Domain

Dominique Delande, Luis Morales-Molina, and Krzysztof Sacha

Phys. Rev. Lett. 119, 230404 – Published 6 December 2017

Abstract

Systems which can spontaneously reveal periodic evolution are dubbed time crystals. This is in analogy with space crystals that display periodic behavior in configuration space. While space crystals are modeled with the help of space periodic potentials, crystalline phenomena in time can be modeled by periodically driven systems. Disorder in the periodic driving can lead to Anderson localization in time: the probability for detecting a system at a fixed point of configuration space becomes exponentially localized around a certain moment in time. We here show that a three-dimensional system exposed to a properly disordered pseudoperiodic driving may display a localized-delocalized Anderson transition in the time domain, in strong analogy with the usual three-dimensional Anderson transition in disordered systems. Such a transition could be experimentally observed with ultracold atomic gases.

Received 17 February 2017

DOI:https://doi.org/10.1103/PhysRevLett.119.230404

Physics Subject Headings (PhySH)

Anderson localization Atom interferometry Cold gases in optical lattices Crystallization Quantum interference effects

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Dominique Delande¹, Luis Morales-Molina², and Krzysztof Sacha^3,4

¹Laboratoire Kastler Brossel, UPMC-Sorbonne Universités, CNRS, ENS-PSL Research University, Collège de France, 4 Place Jussieu, 75005 Paris, France
²Instituto de Física, Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile
³Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11, 30-348 Kraków, Poland
⁴Mark Kac Complex Systems Research Center, Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11, 30-348 Kraków, Poland

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Vol. 119, Iss. 23 — 8 December 2017

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Images

Figure 1
Numerical determination of the mobility edge for $V_{0} = E_{σ}$ . For a given energy, we compute the localization length $λ_{M}$ of a long bar-shaped grid with square section $M \times M$ . (a) Each curve is computed for a single $M$ value from 25 to 55 with step 5 (slope increases with $M$ ) and shows $λ_{M} / M$ vs energy. The various curves cross at the position of the mobility edge $E_{c} / E_{σ} = 0.032 \pm 0.002$ . (b) The localization length $ξ$ vs energy, in the localized regime. It shows an algebraic divergence near the mobility edge, indicated by the red dashed line.
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Figure 2
Spatial probability density for a typical localized eigenstate of Hamiltonian (3) with correlation length of the disorder $σ = 0.1$ , for $V_{0} = E_{σ}$ , and energy $E / E_{σ} = - 0.049 968$ . The upper color 3D plot shows the disordered yet localized character of the state below the mobility edge. Lower plots show how probability densities at a fixed position in $θ$ , $ϕ$ , or $ψ$ in the laboratory frame (integrated along two remaining directions) evolve in time. The semilogarithmic scale indicates an approximate exponential localization. In the rotating frame, the localization length $≃ 0.4 = 4 σ$ is in good agreement with the prediction of the transfer matrix calculation, $ξ = 4.3 σ$ in Fig. 1. It implies that in the laboratory frame the localization length in time reads, e.g., $4 σ / ω_{1}$ if the probability density is integrated over $ψ$ and $ϕ$ .
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Figure 3
Experimental proposal. (a) The red dashed line shows the shape of a sawtooth potential while the black solid line its approximation built with the first three spatial harmonics only. The latter can be created by means of an optical standing wave with wavelength $λ$ and its first two harmonics. (b) Temporal modulation function $f_{1} (t)$ that consists of three harmonics with random phases and with the amplitude $| g_{k} f_{k}^{(1)} | = (1 / \sqrt{2 k_{0}})$ for $| k | \leq k_{0} = 3$ . (c) Schematic plot of the initial stage of the experiment: ultracold bosonic atoms are prepared in a strong optical lattice and shallow trapping potentials. We assume that for the amplitude of the lattice potential of the order of $30 E_{rec}$ (atomic recoil energy), slices of the atomic cloud are formed that consist of well-defined numbers of particles and do not have mutual phase coherence. The average kinetic energy along the $z$ direction is $⟨ p_{0}^{2} ⟩ / 2 m \approx 2 E_{rec}$ . (d) Final stage of the experiment: for a perturbation amplitude $V_{0} = 40 E_{rec}$ , atoms accelerated to the average momentum $⟨ p ⟩ = m ω_{1} λ / 4 π$ will fly over the time-modulated sawtooth potential and do not spread along $z$ due to the predicted Anderson localization. The predictions are valid provided $ω_{1} \geq 400 E_{rec} / ℏ$ .
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