Self-Adapted Floquet Dynamics of Ultracold Bosons in a Cavity

Xi-Wang Luo and Chuanwei Zhang

Phys. Rev. Lett. 120, 263202 – Published 29 June 2018

Abstract

Floquet dynamics of a quantum system subject to periodic modulations of system parameters provides a powerful tool for engineering new quantum matter with exotic properties. While system dynamics is significantly altered, the periodic modulation itself is usually induced externally and independent of Floquet dynamics. Here we propose a new type of Floquet physics for a Bose-Einstein condensate (BEC) subject to a shaken lattice generated inside a cavity, where the shaken lattice and atomic Floquet bands are mutually dependent, resulting in self-adapted Floquet dynamics. In particular, the shaken lattice induces Floquet quasienergy bands for the BEC, whose backaction leads to a self-adapted dynamical normal-superradiant phase transition for the shaken lattice. Such self-adapted Floquet dynamics shows two surprising and unique features: (i) The normal-superradiant phase transition possesses a hysteresis even without atom interactions. (ii) The dynamical atom-cavity steady state could exist at free energy maxima. The atom interactions strongly affect the phase transition of the BEC from zero to finite momenta. Our results provide a powerful platform for exploring self-adapted Floquet physics, which may open an avenue for engineering novel quantum materials.

Received 19 December 2017
Revised 4 May 2018

DOI:https://doi.org/10.1103/PhysRevLett.120.263202

Physics Subject Headings (PhySH)

Atomic & molecular processes in external fields Atoms, ions, & molecules in cavities Cold gases in optical lattices

Atomic, Molecular & OpticalGeneral Physics

Authors & Affiliations

Xi-Wang Luo and Chuanwei Zhang^*

Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080-3021, USA

^*chuanwei.zhang@utdallas.edu

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Vol. 120, Iss. 26 — 29 June 2018

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Images

Figure 1
(a) Schematic of the experimental setup. The cavity is pumped by an external transverse laser (red block arrow), with the cavity mode (red line) shaken by the two EOMs or PZT-driving mirrors. A BEC is prepared inside the cavity with a background lattice (blue line) generated by additional lasers (blue block arrows). The shaken cavity mode may induce interband couplings (dotted arrows) when the band gap matches the shaking frequency. (b) Energy levels of the atom and detunings of the cavity mode and pumping laser.
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Figure 2
(a) Illustration of atomic static bands (solid lines) and phonon-dressed sidebands (dashed lines). (b) Couplings induced by ${\hat{V}}_{c}$ in Eq. (2). The area of the Brillouin zone reduces in half by ${\hat{V}}_{c}$ . The time-independent (time-dependent) term induces intrasideband (intersideband) couplings, indicated by solid arrows (dashed arrows).
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Figure 3
(a), (b) Phase diagrams for dominant intrasideband coupling, with (a) ${\tilde{Δ}}_{c} = 1$ and (b) $η_{t} = 0.5$ . I (II) represents the normal (superradiant) phase. (c),(d) Phase diagrams for dominant intersideband coupling, with (c) ${\tilde{Δ}}_{c} = - 1$ and (d) $η_{t}^{- 1} = 0.1$ , $η_{0 t} = η_{0} η_{t}$ . III represents the hysteresis region. The insets in (a) and (c) show the $s_{0}$ -band dispersion in the normal (red dashed line) and superradiant phases (blue solid line). The three blue (red) dots in (b) [(d)] belong to different phases, and they mark the points where the free energy curves are plotted in the inset. The parameters are $v_{e} = - 6$ , $ω = 4.8$ , and $γ = 2$ (with energy unit $E_{R} = ℏ^{2} k_{0}^{2} / 2 m$ ), and atom-atom interaction is neglected.
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Figure 4
(a), (c) Superradiant order parameters and condensate momenta versus $η_{0}$ and $η_{0 t}$ . Stars and circles mark the transition points between zero and finite condensate momenta for $g = 0$ and $g > 0$ , respectively. The inset of (a) shows the band structure for $| q_{x}^{m} | > 0$ . A clear hysteresis loop appears in (c) with solid lines (dashed lines) corresponding to stable (unstable) steady-state solutions. (b), (d) Difference between $q_{x}^{c}$ (red dotted line) and $q_{x}^{m}$ (blue dashed line) for $g > 0$ . In the gray region, bosons are condensed into the local band maximum at $q_{x} = 0$ . The black solid line shows $q_{x}^{m} = q_{x}^{c}$ for $g = 0$ . (a), (b) [(c), (d)] correspond to dominant intrasideband (intersideband) couplings with $Δ_{c} = 1$ , $η_{t} = 0.5$ and $g \bar{n} = 0.02$ ( $Δ_{c} = - 1$ , $η_{t}^{- 1} = 0.15$ and $g \bar{n} = 0.1$ ). $\bar{n} = N_{a} / L$ is the average density, with $L$ as the system size. Other parameters are the same as in Fig. 3.
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