Dynamics and fragmentation of bosons in an optical lattice inside a cavity using Wannier and position bases

TL;DR

This work addresses how representing the atomic sector in a 1D atom-cavity system with a static lattice affects the predicted dynamics and fragmentation. It compares a single-band Wannier basis with a position basis, using mean-field theory and truncated Wigner approximation to study static and dynamical phases and fragmentation. Key findings show agreement on SF, DW, LC, and ID across bases, but the Wannier basis misses a large irregular-dynamics sector and yields a distinct LC, while photon-mediated interactions drive BEC fragmentation even without native short-range interactions, with stronger fragmentation for Ua>0 and larger system sizes. The results emphasize the need to include higher bands for multimode phenomena and guide modeling choices in dissipative quantum gases in optical cavities.

Abstract

The atom-cavity system is a versatile platform for emulating light-matter systems and realizing dissipation-induced phases, such as limit cycles (LCs) and time crystals. Here, we study the dynamics of a Bose-Einstein condensate (BEC) inside an optical cavity with transverse pumping and an additional intracavity optical lattice along the cavity axis. Specifically, we explore the theoretical predictions obtained from expanding the atomic field operators of the second-quantized Hamiltonian in two ways: (i) position basis and (ii) single-band Wannier basis. Both bases agree on the existence of most types of static and dynamical phases. However, the large sea of irregular dynamical phase, captured within the position basis, is absent in the Wannier basis. Moreover, we show that they predict different types of LCs due to the inherent limitation of the single-band Wannier expansion, highlighting the importance of including higher energy bands to correctly capture certain phenomena. Using truncated Wigner approximation, we investigate the fragmentation dynamics of the BEC. We demonstrate that both position and Wannier bases qualitatively agree on the photon-mediated fragmentation dynamics of the BEC in the density-wave phase, despite the absence of interatomic interactions. The presence of interatomic interaction leads to further fragmentation, which can only be observed in larger system sizes. Finally, we predict a sudden increase in the fragmentation behavior for larger pump intensities.

Figures (20)

• Figure 1: (a) Schematic of the atom-cavity setup with an optical lattice. Ultracold atoms are placed inside a high-finesse optical cavity operating in the good-cavity regime, and are driven using a pump field on the $y$-$z$ plane. The cavity axis is aligned along the $z$-axis. (b) Illustration of the spatial discretization according to the position and (single-band) Wannier basis emphasizing the cavity potential $V(z)$ experienced by a boson. (c) The energies resolved by the (dashed) position and (solid) Wannier bases with $k_\text{c}$ the cavity-mode wave vector. The gray solid curve corresponds to the second band neglected in the standard single-band Wannier expansion.
• Figure 2: Hubbard parameters for the Wannier basis as a function of $V_0$, obtained from numerically obtained Wannier functions. These were obtained using $U_\text{a}/E_\text{rec}=10^{-3}$ and $U_0/\omega_\text{rec}=-10^{-3}$.
• Figure 3: The dynamics of the cavity-photon occupation $\left\lvert {a} \right\rvert^2$ in the SF, DW, LC, and ID phases obtained using the (a) (single-band) Wannier and (b) position basis. The LC phase is labeled as LC$_1$ and LC$_2$ in the Wannier and position basis. The long-time average $\overline{\left\lvert {a} \right\rvert^2_\text{S}}$, standard deviation $\sigma(\left\lvert {a} \right\rvert^2_\text{S})$, and spectral entropy $S$ of $\left\lvert {a} \right\rvert^2_\text{S}$ are used to distinguish between each phase, with (c),(d) showing their behavior in each phase as the pump intensity $V_0$ is increased in the Wannier and position basis, respectively. The black lines in (c),(d) correspond to the threshold spectral entropy values to distinguish the LC, $S_\text{LC}$ and ID, $S_\text{ID}>S_\text{LC}$, phases. Points below the vertical axis in (c),(d) correspond to values close to zero.
• Figure 4: Mean-field phase diagram obtained using the (a)(single-band) Wannier, and position bases in the (b) presence and (c) absence of the static optical lattice. The SF, DW, LC, and ID phases are observed, with LC$_1$ and LC$_2$ corresponding to different LC dynamics in each basis as shown in Fig. \ref{['fig:dynamics']}. The insets in (b),(c) show an enlarged view of the boxed region in the main plot. The dotted line marks the matter-wave superradiance regime. The circle, cross, triangle, and square markers mark the values of $\Delta_\text{c}$ and $V_0$ used to obtain Fig. \ref{['fig:no_dynamics']}.
• Figure 5: Dynamics of the (a),(c) cavity-photon occupation and (b),(d) single-particle atomic distribution according to the (a),(b) Wannier and (c),(d) position bases. For both bases, $\Delta_\text{c}/\omega_\text{rec}=-5.04$, with $V_0/E_\text{rec}=2.58$ in the Wannier basis, and $V_0/E_\text{rec}=0.216$ in the position basis.