Families of localized modes of Bose-Einstein condensates enabled by incommensurate optical lattice and photon-atom interactions

TL;DR

This work shows that a Bose-Einstein condensate inside a 1D optical cavity, subject to mutually incommensurate external and cavity-induced potentials, supports a rich set of nonlinear localized modes even in the absence of two-body interactions. Using a mean-field model with coupled equations for the condensate and the cavity field, the authors map multiple families of localized modes across blue, red, and mixed detunings, reveal mobility edges and pseudo-degeneracies, and analyze stability. Two distinct bistability mechanisms are identified: coexistence of multiple families and hysteresis-like multivalued $A(\mathcal{N})$ within a single family, both enabled by long-range photon-mediated interactions. Dynamics simulations demonstrate excitation of stable localized modes and a cavity-mediated XOR-like gate when two separated modes compete, highlighting potential applications in quantum information processing with driven-dissipative ultracold gases.

Abstract

We consider a Bose-Einstein condensate (BEC) loaded into a one-dimensional optical cavity under the combined action of an external potential and atom-cavity coupling with mutually incommensurate periods. Such configuration enables the localization of matter waves even in the absence of two-body interactions. We study families of localized modes within the mean-field approximation for red and blue detunings from atomic and cavity resonances in relatively shallow quasiperiodic lattices, beyond the validity of the tight-binding approximation. The parameter regimes supporting localization of atomic wave packets are identified. The system exhibits two types of bistability manifested as distinct photon numbers under otherwise identical conditions. One type arises from the coexistence of multiple families of localized modes, typical of conservative nonlinear systems, while the other stems from the multivalued dependence of the families on system parameters, characteristic of systems exhibiting hysteresis. BEC in a cavity may also display pseudodegeneracy, understood as the existence of two distinct atomic-density distributions corresponding to the same atomic and photon numbers (although different chemical potentials). The stability of the localized modes is analyzed. It is shown that, owing to the strong impact of long-range interactions on stability, a two-localized-mode configuration can operate as an XOR logic gate.

Figures (5)

• Figure 1: (a) Lowest 88 families of stationary modes for $\sigma=+1$ and $\Delta=150$ on the diagram $(A,\mathcal{N})$. (b) Examples of three families from the bundle shown in (a) (note that the range of $A$ is changed). The crosses indicate crossing of the families. The colored circles, labeled "A" through "D", indicate specific modes which are used as examples below. (c) The families of the modes shown in (b) but now using the alternative parametrization $\mathcal{N}$vs$\mu$. In (a), (b) and (c) $\tilde{\eta}=141$. In (b) and (c) solid and dashed lines correspond to intervals of dynamical stability and instability. (d) $\mathcal{N}(\mu)$ of the family $n=1$, for three different values of the driving amplitude $\tilde{\eta}=100,122,141$. Here, and in all figures below, the degree of localization is encoded by the color scale indicated in the IPR color-bar.
• Figure 2: Distributions of the localized modes for $\sigma=+1$, $\Delta=150$ and $\tilde{\eta}=141$, with $\mathcal{N}\approx7\times10^2$ on the diagram $(X/L,A)$ (the leftmost panel). The panels on the right show the wavefunctions of the modes, labeled A and B in the left panel and in Figs. \ref{['fig1']} (b) and (c).
• Figure 3: (a) Families of the cavity modes $A(\mathcal{N})$ with intervals of localized solutions for $\sigma=-1$, $\Delta=-150$, and $\tilde{\eta}=141$. Two families (discussed in the text) are highlighted by color specifying IPR, while the rest of the families are shown by light gray dashed lines. The right inset shows $A(\mathcal{N})$ for $n=1$ family for $\tilde{\eta}=103,121,141$. The lower inset magnifies the intersection between families $n=1$ and $n=36$. (b) Number of atoms $\mathcal{N}$vs chemical potential $\mu$ for the chosen families (in color, the rest of the families are shown by light-gray dashed lines). Solid (dashed) lines indicate stable (unstable) intervals.
• Figure 4: Cavity field intensity $A$vs the number of atoms $\mathcal{N}$ for $\sigma=+1$ and $\Delta=-150$ in (a) and for $\sigma=-1$ and $\Delta=150$ in (c). For clarity three families are highlighted by color indication of the IPR, while the remaining 85 families in each bundle are shown by gray color. In (a) and (c), $\tilde{\eta}=364$, and the bottom left insets show zoomed-in regions of their respective panel. In the insets in top right corner of (a) and (c), the respective family bifurcating from the lowest linear states, $n=1$, is shown for three different driving amplitudes, $\tilde{\eta}=264,312,364$. Panels (b) and (d) show the number of atoms ${\cal N}$vs the chemical potential $\mu$ for the same parameters and families shown in panels (a) and (c), respectively. Note that in (c) all families are very close to each other (two of them are zoomed in the left inset). In all panels $\kappa=63$.
• Figure 5: (a) Evolution of the (numerically) exact unstable state C (see Fig. \ref{['fig1']}), for $\sigma=+1$, $\Delta=150$ and $\tilde{\eta}=141$ ($\eta\approx200$), with $\alpha(0)=0.995\alpha_{\rm st}$, where $\alpha_{\rm st}\approx1.92-1.55i$ ($A\approx3.03$) is the amplitude of the cavity mode corresponding to C, and to ${\cal N}\approx 990$. (b) Evolution of a Gaussian wave-packet (\ref{['gaussian']}), with the center of mass and variance equal to those of the mode D (see Fig. \ref{['fig1']}(b)), excited with an initial cavity amplitude equal to the stationary value $\alpha_{\rm st}$ corresponding to mode D. (c) The resulting spatial profile at the end of simulation at $t\approx 95$ (black solid line), over the target mode D (red solid line), and the initial Gaussian spatial profile (gray dashed line). (d) Two stationary modes, from families $n=1$ and $n=44$, with the same stationary cavity amplitude, $\alpha_{\rm st}$, are excited at the same time and for an initial $\alpha(t=0)=\alpha_{\rm st}$.