Fragmented Superradiance of a Bose-Einstein Condensate in an Optical Cavity

The Dicke model and the superradiance of two-level systems in a radiation field have many applications. Recently, a Dicke quantum phase transition has been realized with a Bose-Einstein condensate in a cavity. We numerically solve the many-body Schrödinger equation and study correlations in the ground state of interacting bosons in a cavity as a function of the strength of a driving laser. Beyond a critical strength, the bosons occupy multiple modes macroscopically while remaining superradiant. This fragmented superradiance can be detected by analyzing the variance of single-shot measurements.

Figure 1

Setup for the Dicke quantum phase transition with a Bose-Einstein condensate in an optical cavity. For a cavity pump power $\mathrm{\Omega }$ that is smaller than the critical pump power ${\mathrm{\Omega }}_{\mathrm{cr}}$, the Bose-Einstein condensate (blue) in the cavity formed by the left and right mirrors (gray) is unaffected by the pump laser and no population of cavity photons (red) is built up (left panel). For pump powers $\mathrm{\Omega }$ larger than the critical value ${\mathrm{\Omega }}_{\mathrm{cr}}$, the Bose-Einstein condensate self-organizes as a consequence of the potential which is built up by the population of the cavity with photons (right panel).

Figure 2

Self-organization of the ground state of a Bose-Einstein condensate. The density $\rho (x)$ of the atoms and the potential $V(x)=V_{1\text{−}\mathrm{body}}(x)+V_{\text{cavity}}(x)$ is shown as a function of the cavity pump power in the upper and lower panels, respectively. Once the applied pump power exceeds a critical value, the atoms self-organize because the field which is built up inside the cavity creates a periodic one-body potential (cf. lower and upper panels). As a result of a competition between external and cavity potential as well as interactions, the sign of the cavity amplitude switches one time for $\mathrm{\Omega }\approx 25$ [compare pattern in density and potential with the inset of Fig. 3]. All quantities shown are dimensionless, see text for further discussion.

Figure 3

(a) Buildup of cavity population in the self-organization process. Beyond the critical value ${\mathrm{\Omega }}_{\mathrm{cr}}\approx 12$ the population of photons in the cavity grows with increasing pump power $\mathrm{\Omega }$. The cavity population $|\alpha {|}^2$ drives the self-organization of the atoms, see Eq. (3). Inset: The amplitude $\alpha$ shows a sign change from a positive to a negative value in the superradiant phase at $\mathrm{\Omega }\approx 25$ and continues to decrease monotonically in the fragmented superradiant phase. (b) Fraction $F$ of atoms that are outside the natural orbital with the largest occupation, (c) natural orbital occupations, and (d) variance $\mathcal{V}$ in single shots of the momentum density (1000 samples per data point), all as a function of the pump power $\mathrm{\Omega }$. The variance $\mathcal{V}$ maps the fraction $F$ of atoms in excited orbitals closely, compare panels (b) and (d). The natural occupations (c) show that the dips in the excited fraction $F$ and the variance are due to a redistribution of atoms between excited orbitals. The inset of (c) demonstrates that fragmented superradiance emerges for a range of particle numbers $N$ with a finite-size scaling for $\mathrm{\Omega }=80$. The emergence of fragmentation and the growth of $F$ signals that the system enters a new phase with many-body correlations between the atoms. This is confirmed by comparing (b) and (c) with Figs. 4. All quantities shown are dimensionless, see text.

Figure 4

Tracing the transition from superradiance to fragmented superradiance in the spatial correlation function. The correlation function $|g^{(1)}(x,x^{\prime }){|}^2$ in the superradiant phase is shown for pump powers (a) $\mathrm{\Omega }=30$, (b) $\mathrm{\Omega }=40$, and (c) $\mathrm{\Omega }=100$, wherever the density exceeds 0.001. With increasing pump power, the atoms in distinct wells gradually loose coherence [$|g^{(1)}(x,x^{\prime }\ne x){|}^2$ drops to zero], compare the position of black squares with the upper panel of Fig. 2. The structure of $|g^{(1)}{|}^2$ in (c) demonstrates the threefold fragmentation in the system. All quantities shown are dimensionless.