Control of matter-wave superradiance with a high-finesse ring cavity

We experimentally investigate light scattering from a Bose-Einstein condensate which is positioned inside an optical ring cavity. With a cavity linewidth at the recoil limit, the resulting superradiant scattering behavior is strongly influenced by the cavity detuning. The experimental observations are interpreted with a quantum mean-field description and with a rate-equation model. We discuss applications in the context of quantum phase transitions, Dicke subradiance, and nonconventional superfluidity.

Figure 1

Geometry of the experiment: A BEC is created in a Ioffe-Pritchard trap and then placed in the waist of a TEM${}_{11}$ mode of an optical ring cavity (green). A pump beam is irradiated from the side under the angle $\alpha ={37}^{\circ }$. A single-photon counter records the photons transmitted through one of the cavity mirrors.

Figure 2

Main signatures of the experiment. (a) A single-photon detector counts the photons which are scattered into the left running cavity mode and leak out through one of the high-reflectivity mirrors. The red (light gray) line indicates the power of the pump beam in arbitrary units. The blue (dark gray) spikes represent single-photon counts. (b) Atomic momentum distribution observed via time-of-flight absorption imaging. The populations in the individual momentum states are obtained by summing the pixel values in a fixed area (red rectangle).

Figure 3

Momentum space accessible to the atoms. An incident pump photon with wave vector $k_p$ is scattered by an atom into one of the two cavity modes with wave vectors $k_1$ or $k_2$. Due to the discrete momentum transfer during scattering the possible atomic momentum states $\left(m,n\right)$ form a rectangular lattice in momentum space. The blue (gray) numbers indicate the negative recoil frequency shift of the scattered photon in units of the two-photon recoil frequency of 14.4 kHz. The states with open circles are occupied if photons are scattered from the cavity into the pump beam.

Figure 4

Maximum possible population difference between the two momentum states with $m=1$, $n=0$ and $m=0$, $n=1$ for various pump laser detunings ${\Delta }_c-U_N$ and pump intensities. The solid (dashed) line shows the result of the mean-field (rate) model. Initially, 80 000 atoms are in the zero-momentum state with $m=0$, $n=0$. The simulated momentum space is in the range $-2⩽m⩽3$ and $-2⩽n⩽2$. To start the dynamics each finite-momentum state is initially populated with one atom. The initial number of photons in the resonator is set to zero. The dynamics is simulated for 10 ms after the pump beam has been turned on and the maximum population difference within this time interval is plotted. After 10 ms almost all atoms have scattered more than one photon, and only momentum states with $n,m>1$ are populated.

Figure 5

Time-of-flight absorption images for various pump laser detunings ${\Delta }_c$. The pump duration was $200\mu \mathrm{s}$ with a pump beam intensity of $I_p=50$ mW/cm${}^2$ and an atomic detuning of ${\Delta }_a=-2\pi \times 4.7\mathrm{GHz}$. The solid lines indicate the positions of states with $m=0$ and $n=0$.

Figure 6

Population of selected momentum states $(m,n)$ for various pump durations. The pump intensity amounts to $I=70\mathrm{mW}/{\mathrm{cm}}^2$, the detuning of the pump frequency from the atomic resonance is ${\Delta }_a=2\pi \times 4.3$ GHz, and the detuning from the cavity mode is ${\Delta }_c=2\pi \times 253$ kHz. The solid (dashed) line presents the prediction of the mean-field (rate) model for an initial atom number of $N=80000$ in the condensate. The finite-momentum states are initially occupied with $N_{\mathrm{th}}=1000$ and $N_{\mathrm{th}}=5$ thermal atoms distributed over the discrete-momentum states according to Bose-Einstein statistics. The initial condition for the cavity field is $a_1=a_2=5$, corresponding to 25 photons in each of the two modes. The simulated momentum space is in the range $-2⩽m⩽3$ and $-2⩽n⩽2$. In the simulations the cavity detuning has been set to ${\Delta }_c-N{U}_0$ $=$ $2\kappa$.

Figure 7

Relative population of selected momentum states $(m,n)$ for various pump beam intensities $I$. The pulse duration is 200 $\mu$s, and the detuning from the atomic resonance ${\Delta }_a=-2\pi \times 8.4$ GHz. The laser is resonant to the empty cavity mode, ${\Delta }_c=0$. The solid (dashed) line presents the prediction of the semiclassical (rate) model for an initial atom number of $N=80000$ in the condensate. The finite-momentum states are initially occupied with $N_{\mathrm{th}}=1000$ and $N_{\mathrm{th}}=5$ thermal atoms outside the condensate distributed over the discrete-momentum states according to Bose-Einstein statistics. The initial condition for the cavity field is $a_1=a_2=5$, corresponding to 25 photons in each of the two modes. The simulated momentum space is in the range $-2⩽m⩽3$ and $-2⩽n⩽2$. The effective cavity detuning has been set to ${\Delta }_c$ $-N{U}_0=$ $0.8\kappa$.