Effect of atomic beam alignment on photon correlation measurements in cavity QED

Quantum trajectory simulations of a cavity QED system comprising an atomic beam traversing a standing-wave cavity are carried out. The delayed photon coincident rate for forward scattering is computed and compared with the measurements of Rempe et al. [Phys. Rev. Lett. 67, 1727 (1991)] and Foster et al. [Phys. Rev. A 61, 053821 (2000)]. It is shown that a moderate atomic beam misalignment can account for the degradation of the predicted correlation. Fits to the experimental data are made in the weak-field limit with a single adjustable parameter—the atomic beam tilt from perpendicular to the cavity axis. Departures of the measurement conditions from the weak-field limit are discussed.

Figure 1

Second-order correlation function for ideal coupling [Eq. (30)]: (a) parameter set 1, (b) parameter set 2.

Figure 2

Second-order correlation function with Monte Carlo average over number of atoms $N$ and configuration $\left\{{\mathit{r}}_j\right\}$. The average is taken according to Eq. (39) (thin line) and Eq. (41) (thick line) for (a) parameter set 1 and (b) parameter set 2.

Figure 3

Second-order correlation function with Monte Carlo average, Eq. (41), over number of atoms $N$ and configuration $\left\{{\mathit{r}}_j\right\}$ compared with the experimental data from (a) Fig. 4(a) of Ref. 29 (parameter set 1) and (b) Fig. 4 of Ref. 31 (parameter set 2).

Figure 4

Typical trajectory of the intracavity photon number expectation for parameter set 1: (a) atomic beam aligned perpendicular to the cavity axis, (b) with a $9.6\mathrm{mrad}$ tilt of the atomic beam. The driving field strength is $\mathcal{E}∕\kappa =2.5\times {10}^{-2}$.

Figure 5

As in Fig. 4 but for parameter set 2.

Figure 6

Distribution of intracavity photon number expectation with the atom beam perpendicular to the cavity axis (thin line) and a $9.6\mathrm{mrad}$ tilt of the atomic beam (thick line): (a) parameter set 1 and (b) parameter set 2.

Figure 7

Semiclassical correlation function for parameter set 1, with adiabatic following of the photon number (left column) and without adiabatic following (right column); for atomic beam tilts of [(a) and (e)] $0\mathrm{mrad}$, [(b) and (f)] $4\mathrm{mrad}$, [(c) and (g)] $9\mathrm{mrad}$, and [(d) and (h)] $13\mathrm{mrad}$.

Figure 8

As in Fig. 7 but for parameter set 2 and atomic beam tilts of [(a) and (e)] $0\mathrm{mrad}$, [(b) and (f)] $10\mathrm{mrad}$, [(c) and (g)] $17\mathrm{mrad}$, and [(d) and (h)] $34\mathrm{mrad}$.

Figure 9

Second-order correlation function from full quantum trajectory simulations with a two-quanta truncation: (a) parameter set 1 and $\theta =0\mathrm{mrad}$ (thick line), $7\mathrm{mrad}$ (medium line), $12\mathrm{mrad}$ (thin line); (b) parameter set 2 and $\theta =0\mathrm{mrad}$ (thick line), $10\mathrm{mrad}$ (medium line), $17\mathrm{mrad}$ (thin line).

Figure 10

Best fits to experimental results: (a) data from Fig. 4(a) of Ref. 29 are fitted with parameter set 1 and $\theta =9.7\mathrm{mrad}$ (thick line) and $10\mathrm{mrad}$ (thin line); (b) data from Fig. 4 of Ref. 31 are fitted with parameter set 2 and $\theta =9.55\mathrm{mrad}$. Averages of (a) $200000$ and (b) $50000$ samples were taken with a cavity-mode cutoff $F=0.01$.

Figure 11

Second-order correlation function from full quantum trajectory simulations with a two-quanta basis for parameter set 1 and (a) $\theta =0\mathrm{mrad}$, (b) $\theta =9.7\mathrm{mrad}$, and (c) $\theta =10\mathrm{mrad}$. Averages of (a) $15000$, and (b) and (c) $200000$ samples were taken with a cavity-mode cutoff $F=0.01$.

Figure 12

Doppler-shift compensation for a misaligned atomic beam in (a) ring and (b) standing-wave cavities (parameter set 2). The second-order correlation function is computed with the atomic beam perpendicular to the cavity axis (thin line), a $17.3\mathrm{mrad}$ tilt of the atomic beam (medium line), and a $17.3\mathrm{mrad}$ tilt plus compensating detuning of the cavity and stationary atom resonances $\mathrm{\Delta }\omega ∕\kappa =k{\overline{v}}_{\mathrm{oven}}\sin \theta ∕\kappa =0.916$ (thick line).

Figure 13

Second-order correlation function from full quantum trajectory simulations with a three-quanta truncation and atomic beam tilts as in Fig. 10: (a) parameter set 1, mean intracavity photon number $⟨a^{†}a⟩=6.7\times {10}^{-3}$; (b) parameter set 2, mean intracavity photon numbers $⟨a^{†}a⟩=2.2\times {10}^{-4}$, $5.7\times {10}^{-4}$, $1.1\times {10}^{-3}$, and $1.7\times {10}^{-3}$ (thickest curve to thinnest curve). Averages of $20000$ samples were taken with a cavity-mode cutoff $F=0.1$. Shot noise error bars are added to the data taken from Ref. 29.