Bloch oscillations of cold atoms in a cavity: Effects of quantum noise

We extend our theory of Bloch oscillations of cold atoms inside an optical cavity [Venkatesh et al., Phys. Rev. A 80, 063834 (2009)] to include the effects of quantum noise arising from coupling to external modes. The noise acts as a form of quantum measurement backaction by perturbing the coupled dynamics of the atoms and the light. We take it into account by solving the Heisenberg-Langevin equations for linearized fluctuations about the atomic and optical mean fields and examine how this influences the signal-to-noise ratio of a measurement of external forces using this system. In particular, we investigate the effects of changing the number of atoms, the intracavity lattice depth, and the atom-light coupling strength, and show how resonances between the Bloch oscillation dynamics and the quasiparticle spectrum have a strong influence on the signal-to-noise ratio, as well as heating effects. One of the hurdles we overcome in this paper is the proper treatment of fluctuations about time-dependent mean fields in the context of cold-atom cavity QED.

Figure 1

Schematic of the precision measurement proposal in [15]. A dilute cloud of cold atoms undergoes BOs in the combined intracavity lattice potential and the acceleration due to gravity. The transmitted light field's intensity and phase are modulated at the Bloch frequency. An in situ precise measurement of the Bloch frequency (and hence the force) can be performed by detecting the transmitted light.

Figure 2

Intracavity optical lattice depth $s\left(t\right)\equiv U_0{\left|\alpha \left(t\right)\right|}^2$ in units of the atomic recoil frequency ${\omega }_R$ plotted as a function of time. The curves, which are each for a different value of the collective atom-cavity coupling parameter $N{U}_0/\kappa$, were obtained by solving the mean-field equations of motion Eqs. (6a) and (6b) and illustrate the fact that the change in lattice depth over one BO increases with $N{U}_0/\kappa$. In order to maintain a minimum lattice depth of $3E_{\mathrm{R}}$ as $N{U}_0/\kappa$ was increased by changing $U_0=\{1,3,5\}{u}_0$, where $u_0=7\times {10}^{-3}{\omega }_{\mathrm{R}}$, we also changed the pumping strength as $\eta =\{30.7,24.2,24.3\}\kappa$, giving mean photon numbers $\{458,172,117\}$, respectively. The other parameter values used in this plot are ${\Delta }_{\mathrm{c}}=-0.75\kappa$, $\kappa =345{\omega }_{\mathrm{R}}$, and $N=5\times {10}^4$. For all the plots in this paper the force is such that the Bloch frequency has the value ${\omega }_{\mathrm{B}}={\omega }_{\mathrm{R}}/4$.

Figure 3

The lattice depth $s\left(t\right)$ in units of the atomic recoil frequency ${\omega }_{\mathrm{R}}$ is shown in (a) and its Fourier transform $s\left(\omega \right)$ is given in (b). We have increased the atom-cavity coupling from Fig. 2 to $N{U}_0/\kappa =7.75$. At this larger value some fast fluctuations on top of the slow BO become visible. Their frequency is dominated by a harmonic at $10{\omega }_B$, as can be seen in the inset.

Figure 4

Plot of pump strength (dashed black line) required to maintain a minimum lattice depth of $3E_{\mathrm{R}}$ as a function of $N{U}_0/\kappa$. The red (solid) and blue (dash dotted) lines enclose the values of $\eta$ for which the steady-state photon number in the cavity is bistable for any value of the quasimomentum of the atomic wave function. One sees that for $N{U}_0/\kappa \sim 25$, the pump strength required to maintain the lattice depth leads to bistability. Other parameters for the plot are ${\Delta }_{\mathrm{c}}=-0.75\kappa$, $\kappa =345{\omega }_{\mathrm{R}}$, $N=5\times {10}^4$.

Figure 5

Low-lying levels in the quasiparticle spectrum (elementary excitations). The frequency of the $n$th level is generally complex ${\omega }_n+i{\gamma }_n$. The parameters used in the plots are $U_0=0.01{\omega }_{\mathrm{R}}$, $\kappa =345{\omega }_{\mathrm{R}}$, ${\Delta }_{\mathrm{c}}=-0.75\kappa$, and $N=5\times {10}^4$. The red (solid) and blue (dotted) lines correspond to hybridized atom-cavity modes and generally have nonzero imaginary parts except at certain special points such as at the band center and edges where they can become marginally stable and decouple from the cavity. The green (dash dotted) line corresponds to a cavitylike mode; i.e., the real part of its frequency is close to the effective detuning frequency ${\Delta }_{\mathrm{c}}^{\mathrm{eff}}\left(q\right)$, and the imaginary part is close to $-\kappa$.

Figure 6

Growth of the excited atom fraction over five BO periods for (a) weak and (b) moderately strong atom-cavity coupling. The red (solid) curves are given by solving the full quantum problem in the form of Eq. (29), whereas the black (dashed) curves are the result of treating the atomic modes as independent oscillators plus assuming that the quantum fluctuations in the light come purely from vacuum shot noise, i.e., the coherent state approximation. The mean-field dynamics for (b) is given by the red (solid) curve in Fig. 2. The atom heating rate in these figures oscillates because it is lower at the Brillouin zone edges than at the center. Referring to Fig. 5 we see that at the zone edges the quasiparticle mode with the smallest real part (red solid curve) becomes marginally stable, i.e. the cavity light field part and the atomic part decouple.

Figure 7

Plots of (a) atomic and (b) photonic fluctuation occupation number over 40 BOs calculated using a numerical solution of Eq. (29). The inset in (a) shows $\delta N/N$ as a function of time for the case without an external force, and hence without BOs, whereas the main body of (a) shows the results with an external force: $\delta N/N$ quickly reaches a steady state in the former but not in the latter case. In (b) the lowest curve (red) is for $\beta =1$, the middle curve (blue) is for $\beta =3$, and the highest curve (green) is for $\beta =5$. The inset in (b) shows a closeup of the photonic fluctuation number as a function of time for $\beta =1$. It can be seen how after a transient period the photonic fluctuation number oscillates at the Bloch period, thereby mirroring the mean-field dynamics.

Figure 8

Plots of the SNR as a function of (a) coupling strength $\beta$ and (b) integration time $T$ for different values of $\beta$. In (a) the SNR was computed for an integration time of 10 Bloch periods ($T_B$) and the red (dots) curve gives the mean-field dynamics plus detector shot-noise result, while the black (crosses) curve includes measurement backaction, i.e., the effect of quantum fluctuations upon the coupled atom-cavity dynamics. In (b) the red (solid) and blue (dotted) curves lie almost on top of each other and correspond to values of $\beta$ just before the first dip in the SNR shown in Fig. 8a, whereas the green (dash-dotted) curve corresponds to a value of $\beta$ in the dip. For all plots the minimum lattice depth was 3$E_{\mathrm{R}}$. Other parameters are the typical ones mentioned in the text.

Figure 9

Plots of the SNR as a function of (a) minimum lattice depth $s(t=0)$, and (b) atom number $N$. In both plots the red (dots) curves were computed from mean-field theory plus detector shot noise, and the black (crosses) curves were computed including quantum measurement backaction. For all points $N{U}_0/\kappa =1$ and the signal is integrated over ten Bloch periods. In (a) it is evident that the SNR decreases in both cases for larger lattice depths. In (b) it is evident that the SNR increases linearly as a function of $N$ in both cases.

Figure 10

Plots of the lowest quasiparticle excitation frequency ${\omega }_1$ about the adiabatic solution. In (a) this is given as a function of quasimomentum for three different values of $\beta$: For small $\beta$ (red dashed curve) the minimum of the frequency occurs at $q=\pm 1$, but for larger values of $\beta$ the minimum shifts in to smaller values of $q$. Since each quasiparticle excitation has an energy varying with $q$, in (b) we plot the range of possible excitation frequencies contained in ${\omega }_1$ for all $q$ as a function of $\beta$. The frequency units are the Bloch frequency ${\omega }_{\mathrm{B}}$.

Figure 11

The normalized power in the harmonics of ${\omega }_{\mathrm{B}}$ as calculated from the Fourier transform of the mean-field solution [see Fig. 3b] that lies in the frequency range of the lowest quasiparticle excitation [see Fig. 10b].

Figure 12

Plots of (a) the real part of the quasiparticle energy spectrum and (b) the occupation number as a function of time. The two plots are color coded equivalently. For example, the red (solid) lowest lying level in (a) has occupation number dynamics shown by the red (solid) line in (b). Since the gaps in the spectrum in (a) are smaller than the Bloch frequency the level populations are partially exchanged at the avoided crossings: The gaps become smaller higher up in the spectrum and indeed we see that the exchanges between higher lying states are almost complete. The system parameters are the same as the case with $N{U}_0/\kappa =1$ in Fig. 2.

Figure 13

Plot of quasiparticle (qp) occupation number as a function of time when $N{U}_0/\kappa =5$. The red (solid) line corresponds to the qp band with the smallest energy, followed by the blue (dashed), green (dot dashed), black (dotted), and magenta (dash dotted) lines in ascending order. The inset shows the real part of the qp energy measured in units of ${\omega }_{\mathrm{R}}$ as a function of time over a single Bloch period $T_{\mathrm{B}}$ for the lowest three bands. Since the gap between the lowest two bands [red (solid) and blue (dashed) lines] and the rest of the spectrum is larger than the Bloch frequency, their dynamics is decoupled from the rest. System parameters are as in Fig. 2.

Figure 14

Plot of the quasiparticle occupation number as a function of time for different values of initial lattice depth at the fixed coupling value $\beta =N{U}_0/\kappa =1$. The initial lattice depths $s(t=0)$ are measured in the units of ${\omega }_{\mathrm{R}}$ and are obtained by setting the pump-strength to $\eta =\{44.2,56.1\}\kappa$ for the blue (dashed) and green (dash-dotted) curves, respectively. The inset plots the time $t_l$ at which the linear increase in the quasiparticle number is established as a function of the initial lattice depth.