Classical stochastic measurement trajectories: Bosonic atomic gases in an optical cavity and quantum measurement backaction

We formulate computationally efficient classical stochastic measurement trajectories for a multimode quantum system under continuous observation. Specifically, we consider the nonlinear dynamics of an atomic Bose-Einstein condensate contained within an optical cavity subject to continuous monitoring of the light leaking out of the cavity. The classical trajectories encode within a classical phase-space representation a continuous quantum measurement process conditioned on a given detection record. We derive a Fokker-Planck equation for the quasiprobability distribution of the combined condensate-cavity system. We unravel the dynamics into stochastic classical trajectories that are conditioned on the quantum measurement process of the continuously monitored system. Since the dynamics of a continuously measured observable in a many-atom system can be closely approximated by classical dynamics, the method provides a numerically efficient and accurate approach to calculate the measurement record of a large multimode quantum system. Numerical simulations of the continuously monitored dynamics of a large atom cloud reveal considerably fluctuating phase profiles between different measurement trajectories, while ensemble averages exhibit local spatially varying phase decoherence. Individual measurement trajectories lead to spatial pattern formation and optomechanical motion that solely result from the measurement backaction. The backaction of the continuous quantum measurement process, conditioned on the detection record of the photons, spontaneously breaks the symmetry of the spatial profile of the condensate and can be tailored to selectively excite collective modes.

Figure 1

The system of interest in Sec. 6. A condensate confined by the static external potential of Eq. (42) is placed inside an optical cavity with coupling function $g\left(x\right)=g_{0}sin\left(kx\right)$. The initial condensate density is shown as the blue solid line, and since the optical lattice potential is commensurate with the cavity mode, peaks of density correspond to maxima of $\left|g\right(x\left)\right|$. The atoms are pumped transversely by the spatially constant $h\left(x\right)=h_0$, and photons lost through the cavity mirrors at a rate $\kappa$ are detected by a photon counter. Lattice sites are numbered from left to right, with the centermost sites being numbered 12 and 13.

Figure 2

Quantum measurement-induced phase evolution of the atomic field inside a cavity and the resulting phase decoherence in ensemble-averaged dynamics. (a) Relative phase variation between the atomic field in the two central lattice sites for two distinct measurement trajectories. (b) Ensemble-averaged cosine of the phase between atoms in different lattice sites, averaged over 400 measurement realizations, showing dissipation-induced decoherence. The phases are plotted between site $i=12$ and sites differing in number by $i-j=2,-2,-7,3,-1,1$ (upper to lower curves, respectively). The shaded region for the $i-j=1$ curve represents one standard deviation of the sampling error in the results. A frequency analysis of the rapid oscillations evident in the curves reveals an interplay of several excited collective modes, supporting the need for a multimode treatment such as we present in this paper.

Figure 3

Quantum measurement-induced symmetry breaking in atomic density evolution, conditioned on a measurement record. (a) Time dependence of the stochastic field density ${\left|\psi (x,t)\right|}^2$ for a single measurement trajectory, conditioned on a single measurement record. This trajectory corresponds to the phase evolution shown in Fig. 2 (blue, solid line). (b) The measurement rate of photons at the detector for the single trajectory shown in (a). Ensemble-averaged time dependence of the stochastic field density ${\left|\psi (x,t)\right|}^2$, averaged over 400 realizations, which restores the unbroken spatial pattern.

Figure 4

Steady-state self-organization: steady-state relative population imbalance due to self-organization as a function of pumping strength, ${\tilde{h}}_0$ in the units of $h_0$, the pumping strength used for the classical measurement trajectories in this section and highlighted by the red circle in the figure ($h_0^2{g}_0^2/\kappa {\Delta }_{pa}^2\approx 2.6\times {10}^{-3}\omega$). We solve the steady-state problem using the classical Gross-Pitaevskii equation approach as detailed in Ref. [42], but calculated for our finite system in the harmonic potential trap. Self-organization is seen to occur for factors ${\tilde{h}}_0/h_0\gtrsim 10$.

Figure 5

Ensemble-averaged phase decoherence of the atomic field inside a cavity, when the stochastic phase evolution for a single trajectory is proportional to the spatially varying strength of the quantum measurement backaction. We show Gaussian transverse pump results for a BEC in an optical lattice potential. (a) Initial-state $\psi (x,t=0)$ (solid line) and transverse pump beam profile $h\left(x\right)$ (red dashed line). The circles indicate the pair of sites most illuminated ($16,17$) and the corresponding pair on the other side of the condensate which is not illuminated ($8,9$). (b) Ensemble-averaged cosine of the relative phase between different sites. Lower to upper curves correspond to the following pairs of sites: pair of most illuminated sites ($16,17$) (blue solid line); sites ($16,13$) (black dashed line); sites ($16,12$) (green dash-dotted line); sites ($8,9$) (red dashed line). (c) Ensemble-averaged cosine of the phase relative to site 16 for all other sites at different times, showing the spread of the phase decoherence with time. From the upper to the lower lines, the corresponding times are $(0.025,0.5,1.0,1.5,2.0,2.5)2\pi t_0$, respectively. Parameters as given in Sec. 6a and ensemble averaged over 200 realizations.

Figure 6

A selective excitation of collective modes of a BEC as a result of quantum measurement backaction. (a) Comparison of the shapes of the Kohn mode quasiparticle functions $u_1\left(x\right)$ (blue dashed line) and $v_1\left(x\right)$ (black dash-dotted line) with the cavity mode function $g\left(x\right)$ (red dotted line) and the initial stochastic field for the condensate $\psi \left(x\right)$ (green solid line) for the case where the overlap of the cavity mode and the Kohn mode is maximal. The differing quantities have been scaled into arbitrary units to enable a comparison of their functional form. (b)–(f) The dynamics for a single stochastic realization of a measurement record when maximizing the overlap of the cavity mode with the Kohn mode. (b),(c) The stochastic field density ${\left|\psi (x,t)\right|}^2$ as a function of time for two different classical measurement trajectories. The transverse pump beam is applied at $t=0$ and remains at a constant strength throughout the simulation. (d) The center-of-mass position of the condensate for the two trajectories shown in (b) (solid line) and (c) (dashed line). (e) The measurement rate of photons at the photodetector $r_{\mathrm{meas}}$ for the two trajectories, directly proportional to the measured photocurrent. (f) The variation of $\Delta q$ for the two trajectories.

Figure 7

Condensate response to the transverse pump unconditioned on any particular measurement trajectory, formed by ensemble averaging 400 single trajectories such as are shown in Fig. 6. (a) Ensemble-averaged stochastic field density ${\left|\psi (x,t)\right|}^2$. (b) Dissipation-induced loss of coherence between $x=0$ and $x^{\prime }=3x_0$. The figure shows $|g_1(x,x^{\prime })|\equiv |\langle {\widehat{\Psi }}^{†}\left(x\right)\widehat{\Psi }\left(x^{\prime }\right)\rangle |$; the initial condensate is phase coherent with $|g_1(x,x^{\prime })|=1$.

Figure 8

Quantum measurement-induced excitation of the Kohn mode, simulations with a maximum overlap between $g\left(x\right)$ and the Kohn mode. (a) Decomposition into BdG linearized collective excitations; mode occupation numbers are shown averaged over 400 individual measurement trajectories. (b) Ensemble average of $\langle \left|q_1\right|\rangle$ over 400 realizations. Note that the absolute value must be taken since the measurement backaction generates oscillations with a random phase for different realizations.

Figure 9

Varying the wavelength of the light in the cavity compared to the Kohn mode wavelength. (a) The Kohn mode population and (b) the center-of-mass displacement $\langle \left|q_1\right|\rangle$ after a short time ($0.162\pi t_0$) as a function of the cavity-mode wavelength. Error bars represent the quantum-mechanical uncertainty. The results have been ensemble averaged over 400 realizations for each cavity wavelength.

Figure 10

Response for a cavity wavelength of $k=1.03x_0^{-1}$, corresponding to the point second from the right in Fig. 9. (a) Density response stochastic field ${\left|\psi (x,t)\right|}^2$ for a single stochastic realization of a measurement record. (b) BdG mode populations, ensemble averaged over 400 realizations.

Figure 11

Quantum measurement-induced excitation of the breathing mode: maximal breathing mode overlap results. (a) Density of the stochastic field ${\left|\psi (x,t)\right|}^2$ as a function of time for a single stochastic realization of a measurement record. (b) Ensemble average over 400 realizations of BdG mode populations. (c) Measurement rate for the same single realization as that shown in (a). (d) Ensemble average of $\Delta q=\sqrt{q_2-q_1^2}$. The shaded gray area corresponds to the width of the standard deviation in the result.