Particle-counting statistics of time- and space-dependent fields

The counting statistics give insight into the properties of quantum states of light and other quantum states of matter such as ultracold atoms or electrons. The theoretical description of photon counting was derived in the 1960s and was extended to massive particles more recently. Typically, the interaction between each particle and the detector is assumed to be limited to short time intervals, and the probability of counting particles in one interval is independent of the measurements in previous intervals. There has been some effort to describe particle counting as a continuous measurement, where the detector and the field to be counted interact continuously. However, the formalism based on continuous measurements does not provide a formula applicable to general time- and space-dependent fields. In our work, we derive a fully time- and space-dependent description of the counting process for linear quantum many-body systems, taking into account the back-action of the detector on the field. We apply our formalism to an expanding Bose-Einstein condensate of ultracold atoms, and show that it describes the process correctly, whereas the standard approach gives unphysical results in some limits. The example illustrates that, in certain situations, the back-action of the detector cannot be neglected and has to be included in the description.

Figure 1

Density $|\tilde{\phi }(z_0,t){|}^2$ with respect to time (a) and normalized intensity $\mathcal{I}/N$ with respect to $\tau$ (b) for $z_0=0$ (blue squares), $z_0=0.1/\Gamma$ (green circles), and $z_0=0.4/\Gamma$ (red diamonds). As the distance $z_0$ between the center of the cloud and the detector increases, the wave function spreads and long detector opening times are required to achieve nontrivial intensities. Parameters used: $z_0=0,\epsilon =1,N=100,\Gamma ={10}^5$.

Figure 2

(a) $|\tilde{\phi }{(0,t)|}^2$ with respect to $t$. (b) Normalized intensity $\mathcal{I}/N$ with respect to $\tau$. We compare the exact solution (red triangles), second-order Born approximation (blue squares), exponential Born approximation (light blue diamonds), and the quantum counting formalism (green circles). Both the quantum counting formalism and the Born approximation reach intensities that exceed the total number of particles for long opening times $\tau$. In the exponential Born approximation the intensity is bounded for large $\tau$, however, the asymptotic value exceeds the bound given by the exact solution. Parameters used: $z_0=0,\epsilon =1,N=100,\Gamma ={10}^5$.

Figure 3

Counting distribution $p\left(m\right)$ obtained from exact solution (red triangles), Born approximation (blue squares), and the quantum counting formalism (green circles). For short detection times (a) the approximations agree reasonably well with the exact solution. For longer detection times (b), the approximations are no longer valid. Parameters used: $z_0=0,\epsilon =1,N=100,\Gamma ={10}^5$.