Continuous simultaneous measurement of position and momentum of a particle

We formulate a model of a quantum particle continuously monitored by detectors measuring simultaneously its position and momentum. We implement the postulate of wave-function collapse by assuming that upon detection the particle is found in one of the meters' states chosen as a discrete subset of coherent states. The dynamics, as observed by the meters, is thus a random sequence of jumps between coherent states. We generate such trajectories using the quantum Monte Carlo wave-function method. For sparsely distributed detectors, we use methods from renewal theory of stochastic processes to obtain some semianalytic results. In particular, the different regimes of dynamics of the free particle are identified and quantitatively discussed: from stroboscopic motion in the case of low interrogation frequency, to delayed dynamics reminiscent of the Zeno effect if monitoring is frequent. For a semicontinuous spatial distribution of meters the emergence of classical trajectories is shown. Their statistical properties are discussed and compared to other detection schemes in which the operation on the system due to measurement corresponds to “spatial filtering” of the wave function.

Figure 1

Evolution of the wave function of a particle initially at rest in two dimensions without quantum jumps. Probability density is plotted. Contrary to the case of a free particle, the particle's wave function does not simply broaden, but also tends to concentrate around neighboring detectors.

Figure 2

Sample evolution of the probability density ${\left|\varphi \left(x\right)\right|}^2$ of a particle in free space (top) and an external harmonic potential (bottom). In the first case the particle has some nonzero initial momentum, and in the second it is located initially at rest at some distance from equilibrium. Detections (marked by red dots) are clearly visible as discontinuities in the vertical direction. Both simulations were performed at $\gamma =2$ and a moderately sparse grid.

Figure 3

Left: Histogram of trajectories for a particle in a two-dimensional (2D) harmonic potential and nonzero angular momentum: (a) position space and (b) momentum space. For this histogram we simulate trajectories with the first measurement sampled from the distribution given by Eq. (16), and postselected in position in order to only have runs starting in a circular sector covering the upper region of the distribution centered at $(x,y)=(0,5)$, $(k_x,k_y)=(-5,0)$. Right: (a1)–(a4) Exemplary individual trajectories in position space; (b1)–(b4) the same trajectories in momentum space.

Figure 4

Time dependence of the probability rates $\frac{1}{\gamma }\frac{\delta p_{\alpha }}{dt}=|\langle \alpha |e^{-iHt}{|\alpha \rangle |}^2\equiv \frac{1}{\gamma }{\lambda }_{\alpha }$ (top, blue lines) and $\frac{1}{\gamma }\frac{\delta p_{\beta }}{dt}=|\langle \beta |e^{-iHt}{|\alpha \rangle |}^2\equiv \frac{1}{\gamma }{\lambda }_{\beta }$ (bottom, orange lines) of detecting a particle at $\alpha$ or $\beta$, respectively, provided that at $t=0$ the particle is found at $\alpha$ and assuming no clicks afterwards. At a sufficiently large time both rates are equal: $\frac{\delta p_{\alpha }}{dt}\approx \frac{\delta p_{\beta }}{dt}$. The probability rates are given by the overlaps (squared) of the detectors’ wave functions and the state $|\alpha \rangle$ evolved according to the the non-Hermitian Hamiltonian. These quantities define the intensities of the renewal processes ${\lambda }_{\alpha }$ and ${\lambda }_{\beta }$ corresponding to recapturing of the particle by the detector $\alpha$ and $\beta$, respectively.

Figure 5

The escape time. The blue solid line (the scale at the bottom edge of the figure) shows dependence of the escape time as a function of the detector rate $\gamma$ for a fixed velocity $v=2$. The escape time grows almost linearly with $\gamma$. The larger $\gamma$, the more time it takes to escape. The orange dashed line (the scale at the top edge of the figure) displays dependence of the escape time on the particle velocity for fixed detector rate $\gamma =15$. The larger the velocity, the shorter the time of escape.

Figure 6

Probability distribution of the arrival time of the first click $f_{T_1}\left(t\right)$ for $D=5$, $\gamma =1$, and $v=5$. The colored areas are suitably normalized histograms, with different colors indicating a first click at the $n\mathrm{th}$ detector. The grey noisy line is the sum of these and gives the distribution $f_{T_1}\left(t\right)$ of being caught first at an arbitrary detector after time $t$. This line coincides with the black dotted line, which is the same function obtained numerically from Eq. (32). Because $\gamma$ is not too large, the positions of the first few colored maxima correspond roughly to the time when the particle would arrive classically at the position of the meters. The data for the histogram were obtained from 100 000 single trajectories and grouped into bins of width $\delta t=0.02$. The first three or four maxima are well separated, which corresponds to visible steps in Fig. 8. The inset shows the individual intensities ${\lambda }_i\left(t\right)$ plotted in different colors. Their sum, the total intensity, is depicted by the dashed line.

Figure 7

The mean position of the particle as a function of time observed by two well-separated detectors characterized by $\gamma =1$ (blue line). The first detector is located at ${\alpha }_1=(x=0,v=5)$ and the second at ${\alpha }_2=(x=5,v=5)$. The grey (normalized) histogram shows the frequency of clicks at both detectors (bin width $\delta t=0.02$). At time $t<0.5$ the particle is almost exclusively captured by ${\alpha }_1$ and is thus measured at the position $x=0$. At larger times $t>0.5$ a growing number of trajectories arrives at the second detector. There is a time window around $t=0.5$ when the frequency of clicks at both detectors is significantly lower. This is because at this time, the particle tends on average not to have a large overlap with either of the detectors. It is detected at $x=0$ or at $x=5$ with roughly the same small probability, which is why its average position is between the detectors.

Figure 8

Mean positions as a function of time for a sparse grid of detectors, $v_0\sim 5$, and various $\gamma$. Note how for large $\gamma$ there is a retardation effect. The retardation $t_r$ may well be approximated by Eq. (34). Estimates from a linear asymptotic fit for $\gamma =10$, 20, 50: $t_r=0.24$, 0.51, 1.26, indicated by the colored dots. These curves were obtained using an exact numerical procedure under the assumption that there are no jumps to neighboring momentum states. This assumption is valid for a sparse grid and was tested with results obtained from a full QMCWF simulation—in the extreme case, i.e., for $\gamma =50$, for $N={10}^6$ detections only around 0.1% of all detection events resulted in a change of momentum. A comparison of results from the numerical procedure and the QMCWF calculation for $\gamma =50$ is shown in the inset.

Figure 9

Average position (top) and the dispersion of position (middle) and momentum (bottom) measurements as a function of time for the particle launched with initial velocity $k_0=5$ observed by detectors characterized by $\gamma =1$ and distributed along a line with spacing $D=5.1$ (left panels) and $D=0.73$ (right panels). The thin orange line corresponds to the jump approach from the present study. Results from the filtering method are shown for comparison as a thick blue line. The black dotted lines indicate classical trajectories (top row), quadratic $\sim t^2$ (thick blue) and $\sim t^{3/2}$ (thin orange) fits (middle row), and linear $\sim t$ (thick blue) and $\sim t^{1/2}$ (thin orange) fits (bottom row).