Semitransparency in interaction-free measurements

We discuss the effect of semitransparency in a quantum-Zeno-like interaction-free measurement setup, a quantum-physics based approach that might significantly reduce sample damage in imaging and microscopy. With an emphasis on applications in electron microscopy, we simulate the behavior of probe particles in an interaction-free measurement setup with semitransparent samples, and we show that the transparency of a sample can be measured in such a setup. However, such a measurement is not possible without losing (i.e., absorbing or scattering) probe particles in general, which causes sample damage. We show how the amount of lost particles can be minimized by adjusting the number of round trips through the setup, and we explicitly calculate the amount of lost particles in measurements which either aim at distinguishing two transparencies or at measuring an unknown transparency precisely. We also discuss the effect of the sample causing phase shifts in interaction-free measurements. Comparing the resulting loss of probe particles with a classical measurement of transparency, we find that interaction-free measurements only provide a benefit in two cases: first, if two semitransparent samples with a high contrast are to be distinguished, interaction-free measurements lose less particles than classical measurements by a factor that increases with the contrast. This implies that interaction-free measurements with zero loss are possible if one of the samples is perfectly transparent. A second case where interaction-free measurements outperform classical measurements is if three conditions are met: the particle source exhibits Poissonian number statistics, the number of lost particles cannot be measured, and the transparency is larger than approximately $\frac{1}{2}$. In all other cases, interaction-free measurements lose as many probe particles as classical measurements or more. Aside from imaging of gray levels, another possible application for interaction-free measurements is the detection of arbitrarily small phase shifts in transparent samples.

Figure 1

Sketches of quantum-Zeno-like IFM setups. (a) The light mode in the left reference cavity is coupled via a semitransparent mirror to a mode in the sample cavity on the right, as proposed in Ref. [5]. Panel (b) shows a possible realization of two coupled cavities for electrons with a diffraction-based beam splitter. (c) With a transparent sample in the sample beam, the probability of finding the electron will coherently oscillate between the reference and sample beam. (d) With a lossy sample in the sample beam, the coherent buildup of probability amplitude in the sample beam is prevented and the electron stays in the reference beam. Depending on whether we measure the probe particle in the sample or in the reference beam, we determine to have a transparent or lossy sample, respectively, in the sample beam. The same concept as in panels (c) and (d) applies to panel (a).

Figure 2

Probabilities $P_R,P_S$, and $P_L$ of finding the probe particle in the reference state $|R\rangle$ (dotted blue line), sample state $|S\rangle$ (dashed red line), or loss state $|L\rangle$ (black line). (a) Probabilities versus time for a perfectly transparent sample, as in Fig. 1. At time $T$ the particle is in the sample state. (b) Probabilities versus time for $N=10$ with an opaque sample, as in Fig. 1. At time $T$, the particle is in the reference state with probability $\sim 0.78$ and lost with probability $\sim 0.22$. (c) Probability $P_L$ at time $T$ as a function of $N$ with an opaque sample. The inset shows the same on a double-logarithmic scale.

Figure 3

Time evolution of the probe particle state probabilities for different transparencies: (a) $\alpha =0.2$, (b) $\alpha =0.5$, (c) $\alpha =0.95$. Shown are the probabilities $P_R$ (dotted blue line), $P_S$ (dashed red line), and $P_L$ (black line) for $N=10$.

Figure 4

Probabilities for the probe particle to be detected in $|R\rangle ,|S\rangle$, or $|L\rangle$ at time $T$ as a function of $\alpha$ for (a) $N=10$, (b) 50, and (c) 200. Shown are $P_R$ (dotted blue lines), $P_S$ (dashed red lines), and $P_L$ (black lines). For low transparencies, the particle is most likely found in the reference state, while the particle is most likely to enter the sample state for high transparencies. Between low and high transparencies, the loss probability has a maximum and the two other probabilities change swiftly.

Figure 5

Same as Fig. 4 but for (a) $N=200$, (b) 2000, and (c) $20000$, and on a logarithmic scale. For large $N$, a change of $N$ simply corresponds to a shift of the probability curves along the logarithmic axis.

Figure 6

Loss probability as a function of the contrast $(1-{\alpha }_1)/(1-{\alpha }_2)$ if two transparencies ${\alpha }_1$ and ${\alpha }_2$ are to be distinguished in an IFM with $N\gg 1$, assuming that $N$ is always chosen so as to minimize the average loss probability at ${\alpha }_1$ and ${\alpha }_2$.

Figure 7

Relationship between the average number of lost particles and the error probability in a measurement to distinguish two transparencies. Shown here are results from simulations of a classical transmission measurement (blue circles), IFM with various parameters (other symbols), and the minimum amount of loss in such a measurement according to Eq. (8) (black line). The task is to distinguish the two transparencies (a) ${\alpha }_1=0.2$ and ${\alpha }_2=0.5$, (b) ${\alpha }_1=0.04$ and ${\alpha }_2=0.64$, as in Fig. 3 of Ref. [17], (c) ${\alpha }_1=0.5$ and ${\alpha }_2=0.99$, and (d) ${\alpha }_1=0.001$ and ${\alpha }_2=0.999$. Each data point was obtained by $40000$ simulation runs. For example, to distinguish the two transparencies in panel (c) a classical measurement loses $\sim 0.45$ particles on average to reach an error probability of $\sim 0.08$, while an IFM with $N=100$ can reach the same error probability with only $\sim 0.25$ lost particles and the minimum amount of loss for this error probability is $\sim 0.15$.

Figure 8

Expected number of lost particles in a measurement of $\alpha$ with uncertainty $\Delta \alpha \le 0.01$ as a function of transparency using (a) the reference state or (b) the sample state as signal. The curves for a classical measurement (blue line), and IFM with $N=10$ (dash-dotted green line), $N=100$ (dashed cyan line), and $N=500$ (dotted red line).

Figure 9

Expected number of lost particles in a measurement of $\alpha$ with uncertainty $\Delta \alpha \le 0.01$ as a function of transparency using the (a) reference state or the (b) sample state as signal and using Poissonian statistics. The curves shown here are for a classical measurement (blue line), and IFM with $N=10$ (dash-dotted green line), $N=100$ (dashed cyan line), and $N=500$ (dotted red line). For comparison to binomial statistics, the result of a binomial classical measurement is shown as a light gray line (same as the blue solid line in Fig. 8). The binomial measurement outperforms all Poissonian measurements in terms of lost particles.

Figure 10

Results for an IFM process of a fully transparent sample as a function of the phase shift $\varphi$ induced by the sample for (a) $N=2$, (b) $N=5$, and (c) $N=50$. Shown here are $P_R$ (dotted blue line) and $P_S$ (dashed red line).