---
author:
- Guang Ping He
title: 'Realizing the fictitious beam splitter - A stationary implementation of semi-counterfactual interaction-free imaging'
---

**Based on quantum counterfactual interaction-free measurement, we propose an implementation scheme for a beam splitter with anomalous reflection and transmission properties that looks impossible at first glance. Our scheme is stationary without requiring switchable mirrors and polarization rotators. Using the scheme for imaging will ensure that the optical radiation received by the object being imaged can be arbitrarily low. Thus it enables applications such as stealthy night vision devices that can work without detectable ambient light, or being used as a hackware against some counterfactual quantum cryptographic protocols.**

Introduction
============

Interaction-free measurement (IFM), first proposed by Elitzur and Vaidman in 1993 @qi1746, is a quantum method to detect the existence of objects without interacting with them. For example, when a light-triggered bomb is put in one of the two arms of a Mach-Zehnder interferometer, there will be a non-trivial probability for detecting its existence via the interference pattern without activating the bomb. This result highlights yet another counterfactual phenomenon that absents in the classical world. But from a practical point of view there is still much room to improve, because in the original setup in Ref. <span>qi1746</span>, the maximum probability for detecting the bomb without causing it to explode is merely $1/2$. With the rest probability $1/2$ the bomb will be triggered. Thus it is too low in efficiency for picking good bombs (which was the original purpose of the example illustrated in Ref. @qi1746) and too danger if we want to use the method for real-life security checks. Later, Kwiat et al. @qi1761 proposed a scheme aimed to raise the fraction of IFM arbitrarily close to $1$, but did not discuss the experimental implementation in details.

In 2009, based on similar ideas, Noh proposed a more ingenious setup of the interferometer and applied it to quantum key distribution (QKD) @qi801. It immediately caught great interests, most of which focused on whether there can be fully counterfactual protocols for transferring classical <span>qi1026,qi1025,qi1126,qi1127,qi1221,qi1239,qi1222,qi1269,qi1381,qi1383,qi1748,qi1749,qi1750,qi1751,qi1752,qi1523,qi1531,qi1621,qi1720</span> or quantum information @qi1760 [@qi1382; @qi1389; @qi1219; @qi1223]. Besides using visible light, IFM scheme that works in the microwave domain was also proposed @qi1849.

While there is no doubt that the counterfactuality of the transferred information has deep philosophical significance, here we are more interested on its practical application on interaction-free imaging. The apparatuses in Refs. @qi1026 [@qi1025; @qi1126] seem to fit this purpose well, despite that they were designed for communications instead of imaging, because in these proposals, the probability for photons to be detected at the receiver Bob's side can all be made arbitrarily close to $1$ in theory. A proof-of-principle experiment of the protocol in Ref. @qi1126 was already performed @qi1762. Very recently, an interaction-free ghost-imaging experiment was reported @qi1745, showing that it is indeed advantageous to develop the imaging technology using the counterfactual effect, e.g., image quality can be maintained with reduced photon illumination, and direct imaging of birefringent and phase-only objects is made possible too.

However, these works still have their own shortcomings. As stated in Ref. @qi1762, the apparatus in Ref. @qi1126 requires high-speed switchable mirrors and switchable polarization rotators, whose status needs to be switched within nanoseconds. It is still a technical challenge today, so that even the experiment in Ref. @qi1762 had not implemented these parts faithfully. The fraction of IFM in the experiment in Ref. @qi1745 is much less than $1$. Ref. @qi1025 avoided the use of switchable devices. Instead, it came up with the concept fictitious beam splitter (FBS) which, if turned into reality, can be used for eavesdropping in the QKD protocol in Ref. <span>qi801</span>. If we apply this FBS to imaging, a fraction of IFM arbitrarily close to $1$ can be reached. But the FBS was considered fictitious, because it is supposed to have anomalous reflection and transmission properties (which will be elaborated detailedly in the next section) that cannot be actually achieved. Fortunately, following the proposal in Ref. @qi1026 (which was designed to increase the key rate of Noh's QKD protocol @qi801), the FBS can be approximated by combining an infinite number of ordinary beam splitters (BSs) and mirrors. Nevertheless, it is obvious that using such a great number of devices is still very impractical.

In this paper, we give a stationary design of the FBS without using switchable devices, and it contains a single ordinary BS and a very limited number of mirrors only. Thus the two difficulties in previous works are both evaded. While the BS and mirrors still need to be manufactured and adjusted to a very high precision which may still be somehow challenging in practice, we believe that the hardest parts were already solved so that practical interaction-free imaging is no longer out-of-reach.

The remainder of the paper is organized as follows. The concept of FBS is introduced in the next section. In Sec. III, we review briefly the implementation scheme in Refs. @qi1026 [@qi1025], which formes the base of our proposal. Section IV elaborates how it works for interaction-free imaging, and addresses the remaining difficulties. Then we present our scheme in Sec. V, and compare it with these in Refs. @qi1126 [@qi1760]. Finally, we summary the result and discuss its possible applications and advantages in Sec. VI.

The fictitious beam splitter
============================

In Fig.1, we illustrate an interferometer with the FBS proposed in Ref. <span>qi1025</span> which is supposed to have the following bizarre properties:

\(i) Any photon hitting the FBS from path $c$ will be reflected with certainty.

\(ii) Like an ordinary BS in a balanced Michelson-type interferometer, when no object presented before Mirror $X$, if a photon is split by the FBS into two wave packets that travel along paths $a$ and $b$ respectively, they will be reflected back and recombine at the FBS, where constructive interference occurs so that the combined wave packet will enter path $c$ with certainty and the detector $D_{1}$ will not click.

\(iii) Any photon hitting the FBS from path $a$ alone will pass through the FBS and enter path $d$ with certainty.

![Interferometer with a fictitious beam splitter (FBS). The optical delay OD is adjusted precisely to make the optical length of path $a$ match that of path $b$. When there is an object blocking the path to Mirror $X$, the FBS is supposed to be able to ensure that the photons sent from the source along path $c$ will finally enter path $d$ and reach the detector $D_{1}$ with certainty.<span data-label="fig:epsart"></span>](he_fig1)

Such an apparatus can detect the existence of the object, because by combining properties (i) and (iii), we can see that any photon sent from the source will eventually make $D_{1}$ click when path $b$ is blocked. Meanwhile, property (ii) showed that $D_{1}$ never clicks when path $b$ is not blocked. Thus the response of $D_{1}$ will serve as a signal indicating whether there is an object in the path to Mirror $X$.

But this FBS is called fictitiousbecause in principle, the above three properties conflict with each other by definition, so that they cannot be achieved simultaneously. Any device satisfying property (i) rigorously can never display property (ii), because a photon hitting the FBS from path $c$ will not be hopkinssteven@example.org. Also, property (i) implies that the reflectivity of the FBS is $r=1$, while property (iii) indicates that the transmissivity of the FBS is $t=1$. But for any ordinary BS actually exists, $r+t>1$ is simply impossible.

An existing implementation scheme
=================================

Intriguingly, Ref. @qi1025 pointed out that the FBS can be implemented approximately by using a chained interferometer structure proposed in Ref. @qi1026, which was designed for counterfactual QKD. Fig.2 shows a slightly simplified version of their proposal, where we reduced Bob's devices in Fig.2 of Ref. @qi1026 down to a single Mirror $X$ and discarded Alice's optical circulator $C$ and detector $D_{0}$, because they are irrelevant to our purpose of imaging. We also added back the quarter-wave plates (QWPs) which were omitted purposely in Ref. @qi1026 for simplicity. Now let us elaborate how the design can approximate the FBS.

\[tb\] ![image](he_fig2)

When a photon is emitted from the source and travels from the left to the right, it will be split by the beam splitters $BS_{i}$ ($i=1,...,N$) into wave packets that travel along different paths $x_{i}y_{i}$ ($i=1,...,N$), $%
x_{i}x_{i+1}$ ($i=1,...,N-1$) and $x_{N}z$, and are reflected back by Mirror $Y_{i}$ ($i=1,...,N$) and Mirror $X$, respectively. Fig.2 illustrated the case for $N=4$. When $N=1$ it will reduce to the counterfactual QKD apparatus in Ref. @qi801 (except that some of Alice's and Bob's devices in Ref. @qi801 were discarded as stated above).

The last optical delay $OD_{N}$ is carefully adjusted to ensure that the optical path length $\overline{x_{N}y_{N}}$ equals to $\overline{x_{N}z}$, so that they form a balanced Michelson-type interferometer. Consequently, when no object presented before Mirror $X$, a wave packet reaching the last beam splitter $BS_{N}$ from the left will be split into two wave packets that travel along the paths $x_{N}y_{N}$ and $x_{N}z$, respectively. After being reflected by Mirror $Y_{N}$ and Mirror $X$, the two wave packets will recombine at $BS_{N}$ and interfere. Now let us calculate the result of this interference. Following the occupation number representation wildly used in quantum optics, let $\left\vert m\right\rangle _{\alpha \beta }$ ($m=0,1$) denote that there is $m$ photon on the path $\alpha \beta $, where $\alpha $ and $\beta $ can be any of the points $x_{i}$, $y_{i}$, $z$, the detector $%
D_{1}^{(i)}$ ($i=1,...,N$) and the source $S$. According to Eq. (3) of Ref. @qi917, in our Fig.2 when a photon hits $BS_{N}$ from the path $%
x_{N-1}x_{N}$, the transformation caused by $BS_{N}$ is$$\begin{aligned}
\left\vert 1\right\rangle _{x_{N-1}x_{N}}\left\vert 0\right\rangle
_{D_{1}^{(N)}x_{N}} &\rightarrow &\sqrt{t_{N}}\left\vert 0\right\rangle
_{x_{N}y_{N}}\left\vert 1\right\rangle _{x_{N}z}  \nonumber \\
&&-i\sqrt{r_{N}}\left\vert 1\right\rangle _{x_{N}y_{N}}\left\vert
0\right\rangle _{x_{N}z},  \label{bs1}\end{aligned}$$where $t_{N}$ and $r_{N}=t_{N}-1$ are the transmissivity and reflectivity of $BS_{N}$, respectively. The quarter-wave plate $QWP_{N}$ will give the wave packet a phase factor $e^{i(2\pi /4)}=i$ each time it passes. Consequently, after being reflected back by Mirror $Y_{N}$, $\left\vert
1\right\rangle _{x_{N}y_{N}}$ will become $-\left\vert 1\right\rangle
_{y_{N}x_{N}}$ because it has passed $QWP_{N}$ twice. Thus, at the moment right before hitting $BS_{N}$ for the second time, the form of the wave packets in Eq. (\[bs1\]) turned into$$\sqrt{t_{N}}\left\vert 0\right\rangle _{y_{N}x_{N}}\left\vert 1\right\rangle
_{zx_{N}}+i\sqrt{r_{N}}\left\vert 1\right\rangle _{y_{N}x_{N}}\left\vert
0\right\rangle _{zx_{N}}.  \label{bsN}$$Again, from Eq. (3) of Ref. @qi917 we know that when a photon hits $%
BS_{N}$ from the path $zx_{N}$ or $y_{N}x_{N}$, respectively, the transformation caused by $BS_{N}$ is$$\begin{aligned}
\left\vert 0\right\rangle _{y_{N}x_{N}}\left\vert 1\right\rangle _{zx_{N}}
&\rightarrow &\sqrt{t_{N}}\left\vert 1\right\rangle
_{x_{N}x_{N-1}}\left\vert 0\right\rangle _{x_{N}D_{1}^{(N)}}  \nonumber \\
&&-i\sqrt{r_{N}}\left\vert 0\right\rangle _{x_{N}x_{N-1}}\left\vert
1\right\rangle _{x_{N}D_{1}^{(N)}}, \label{bs2}\end{aligned}$$or$$\begin{aligned}
\left\vert 1\right\rangle _{y_{N}x_{N}}\left\vert 0\right\rangle _{zx_{N}}
&\rightarrow &\sqrt{t_{N}}\left\vert 0\right\rangle
_{x_{N}x_{N-1}}\left\vert 1\right\rangle _{x_{N}D_{1}^{(N)}}  \nonumber \\
&&-i\sqrt{r_{N}}\left\vert 1\right\rangle _{x_{N}x_{N-1}}\left\vert
0\right\rangle _{x_{N}D_{1}^{(N)}}. \label{bs3}\end{aligned}$$Substituting Eqs. (\[bs2\]) and (\[bs3\]) into Eq. (\[bsN\]), we know that after passing $BS_{N}$ again, the status of the wave packets in Eq. (\[bsN\]) becomes$$\begin{aligned}
&&\sqrt{t_{N}}(\sqrt{t_{N}}\left\vert 1\right\rangle
_{x_{N}x_{N-1}}\left\vert 0\right\rangle _{x_{N}D_{1}^{(N)}}  \nonumber \\
&&-i\sqrt{r_{N}}\left\vert 0\right\rangle _{x_{N}x_{N-1}}\left\vert
1\right\rangle _{x_{N}D_{1}^{(N)}})  \nonumber \\
&&+i\sqrt{r_{N}}(\sqrt{t_{N}}\left\vert 0\right\rangle
_{x_{N}x_{N-1}}\left\vert 1\right\rangle _{x_{N}D_{1}^{(N)}}  \nonumber \\
&&-i\sqrt{r_{N}}\left\vert 1\right\rangle _{x_{N}x_{N-1}}\left\vert
0\right\rangle _{x_{N}D_{1}^{(N)}})  \nonumber \\
&=&\left\vert 1\right\rangle _{x_{N}x_{N-1}}\left\vert 0\right\rangle
_{x_{N}D_{1}^{(N)}}.\end{aligned}$$That is, constructive interference occurs at $BS_{N}$ so that the combined wave packet will always travel back along the path $x_{N}x_{N-1}$ and will not be detected by $D_{1}^{(N)}$.

Similarly, for any $i=1,...,N-1$, $OD_{i}$ is adjusted so that the optical path lengths satisfy $\overline{x_{i}y_{i}}=\overline{x_{i}z}$, which ensures that constructive interference occurs at every $BS_{i}$. Therefore, after having been split and recombined at all the beam splitters, the resultant wave packet will finally return to the path from $BS_{1}$ to the source with certainty and none of the detectors $D_{1}^{(i)}$ ($i=1,...,N$) will click. Thus, property (ii) of the FBS is satisfied rigorously.

On the contrary, if there is a non-reflective object in the middle of the path $x_{N}z$, any wave packet entered $x_{N}z$ will not return from Mirror $%
X$, so that no constructive interference at $BS_{N}$ will occur anymore. Or if the object is a reflective one, then the length of the path is shortened so that this wave packet will return to $BS_{N}$ earlier than the one reflected back by Mirror $Y_{N}$ does. There will be no more constructive interference at $BS_{N}$ either, as long as the distance between this object and Mirror $X$ is greater than the coherence length of the photons. Either way, $D_{1}^{(N)}$ could click with a non-vanishing probability. For the same reason, the constructive interference at other $BS_{i}$ ($i=1,...,N-1$) will also be destroyed, so that the photon may be detected by other $%
D_{1}^{(i)}$ too.

Now let us calculate the probabilities for this case to happen. Let $t_{i}$and $r_{i}=1-t_{i}$ denote the transmissivity and reflectivity of $BS_{i}$($i=1,...,N$), respectively. When there is a non-reflective object blocking the path $x_{N}z$, the photon will finally be reflected back to the source by $BS_{i}$ and Mirror $Y_{i}$ ($i=1,...,N$) with probability$$\begin{aligned}
p_{0} &=&r_{1}r_{1}+t_{1}r_{2}r_{2}t_{1}+t_{1}t_{2}r_{3}r_{3}t_{2}t_{1}+...
\nonumber \\
&&+(t_{1}t_{2}...t_{N-1}r_{N})^{2}  \nonumber \\
&=&r_{1}^{2}+\sum\limits_{j=2}^{N}\left(
r_{j}\prod\limits_{i=1}^{j-1}t_{i}\right) ^{2}.\end{aligned}$$The probability for the photon to pass all $BS_{i}$ ($i=1,...,N$) and hit the object is$$p_{2}=\prod\limits_{i=1}^{N}t_{i}.  \label{p2}$$Thus the probability for the photon to be detected by any of $D_{1}^{(i)}$ ($%
i=1,...,N$) is$$\begin{aligned}
p_{1} &=&1-p_{0}-p_{2}  \nonumber \\
&=&1-\left( r_{1}^{2}+\sum\limits_{j=2}^{N}\left(
r_{j}\prod\limits_{i=1}^{j-1}t_{i}\right) ^{2}\right)
-\prod\limits_{i=1}^{N}t_{i}.  \label{p1}\end{aligned}$$

For simplicity, assume that all $BS_{i}$ ($i=1,...,N$) have the equal transmissivity $t$ and reflectivity $r=1-t$. Then$$p_{0}=(1-t)^{2}\sum\limits_{j=1}^{N}t^{2(j-1)},$$$$p_{1}=1-(1-t)^{2}\sum\limits_{j=1}^{N}t^{2(j-1)}-t^{N},  \label{p1s}$$$$p_{2}=t^{N}.  \label{p2s}$$

On the other hand, for a reflective object, $p_{0}$ should be replaced by$$p_{0}^{\prime }=p_{0}+p_{2}^{2},$$and $p_{1}$ should be replaced by$$p_{1}^{\prime }=1-p_{0}^{\prime }=p_{1}+p_{2}-p_{2}^{2}.$$

When $N\rightarrow \infty $ we have$$p_{2}=0,$$so that property (i) of the FBS is satisfied approximately. Meanwhile, there will be$$p_{1}=p_{1}^{\prime }=\frac{2}{1+\frac{1}{t}},$$showing that non-reflective and reflective objects now reach the same result. Thus we will no longer differ $p_{1}^{\prime }$ from $p_{1}$ in this limit from now on.

When there is also $t\rightarrow 1$, we get$$p_{1}=1,$$which means that property (iii) is satisfied approximately too. Therefore, if we take both $N\rightarrow \infty $ and $t\rightarrow 1$ simultaneously, i.e., by using a sufficiently large number of beam splitters with a sufficiently low (but non-vanishing) reflectivity $r$, in principle the apparatus in Fig.2 can approximate the FBS in Fig.1 with arbitrarily high precision.

Interaction-free imaging
========================

Mechanism of the imaging
------------------------

In the apparatus in Fig.2, as stated above, when no object presented before Mirror $X$, property (ii) of the FBS ensures that all $D_{1}^{(i)}$ ($%
i=1,...,N$) will never click. But when there is an object blocking the path $x_{N}z$, with probability $p_{1}$ the photon will reach one of these detectors. Consequently, once any of $D_{1}^{(i)}$ ($i=1,...,N$) clicks, the existence of the object is revealed unambiguously. By moving the object in the plane perpendicular to the path $x_{N}z$ while constantly sending photons from the source and recording whether any of $D_{1}^{(i)}$ ($%
i=1,...,N$) clicks, an image of the object can be obtained.

Counterfactuality
-----------------

The most remarkable feature of this imaging process is that we can learn the presence of the object even when it does not interact with the photons. This is because in the above analysis, we showed that the object can be either reflective or non-reflective. Therefore, even if it is another detector or something that absorbs photons, we still know that it exists once we find any of $D_{1}^{(i)}$ clicks. In this case, it is clear that the photon was not detected or absorbed by the object, or had any energy-related interaction with the object. Otherwise, due to the quantumness of energy, the same photon should no longer be able to make $D_{1}^{(i)}$ click. This is why the process is called interaction-free @qi1746 or counterfactual @qi801.

Note that when no object presented and the photon returns to the source, the process is no longer counterfactual @qi801. Also, if a photon left a weak trace in the path $x_{N}z$ (e.g., in the apparatus proposed in Ref. @qi1126), it is arguable whether we can still call it counterfactual even when the photon is finally found by $D_{1}^{(i)}$ <span>qi1127,qi1221,qi1269,qi1381,qi1748,qi1749,qi1750,qi1720</span>. Thus it seems safer to call the whole process semi-counterfactual. Fortunately, these downsides are not important for our propose of imaging, because as long as the energy of the photon is not absorbed by the object, we can be sure that no harmful effect on the object can be measured.

Performance
-----------

To approximate the FBS using the apparatus in Fig.2, it was showed above that we should take both $N\rightarrow \infty $ and $t\rightarrow 1$. But if the purpose is for imaging only, then satisfying properties (i) and (ii) of the FBS is already sufficient. Thus we merely need $N\rightarrow \infty $ while $t\rightarrow 1$ is no longer required. That is, the beam splitters do not have to have an extremely low but non-vanishing reflectivity $r $. Instead, ordinary beam splitters with a finite reflectivity (e.g., common half-silvered mirrors where $r=t=1/2$) can be used so that the scheme becomes more feasible.

This is because by taking $N\rightarrow \infty $ alone, Eqs. (\[p1s\]) and (\[p2s\]) give $p_{1}=2/(1+1/t)$ and $p_{2}=0$. Since we take the clicking of any $D_{1}^{(i)}$ ($i=1,...,N$) as the evidence of the presence of the object, with probability $p_{1}$ the apparatus can detect the object correctly. In the case $t=r=1/2$, we have $p_{1}=2/3$. Meanwhile, $p_{2}=0$implies that the energy received by the object can be made arbitrarily low, thus the imaging is interaction-free.

The efficiency of successful imaging can be further improved by sending more than one photon, and we conclude that the object is not presented only if none of the photons is detected by $D_{1}^{(i)}$ ($i=1,...,N$). For example, suppose that we send $m=10$ photons. Then the error rate for us to conclude mistakenly that the object is absent while it actually exists is merely$$\varepsilon =(1-p_{1})^{m}=\left( \frac{1}{3}\right) ^{10}\simeq 1.7\times
10^{-5}.$$Meanwhile, the energy that the object received still remains very low even for finite $N$. Taking $N=20$ as an instance, the probability that at least one of the photon is absorbed by the object is$$\begin{aligned}
P_{2}^{(m,N)} &=&1-(1-p_{2})^{m}=1-(1-t^{N})^{m}  \nonumber \\
&=&1-\left( 1-\left( \frac{1}{2}\right) ^{20}\right) ^{10}\simeq 9.5\times
10^{-6}.\end{aligned}$$

Additionally, though we assumed above that all $BS_{i}$ have the equal transmissivity $%
t_{i}=t$, from Eqs. (\[p2\]) and (\[p1\]) it is not hard to see that $%
\varepsilon $ and $P_{2}^{(m,N)}$ can remain low even if $t_{i} $ varies. Therefore, in practice all $BS_{i}$ need not to be perfectly identical.

Remaining difficulties
----------------------

Unfortunately, the apparatus in Fig.2 is still very infeasible, because to reach a high $N$ value, a large number of $BS_{i}$ and $OD_{i}$ is required which is obviously impractical and very costly. The delay time of each $%
OD_{i}$ has to be different too, making the implementation even harder.

Our solution
============

The scheme
----------

To circumvent these difficulties, we propose an apparatus for interaction-free imaging, as shown in Fig.3. Our proposal needs only $1$ photon source, $1$ detector, $1$ beam splitter, and $10$ mirrors for any $N$ value. No switchable mirror nor polarization rotator required. To understand how it works, let us explain it part by part.

\[t\] ![image](he_fig3)

**(1) Beam Splitter (BS) Unit:** This unit contains a single ordinary BS and $4$ mirrors. A top view of BS Unit is given in Fig.4. The paths of the outgoing (from the source to Mirror $X$) and returning (from Mirror $X$ to the detector $D_{1}$) beams are much the same, except for these from BS to $D_{1}$, which are for the returning beam only.

Let us look at the outgoing beams first. The photon coming from the source along the path shown in purple meets BS at point $x_{1}$, and is split into two wave packets (shown as the red lines). One packet goes upwards to Delay Unit. The other goes to the right and meets Mirror $B$ at point $b_{1}$. Then it is reflected to points $c_{1}$, $d_{1}$, $a_{1}$, until it reaches BS again at point $x_{2}$, where it is split for the second time into two wave packets (shown as the pink lines). Again, one packet goes upwards to Delay Unit, and the other goes to the right to points $b_{2}$, then to points $c_{2}$, $d_{2} $, $a_{2}$, until it reaches BS for the third time at point $x_{3}$ and is split into two wave packets (shown as the orange lines) which go to Delay Unit or to points $b_{3}\rightarrow c_{3}\rightarrow
d_{3}\rightarrow a_{3}$, respectively. The latter hits BS at point $x_{4}$, and is split into two wave packets (shown as the yellow lines). One goes to Delay Unit as before. But the other passes Mirror $B$ by the edge and goes straight to the right towards Mirror $X$. That is, by making the beam hit the same BS for $4$ times, we manage to implement the $N=4$ case in Fig.2 without actually using four BS. The method for further increasing $N$ will be discussed later.

The angle of Mirror $A$ should be precisely adjusted so that the path $%
a_{1}x_{2}$ is parallel to the purple path. Then it automatically guarantees that all the rest paths will always be parallel to those in the previous circle, and the lengths satisfy $\overline{x_{1}b_{1}c_{1}d_{1}a_{1}x_{2}}=%
\overline{x_{2}b_{2}c_{2}d_{2}a_{2}x_{3}}=\overline{%
x_{3}b_{3}c_{3}d_{3}a_{3}x_{4}}$.

![Top view of BS Unit.<span data-label="fig:epsart"></span>](he_fig4)

The paths of the returning beam are basically in the reverse direction. The path returned from Mirror $X$ will rejoin the one returned from Delay Unit (both are shown in yellow in Fig.4) at point $x_{4}$. Delay Unit is strategically designed (see point (2) below) so that when no object presented before Mirror $X$, the wave packets returned from the above two paths will arrive at $x_{4}$ simultaneously and lead to constructive hopkinssteven@example.org. Thus the combined wave packet will always travel along the orange line to point $a_{3}$. After being reflected to points $d_{3}$, $%
c_{3}$ and $b_{3}$, it rejoins the wave packet returned from Delay Unit along the orange line at point $x_{3}$. Constructive interference occurs at BS again and the combined wave packet will always travel along the pink line to point $a_{2}$ then $d_{2}$, $c_{2}$, $b_{2}$ and $x_{2}$. Similar process continues, until finally the two red paths rejoin at point $x_{1}$ and the final resultant wave packet will always travel along the purple path to the left towards the source, so that $D_{1}$ will not click. On the contrary, when there is an object blocking the yellow path from Mirror $X$, no wave packet can return from this path so that constructive interference can no longer be guaranteed at points $x_{4}$, $x_{3}$, $x_{2}$ and $x_{1}$. As a consequence, the wave packets returned from Delay Unit stand a non-vanishing probability to pass straight through BS and make detector $D_{1}$ click, thus revealing the existence of the object.

**(2) Delay Unit:** Fig.5 shows the top view of Delay Unit. It also features circling paths similar to these of BS Unit, except for the absence of BS.

![Top view of Delay Unit.<span data-label="fig:epsart"></span>](he_fig5)

As shown in Fig.3, the placement of Delay Unit is not parallel to BS Unit. This makes the wave packets coming along the red/pink/orange/yellow paths from BS Unit actually travel at different layer in Delay Unit, though they look overlapped in Fig.5. When the spacing between the paths in BS Unit is $%
s_{b}$, and the horizontal (vertical) spacing between the paths in Delay Unit is $s_{d}$ ($s_{v}$), there should be $s_{b}^{2}=s_{d}^{2}+s_{v}^{2}$. Thus the tilt angle between the $xy$ plane of BS Unit and the $x^{\prime
}y^{\prime }$ plane of Delay Unit is$$\theta =\arcsin \frac{s_{v}}{s_{b}}.$$In Fig.3 we chose $s_{b}:s_{d}:s_{v}=17:8:15$, therefore $\theta \simeq 61.9%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$.

Unlike other mirrors, the shape of Mirror $D^{\prime }$ of Delay Unit is trapezoid, as shown in Fig.6. The bottom edge of the trapezoid is parallel to, and just a little higher than the $xy$ plane of the paths from/to BS Unit. Taking the outgoing beams as an example, we can see that such a shape allows the red/pink/orange/yellow paths from BS Unit to pass Mirror $%
D^{\prime }$ by the bottom edge, and reach Mirror $A^{\prime }$ directly at points $a_{1}^{\prime }$, $a_{2}^{\prime }$, $a_{3}^{\prime }$ and $%
a_{4}^{\prime }$, respectively. Then the red path will be reflected from point $a_{1}^{\prime }$ to $b_{1}^{\prime }$, then to $c_{1}^{\prime
}\rightarrow d_{1}^{\prime }\rightarrow a_{2}^{\prime }\rightarrow
b_{2}^{\prime }\rightarrow c_{2}^{\prime }\rightarrow d_{2}^{\prime
}\rightarrow a_{3}^{\prime }\rightarrow b_{3}^{\prime }\rightarrow
c_{3}^{\prime }\rightarrow d_{3}^{\prime }\rightarrow a_{4}^{\prime }$, and finally be directed to Mirror $Y$. But the pink path, as Delay Unit is tilted, will travel through $a_{2}^{\prime }\rightarrow b_{2}^{\prime
}\rightarrow c_{2}^{\prime }\rightarrow d_{2}^{\prime }\rightarrow
a_{3}^{\prime }\rightarrow b_{3}^{\prime }\rightarrow c_{3}^{\prime
}\rightarrow d_{3}^{\prime }\rightarrow a_{4}^{\prime }$ only (In Fig.5 the points $a_{2}^{\prime }$, $b_{2}^{\prime }$, ..., $a_{4}^{\prime }$ of the pink path look overlapped with these of the red path. But they actually belong to a lower layer, i.e., their $z^{\prime }$ coordinates are different, as shown in Fig.3 and Fig.6). That is, it is shorter than the red path by one circle. Similarly, the orange path travels through $%
a_{3}^{\prime }\rightarrow b_{3}^{\prime }\rightarrow c_{3}^{\prime
}\rightarrow d_{3}^{\prime }\rightarrow a_{4}^{\prime }$ only (again, the $%
z^{\prime }$ coordinates of these points are lower), and the yellow path goes to Mirror $Y$ directly after being reflected at point $a_{4}^{\prime }$ (with an even lower $z^{\prime }$ coordinate). Each of them is one circle shorter than the previous one.

![Close-up angled view of the trapezoid mirror in Delay Unit.<span data-label="fig:epsart"></span>](he_fig6)

The angle of Mirror $D^{\prime }$ in the $x^{\prime }y^{\prime }$ plane is also precisely adjusted to ensure the reflected paths parallel to each other. In this case, it is easy to prove that the lengths of the paths satisfy $\overline{a_{1}^{\prime }b_{1}^{\prime }c_{1}^{\prime
}d_{1}^{\prime }a_{2}^{\prime }}=\overline{a_{2}^{\prime }b_{2}^{\prime
}c_{2}^{\prime }d_{2}^{\prime }a_{3}^{\prime }}=\overline{a_{3}^{\prime
}b_{3}^{\prime }c_{3}^{\prime }d_{3}^{\prime }a_{4}^{\prime }}$. Let us denote this length as $L$. Then the length of the red/pink/orange/yellow paths within the mirrors in Delay Unit is $3L$, $2L$, $L$ and $0$, respectively. Thus, with a single Delay Unit, we manage to create different amount of delay for the paths that left BS at different times.

Finally, by adjusting the distance between these mirrors, we can make $L=%
\overline{x_{1}b_{1}c_{1}d_{1}a_{1}x_{2}}$ (where the latter also includes the extra wave path length caused by the thickness of BS). Meanwhile, the distance between Delay Unit and Mirror $Y$ should be adjusted so that the sum of the path length between point $x_{4}$ in Fig.4 and $a_{4}^{\prime }$ in Fig5 plus the length between $a_{4}^{\prime }$ and Mirror $Y$ equals to the one between $x_{4}$ and Mirror $X$. Then we can see that in the case where no object is presented before Mirror $X$, when the wave packets returned from Mirror $X $ reach BS at points $x_{4}$, $x_{3}$, $x_{2}$ and $%
x_{1}$ along the yellow/orange/pink/red paths at different times, respectively, the wave packets returned from Delay Unit will also reach these points at the corresponding times. Thus constructive interference is always guaranteed at these points, so that the apparatus in Fig.3 has exactly the same performance as that of Fig.2, but requires significantly less devices.

Methods for increasing $N$
--------------------------

Eqs. (\[p2\]) and (\[p2s\]) imply that the number of photons received by the object drops as $N$ increases. There are two methods to increase $N$ so that the wave packets can bounce between the mirrors in BS Unit and Delay Unit for more rounds before they exit:

\(A) Moving Mirror $A$ (and moving Mirror $D^{\prime }$ simultaneously) along the negative $y$ axis (the negative $x^{\prime }$ axis) while keeping its angle unchanged. This can reduce the spacing between the parallel paths.

\(B) Increasing the size of the whole apparatus (including the area of the mirrors and the distance between them) while keeping the spacing between the paths unchanged.

Method (A) seems harder because it requires the mirrors in BS Unit and Delay Unit to be very precisely crafted, so that they remain perfectly flat to keep the reflected paths parallel. The edge of Mirrors $A$, $B$, $B^{\prime
} $ and $D^{\prime }$ should also be very thin. Otherwise, when the paths have a very narrow spacing, there will inevitably be incident photons that reach the edge area of the mirrors and be reflected or refracted to an unwanted direction. Thus we believe that method (B) is relatively more practical.

Either way, our proposal has the advantage that there is no need to add additional device in our design to increase $N$. All we need is higher precision in the manufacturing and adjustment of the experimental devices.

Note that Eq. (\[p2\]) shows that when $N\rightarrow \infty $, there will always be $p_{2}=0$ even if each $t_{i}$ ($i=1,...,N$) has a different value. Thanks to this property, the BS in our apparatus does not have to be precisely plated to obtain consistent transmissivity and reflectivity everywhere across the entire surface. The optical radiation on the object will always remain low.

Comparing with the nested interferometer design
-----------------------------------------------

Ref. @qi1126 proposed a nested interferometer for counterfactual communication. The design can be used for imaging too, and also has the advantage that it only takes a constant number of devices even for very high $N$ value. But to control when to allow the photons to enter or leave the nested interferometer, the proposal requires high-speed switchable mirrors and switchable polarization rotators. Ref. @qi1762 showed that the switching needs to be accomplished within nanoseconds, which is challenging for currently available devices. Therefore, even though Ref. @qi1762 performed a brilliant proof-of-principle experimental implementation of the proposal in Ref. @qi1126, these switchable devices were not implemented. Instead, half-mirrors were used strategically as replacements. Consequently, the photons stand a considerable probability to escape from the nested interferometer every time they meet the half-mirror, which reduces the remaining useful signal significantly, so that the implementation cannot work for really high $N$.

On the contrary, our apparatus is completely stationary. No switchable devices required. Thus it avoids both the complexity problem of Ref. <span>qi1126</span> and the efficiency problem of Ref. @qi1762.

Furthermore, in Ref. @qi1126 the path to the object (like our $x_{N}z$ in Fig.2) needs to be reused for many times. Thus it has the shortcoming pointed out in the 3rd paragraph of page 5 of Ref. @qi1523, that the method requires a promise ... that either a particular place is empty at two times or it is blocked at two times. It does not achieve the dual task to the original IFM of finding an object at a particular place at a particular time without the particles being there. Our proposal does not have this shortcoming because the wave packet of the photon meets the object only once.

If our scheme is expanded for transferring a quantum state (instead of imaging classical objects) following the idea in Ref. @qi1760, then it has the advantage that the transferred state is exact the quantum state at a single time moment (while the method in Ref. @qi1760 is based on Ref. @qi1126 so that it actually transfers the states of different time moments).

Absorption problems
-------------------

In practice, as the photons pass the mirrors and BS so many times, the absorption could be very high so that the final signal will be weak. Fortunately, it is worth noting that our scheme does not require single photon source. Strong light pulse can be used instead. Since each photon obeys the analyses of the apparatuses in Fig.2 and Fig.3, the counterfactual character still remains for strong pulse. That is, any photon survived through all BS and mirrors still has an extremely low probability to be detected by and interact with the object that blocks the path to Mirror $X$. Thus, we can feel free to increase the intensity of the input light to improve the visibility of the imaging, without needing to worry that the object being imaged will be exposed to strong optical radiation.

Meanwhile, when $D_{1}$ does not click after a photon was sent, it is hard to tell whether it was absorbed by BS and mirrors or the object being imaged, or returned to the source. But note that we only take the clicking of $D_{1}$ as the sign of the presence of the object, while the lost photons should not be counted. For this reason, the absorption from BS and mirrors will not be treated mistakenly as the presence of the object, so that they will not hurt the result of the imaging.

Indeed, the proposal in Ref. @qi1126 also requires the photons to travel back and forth within the nested interferometer for many times. Thus the same absorption problem exists too. But the experiment in Ref. <span>qi1762</span> managed to show that the proposal still works as desired. Therefore, our scheme should also be workable despite of the absorption in practice.

Summary and applications
========================

In summary, we proposed the apparatus in Fig.3, and showed that when using it for imaging, the probability for a photon to interact with the object being imaged is merely $p_{2}=t^{N}$ where $t$ is the transmissivity of the BS. By raising the precision in the adjustment of the position and angle of the mirrors, higher $N$ value can be achieved so that $p_{2}\rightarrow 0$, meaning that the optical radiation on the object can be made arbitrarily low.

This feature make our proposal advantageous for many applications. Besides detecting light-triggered bombs without making them explode <span>qi1746</span>, it is also a good solution for photographing ancient arts where shining a light directly may cause damage, as pointed out in Ref. <span>qi1762</span>. It can even image unexposed photographic films (maybe a spy will like it).

The apparatus can also serve as a night vision device. The photons sent from the apparatus work as illumination light, so that no ambient light needed. Meanwhile, the interaction-free feature ensures that the object cannot detect these photons. Therefore, for wildlife studies at night, there will be no risk of disturbing the animals. From a military point of view, such a night vision device can hide itself from being found by the enemy. Moreover, as stated above, non-reflective objects can also be viewed. In contrast, existing night vision devices receive passively the light emitted or reflected by the objects, and amplify it dramatically for viewing. Thus they cannot operate in total darkness. Non-reflective objects will remain hard to observe in dim ambient light too. If the user shreds illumination light on the object, it also raises the chance of being discovered, because traditional illumination is not interaction-free.

Our method is also capable of imaging phase-only objects. These objects are semitransparent, such that a photon passing through it will acquire a phase, even though there is no energy exchange. It was pointed out in Ref. <span>qi1745</span> that a major disadvantage of ghost-imaging is its inability to directly image birefringent and phase-only objects. Ref. @qi1782 also showed that any kind of coincidence imaging technique using a bucket detector in the test arm is incapable of imaging such objects, whether a classical or quantum source is employed. But in our apparatus, once the phase of the wave packet returned from Mirror $X$ was altered, constructive interference on BS will no longer be guaranteed. Thus $D_{1}$ stands a non-trivial probability to click. Sending many photons for imaging each single position will then reveal the presence of such an object. Note that it is arguable whether this case can still be considered interaction-free due to the phase change @qi1761. But at least it is clear that there is still no energy transferred from the photons to the phase-only object.

Also, in practical quantum cryptography, our apparatus makes the conception of FBS in Refs. @qi1026 [@qi1025] come true, so that it can serve as a hacking device against specific QKD @qi801 and other protocols <span>qbc228</span>. In these protocols, the participants encode their secret bits by either measuring the photon they received or reflecting the photon back. Using our apparatus as the FBS, an eavesdropper can detect whether a participant is placing a reflector on the transmission channel or not, thus learn his choice of the secret bit. The counterfactual feature ensures that the probability for detecting this eavesdropping can be made arbitrarily small, as elaborated in Ref. @qi1025.

In principle, if our apparatus can be further adapted to use radar wave, it could enable stealth radars that are invisible to anti-radar missiles, since they merely emit an arbitrarily small amount of radiation. Similarly, if X-ray can be used, then the apparatus can have even more practical applications, e.g., medical examination for human body. In this case, we can be benefited from the counterfactual feature of the apparatus that the X-ray radiation on the biological cells can be arbitrarily low. The same feature also makes it a healthy choice for security gates at stations, airports and other public venues.

Acknowledgements {#acknowledgements .unnumbered}
================

The author thanks Cheng-Long You for his help on drawing the $3$-D figure. The work was supported in part by Guangdong Basic and Applied Basic Research Foundation under Grant No. 2019A1515011048.

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{'source': 'arXiv_out/2007'}