The Experimental Demonstration of High Efficiency Interaction-free Measurement for Quantum Counterfactual-like Communication

We present an interaction-free measurement with quantum Zeno effect and a high efficiency η = 74.6% ± 0.15%. As a proof-of-principle demonstration, this measurement can be used to implement a quantum counterfactual-like communication protocol. Instead of a single photon state, we use a coherent light as the input source and show that the output agrees with the proposed quantum counterfactual communication protocol according to Salih et al. Although the counterfactuality is not achieved due to the presence of a few photons in the public channel, we show that the signal light is nearly absent in the public channel, which exhibits a proof-of-principle quantum counterfactual-like property of communication.

photon in the public channel is strictly zero when the numbers of chained MZIs are going to be quite large. If this probability is not zero, then the protocol is not counterfactual. The experimental demonstration of the principle of counterfactual communication for SLAZ scheme with M = 4 and N = 2 was performed using a single photon source 19 . By entangling and disentangling a photon and an atom via nonlocal interaction, a new protocol for transferring an unknown quantum state counterfactually was also proposed via nonlocal interaction 20 .
An experimental implementation of the counterfactual protocol with a single photon source was usually difficult 21 , and it also requires a weak coherent light as a reference to lock the phase of the system. Therefore, the demonstration of a single-photon-based counterfactual scheme needs to be carried out with the help of a weak coherent light for phase stabilization. In this paper, we perform an experiment on high-efficiency IFM using an interlinked structure of MZIs, by this we demonstrate a proof-of-principle experiment for quantum counterfactual-like communication with coherent light. In this scheme, the quantum counterfactual property is not reached due to the presence of a few portion of light in the public channel, the scheme supplies the technology for phase stabilization of MZIs for single photon operation.
In the scheme proposed in ref. 18, a faithful interaction-free measurement system can be obtained with multiple MZIs e.g. there are M − 1 outer MZIs (i.e., there are M beamsplitters in these MZIs) connected in series, while there are N − 1 small MZIs (i.e. N beamsplitters are included) connected in series in one arm of each outer MZI. An efficiency approaching 100% can be realized if M and N are large enough. However, for a practical setup, the inevitable loss resulting from the optical elements will be introduced and it will increase with the number of the MZIs. As a result, the efficiency will decrease correspondingly. One must have a balanced consideration for accomplishing this protocol with a proper number of MZI.
In our experimental setup, the structure of MZIs is designed with a multiple-series connection, consisting of two outer MZIs and seven inner MZIs in one arm of each outer MZI. The reflectivity of the beamsplitters in MZIs is specially designed to satisfy the effect of IFMs, which gives the possibility of detecting the presence of an object without direct interaction with the object. The detection of the intensity ratio of the two final outputs can also tells the possible operation of quantum counterfactual-like communication in principle.

Results
Interaction-free measurement with two outputs for quantum counterfactual-like communication. We design the experiment with M = 3 (i.e two big MZIs) and N = 8 (i.e. seven small MZIs), as illustrated in Fig. 1. The number of MZIs is optimized to get the maximum outputs when considering the possible loss, the theoretical discussion is given below. Two outer MZIs are connected in series. In each of the two big MZIs, there are seven small MZIs connected in series. Thereforehree beam splitters BS Mj(j=1, 2,3) in both the MZIs are designed with the same reflectivity R cos ( /6) M 2 π = , corresponding to the experimental coating of 75% ± 1%. In addition, sixteen (2 × 8) beam splitters BS N in small MZIs have the reflectivity R cos ( /16) , with the experimental value being 96.2% ± 0.5%. The mirrors HR M and HR N are high reflection mirrors with reflectivity larger than 99.99%. The wavelength band selection for the optics in our setup is 790 nm~950 nm since the wavelength of light source is 850 nm, and each of the optics are coated for horizontally polarized light. To extend the scheme in C + L wavelength band, one can simply change the light source and optics for suitable wavelength. The piezoelectric transducers (PZT M1,M2 , PZT Ni(i=1,…, 7) and ′ = … PZT Ni i ( 1, ,7) ) are used to adjust the path difference (the phase difference is set to be 2nπ) of each MZI. D 3 , D 3 ′, D 1 and D 2 are the photon detectors with the same sensitivity and 85% quantum efficiency at 852 nm wavelength. A 490 μW coherent light represented by the annihilation operator â in is incident into the system through the first beamsplitter (BS M1 ), while the other input port of BS M1 is in vacuum represented by the operator â v . There are four outputs represented by the operators a out 3 , a out 3′ , a out 1 and â out 2 corresponding to the detectors by D 3 , ′ D 3 , D 1 and D 2 , respectively. Fourteen (2 × 7) blocks indicated by triangles in one arm (as the transmission channel) of each small MZIs are served as switch (or logic gate) to let the light be absorbed by the blocks (corresponding to logic 1 in Fig. 1a) or pass through the MZIs (logic 0 in Fig. 1b).
In order to create the communication between Alice and Bob, this system is assigned to have two separated parts at Alice's and Bob's sides. The input light (represented by annihilation operator â in ), the detectors and optics below the black dash line are at Alice's side, while the blocks and optics above the black dashed line are at Bob's side. Bob's selection of Logic 1 or 0 leads the input light to detectors D 2 or D 1 , i.e. communication is created between Alice and Bob. In order to have direct counterfactual quantum communication, almost no light should pass through the public or transmission channel (see the yellow line in Fig. 1).
We use the transfer-matrix method to verify the propagation of the input light of the system. If we denote the input light by the column vector ˆâ a ( ) , in v T , the output light can be written as The transformation T t(logic) takes the different forms for two cases of logic 1 and logic 0. We consider the two cases separately.
(a) Logic 1 For the case of logic 1 in Fig. 1a with absorbers at the Bob's ends in all (N − 1) × (M − 1) MZIs, the transfer matrix of BS Mj(j=1,2,3) is described as where r m = cos π/2M, t m = sin π/2M are the reflection and transmission amplitudes of the BS Mj . The reflection matrix of "N" BS N is described as Ni and the effect of the phase difference ϕ j via the HR M is represented by Finally, the total transfer matrix is expressed as The output intensity I(D 1 ) and I(D 2 ) at detectors D 1 and D 2 are given by a a A A a a out out in in A A a a out out in in 2 2 21 21 . In Table 1, we present the theoretical calculation of intensity ratio I(D 2 )/I(D 1 ) for different choices of N and M when the phase differences of big MZIs are ϕ j = 0 or 2nπ. We note that most of the light exits via the output port at D 2 due to constructive interference, while a small intensity of light is detected at port D 1 due to destructive interference. We also note that the higher ratio is obtained when the number M of big MZIs is smaller than the number N of small MZIs, i.e., M < N. According to this discussion, we take M = 3, N = 8 in the experiment as shown in Fig. 1. In Figs 2 and 3, we plot the theoretical and experimental interference fringes of each of the two big MZIs for N = 8, M = 3, in which, the solid lines (constructive) and dashed line (destructive) are obtained at outputs of D 2 and D 1 , respectively. The blue and black lines are for the results of first (with the second big MZI locked) and second big MZI (with the first big MZI locked), respectively. In the experiment, the intensity ratio of I(D 2 )/I(D 1 ) is obtained when we lock the phase differences of two big MZIs to be ϕ 1 = ϕ 2 = 0. During the locking time, the stable intensities at D 2 (solid line) and D 1 (dashed line) as a function of the time are obtained as shown in Fig. 4, corresponding to the normalized intensities I(D 2 )/I 0 = 65% at D 2 and I(D 1 )/I 0 = 5% at D 1 respectively, where I 0 is the initial light intensity. It gives the ratio of I(D 2 )/I(D 1 ) = 13.0, which is lower than the theoretical predication of 51.81 (see Table 1) with I(D 2 ) = 74.6% and I(D 1 ) = 1.44%. The deviation mainly comes from the loss in the arm with 2N BS N , leading to the unbalanced intensity of light in two arms of big MZIs, as a result decrease in the visibility of fringes. Note that, in order to lock the phase of the two big MZIs, we inject a relative strong reference light from the vacuum input part, the first MZI is locked using the interference fringe at the upper output from the first HR M at the second big MZI, while the second MZI is locked using the interference fringe at D 1 . The measurement is carried on untill the reference light beam and the Lock-in-system switch off via computer controlled devices. (b) Logic 0 Next we consider the case of logic 0 in Fig. 1b without blocks at Bob's end. First let us discuss the array of N − 1 small MZIs in the inner loops. Once the input light enter the small MZIs, the constructive interference makes the light exit at the ports of detectors D 3 , ′ D 3 (see the black line in Fig. 5a at φ i = 0), while the destructive interference as a result no light enters the big MZIs from the small MZIs (see the black line in Fig. 5b at φ i = 0).
The transfer matrix of BS N is described as BS n n n n N where r n = cos π/2N, t n = sin π/2N are the reflection and transmission amplitudes of the BS N , respectively. the effect of the phase difference φ i via the HR N can be represented as Ni therefore, we can consider the transfer matrix for small MZIs as   When all the seven small MZIs work with the phase difference φ Ni = 0(i = 1, …, 7), the coherent state will completely exit to D 3 . Theoretically, we should have I(D 3 ) = I( ′ D 3 ) = 100% without loss and dissipation. Here due to the dissipation, non-perfect mirrors and loss of other optical elements, the experimental value for the normalized intensity detected at D 3 is 91.7%, as shown in Fig. 6a. Almost the same result is obtained for the detection at ′ D 3 , which is 91.5%. Experimentally, we find the intensity leakage from the small MZIs back to the second big MZI (from last BS N to BS M3 ) approximately 0.7% of the intensity just before the first BS N .
Note that the measurement of each interference is done with the condition that the (N − 1)th fringe is obtained when the phase of the other small MZIs is φ i = 0, as shown in Fig. 6. The phase is controlled via a computer controlled series of lock-in systems. This process is similar with the lock-in system for two big MZIs, we inject a strong reference beam from vacuum input part, then with the aid of the interference fringes from upper HR N s, HR M s and D 1 , the interlinked MZIs are locked. Thereafter, we switch off the injected beam and lock-in system to detect the results.
After finishing the adjustment of the small chains of MZIs, we consider the final output intensity I(D 1 ) and I(D 2 ) at detectors D 1 and D 2 for the case of logic 0, that is Here G lk (l, k = 1, 2) are the matrix elements of T t0 , given by ( ) For logic 1, most of light comes throughout the output port at D 2 , on the contrary for logic 0, most of light comes throughout the output port at D 1 . Therefore, Alice can read the information of Bob's logic gates by measuring the intensities at the two detectors, D 1 and D 2 .
It is important to note that, although the information of logic encoded in the light of transmission channel (yellow line) is inferred from the detections at D 1 and D 2 , the light in the transmission channel actually do not reach to the outputs of D 1 and D 2 , it goes out from the output port at D 3 and D 3 ′. The results demonstrate, in principle, that the quantum communication could be realized via interaction-free measurement of quantum logic. It also shows that few light is transferring through the transmission channel, proving the idea of quantum counterfactual-like communication to be accessible.
Our experiment is the realization of interaction-free measurement with a large efficiency. We note that when light enters the small MZIs for logic 1, it exits from the output port of D 3 or ′ D 3 which is 0.7% and 1.7% respectively as shown in Fig. 8 (note D 3 and ′ D 3 are in the hands of Alice), which is not detected by the detectors D 1 and D 2 (in the hands of Alice). This process corresponds to interaction-free measurement, which is clearly seen from Eq. (5) and evident from Figs 2 and 3. For our experimental system, we use the fraction η, defined by η = P det / (P det + P abs ) in ref. 2, to characterize the quality of the interaction-free measurement. Here P det is the probability of an interaction-free measurement and P abs is the probability that the light is absorbed by the block (including other loss factors). For a perfect interaction-free measurement, we have P abs = 0, i.e. η = 1. The range 0 < η < 1 represents the system which accomplishes interaction-free measurement with finite efficiency, e.g., η = 1/2 in ref.
For our scheme, η is obtained from the evolution of the input state a in  In Fig. 9, The solid lines show the dependence of probability P det and P abs on N with the number of M = 3 for the case of logic 1. It is shown that P det increases and P abs decreases with the increase of N. As a result, the efficiency η increases when N increases, it approaches complete interaction free when N → ∞. The triangle points show the experimental results for P det , P abs and η with M = 3, N = 8 for logic 1. P det represents the detection of light probability at D 2 , and P det = P(D 2 ) = 65% is obtained from the experimental data in Fig. 3a for the maximum value, while P abs is the probability of light absorbed or lossed by all the mirrors above the black dashed line, it is read from the detectors D 3 , ′ D 3 in Fig. 8  Note that, in this case, P(D 1 ) = 5% (see Fig. 3b), the total loss of the system induced by the mirrors and other elements is the 8%. Finally, we obtain the values for our system with N = 8, M = 3, corresponding to η = 0.76 for theory and η = 74.6% ± 0.15% for experiment in repeated measurements, which is larger than the predicted and reported results of η = 1/2 in refs 1 and 2 or improved value of η = 2/3 in ref. 2.

Discussion
In conclusion, we performed an experiment of the high-efficiency interaction-free measurement. Based on the measurement, the quantum counterfactual-like communication with few portion of light involved in the transmission channel can be reached in this scheme. we analysed and implemented a principle scheme with finite M and N of linked interferometers, in which the inevitable loss of optics is involved and increased with the increases of the number of M and N, causing an unexpected decrease of the quantum efficiency of the system. We showed that the practical scheme of high-efficiency interaction-free measurement with low number of M and relatively higher number N is accessible.