## Abstract

We present an experimental study of controlled-NOT (CNOT) gate through four-wave mixing (FWM) process in a Rubidium vapor cell. A degenerate FWM process in a two level atomic system is directly excited by a single diode laser, where backward pump beam and probe beam are Laguerre Gaussian mode. By means of photons carrying orbital angular momentum, we demonstrate the ability to realize CNOT gate with topological charges transformation in this nonlinear process. The fidelity of CNOT gate for a superposition state with different topological charge reaches about 97% in our experiment.

© 2014 Optical Society of America

## 1. Introduction

Optical vortices have attracted a great deal of attention in recent years. The field of optical vortices usually has singular points where the field goes to zero and around which the phase varies as *l* · 2*π*, with *l* the topological charge. The Laguerre-Gaussian (LG) beam carrying orbital angular momentum (OAM) is one such optical field with a doughnut-shaped intensity distribution and zero intensity at the beam center [1], which has various applications, such as laser trapping [2], optical manipulation [3], and sensitive imaging [4]. Moreover, the OAM of light can be identified by a series of integer quantum number, *l*, which form a complete basis set in Hilbert space. Contrasting to the traditional degree of freedom with spin angular momentum, the OAM provides an infinite-dimensional degree of freedom. This paves the way in information coding [5] and quantum computation [6]. On the other hand, four-wave mixing (FWM) has been largely investigated in alkali atomic vapors in past several decades, which can be used for quantum storage [7] and photon pairs generation [8], etc. Recently, plenty of research based on OAM transferring has been explored in FWM process [9–13]. The quantum number of OAM has been proven to obey OAM conservation during the nonlinear process [12–14]. Based on above transformations, the computation of topological charges of two optical vortices has been achieved via a non-degenerate FWM process in [15], which presents some potential applications in quantum computing, such as quantum Deutsch’s algorithm [16].

Controlled-NOT (CNOT) gate lies at the heart of computing algorithm [17, 18], which has been demonstrated in several different physical systems including trapped ions [19], superconducting circuits [20], Rydberg atoms [21], cavity QED [22], and linear optics [23]. In our previous work [24], we have presented an experimental scheme to realize the logical operations of Deutsch’s algorithm in atomic ensembles through a non-degenerate FWM process. By employing OAM as a qubit, we have demonstrated the realization of four logic gates used for implementing Deutsch’s algorithm. In this work, we experimentally demonstrate the CNOT gate with OAM states through a degenerate FWM process in ^{87}*Rb* atom vapor. The OAM states transformation are proved to follow the CNOT logical rule. Our experimental results show that the transformed states can be well maintained and the fidelity reaches about 97% in the transferring process.

## 2. Theory

The *LG* beam carrying different OAM can be expressed as

*z*and

*ω*(0) is the width at the beam waist,

*z*is Rayleigh range, (2

_{R}*p*+ |

*l*|+1)

*tan*

^{−1}(

*z/z*) denotes Gouy phase. ${L}_{p}^{\left|l\right|}(x)$ is a generalised Laguerre polynomial.

_{R}*l*is the azimuthal index representing OAM of

*lh̄*per photon, and

*p*+ 1 is the number of radial nodes in the intensity distribution. For simplicity and without loss of generality, we set

*p*= 0 all through the text and only consider OAM number

*l*. In our following experiment we also mainly concern OAM changes in the degenerate FWM process, thus only the transverse phase term

*exp*(−

*ilϕ*) of laser field is taken into account. Therefore, the forward pump (

*F*), probe (

*P*) and backward pump (

*B*) beams with different OAMs can be written as:

*i*=

*F*,

*P*,

*B*. Since

*A*only contributes to the amplitude of the field, it can be departed from the transverse phase. The generated signal field from the FWM process is determined by both the third-order nonlinearity

_{i}*χ*

^{(3)}and the input field. We note that the nonlinearity

*χ*

^{(3)}can be effectively enhanced at the cost of greatly reducing

*χ*

^{(1)}due to the electromagnetically induced transparency (EIT) effect [25]. Thus it offers us a great advantage to obtain higher transform efficiency during the FWM process. The FWM signal field

*S*follows [26]: Substituting Eq. (2) into Eq. (3), the generated signal field

*E*can be obtained as:

_{S}*l*=

_{S}*l*+

_{F}*l*−

_{P}*l*, which is the fundamental relationship used to realize CNOT gate in our following experiment.

_{B}To realize CNOT gate, we choose different OAMs to encode the qubit states as described in [24]. The encoding protocol of control and target qubits is shown in Table 1. We choose photon’s OAM of probe field (*P*) as the control qubit, where the photon carrying OAMs of 0*h̄* is defined as |0〉, and photon carrying +*h̄* is |1〉. The photon’s OAM of backward field (*B*) and signal field (*S*) are selected as the input and output target qubits, where the photon carrying OAM of 0*h̄* is defined as |0〉, and photon carrying *±h̄* is |1〉.

If we set the OAM of *F* field to 0*h̄*, then the OAM transition relation in the FWM process is *l _{S}* =

*l*−

_{P}*l*. When the control qubit is |0〉, it can be easily obtained from transition relation that the generated signal photon maintains the same state as the input target qubit. Noting that OAM of −

_{B}*h̄*is also assigned as |1〉 state according to the encoding rule. On the other hand, when the control qubit is |1〉, the generated signal photon acquires the opposite state against the state of the input target qubit. The above operation follows the CNOT logic transformation rule as shown in the following.

## 3. Experimental results and discussion

Our experiment to realize CNOT gate utilizes an degenerate FWM process, in which three beams are sent to atom vapor and a fourth beam is generated by the induced polarization [25]. A typical FWM process is schematically shown in Fig. 1(a). The forward (*F*) and the backward pump Beams (*B*) counter-propagate along the cell. The *F* beam is circularly polarized *σ*^{+} and the *B* beam is circularly polarized *σ*^{−}. The probe (*P*) beam is sent to the atomic medium and crossed with pump beams by a small angle. The probe beam has the same circular polarization *σ*^{−} with backward beam. According to the phase matching condition, FWM signal (*S*) can be generated and counter-propagates with *P*. The polarization of *S* field is *σ*^{+} which is orthogonal with *P* field. Figure 1(b) depicts the corresponding energy levels in our experiment. The two degenerated ground states correspond to the *F* = 2 hyperfine level of ^{87}*Rb* 5*S*_{1/2} state, the excited state corresponds to the *F′* = 1 hyperfine level of ^{87}*Rb* 5*P*_{1/2} state. In our experiment, forward and backward pump beams and probe beam all come from one semiconductor laser and couple with ground and excited states.

The detailed experimental setup is schematically shown in Fig. 2. An external cavity diode laser (TOPTIC, DL100) with a central wavelength of 795nm and output power of 100mw is used. The main beam of laser is split into two parts, one part is used for frequency locking and the other part is coupled into a single mode fiber (SMF). The beam out from SMF is also divided into two parts with the opposite polarizations. The horizontal polarization part is selected as *F* beam with the size expanded to three times by two lenses (the focus length is 50mm and 150mm, respectively) in front of the atom cell. The vertical polarization part reflects from the front facet of a computer-controlled liquid crystal spatial light modulator (SLM1, Hamamatsu, x10468) to generated a desired *LG* mode beam with a size of about 2mm, which is severed as the probe beam. Then the probe beam carrying OAM and the expanded forward pump beam are combined on the polarizing beam-splitter (PBS1) with an intercross angle about 0.4 degree. The combined beams pass through a quarter-wave plate (QWP) and incident into the Rubidium vapor cell with the opposite polarization. The temperature of the cell is set to 70°*C*. The vapor cell is 5cm long and filled with enriched ^{87}*Rb*, which is mounted inside a three-layer *μ*-metal shield in order to reduce stray magnetic fields. The transmitted beams are converted back to the linear polarization by a second quarter-wave plate and then separated by PBS2. After PBS2, *F* beam is converted to the vertical polarization and injected on the second SLM (SLM2). The generated *LG*_{00} or *LG*_{01} beam propagates in an opposite direction with *F*, which is selected as the backward pump beam (*B*). The generated signal field *S* from FWM process is detected by placing a beam-splitter (BS) in the path of *P* field. In our experiment, the power of forward beam is 6.8mW and the probe beam is 0.7mW. The *LG*_{01} mode diffraction efficiency of SLM2 is 75%, thus the power of *B* beam is about 5mW. Under EIT condition, the generated *S* beam reflected by the BS is about 0.24mW, which gives the FWM efficiency 69.1%. Here, we only consider the photons that participate in the FWM process, and photons for optical pumping have not been taken into account in gate operation.

Now we proceed to the CNOT gate experiment using our previous scheme. Figure 3(a) and 3(b) are the experimental results where the control qubit (*P*) is logic 0, i.e. *LG*_{00} mode. There are two beam patterns recorded in each image. The left part of images are the leakages from backward pump beam (*B*), which is served as reference input target qubit mode. While the right part of images are the pattern of generated *S* beam from FWM process. It is clear to see that when control qubit is logic 0, the generated *S* beam maintains the same mode as input target qubit (*B*). On the contrary, Figs. 3(c) and 3(d) are recorded on the condition that the control qubit (*P*) is logic 1, which means *LG*_{01} mode. In this case, the target qubit *B* is converted from a central filled non-doughnut mode to a doughnut mode in Fig. 3(c), and from a doughnut mode to a non-doughnut mode in Fig. 3(d).

The mode of *S* beam can be analyzed by sending it into an interferometer to interfere with a plane wave. Figures 4(a) and 4(b) show our experimental results when *S* beam are non-doughnut and doughnut patterns, respectively. While Figs. 4(c) and 4(d) are the corresponding results of theoretical simulations when *S* beam are *LG*_{00} and *LG*_{01} modes, respectively. From the comparisons we clearly see that the non-doughnut and doughnut patterns are indeed *LG*_{00} and *LG*_{01} modes, correspondingly. Thus we can conclude that the operations in Fig. 3 follow the CNOT logic transformation rule.

To show how the CNOT gate works in a FWM process, we need input an arbitrary initial
superposition state as *a*|0〉 +
*b*|1〉 for the target photon, where
*a*^{2} + *b*^{2}
= 1. The generated photon state can be obtained as:

Here, the operator *P̂* denotes the FWM operation where the initial control qubit is |0〉. In this case, the generated qubit would be the same as initial superposition state *a*|0〉 + *b*|1〉. While *P̂′* denotes the FWM operation where the initial control qubit is |1〉, and the generated qubit would flip to *a*|1〉 + *b*|0〉. In both cases, *a* and *b* should maintain the same value during the operation. In order to test the generated states that fit the transformation rule, we use an OAM sorter to sort |0〉 and |1〉 state separatively. In this way, the testing superposition states could be accurately measured. In the experiment, the OAM sorter is formed by Mach-Zehnder interferometer with a dove prism in each arm, as is shown in the box of Fig. 2. The two dove prisms are placed with an angle of 90 degree, so the beams in two arms are rotated with respect to each other through an angle of 180 degree. By correctly adjusting the path length of the interferometer, we can ensure photons with *l* = 0 appear in one port that is recorded by detector D1, and photons with *l* = 1 would appear in another port that is recorded by detector D2.

The results depicted the quantum CNOT logic transformation are shown in Fig. 5. The initial input superposition states are set as
|0〉, $\frac{\sqrt{5}}{5}|0\u3009+\frac{2\sqrt{5}}{5}|1\u3009$, $\frac{1}{2}|0\u3009+\frac{2\sqrt{3}}{4}|1\u3009$, $\frac{\sqrt{3}}{3}|0\u3009+\frac{\sqrt{6}}{3}|1\u3009\frac{\sqrt{2}}{2}|0\u3009+\frac{\sqrt{2}}{2}|1\u3009$. These five input superposition states are plotted
in Fig. 5 with the black symbols. The dashed
line (blue color) satisfies the normalization condition
*a*^{2} + *b*^{2}
= 1. The testing results of control qubit 0 is depicted in Fig. 5(a), where the generated signal states corresponding
to the initial superposition states are plotted with red symbols. The generated
qubit states almost maintain the same value as the initial states in the operation.
We calculate the fidelities of the five superposition states, the value are
96.1%, 97.1%, 98.2%, 97.8%, 96.8%,
respectively. The testing results of control qubit 1 is depicted in Fig. 5(b), the red points on the other side mean that the
generated qubit states have flipped in the operation. The fidelities are calculated
as 98.5%, 96.7%, 97.5%, 98.5%, and 96.9%,
respectively.

## 4. Conclusion

We experimentally realized the CNOT gate through a degenerate FWM process in a rubidium vapor cell. This is due to the conservation of OAM in the nonlinear process. We found that the fidelity of superposition states with different topological charges could reach about 97% in the process. Although we performed the experiment with *LG*_{00} or *LG*_{01} mode, we confirmed that the experimental scheme paves the way for higher-dimensional system using OAM of photons.

## Acknowledgments

We acknowledge financial support from the National Natural Science Foundation of China (NSFC) under grants 11374238, 11074198, 11204235, 91336101, 61127901, and NSFC for Distinguished Young Scholars of China under grant 61025023.

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