Demonstration of CNOT gate with Laguerre Gaussian beams via four-wave mixing in atom vapor

Mingtao Cao; Ya Yu; Liyun Zhang; Fengjuan Ye; Yunlong Wang; Dong Wei; Pei Zhang; Wenge Guo; Shougang Zhang; Hong Gao; Fuli Li

doi:10.1364/OE.22.020177

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 · 2π, 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 ⁸⁷Rb 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

\begin{array}{l} {LG}_{p}^{l} = & \sqrt{\frac{2 p!}{π (p + | l |!)}} \frac{1}{ω (z)} {[\frac{r \sqrt{2}}{ω (z)}]}^{| l |} \exp [\frac{- r^{2}}{ω^{2} (z)}] L_{p}^{| l |} (\frac{2 r^{2}}{ω^{2} (z)}) \\ \exp [- \frac{i k_{0} r^{2} z}{2 (z^{2} + z_{R}^{2})}] \exp [i (2 p + | l | + 1) \tan^{- 1} (\frac{z}{z_{R}})] \exp (- i l ϕ), \end{array}

where

ω (z) = ω (0) \sqrt{1 + {(z_{R} / z)}^{2}}

is the radius of the beam at position z and ω(0) is the width at the beam waist, z_R is Rayleigh range, (2p + |l|+1)tan⁻¹(z/z_R) denotes Gouy phase.

L_{p}^{| l |} (x)

is a generalised Laguerre polynomial. 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:

E_{F} = A_{F} e^{- i l_{F} ϕ}, E_{P} = A_{P} e^{- i l_{P} ϕ}, E_{B} = A_{B} e^{- i l_{B} ϕ},

where

A_{i} = \sqrt{\frac{2}{π | l_{i} |!}} \frac{1}{ω (z)} {[\frac{r \sqrt{2}}{ω (z)}]}^{| l_{i} |} \exp [\frac{- r^{2}}{ω^{2} (z)}] L_{0}^{| l_{i} |} (\frac{2 r^{2}}{ω^{2} (z)}) \exp [- \frac{i k_{0} r^{2} z}{2 (z^{2} + z_{R}^{2})}] \exp [i (| l_{i} | + 1) \tan^{- 1} (\frac{z}{z_{R}})]

, for i = F, P, B. Since A_i 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 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]:

E_{S} \propto χ^{(3)} E_{F} E_{P} E_{B}^{*} .

Substituting Eq. (2) into Eq. (3), the generated signal field E_S can be obtained as:

E_{S} \propto χ^{(3)} A_{F} A_{P} A_{B}^{*} \exp [- i (l_{F} + l_{P} - l_{B}) ϕ] .

Thus, the OAM transferring in the process can be described as: l_S = l_F + l_P − l_B, which is the fundamental relationship used to realize CNOT gate in our following experiment.

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 0h̄ 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 0h̄ is defined as |0〉, and photon carrying ±h̄ is |1〉.

Table 1. Encoding protocol using quantum states of OAM to realize the CNOT gate.

View Table

If we set the OAM of F field to 0h̄, then the OAM transition relation in the FWM process is l_S = l_P − l_B. 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 −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 ⁸⁷Rb 5S_1/2 state, the excited state corresponds to the F′ = 1 hyperfine level of ⁸⁷Rb 5P_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.

Fig. 1 (a) The schematic of the four-wave mixing. Forward pump, probe and backward pump beam are denoted as F, P, B respectively, the S is FWM signal field. (b)The energy diagram of atom levels coupled by different laser fields in the scheme.

Download Full Size | PDF

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 ⁸⁷Rb, 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 LG₀₀ or LG₀₁ 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 LG₀₁ 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.

Fig. 2 Experimental setup of the four-wave mixing scheme. SLM, spatial light modulator, PBS, polarizing beam-splitter, QWP, quarter waveplate, HWP, half waveplate.

Download Full Size | PDF

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. LG₀₀ 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 LG₀₁ 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).

Fig. 3 Experimental results. The left part of images are the backward beam profile and the right part of images are the pattern of generated FWM signal, S. (a) and (b) are the operation of control qubit 0 with target qubit 0 and 1. (c) and (d) are the operation of control qubit 1 with target qubit 0 and 1, respectively.

Download Full Size | PDF

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 LG₀₀ and LG₀₁ modes, respectively. From the comparisons we clearly see that the non-doughnut and doughnut patterns are indeed LG₀₀ and LG₀₁ modes, correspondingly. Thus we can conclude that the operations in Fig. 3 follow the CNOT logic transformation rule.

Fig. 4 Interference pattern of qubit S with a plane wave. (a) interference pattern of Gaussian mode with plane wave. (b) interference pattern of LG01 mode with plane wave. (c), (d) are theoretical results on corresponding conditions.

Download Full Size | PDF

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 a² + b² = 1. The generated photon state can be obtained as:

\begin{array}{l} \hat{P} [a | 0 〉 + b | 1 〉] = a | e^{- i (0 + 1 - 1) ϕ} 〉 + b | e^{- i (0 + 0 - 1) ϕ} 〉 = a | 0 〉 + b | 1 〉, \\ {\hat{P}}^{'} [a | 0 〉 + b | 1 〉] = a | e^{- i (0 + 1 - 0) ϕ} 〉 + b | e^{- i (0 + 0 - 0) ϕ} 〉 = a | 1 〉 + b | 0 〉 . \end{array}

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〉, $\frac{\sqrt{5}}{5} | 0 〉 + \frac{2 \sqrt{5}}{5} | 1 〉$ , $\frac{1}{2} | 0 〉 + \frac{2 \sqrt{3}}{4} | 1 〉$ , $\frac{\sqrt{3}}{3} | 0 〉 + \frac{\sqrt{6}}{3} | 1 〉 \frac{\sqrt{2}}{2} | 0 〉 + \frac{\sqrt{2}}{2} | 1 〉$ . These five input superposition states are plotted in Fig. 5 with the black symbols. The dashed line (blue color) satisfies the normalization condition a² + b² = 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.

Fig. 5 The experimental results of CNOT gate for superposition state. The initial five superposition states are |0〉, $\frac{\sqrt{5}}{5} | 0 〉 + \frac{2 \sqrt{5}}{5} | 1 〉$ , $\frac{1}{2} | 0 〉 + \frac{2 \sqrt{3}}{4} | 1 〉$ , $\frac{\sqrt{3}}{3} | 0 〉 + \frac{\sqrt{6}}{3} | 1 〉 \frac{\sqrt{2}}{2} | 0 〉 + \frac{\sqrt{2}}{2} | 1 〉$ , which are presented by black color with the shape of half-gap triangle, circle, square, triangle, and star. The red symbols are the corresponding experimental results. The dashed line satisfies a² + b² = 1. (a) and (b) correspond to the control qubit of 0 and 1, respectively.

Download Full Size | PDF

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 LG₀₀ or LG₀₁ 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.

References and links

1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef] [PubMed]

2. T Kuga, Y Torii, N Shiokawa, T Hirano, Y Shimizu, and H Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997). [CrossRef]

3. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef] [PubMed]

4. N. Uribe-Patarroyo, A. Fraine, D. S. Simon, O. Minaeva, and A. V. Sergienko, “Object identification using correlated orbital angular momentum states,” Phys. Rev. Lett. 110, 043601 (2013). [CrossRef]

5. J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. 6, 488–496 (2012). [CrossRef]

6. P. Zhang, B. H. Liu, R. F. Liu, H. R. Li, F. L. Li, and G. C. Guo, “Implementation of one-dimensional quantum walks on spin-orbital angular momentum space of photons,” Phys. Rev. A 75, 052310 (2010). [CrossRef]

7. N. B. Phillips, A. V. Gorshkov, and I. Novikova, “Light storage in an optically thick atomic ensemble under conditions of electromagnetically induced transparency and four-wave mixing,” Phys. Rev. A 83, 063823 (2011). [CrossRef]

8. S.W. Du, P. Kolchin, C. Belthangady, G.Y. Yin, and S. E. Harris, “Subnatural Linewidth Biphotons with Controllable Temporal Length,” Phys. Rev. Lett. 100, 183603 (2008). [CrossRef] [PubMed]

9. Q. F. Chen, B. S. Shi, Y. S. Zhang, and G. C. Guo, “Entanglement of the orbital angular momentum states of the photon pairs generated in a hot atomic ensemble,” Phys. Rev. A 78, 053810 (2008). [CrossRef]

10. S. Barreiro and J. W. R. Tabosa, “Generation of light carrying orbital angular momentum via induced coherence grating in cold atoms,” Phys. Rev. Lett. 90, 133001 (2003). [CrossRef] [PubMed]

11. G. Walker, A. S. Arnold, and S. Franke-Arnold, “Trans-spectral orbital angular momentum transfer via four-wave mixing in Rb vapor,” Phys. Rev. Lett. 108, 243601 (2012). [CrossRef] [PubMed]

12. A. M. Marino, V. Boyer, R. C. Pooser, P. D. Lett, K. Lemons, and K. M. Jones, “Delocalized correlations in twin light beams with orbital angular momentum,” Phys. Rev. Lett. 101, 093602 (2008). [CrossRef] [PubMed]

13. D. Akamatsu and M. Kozuma, “Coherent transfer of orbital angular momentum from an atomic system to a light field,” Phys. Rev. A 67, 023803 (2003). [CrossRef]

14. H. H. Arnaut and G. A. Barbosa, “Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion,” Phys. Rev. Lett. 85, 286–289 (2000). [CrossRef] [PubMed]

15. W. Jiang, Q. F. Chen, Y. S. Zhang, and G. C. Guo, “Computation of topological charges of optical vortices via nondegenerate four-wave mixing,” Phys. Rev. A 74, 043811 (2006). [CrossRef]

16. D. Deutsch, “Quantum theory, the Church-Turing principle and the universal quantum computer,” Proc. R. Soc. London A 400, 97–117 (1985). [CrossRef]

17. G. J. Milburn, “Quantum optical Fredkin gate,” Phys. Rev. Lett 62, 2124–2127 (1989). [CrossRef] [PubMed]

18. A. Barenco, D. Deutsch, and A. Ekert, “Conditional quantum dynamics and logic gates,” Phys. Rev. Lett. 74, 4083–4086 (1995). [CrossRef] [PubMed]

19. J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,” Phys. Rev. Lett. 74, 4091–4094 (1995). [CrossRef] [PubMed]

20. T. Yamamoto, “Demonstration of conditional gate operation using superconducting charge qubits,” Nature 425, 941–944 (2003). [CrossRef] [PubMed]

21. L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a neutral atom controlled-not quantum gate,” Phys. Rev. Lett. 104, 010503 (2010). [CrossRef] [PubMed]

22. C. Bonato, C. Bonato, F. Haupt, S. S. R. Oemrawsingh, J. Gudat, D.P. Ding, M. P. van Exter, and D. Bouwmeester, “CNOT and Bell-state analysis in the weak-coupling cavity QED regime,” Phys. Rev. Lett. 104, 160503 (2010). [CrossRef] [PubMed]

23. J. L. OBrien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003). [CrossRef]

24. M. T. Cao, L. Han, R. F. Liu, H. Liu, D. Wei, P. Zhang, Y. Zhou, W. G. Guo, S. G. Zhang, H. Gao, and F. L. Li, “Deutsch’s algorithm with topological charges of optical vortices via non-degenerate four-wave mixing,” Opt. Express 20, 24263–24271 (2012). [CrossRef] [PubMed]

25. T. Passerat de Silans, C. S. L. Goncalves, D. Felinto, and J. W. R. Tabosa, “Enhanced four-wave mixing via crossover resonance in cesium vapor,” J. Opt. Soc. Am. B 28, 2220–2226 (2011). [CrossRef]

26. R. W. Boyd, Nonlinear Optics (Academic, 1992).