Transporting continuous quantum variables of individual light pulses

Yujiro Eto; Yun Zhang; Takuya Hirano

doi:10.1364/OE.19.001360

1. Introduction

Quantum entanglement is an important ingredient for future information and communication technology, because protocols exceeding the capabilities of classical technology can be implemented with the help of entanglement, e.g., quantum teleportation [1, 2], quantum dense coding [3, 4], and entanglement based quantum cryptography [5]. These techniques have been successfully demonstrated in both discrete and continuous variable (CV) optical experiments. Continuous wave (cw) laser have usually been used as light sources in the CV experiments. Deterministic quantum teleporatation has been experimentally reported by some groups [6–8], who transferred the input states of cw light by using quadrature entanglement and homodyne detection. CV quantum dense coding has also been demonstrated using cw entanglement [9,10]. An important experimental challenge is to develop entanglement-based communication technology using pulsed light sources. Pulsed light is preferable because it is able to encode relevant information into individual optical pulses in the pulse train and decode each piece of information independently by using a pulse-resolved detector [11, 12]. This feature is important for many applications such as quantum key distribution and probabilistic generation of quantum states where a controlled and isolated manipulation of quantum systems is required.

This paper reports the transportation of continuous quantum variables of individual vacuum light pulses at telecommunication wavelengths. Entangled light beams in our scheme are generated by an optical parametric amplifier placed in a ring interferometer [13]. The input vacuum state interferes with one entangled pair, then simultaneous measurements of a pair of conjugate observables, the so called “CV Bell measurements”, are carried out by using two time-domain pulsed homodyne detectors (PHDs). The PHD yields one quadrature value for one incident laser pulse [14]. Consequently, it is possible to individually detect each pulse. The other entangled beam is detected by another PHD and its measured quadrature values are displaced based on the results obtained from CV Bell measurements. The variances in the displaced data are beyond the limitations of classical transportation without entanglement, and the obtained fidelity of F = 0.57 ± 0.03 is higher than the 0.5 of the classical case without entanglement.

2. Scheme for transporting continuous quantum variables

Here, we will briefly explain our scheme. Figure 1 illustrates the scheme for transporting continuous quantum variables. We represent the amplitude and phase quadratures for one mode as x̄ and p̂, which obey the commutation relationship, [x̄, p̂] = i/2.

Fig. 1 Transportation of continuous quantum variables.

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The entangled pair (the entangled beams a and b in Fig. 1) is created by combining two squeezed states, which are generated by optical parametric amplification (OPA), on a half beamsplitter (50/50 BS) [6]. When the relative phase between squeezed beams is set to π/2, the amplitude and phase quadratures of the two output beams can be expressed as ${\hat{x}}_{a} = (e^{r_{1}} {\hat{x}}_{1} + e^{- r_{2}} {\hat{x}}_{2}) / \sqrt{2}$ , ${\hat{p}}_{a} = (e^{- r_{1}} {\hat{p}}_{1} + e^{r_{2}} {\hat{p}}_{2}) / \sqrt{2}$ , ${\hat{x}}_{b} = (e^{r_{1}} {\hat{x}}_{1} + e^{- r_{2}} {\hat{x}}_{2}) / \sqrt{2}$ , and ${\hat{p}}_{b} = (e^{- r_{1}} {\hat{p}}_{1} + e^{r_{2}} {\hat{p}}_{2}) / \sqrt{2}$ , where the subscripts (1 and 2) indicate the vacuum fields injected into the OPA and r is a squeezing parameter. The following correlations are derived as the features of quadrature entanglement:

{\hat{x}}_{a} - {\hat{x}}_{b} = \sqrt{2} e^{- r_{2}} {\hat{x}}_{2},

{\hat{p}}_{a} + {\hat{p}}_{b} = \sqrt{2} e^{- r_{1}} {\hat{p}}_{1} .

Maximum correlations are obtained in the limit of infinite squeezing.

After an unknown input state is combined with entangled beam a, the canonically conjugate quadratures are jointly measured. They can be expressed as ${\hat{x}}_{u} = ({\hat{x}}_{in} - {\hat{x}}_{a}) / \sqrt{2}$ and ${\hat{p}}_{v} = ({\hat{p}}_{in} + {\hat{p}}_{a}) / \sqrt{2}$ , where the subscript “in” indicates the unknown input state. This CV Bell measurements yield various classical values for x_u and p_v that are random for each light pulse. The instruments used in these CV Bell measurements consist of the 50/50 BS and two PHDs, as shown in Fig. 1. The quadrature component of entangled beam b in any local oscillator (LO) phase ϕ_b can be expressed by using the measured values x_u and p_v as [15]:

{\hat{X}}_{b} (ϕ_{b}) = ({\hat{x}}_{in} - \sqrt{2} e^{- r_{2}} {\hat{x}}_{2} - \sqrt{2} x_{u}) cos ϕ_{b} + ({\hat{p}}_{in} + \sqrt{2} e^{- r_{1}} {\hat{p}}_{1} - \sqrt{2} p v) sin ϕ_{b}

where X̂_b(ϕ_b) is defined by x̄_b cosϕ_b + p̂_b sinϕ_b. This state is identical to the input state, x̄_in cosϕ_b + p̂_in sinϕ_b, for the ideal case (r₁, r₂ → ∞) except for the classical phase-space displacement of

- \sqrt{2} (x_{u} cos ϕ_{b} + p_{v} sin ϕ_{b})

. In a standard quantum teleportation scheme, entangled beam b is displaced in phase space by amplitude and phase modulators. We did not displace entangled beam b in the present experiment, but we displaced the measured value, X_b(ϕ_b), for individual light pulses by using the random x_u and p_v obtained from CV Bell measurements after measuring X̂_b(ϕ_b) by using PHD. The values after displacement X_out(ϕ_b) can be expressed as

X_{out} (ϕ_{b}) = X_{b} (ϕ_{b}) + \sqrt{2} (x_{u} cos ϕ_{b} + p_{v} sin ϕ_{b}) .

This posterior displacement can easily be achieved for pulsed-light CV entanglement where each pulse can be regarded as a single mode and the correspondence between entangled pairs is well identified. The quality of transportation is evaluated by using the variance of X_out(ϕ_b). When a vacuum state or coherent state is used as the input state, 〈Δ²X_out(ϕ_b)〉 at any ϕ_b is at least three times larger than the vacuum state variance for classical transportation without entanglement [2].

The quantum state itself is not transported in our experimental scheme for transporting CV quantum variables with posterior displacement of measured values (without displacement operation on the entangled beam b). Therefore, measurements of observables other than quadrature-phase amplitudes will give completely different results from measurements on the input state except for when the results of the CV Bell measurements (x_u and p_v) are close to zero. When both x_u and p_v are close to zero, there is no need for unitary operation to transport quantum states analogous to obtaining |ψ⁻ > out of four Bell states for quantum teleportation of discrete variables [16]. However, if quadrature-phase amplitudes are measured just after displacement operation on one of the entangled pairs as has been done in previous CV experiments, posterior-displaced measured-values will be identical to the quantum teleportation results with an ideal displacement operation. Therefore, the present method may be useful for CV quantum communication protocols such as the quantum repeaters for entanglement-based CV quantum cryptography [17].

3. Experimental setup

Figure 2 shows the schematics of the main experimental apparatus. The fundamental wave (FW) is delivered from a passively Q-switched erbium-doped glass laser, which produces optical pulses at a wavelength of 1.535 μm with a duration of 3.9 ns and a repetition rate of 2.7 kHz (Cobolt model Tango). A small fraction of FW is used as the LO for homodyne detection, and the most of this is frequency doubled in a periodically poled LiNbO₃ waveguide (PPLN-WG, not shown in Fig. 2) [18,19]. A second harmonic wave (SHW) at 767 nm is employed as pump light for OPA.

Fig. 2 Schematic of experimental apparatus. Blue solid line and red dotted line correspond to light beams at wavelengths of 1.535 μm and 767 nm. Blue dashed line indicates the propagation of two squeezed beams and entanglement.

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The ring interferometer is employed to generate entanglement, and the method of generating entanglement is described in detail in Ref. [13]. The three dielectric polarizing beamsplitters (DPBS1-3) only act as polarizing beam splitter for 1.535 μm. The two harmonic wave plates (HWP1 and 2) serve as λ/2 plates at a wavelength of 1.535 μm and as a λ plate at 767 nm. The two dual-wavelength beamsplitters (DBS1 and 2) divide the horizontally polarized beam at wavelengths of both 1.535 μm and 767 nm (transmissivity/reflectivity for 1.535 μm is 50/50, and that for 767 nm is 52/48). After passing through the DPBS1 and HWP1, the horizontally polarized SHW (the red dotted line in Fig. 1) and the vertically polarized FW (the solid line) are introduced into the ring interferometer. The SHW is divided into two beams, and is used to pump the PPLN-WG from both sides. The two counter-propagating squeezed beams (dashed line) are generated by bidirectional pumping. Entanglement is generated from the DBS1 by controlling the relative phase between squeezed beams θ to π/2 using BK7 plates. The vertically polarized FW is reflected by DBS1 and propagated into three PHDs (PHD1-3) after passing through the ring interferometer.

PHD1 and 2 are used to carry out the CV Bell measurements. One of the entangled beams and the LO, which have orthogonal polarizations, propagate collinearly (upward direction after DBS1 in Fig. 2) and they are divided into two and combined with input vacuum state by using 50/50 BS. The conjugate quadratures, x_u and p_v, are simultaneously measured in PHD1 and PHD2 by shifting the relative phase using a properly designed birefringence plate (BP). The quadrature values, X_b(ϕ_b), of the other entangled beam are measured in PHD3. The relative phase, ϕ_b, between the other entangled beam and LO can be adjusted by displacing one of the wedged BK7 plates.

4. Results and discussion

The correlation properties between x_u or p_v and X_b(ϕ_b) are measured, before the quantum transportation experiment was done. Figure 3 shows the measured noise variance in the sum and difference in quadratures while the LO phase in PHD3 is scanned. Each point is calculated from the quadrature values of pairs of 2000 pulses. As seen in Fig. 3(a), ϕ_b = 0 and 2π are defined for the minimum variance of 〈Δ²[x_u + X_b(ϕ_b)]〉. The quantum correlation of 〈Δ²[x_u + X_b(0)]〉 = 0.40 (−0.9 dB compared to the corresponding vacuum variances of 0.5) is obtained after correcting for electronic noise. The electronic noise levels of PHD1, PHD2, and PHD3 are 8.8 dB, 9.5 dB, and 11.0 dB below the shot noise levels (SNLs). When ϕ_b is chosen as π/2, π, and 3/2π, the quantum correlations of 〈Δ²[p_v + X_b(π/2)]〉 = 〈Δ²[x_u − X_b(π)]〉 = 〈Δ²[p_u − X_b(3π/2)]〉 = 0.40 (−0.9 dB) are also respectively observed. The output pump average powers from PPLN-WG are 80 and 70 μW (corresponding to peak powers of 7.6 and 6.6 W). The squeezing parameters of r₁ = 1.16 and r₂ = 1.38 are estimated from these powers. Taking into account the loss in ring interferometer (ξ₁ = 0.77 and ξ₂ = 0.68) and the total detection efficiencies (η₁ = η₂ = 0.60, η ₃ = 0.58 for PHD1, 2, and 3, respectively), 〈Δ² [x_u + X_b(0)]〉 = 〈Δ²[p_v + X_b(π/2)]〉 = 〈Δ²[x_u − X_b(π)]〉 = 〈Δ²[p̂_u − X_b(3π/2)]〉 = 0.41 (−0.9 dB) are calculated, and the black solid lines in Fig. 3 represent theoretical curves (no fitting parameters) [13]. These theoretical predictions are in good agreement with the experimental values. In both of Fig. 3(a) and (b), the noise levels oscillates asymmetrically. These asymmetry is caused by the different squeezing levels r ₁ ≠ r₂). By using these parameters, the inseparability, $Δ_{I}^{2} = 〈 Δ^{2} ({\hat{x}}_{a} - {\hat{x}}_{b}) 〉 + 〈 Δ^{2} ({\hat{p}}_{a} + {\hat{p}}_{b}) 〉 = 0.30 + 0.31 = 0.61$ , is also calculated. This means that the sufficient condition for entanglement is satisfied ( $Δ_{I}^{2} < 1$ ) [20, 21].

Fig. 3 Measured correlation properties between x_u or p_v and X_b(ϕ_b). (a) Blue squares indicate 〈Δ²[x_u + X_b(ϕ_b)]〉 and red circles indicate 〈Δ²[x_u − X_b(ϕ_b)]〉. (b) Blue squares indicate 〈 Δ²[p_v + X_b(ϕ_b)]〉 and red circles indicate 〈Δ²[p_v − X_b(ϕ_b)]〉.

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Figure 4 shows the noise variance of X_out(ϕ_b) as a function of ϕ_b. With the use of entanglement, the variance 〈Δ²X_out〉 obtained by averaging over all ϕ_b is 0.63 ± 0.04 (4.0 ±0.3 dB compared to the corresponding vacuum variance of 0.25), and fidelity F is achieved as 0.57 ± 0.03. For classical transportation in the absence of entanglement, the measured noise variance, 〈Δ²X_out〉 = 0.75 ± 0.01 (4.8 ± 0.1 dB), is measured, which corresponds to F_c = 0.50 ± 0.01. The fidelity of F = 0.57 ± 0.03 is beyond the classical bound of F_c. Thus, quantum transportation for individual pulses is experimentally demonstrated.

Fig. 4 Noise variances of X_out(ϕ_b) with (blue squares) and without (red circles) entanglement.

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The fidelity of the transport state depends directly on the squeezing levels, which can be expressed as [22]

F_{eff} = 1 / (1 + e^{- 2 r_{eff}}),

where r_eff is the effective squeezing parameter. In our experiments, considering the loss in the ring interferometer and the total detection efficiencies, r_eff = 0.25, and thus F_eff = 0.62 was expected. This value is slightly higher than the experimentally obtained fidelity of F = 0.57 ± 0.03. One possible cause for this discrepancy is due to the temporal fluctuations in ϕ_b during the measuring time of 2 minutes when measurement were done [8].

5. Conclusion

In summary, we reported the transportation of continuous variables of individual vacuum light pulses. Time-domain PHDs and pulsed entanglement were used to transport individual pulses. The estimated fidelity was beyond the classical bound without entanglement. This is the first experimental report on the quantum transportation of continuous variables of individual pulses. Our scheme using pulsed sources at telecommunication wavelengths should prove useful for practical applications of CV QICT.

Acknowledgments

The authors gratefully acknowledge by M. Kozuma, K. Honda, R. Inoue, and A. Noguchi for their valuable comments. This work was supported by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology of Japan, and CREST, Japan Science and Technology (JST) Corporation. One of the authors (Y. E.) was supported by a Research Fellowship of the Japan Society for the Promotion of Science for Young Scientists.

References and links

1. L. Vaidman, “Teleportation of quantum states,” Phys. Rev. A 49, 1473–1476 (1994). [CrossRef] [PubMed]

2. S. L. Braunstein and H. J. Kimble, “Teleportation of Continuous Quantum Variables,” Phys. Rev. Lett. 80, 869–872 (1998). [CrossRef]

3. C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69, 2881–2884 (1992). [CrossRef] [PubMed]

4. S. L. Braunstein and H. J. Kimble, “Dense coding for continuous variables,” Phys. Rev. A 61, 042302 (2000). [CrossRef]

5. A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991). [CrossRef] [PubMed]

6. A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional Quantum Teleportation,” Science 282, 706–709 (1998). [CrossRef] [PubMed]

7. W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph, H.-A. Bachor, T. Symul, and P. K. Lam, “Experimental investigation of continuous-variable quantum teleportation,” Phys. Rev. A 67, 032302 (2003). [CrossRef]

8. T. C. Zhang, K. W. Goh, C. W. Chou, P. Lodahl, and H. J. Kimble, “Quantum teleportation of light beams,” Phys. Rev. A 67, 033802 (2003). [CrossRef]

9. X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, “Quantum Dense Coding Exploiting a Bright Einstein-Podolsky-Rosen Beam,” Phys. Rev. Lett. 88, 047904 (2002). [CrossRef] [PubMed]

10. J. Mizuno, K. Wakui, A. Furusawa, and M. Sasaki, “Experimental demonstration of entanglement-assisted coding using a two-mode squeezed vacuum state,” Phys. Rev. A 71, 012304 (2005). [CrossRef]

11. H. Hansen, T. Aichele, C. Hettich, P. Lodahl, A. I. Lvovsky, J. Mlynek, and S. Schiller, “Ultrasensitive pulsed, balanced homodyne detector: application to time-domain quantum measurements,” Opt. Lett. 26, 1714–1716 (2001). [CrossRef]

12. R. Okubo, M. Hirano, Y. Zhang, and T. Hirano, “Pulse-resolved measurement of quadrature phase amplitudes of squeezed pulse trains at a repetition rate of 76 MHz,” Opt. Lett. 33, 1458–1460 (2008). [CrossRef] [PubMed]

13. Y. Eto, A. Nonaka, Y. Zhang, and T. Hirano, “Stable generation of quadrature entanglement using a ring interferometer,” Phys. Rev. A 79, 050302 (2009). [CrossRef]

14. D. T. Smithey, M. Beck, and M. G. Raymer, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993). [CrossRef] [PubMed]

15. P. van Loock, S. L. Braunstein, and H. J. Kimble, “Broadband teleportation,” Phys. Rev. A 62, 022309 (2000). [CrossRef]

16. D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental Quantum Teleportation,” Nature 390, 575–579 (1997). [CrossRef]

17. Ch. Silberhorn, N. Korolkova, and G. Leuchs, “Quantum Key Distribution with Bright Entangled Beams,” Phys. Rev. Lett. 88, 167902 (2002). [CrossRef] [PubMed]

18. Y. Eto, T. Tajima, Y. Zhang, and T. Hirano, “Pulsed Homodyne Detection of Squeezed Light at Telecommunication Wavelength,” Jpn. J. Appl. Phys. 45(Part 2), L821–L823 (2006). [CrossRef]

19. Y. Eto, T. Tajima, Y. Zhang, and T. Hirano, “Observation of squeezed light at 1.535 μm using a pulsed homodyne detector,” Opt. Lett. 32, 1698–1700 (2007). [CrossRef] [PubMed]

20. L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability Criterion for Continuous Variable Systems,” Phys. Rev. Lett. 84, 2722–2725 (2000). [CrossRef] [PubMed]

21. R. Simon, “Peres-Horodecki Separability Criterion for Continuous Variable Systems,” Phys. Rev. Lett. 84, 2726–2729 (2000). [CrossRef] [PubMed]

22. S. Suzuki, H. Yonezawa, F. Kannari, M. Sasaki, and A. Furusawa, “7 dB quadrature squeezing at 860 nm with periodically poled KTiOPO₄,” Appl. Phys. Lett. 89, 061116 (2006). [CrossRef]

Transporting continuous quantum variables of individual light pulses

Abstract

1. Introduction

2. Scheme for transporting continuous quantum variables

3. Experimental setup

4. Results and discussion

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (5)

Optics Express