March 15, 2015 / Vol. 40, No. 6 / OPTICS LETTERS
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Continuous-variable entanglement measurement using an unbalanced Mach–Zehnder interferometer Chuanqing Xia,1 Dong Wang,1,2,4 Yang Wu,1 Jiale Guo,1 Fang Liu,1 Yong Zhang,1,* and Min Xiao1,3 1
National Laboratory of Solid State Microstructures, College of Engineering and Applied Sciences, School of Physics, Nanjing University, Nanjing 210093, China 2
School of Mathematics and Physics, Anhui University of Technology, Maanshan, 243032, China 3
Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA 4 e-mail: [email protected] *Corresponding author: [email protected] Received November 20, 2014; accepted February 3, 2015; posted February 11, 2015 (Doc. ID 228263); published March 13, 2015
We propose a simple scheme using an unbalanced Mach–Zehnder interferometer to directly measure the entangled beams with correlation of amplitude quadratures and anticorrelation of phase quadratures. In the experiment, we use two fibers with lengths of 2 and 50 m to construct the interferometer. The correlation variances of amplitude difference and phase sum of the entangled beams from a nondegenerate optical parametric amplifier are simultaneously measured to be 1.79 and 1.62 dB below the shot noise limit, respectively. Such a simple and convenient method has potential applications in quantum measurements. © 2015 Optical Society of America OCIS codes: (270.0270) Quantum optics; (270.6570) Squeezed states; (190.4410) Nonlinear optics, parametric processes. http://dx.doi.org/10.1364/OL.40.001121
The entanglement of continuous variables (CV) plays a very important role in quantum information [1,2]. The Einstein–Podolsky–Rosen (EPR) paradox [3], quantum teleportation [4], quantum-dense coding [5,6], quantum image [7], quantum key distribution [8], and quantum logic gate [9] have been experimentally demonstrated. Among these experiments, a typical balanced-homodynedetection (BHD) scheme has been widely used for the measurement of various CV quantum states. One set of BHD can measure the amplitude or phase quadrature of a signal beam with the help of an intensive local oscillator (LO) [10]. By using two sets of BHDs, one can measure the two-party entanglement [3]. However, it is crucially important to develop simpler and more efficient methods to measure the quantum entanglement for future quantum processes. A direct Bell-state measurement (DBSM) without LO proposed by Zhang and Peng [11] has greatly simplified the measurement of such quantum entanglement, which has been used in quantum-dense coding [5,6] and entanglement swapping [12]. However, it works only for the EPR beams with anticorrelation of amplitude quadratures (hδ2 Xˆ 1 Xˆ 2 i < SNL), and correlation of phase quadratures (hδ2 Yˆ 1 − Yˆ 2 i < SNL). Here, Xˆ 1 (Yˆ 1 ) and Xˆ 2 (Yˆ 2 ) are the quadrature-amplitude (quadrature-phase) operators of the entangled beams, and SNL stands for the shot noise limit. There exists another important CV EPR entanglement source, i.e., twin beams with correlation of amplitude quadratures (hδ2 Xˆ 1 − Xˆ 2 i < SNL) and anticorrelation of phase quadratures (hδ2 Yˆ 1 Yˆ 2 i < SNL). Such entanglement can be generated from a nondegenerate optical parametric oscillator (NOPO) above threshold [13], from a nondegenerate optical parametric amplifier (NOPA) operating in an amplification mode below threshold [14], or by combining two phase-squeezed lights at a 50/50 beam splitter [4]. In the NOPO case, the entangled twin beams are frequency-nondegenerate, which can be measured by two BHD systems with LOs at different 0146-9592/15/061121-04$15.00/0
frequencies [15], by synchronously scanning a pair of analysis cavities [16], or by two sets of unbalanced Mach–Zehnder interferometers (UMZI) [17,18]. For the NOPA case, the bright twin beams have degenerate frequencies and orthogonal polarizations, which, however, cannot be directly measured up to now. Due to high intensity of bright twin beams (usually in a range from several hundred microwatts to several milliwatts [19]), homodyne detectors are not applicable, because the photodiode is very likely to be saturated considering a far more intensive LO involved. To avoid this problem, Ref. [19] has proposed an indirect method to infer the entanglement from the coupled bright mode and dark mode. However, it is inconvenient. In this Letter, we experimentally realize a novel DBSM scheme based on UMZI to simultaneously measure amplitude correlations and phase anti-correlations of twin beams. Mach–Zehnder interferometer has been exploited in many quantum-enhanced measurements to beat the standard quantum limits [20–23]. Its unbalanced configuration (i.e., UMZI) is extremely useful in quantum interference [24], continuous-variable quantum erasing [25], squeezed-state purification [26], and three-color entanglement [27]. Comparing to the indirect homodyne method in Ref. [19], our UMZI scheme can measure quantum correlations of amplitude quadratures and phase quadratures simultaneously, which is essential to demonstrate the quantum entanglement from a NOPA. Moreover, our setup is much simpler because only one interferometer is required and no LO needed. In the experiment, we observed 1.79 dB of amplitude correlation and 1.62 dB of phase correlation below the SNL between the twin beams from a NOPA operating in an amplification mode. It should be noted that this method also works for the entangled modes generated by combining two phase-squeezed lights at a beam splitter [4]. The theoretical model of the UMZI is shown in Fig. 1. We define annihilation operators aˆ and bˆ denoting the © 2015 Optical Society of America
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1 δiˆc Ω αδXˆ a Ω δYˆ a Ω − δXˆ b Ω δYˆ b Ω; 2 1 δiˆd Ω αδXˆ a Ω − δYˆ a Ω − δXˆ b Ω − δYˆ b Ω: 2
Fig. 1. Theoretical measurement scheme for the entanglement with correlation of amplitude quadratures and anticorrelation of phase quadratures. PBS: polarization beam splitter; BS: 50/ 50 beam splitter; D: detector; RFS: radio-frequency splitter.
entangled modes, which are generated from a NOPA and are split by a polarization beam splitter (PBS). Using the linear operator method, the entangled modes can be writˆ α δbt. ˆ ˆ α δat ˆ and bt ten as at Here, α is the classical real amplitude of the modes, which is assumed ˆ to be the same for both the entangled modes. δat and ˆ δbt are the corresponding fluctuations, which have mean values of zero. aˆ and bˆ travel through the short arm and the long arm of the UMZI, respectively. Due to the optical length difference, a time delay of τ ˆ Here, ΔL is the length ΔL∕c is introduced in the mode b. difference of the UMZI and c is the speed of light in vacˆ is phase-shifted by uum. Then, one mode (for example, b) a value of π∕2 and interferes with the other one at a 50/50 beam splitter. The output modes are given by 1 π ˆ − τ ; ˆ ei2 bt cˆ t p at 2 π ˆ p1 at ˆ − τ : ˆ − ei2 bt dt 2
(1)
(3)
The AC component of each measured photocurrent is divided into two parts through a radio frequency splitter (RFS). The sum and difference of the divided photocurrents, which correspond to amplitude-difference and phase-sum noises between the twin beams, respectively, can be written as α2 2 ˆ δ X a Ω − Xˆ b Ω; 2 α2 δiˆ2− Ω δ2 Yˆ a Ω Yˆ b Ω: 2
δiˆ 2 Ω
(4)
Equation (4) decides whether the modes aˆ and bˆ are entangled with correlation of amplitude quadratures and anticorrelation of phase quadratures, according to the nonseparability criterion for Gaussian system [28], ˆ ˆ X Ω − Xˆ Ω Y Ω Yˆ Ω δ2 a p b δ2 a p b < 2. 2 2 (5) The experimental setup is shown in Fig. 2. The quantum light is provided by a CV entanglement source (CV-Entanglor, Yuguang Co. Ltd., Taiyuan, China). An all-solid-state, intracavity-frequency-doubled, frequencystabilized, continuous wave Nd:YAP/KTP laser simultaneously outputs 540 nm green light and 1080 nm near-infrared light, which serve as the pump and the seed lights for the NOPA, respectively. The NOPA is built in a semimonolithic configuration with a 10-mm-long, α-cut,
The photon number operators iˆ c cˆ † cˆ and iˆ d dˆ † dˆ can be calculated by 1 iˆc t α2 αδXˆ a t δYˆ a t 2 δXˆ b t − τ − δYˆ b t − τ; 1 iˆd t α2 αδXˆ a t − δYˆ a t 2 ˆ δX b t − τ δYˆ b t − τ;
(2)
where the quadrature components Xˆ o and Yˆ o (o a; b) are defined as Xˆ o oˆ oˆ † and Yˆ o oˆ − oˆ † ∕i, respectively. Through Fourier transformation, we can obtain the corresponding noise spectrum. Noting that Fourier transformation F gives Ff t − τ eiΩτ Ff t, the spectral component of a signal with a delay of τ at the sideband frequency of Ω is phase shifted by θ Ωτ. By adjusting the length difference between the two arms of the UMZI to realize θ π, the noise components at sideband frequency Ω can be written as
Fig. 2. Experimental setup. DBS: dichroic beam splitter; BS: 50/50 beam splitter; PBS: polarization beam splitter; EOM: electro-optical modulator; IO: optical isolator; PZT: piezoelectric transducer; HWP: half-wave plate; QWP: quarter-wave plate; FC: fiber coupler; PD: photodiode; RFS: radio-frequency splitter; SA: spectral analyzer.
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type-II-phase-matched KTP crystal. The coated front face of the KTP crystal has transmissivities of >95% at 540 nm and ∼0.5% at 1080 nm, which is used as the input coupler of the NOPA cavity. The end face of the crystal is coated with dual-band antireflection film at both 540 and 1080 nm. A concave mirror with a 50-mm curvature radius is used as the output coupler for the generated EPR beams. It has a ∼4% transmissivity at 1080 nm and a high reflectivity at 540 nm. The measured fineness and the free spectral range of the cavity at 1080 nm are 145 GHz and 2.5 GHz, respectively. The output coupler is mounted on a piezoelectric transducer (PZT) to actively lock the cavity on resonance at 1080 nm by means of frequency modulation (FM) sideband technique with an EOM [29]. The pump at 540 nm is polarized along the b axis of the KTP crystal. The injected seed at 1080 nm is polarized at 45° relative to the b axis of the KTP crystal, which can be decomposed to frequency-degenerate but orthogonally-polarized signal and idler modes. By carefully tuning the crystal temperature, the birefringence between signal and idler waves in the KTP crystal is compensated and their simultaneous resonances in the cavity are realized. PZT1 and PZT2 in Fig. 2 are used to sweep and lock the relative phase between the pump at 540 nm and the seed at 1080 nm. When the injected seed and pump are in phase, the NOPA operates in a state of parametric amplification, which outputs twin beams with correlated amplitude quadratures and anticorrelated phase quadratures [14,19]. In the experiment, the powers of the pump, the injected seed, and the output twin beams are 196 mW, 1.15 mW, and 324 uW, respectively. Next we set up the UMZI to measure the generated entangled beams. At an analysis frequency of v 2 MHz, ΔL will be 75 m in free space according to the theoretical model above, which is hard to achieve in experiment. Therefore, we choose fibers to construct the interferometer. Considering that the refractive index of fiber is 1.55, the length difference ΔL is reduced to be 48 m. In experiment, we choose two single-mode fibers with lengths of 2 and 50 m as the two arms of the UMZI. After the twin beams travel through the telescope in Fig. 2, which is used for mode matching with fibers, the orthogonally polarized signal and idler beams are separated by a PBS and then are coupled into short and long arms of the UMZI, respectively. The coupling efficiencies of the two fibers are measured to be 82.5% and 81.3%, respectively. Due to fiber birefringence, the output light from a singlemode fiber is usually elliptically polarized [30]. So two quarter-wave plates (QWP1 and QWP2 in Fig. 2) are used in our experiment to turn the light polarizations back to linear. Then, two half-wave plates (HWP3 and HWP4 in Fig. 2) are used to rotate their polarizations along the same orientation before the interference at a 50/50 beam splitter. Their relative phase difference of π∕2 is controlled by PZT3 in Fig. 2. The interference efficiency is 97.4%. The experimental result is shown in Fig. 3. The correlation variances of amplitude difference and phase sum at 2 MHz analysis frequency are measured to be 1.79 0.11 dB and 1.62 0.15 dB below the SNL, respectively. To measure the corresponding SNL, we block the pump light before the NOPA and adjust the near-infrared powers at detectors to be the same as the entangled
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Fig. 3. Normalized noise power of correlation variances: (a) amplitude difference hδ2 Xˆ a − Xˆ b i, (b) phase sum hδ2 Yˆ a Yˆ b i. Trace i is the SNL. Trace ii is the measured correlation variance. The analysis frequency, the resolution bandwidth, and the video bandwidth of the SA are 2 MHz, 100 kHz, and 30 Hz, respectively. Here, we have omitted the electric noise, which is more than 10 dB below SNL.
beams. After counting the total propagation efficiency (70.2%) and the quantum efficiencies of detectors (80.8%), one can infer that the correlation degrees of amplitude difference and phase sum after the output coupler of NOPA are 4.11 dB and 3.60 dB, respectively. The squeezing degree of phase sum is smaller than that of amplitude difference because of pump fluctuation and Brillouin scattering. According to the nonseparability criterion for Gaussian system [28], ˆ ˆ X a Ω − Xˆ b Ω Y a Ω Yˆ b Ω 2 2 p p δ δ 2 2 1.352 < 2;
(6)
we can confirm that the signal and idler beams from the NOPA are entangled. In conclusion, we have experimentally realized a simple and direct measurement scheme for frequencydegenerate twin beams with correlation of amplitude quadratures and anticorrelation of phase quadratures. An UMZI consisting of two fibers with different lengths is employed. No local oscillator is required. It should be noted that the work mechanism of our UMZI is totally different from those in Refs. [17,18]. In our work, the phase and amplitude correlations can be simultaneously measured while the phase and amplitude quadratures have to be separately measured in Refs. [17,18]. Quantum entanglement of the twin beams from a NOPA operating in an amplification mode is experimentally demonstrated by this method. The directly measured correlation variances of amplitude difference and phase sum are 1.79 dB and 1.62 dB below the SNL, respectively. Such fiberbased UMZI scheme has high extendibility and applicability in various quantum measurement experiments. This work was supported by the National Basic Research Program of China (Nos. 2012CB921804 and 2011CBA00205), the National Natural Science Foundation of China (Nos. 11321063, 61205115, 11274162, and 61222503) and the Natural Science Research Program
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of Higher Education Institution of Anhui Province (No. KJ2012Z023). The authors also thank Yuguang Co. for the help in the CV-entanglement source. References 1. S. L. Braunstein and P. Van Loock, Rev. Mod. Phys. 77, 513 (2005). 2. M. Reid, P. Drummond, W. Bowen, E. G. Cavalcanti, P. Lam, H. Bachor, U. L. Andersen, and G. Leuchs, Rev. Mod. Phys. 81, 1727 (2009). 3. Z. Ou, S. F. Pereira, H. Kimble, and K. Peng, Phys. Rev. Lett. 68, 3663 (1992). 4. A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, Science 282, 706 (1998). 5. X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, Phys. Rev. Lett. 88, 047904 (2002). 6. J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, Phys. Rev. Lett. 90, 167903 (2003). 7. V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, Science 321, 544 (2008). 8. L. S. Madsen, V. C. Usenko, M. Lassen, R. Filip, and U. L. Andersen, Nat. Commun. 3, 1083 (2012). 9. X. Su, S. Hao, X. Deng, L. Ma, M. Wang, X. Jia, C. Xie, and K. Peng, Nat. Commun. 4, 2828 (2013). 10. H. P. Yuen and V. W. Chan, Opt. Lett. 8, 177 (1983). 11. J. Zhang and K. Peng, Phys. Rev. A 62, 064302 (2000). 12. X. Jia, X. Su, Q. Pan, J. Gao, C. Xie, and K. Peng, Phys. Rev. Lett. 93, 250503 (2004). 13. M. Reid and P. Drummond, Phys. Rev. Lett. 60, 2731 (1988). 14. Y. Zhang, H. Su, C. Xie, and K. Peng, Phys. Lett. A 259, 171 (1999).
15. J. Jing, S. Feng, R. Bloomer, and O. Pfister, Phys. Rev. A 74, 41804 (2006). 16. A. Villar, L. Cruz, K. Cassemiro, M. Martinelli, and P. Nussenzveig, Phys. Rev. Lett. 95, 243603 (2005). 17. O. Glöckl, U. Andersen, S. Lorenz, C. Silberhorn, N. Korolkova, and G. Leuchs, Opt. Lett. 29, 1936 (2004). 18. X. Su, A. Tan, X. Jia, Q. Pan, C. Xie, and K. Peng, Opt. Lett. 31, 1133 (2006). 19. Y. Zhang, H. Wang, X. Li, J. Jing, C. Xie, and K. Peng, Phys. Rev. A 62, 023813 (2000). 20. M. Xiao, L. A. Wu, and H. J. Kimble, Phys. Rev. Lett. 59, 278 (1987). 21. M. I. Kolobov and P. Kumar, Opt. Lett. 18, 849 (1993). 22. V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330 (2004). 23. L. Pezze and A. Smerzi, Phys. Rev. Lett. 100, 073601 (2008). 24. Y. Guo, W. Zhang, S. Dong, Y. Huang, and J. Peng, Opt. Lett. 39, 2526 (2014). 25. U. L. Andersen, O. Glöckl, S. Lorenz, G. Leuchs, and R. Filip, Phys. Rev. Lett. 93, 100403 (2004). 26. O. Glöckl, U. Andersen, R. Filip, W. Bowen, and G. Leuchs, Phys. Rev. Lett. 97, 053601 (2006). 27. X. Jia, Z. Yan, Z. Duan, X. Su, H. Wang, C. Xie, and K. Peng, Phys. Rev. Lett. 109, 253604 (2012). 28. L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 84, 2722 (2000). 29. R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983). 30. S. Rashleigh, IEEE J. Lightwave Technol. 1, 312 (1983).
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Continuous-variable entanglement measurement using an unbalanced Mach–Zehnder interferometer Chuanqing Xia,1 Dong Wang,1,2,4 Yang Wu,1 Jiale Guo,1 Fang Liu,1 Yong Zhang,1,* and Min Xiao1,3 1
National Laboratory of Solid State Microstructures, College of Engineering and Applied Sciences, School of Physics, Nanjing University, Nanjing 210093, China 2
School of Mathematics and Physics, Anhui University of Technology, Maanshan, 243032, China 3
Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA 4 e-mail: [email protected] *Corresponding author: [email protected] Received November 20, 2014; accepted February 3, 2015; posted February 11, 2015 (Doc. ID 228263); published March 13, 2015
We propose a simple scheme using an unbalanced Mach–Zehnder interferometer to directly measure the entangled beams with correlation of amplitude quadratures and anticorrelation of phase quadratures. In the experiment, we use two fibers with lengths of 2 and 50 m to construct the interferometer. The correlation variances of amplitude difference and phase sum of the entangled beams from a nondegenerate optical parametric amplifier are simultaneously measured to be 1.79 and 1.62 dB below the shot noise limit, respectively. Such a simple and convenient method has potential applications in quantum measurements. © 2015 Optical Society of America OCIS codes: (270.0270) Quantum optics; (270.6570) Squeezed states; (190.4410) Nonlinear optics, parametric processes. http://dx.doi.org/10.1364/OL.40.001121
The entanglement of continuous variables (CV) plays a very important role in quantum information [1,2]. The Einstein–Podolsky–Rosen (EPR) paradox [3], quantum teleportation [4], quantum-dense coding [5,6], quantum image [7], quantum key distribution [8], and quantum logic gate [9] have been experimentally demonstrated. Among these experiments, a typical balanced-homodynedetection (BHD) scheme has been widely used for the measurement of various CV quantum states. One set of BHD can measure the amplitude or phase quadrature of a signal beam with the help of an intensive local oscillator (LO) [10]. By using two sets of BHDs, one can measure the two-party entanglement [3]. However, it is crucially important to develop simpler and more efficient methods to measure the quantum entanglement for future quantum processes. A direct Bell-state measurement (DBSM) without LO proposed by Zhang and Peng [11] has greatly simplified the measurement of such quantum entanglement, which has been used in quantum-dense coding [5,6] and entanglement swapping [12]. However, it works only for the EPR beams with anticorrelation of amplitude quadratures (hδ2 Xˆ 1 Xˆ 2 i < SNL), and correlation of phase quadratures (hδ2 Yˆ 1 − Yˆ 2 i < SNL). Here, Xˆ 1 (Yˆ 1 ) and Xˆ 2 (Yˆ 2 ) are the quadrature-amplitude (quadrature-phase) operators of the entangled beams, and SNL stands for the shot noise limit. There exists another important CV EPR entanglement source, i.e., twin beams with correlation of amplitude quadratures (hδ2 Xˆ 1 − Xˆ 2 i < SNL) and anticorrelation of phase quadratures (hδ2 Yˆ 1 Yˆ 2 i < SNL). Such entanglement can be generated from a nondegenerate optical parametric oscillator (NOPO) above threshold [13], from a nondegenerate optical parametric amplifier (NOPA) operating in an amplification mode below threshold [14], or by combining two phase-squeezed lights at a 50/50 beam splitter [4]. In the NOPO case, the entangled twin beams are frequency-nondegenerate, which can be measured by two BHD systems with LOs at different 0146-9592/15/061121-04$15.00/0
frequencies [15], by synchronously scanning a pair of analysis cavities [16], or by two sets of unbalanced Mach–Zehnder interferometers (UMZI) [17,18]. For the NOPA case, the bright twin beams have degenerate frequencies and orthogonal polarizations, which, however, cannot be directly measured up to now. Due to high intensity of bright twin beams (usually in a range from several hundred microwatts to several milliwatts [19]), homodyne detectors are not applicable, because the photodiode is very likely to be saturated considering a far more intensive LO involved. To avoid this problem, Ref. [19] has proposed an indirect method to infer the entanglement from the coupled bright mode and dark mode. However, it is inconvenient. In this Letter, we experimentally realize a novel DBSM scheme based on UMZI to simultaneously measure amplitude correlations and phase anti-correlations of twin beams. Mach–Zehnder interferometer has been exploited in many quantum-enhanced measurements to beat the standard quantum limits [20–23]. Its unbalanced configuration (i.e., UMZI) is extremely useful in quantum interference [24], continuous-variable quantum erasing [25], squeezed-state purification [26], and three-color entanglement [27]. Comparing to the indirect homodyne method in Ref. [19], our UMZI scheme can measure quantum correlations of amplitude quadratures and phase quadratures simultaneously, which is essential to demonstrate the quantum entanglement from a NOPA. Moreover, our setup is much simpler because only one interferometer is required and no LO needed. In the experiment, we observed 1.79 dB of amplitude correlation and 1.62 dB of phase correlation below the SNL between the twin beams from a NOPA operating in an amplification mode. It should be noted that this method also works for the entangled modes generated by combining two phase-squeezed lights at a beam splitter [4]. The theoretical model of the UMZI is shown in Fig. 1. We define annihilation operators aˆ and bˆ denoting the © 2015 Optical Society of America
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1 δiˆc Ω αδXˆ a Ω δYˆ a Ω − δXˆ b Ω δYˆ b Ω; 2 1 δiˆd Ω αδXˆ a Ω − δYˆ a Ω − δXˆ b Ω − δYˆ b Ω: 2
Fig. 1. Theoretical measurement scheme for the entanglement with correlation of amplitude quadratures and anticorrelation of phase quadratures. PBS: polarization beam splitter; BS: 50/ 50 beam splitter; D: detector; RFS: radio-frequency splitter.
entangled modes, which are generated from a NOPA and are split by a polarization beam splitter (PBS). Using the linear operator method, the entangled modes can be writˆ α δbt. ˆ ˆ α δat ˆ and bt ten as at Here, α is the classical real amplitude of the modes, which is assumed ˆ to be the same for both the entangled modes. δat and ˆ δbt are the corresponding fluctuations, which have mean values of zero. aˆ and bˆ travel through the short arm and the long arm of the UMZI, respectively. Due to the optical length difference, a time delay of τ ˆ Here, ΔL is the length ΔL∕c is introduced in the mode b. difference of the UMZI and c is the speed of light in vacˆ is phase-shifted by uum. Then, one mode (for example, b) a value of π∕2 and interferes with the other one at a 50/50 beam splitter. The output modes are given by 1 π ˆ − τ ; ˆ ei2 bt cˆ t p at 2 π ˆ p1 at ˆ − τ : ˆ − ei2 bt dt 2
(1)
(3)
The AC component of each measured photocurrent is divided into two parts through a radio frequency splitter (RFS). The sum and difference of the divided photocurrents, which correspond to amplitude-difference and phase-sum noises between the twin beams, respectively, can be written as α2 2 ˆ δ X a Ω − Xˆ b Ω; 2 α2 δiˆ2− Ω δ2 Yˆ a Ω Yˆ b Ω: 2
δiˆ 2 Ω
(4)
Equation (4) decides whether the modes aˆ and bˆ are entangled with correlation of amplitude quadratures and anticorrelation of phase quadratures, according to the nonseparability criterion for Gaussian system [28], ˆ ˆ X Ω − Xˆ Ω Y Ω Yˆ Ω δ2 a p b δ2 a p b < 2. 2 2 (5) The experimental setup is shown in Fig. 2. The quantum light is provided by a CV entanglement source (CV-Entanglor, Yuguang Co. Ltd., Taiyuan, China). An all-solid-state, intracavity-frequency-doubled, frequencystabilized, continuous wave Nd:YAP/KTP laser simultaneously outputs 540 nm green light and 1080 nm near-infrared light, which serve as the pump and the seed lights for the NOPA, respectively. The NOPA is built in a semimonolithic configuration with a 10-mm-long, α-cut,
The photon number operators iˆ c cˆ † cˆ and iˆ d dˆ † dˆ can be calculated by 1 iˆc t α2 αδXˆ a t δYˆ a t 2 δXˆ b t − τ − δYˆ b t − τ; 1 iˆd t α2 αδXˆ a t − δYˆ a t 2 ˆ δX b t − τ δYˆ b t − τ;
(2)
where the quadrature components Xˆ o and Yˆ o (o a; b) are defined as Xˆ o oˆ oˆ † and Yˆ o oˆ − oˆ † ∕i, respectively. Through Fourier transformation, we can obtain the corresponding noise spectrum. Noting that Fourier transformation F gives Ff t − τ eiΩτ Ff t, the spectral component of a signal with a delay of τ at the sideband frequency of Ω is phase shifted by θ Ωτ. By adjusting the length difference between the two arms of the UMZI to realize θ π, the noise components at sideband frequency Ω can be written as
Fig. 2. Experimental setup. DBS: dichroic beam splitter; BS: 50/50 beam splitter; PBS: polarization beam splitter; EOM: electro-optical modulator; IO: optical isolator; PZT: piezoelectric transducer; HWP: half-wave plate; QWP: quarter-wave plate; FC: fiber coupler; PD: photodiode; RFS: radio-frequency splitter; SA: spectral analyzer.
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type-II-phase-matched KTP crystal. The coated front face of the KTP crystal has transmissivities of >95% at 540 nm and ∼0.5% at 1080 nm, which is used as the input coupler of the NOPA cavity. The end face of the crystal is coated with dual-band antireflection film at both 540 and 1080 nm. A concave mirror with a 50-mm curvature radius is used as the output coupler for the generated EPR beams. It has a ∼4% transmissivity at 1080 nm and a high reflectivity at 540 nm. The measured fineness and the free spectral range of the cavity at 1080 nm are 145 GHz and 2.5 GHz, respectively. The output coupler is mounted on a piezoelectric transducer (PZT) to actively lock the cavity on resonance at 1080 nm by means of frequency modulation (FM) sideband technique with an EOM [29]. The pump at 540 nm is polarized along the b axis of the KTP crystal. The injected seed at 1080 nm is polarized at 45° relative to the b axis of the KTP crystal, which can be decomposed to frequency-degenerate but orthogonally-polarized signal and idler modes. By carefully tuning the crystal temperature, the birefringence between signal and idler waves in the KTP crystal is compensated and their simultaneous resonances in the cavity are realized. PZT1 and PZT2 in Fig. 2 are used to sweep and lock the relative phase between the pump at 540 nm and the seed at 1080 nm. When the injected seed and pump are in phase, the NOPA operates in a state of parametric amplification, which outputs twin beams with correlated amplitude quadratures and anticorrelated phase quadratures [14,19]. In the experiment, the powers of the pump, the injected seed, and the output twin beams are 196 mW, 1.15 mW, and 324 uW, respectively. Next we set up the UMZI to measure the generated entangled beams. At an analysis frequency of v 2 MHz, ΔL will be 75 m in free space according to the theoretical model above, which is hard to achieve in experiment. Therefore, we choose fibers to construct the interferometer. Considering that the refractive index of fiber is 1.55, the length difference ΔL is reduced to be 48 m. In experiment, we choose two single-mode fibers with lengths of 2 and 50 m as the two arms of the UMZI. After the twin beams travel through the telescope in Fig. 2, which is used for mode matching with fibers, the orthogonally polarized signal and idler beams are separated by a PBS and then are coupled into short and long arms of the UMZI, respectively. The coupling efficiencies of the two fibers are measured to be 82.5% and 81.3%, respectively. Due to fiber birefringence, the output light from a singlemode fiber is usually elliptically polarized [30]. So two quarter-wave plates (QWP1 and QWP2 in Fig. 2) are used in our experiment to turn the light polarizations back to linear. Then, two half-wave plates (HWP3 and HWP4 in Fig. 2) are used to rotate their polarizations along the same orientation before the interference at a 50/50 beam splitter. Their relative phase difference of π∕2 is controlled by PZT3 in Fig. 2. The interference efficiency is 97.4%. The experimental result is shown in Fig. 3. The correlation variances of amplitude difference and phase sum at 2 MHz analysis frequency are measured to be 1.79 0.11 dB and 1.62 0.15 dB below the SNL, respectively. To measure the corresponding SNL, we block the pump light before the NOPA and adjust the near-infrared powers at detectors to be the same as the entangled
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Fig. 3. Normalized noise power of correlation variances: (a) amplitude difference hδ2 Xˆ a − Xˆ b i, (b) phase sum hδ2 Yˆ a Yˆ b i. Trace i is the SNL. Trace ii is the measured correlation variance. The analysis frequency, the resolution bandwidth, and the video bandwidth of the SA are 2 MHz, 100 kHz, and 30 Hz, respectively. Here, we have omitted the electric noise, which is more than 10 dB below SNL.
beams. After counting the total propagation efficiency (70.2%) and the quantum efficiencies of detectors (80.8%), one can infer that the correlation degrees of amplitude difference and phase sum after the output coupler of NOPA are 4.11 dB and 3.60 dB, respectively. The squeezing degree of phase sum is smaller than that of amplitude difference because of pump fluctuation and Brillouin scattering. According to the nonseparability criterion for Gaussian system [28], ˆ ˆ X a Ω − Xˆ b Ω Y a Ω Yˆ b Ω 2 2 p p δ δ 2 2 1.352 < 2;
(6)
we can confirm that the signal and idler beams from the NOPA are entangled. In conclusion, we have experimentally realized a simple and direct measurement scheme for frequencydegenerate twin beams with correlation of amplitude quadratures and anticorrelation of phase quadratures. An UMZI consisting of two fibers with different lengths is employed. No local oscillator is required. It should be noted that the work mechanism of our UMZI is totally different from those in Refs. [17,18]. In our work, the phase and amplitude correlations can be simultaneously measured while the phase and amplitude quadratures have to be separately measured in Refs. [17,18]. Quantum entanglement of the twin beams from a NOPA operating in an amplification mode is experimentally demonstrated by this method. The directly measured correlation variances of amplitude difference and phase sum are 1.79 dB and 1.62 dB below the SNL, respectively. Such fiberbased UMZI scheme has high extendibility and applicability in various quantum measurement experiments. This work was supported by the National Basic Research Program of China (Nos. 2012CB921804 and 2011CBA00205), the National Natural Science Foundation of China (Nos. 11321063, 61205115, 11274162, and 61222503) and the Natural Science Research Program
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