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Dispersion-free quantum clock synchronization via fiber link
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Advances in Space Research 50 (2012) 1489–1494 www.elsevier.com/locate/asr
Dispersion-free quantum clock synchronization via fiber link Fei-yan Hou a, Rui-fang Dong a,⇑, Run-ai Quan a, Yu Zhang a,b, Yun Bai a,b, Tao Liu a, Shou-gang Zhang a, Tong-yi Zhang c a
Key Laboratory of Time and Frequency Primary Standards, National Time Service Center, Chinese Academy of Sciences, Xian 710600, China b Graduate University of Chinese Academy of Sciences, Beijing 100039, China c State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xian 710119, China Received 24 February 2012; received in revised form 24 July 2012; accepted 26 July 2012 Available online 4 August 2012
Abstract We proposed a fiber-based quantum clock synchronization protocol by employing the dispersion cancellation feature of the frequency anti-correlated entangled source under the quantum interference measurement. It is shown that, the accuracy of the synchronization is mainly dependent on the bandwidth of the frequency entangled biphoton spectrum and on the temperature variation induced fluctuations in both the reference fiber coiling and the fiber link. With this proposal, synchronization between two clocks at a distance of ten kilometers can be implemented with an accuracy below a picosecond. Ó 2012 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Frequency anti-correlated entanglement; Dispersion-free; Quantum clock synchronization via fiber link
1. Introduction High-accuracy synchronization of spatially separated atomic clocks plays an important role in fundamental physics and in a wide range of applications such as communications, message encryption, navigation, geolocation and homeland security. Einstein synchronization is the most applicable method for clock synchronization which is implemented by an exchange of electromagnetic pulses between two spatially separated clocks and measuring their time of arrival (Einstein, 1905). Based on this method, atomic clocks are compared by means of the satellite-based Global Positioning System (GPS) (Parkinson and Spilker 1996; Kaplan 1996; Hofmann-Wellenhof et al., 1993), and by employing telecommunication satellites for twoway satellite time and frequency transfer (TWSTFT) (Kirchner, 1999; Bauch et al., 2009). The time scales can
⇑ Corresponding author.
E-mail address: [email protected] (R.-f. Dong).
be synchronized with an uncertainty up to 1nanosecond. However, such accuracy is far from enough to satisfy the growing requirement for comparison of the new generation of high-precision atomic clocks, meanwhile limits many other prospective applications, such as coherent detection of electromagnetic signals and fundamental physics principle tests. In a recent future, the ACES mission (Atomic Clock Ensemble in Space) based on an elaborated manipulation of radiofrequency signals will demonstrate an improved accuracy as low as 100 ps (Cacciapuoti and Salomon, 2009). To further improve the synchronization accuracy, the time transfer by laser link (T2L2) was proposed and applied in the real satellite-based synchronization system (Fridelance and Veillet, 1995; Fridelance et al., 1997; Guillemot et al., 2008; Vrancken, 2008). With a better control of light pulses than radio wave propagation, the recent T2L2 experiment gives the best clock synchronization accuracy within 50 ps (Guillemot et al., 2008; Vrancken, 2008), though it has a strong dependence on the weather condition. Owing to the low loss, strong anti-jamming ability, high transfer security, and wider transmission
0273-1177/$36.00 Ó 2012 COSPAR. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.asr.2012.07.029
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bandwidth of optical fiber, time transfer through optical fibers has demonstrated a further reduction of uncertainty (Czubla et al., 2006; Sliwczynski et al., 2010). Therefore, synchronization via optical fibers becomes a promising candidate as a part of the infrastructure connecting the ground station setups during forthcoming T2L2 and ACES experiments, supporting deep-space network and antenna arrays in astronomy (Calhoun et al., 2007) or in accelerators for particle physics (Kartner et al., 2010). One of the most important characteristics among the above mentioned methods is the uncertainty evaluation of atomic clocks and time scales. Upon all kinds of noises contributing to the achievable accuracy, the noise resulting from the time-of-arrival measurement is the fundamental noise and is classically limited by the available power and bandwidth, which is the so called shot noise limit. For example, according to the settings of the T2L2 instrument on board the Jason-2 space vehicle, this limit contributes an accuracy of about 3.5 ps over an integration time of 100 s. Quantum clock synchronization is a new technology which utilizes quantum sources and quantum measurement technique in the clock synchronization system to break through the shot noise limit on the classical system, finally reaches the fundamental limit for quantum mechanics, that is, the Heisenberg limit. Since it was proposed in the beginning of this century (Giovannetti et al., 2001a), quantum clock synchronization has attracted much research interests (Giovannetti et al., 2002; Giovannetti et al., 2001b; Giovannetti et al., 2004; Valencia et al., 2004; Bahder and Golding, 2004). According to recent investigations, quantum clock synchronization, as a highly promising technology to improve the classical synchronization accuracy dramatically, will not only be capable of quantum encrypted (Giovannetti et al., 2002), but also have the ability of cancelling propagation noise induced by dispersion in the optical path (Giovannetti et al., 2001b; Giovannetti et al., 2004). As light pulses will be broadened by the group velocity dispersion (GVD) after passing through dispersive medium, dispersion is one of the main restrictions to improve the accuracy of the fiber-link based clock synchronization systems. In this paper we propose a fiber-based quantum clock synchronization scheme to eliminate the timing error due to the GVD in the fiber, based on the quantum feature that the dispersion experienced by the frequency anticorrelated entangled source can be cancelled by the quantum interference measurement in the frame of reference of a Hong-Ou-Mandel (HOM) interferometer (Bahder and Golding, 2004; Hong et al., 1987; Steinberg et al., 1992a,b). Through a thorough analysis, we show that the final synchronization accuracy is jointly determined by the bandwidth of the frequency entangled biphoton spectrum as well as the temperature variation induced fluctuations in both the reference fiber coiling and the fiber link. The rest of the paper is organized as follows. Section 2 is a brief description of the scheme. Section 3 gives the theoretical derivation. Section 4 is the conclusion.
2. Scheme description The dispersion-free fiber-link based quantum clock synchronization scheme is sketched in Fig. 1. Positions of clock A and clock B are linked by a fiber with an unknown length d. To synchronize clock A and clock B, the following procedures are implemented. At clock A position, frequency anti-correlated photon pairs are generated by spontaneous parametric down-conversion (SPDC) in a nonlinear crystal (Louisell et al., 1961; Mollow, 1973; Rubin et al., 1994). In the SPDC process, a photon from an intense cw pump beam is occasionally annihilated due to the nonlinear interaction inside the crystal and two photons with lower frequencies, named as signal and idler, are created. The signal and idler photons are then departed spatially and disseminated through different fiber paths. The signal is traveled from A to B and then reflected back through the fiber link, while the idler is held at A and sent through the reference fiber coiling with an adjustable length of l0. After traveling through different fiber paths, the signal and idler photons are interfered at a 50/50 beam splitter. The two outputs of the beam splitter are subsequently detected by two single-photon counters D1 and D2. The photon count outputs are then sent into a high-speed AND-logic circuit for coincidence measurement. Such measurement is the so-called Hong-Ou-Mandel (HOM) interferometric measurement (Hong et al., 1987), which is also named as second-order quantum interference measurement. The length of the reference fiber coiling l0 is adjusted until the interferometer is balanced, which corresponds to the minimum coincidence. Once the HOM interferometer is balanced, the time delay in the fiber coiling l0 will be equal to the round-trip time delay in the fiber link 2d , that is, b0 12d = b00 1l0, where b0 1 and b00 1 denote the inverse group velocities in the fiber link and the adjustable fiber coiling, respectively. The achievable accuracy of such balance is then determined by the resolution of the HOM interferometer, which is shown limited by the spectral bandwidth of the entangled photons and unaffected by group velocity dispersion in the fiber (for details see Section 3). Since the adjustable fiber coiling l0 is in the laboratory, standard techniques can be applied to determine the delay in the fiber coiling accurately and, hence, the delay in the fiber link. Maintaining the balance of the HOM interferometer, clocks A and B are used to record the SPDC photons creation times at position A {tA(i)}, and the arrival times of the signal photons at position B {tB(i)} after traveling through the fiber link, where i = 1. . .N denotes a series of N photon pairs. The time data are brought together through a classical communication channel for comparison, and the registration time difference tB tA is obtained by maximum “coincidences” of the photon registration time records. When the path length d is known, the measurement resolution is determined by the natural width of G(2) for SPDC (Valencia et al., 2004), which is given by the Fourier transform of the
F.-y. Hou et al. / Advances in Space Research 50 (2012) 1489–1494
A
B
Signal
Frequency entanglement source Idler
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BS D1
D2
C Classical Channel Fig. 1. Sketch of the fiber-link quantum clock synchronization scheme. Positions of clock A and clock B is linked by a fiber path with an unknown length d. The Signal and Idler are a pair of frequency anti-correlated entangled beams. The length l0 of the fiber coiling is adjustable and measurable. After passing through the different fiber paths, the Signal and Idler are interfered on a 50/50 beam splitter. D1 and D2 are two single photon detectors for counting the two output photons of the beam splitter. The C box represents the logic unit for coincidence detection. Clock A records the SPDC photon pair creation time and clock B the time when the Signal photon arrive at B after traveling through the fiber link. The registration time tA(i) and tB(i) are brought together via a classical channel to measure tB tA.
biphoton spectrum and typically on the order of a few femtoseconds to hundreds of femtoseconds. Assume the real time difference between clock A and B is t0 , it is then estimated by t0 = (tB tA) – b0 1d (Valencia et al., 2004). Informed this time offset, the clocks are then synchronized accordingly. 3. Theoretical analysis To investigate the achievable accuracy of the above fiber-linked clocks synchronization scheme, the theoretical analysis is presented as follows. Consider a cw pump at the frequency of xp and the degenerate phase matching such that the center frequencies of the SPDC generated signal and idler both equal half the pump frequency. This system produces a biphoton source of the form Z x x p p 0 þ 0 ^ jwi ¼ dx0 f ðx0 Þ^ þ x x aþ a j0i; ð1Þ s i 2 2 ^þ where ^ aþ s ðxs Þ and a i ðxi Þ are the creation operators for the signal and idler, respectively; j0i is the vacuum state. |f(x0 )|2 is the frequency spectrum of the biphoton state. As a signal photon at frequency xp/2 + x0 is accompanied by an idler photon at frequency xp/2 x0 , the signal and idler are frequency anti-correlated entangled. The signal and idler photons are then separated and traveled through the fiber link and the adjustable reference fiber coiling, respectively. Due to the frequency dependence of the refractive index n(x) in optical fibers, chromatic dispersion is induced during propagation of the signal and idler photons. This effect is described by the mode-propagation constant b(x), which can be expanded as Taylor series about the frequency x0 = xp/2 (Agrawal, 2001)
bðxÞ ¼
nðxÞx c
¼ b0 þ b1 ðx x0 Þ þ
1 b ðx x0 Þ2 þ ; 2! 2
ð2Þ
where c is the velocity of light in vacuum, bm is defined as m d b bm ¼ with m ¼ 1; 2; 3; . . . ð3Þ dxm x¼x0 The first order term b1 can be interpreted as the inverse of the group velocity tg, the speed at which the pulse envelop travels (Agrawal, 2001): 1 dn 1 dn ng 1 nþx nk b1 ¼ ð4Þ ¼ ¼ ¼ ; c dx c dk tg c where k is the wavelength corresponding to x, and ng is the group refractive index. The second order term describes the frequency dependence of the propagation speed (Agrawal, 2001): 1 dn d 2n x d 2n k3 d 2 n b2 ¼ 2 þx 2 c dx dx c dx2 2pc2 dk2 ¼
k2 D; 2pc
ð5Þ
which represents the dispersion of the group velocity and is responsible for pulse broadening or distortion. This phenomenon is known as group velocity dispersion (GVD), and D ¼ 2pc b is the GVD parameter. b1 and b2 are nork2 2 mally specified by manufacturer of the fiber. After passing through the fiber link d and the fiber coiling l0 respectively, the signal and idler photons are then interfered on a 50/50 beam splitter, as shown in Fig. 1. The annihilation operators for the two outputs 1 and 2 of the beam splitter are related to those for the signal and idler by the following formula:
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00
^ a1 ðxÞ ¼ piffiffi2 ^ as ðxÞeib ðxÞ2d þ p1ffiffi2 ^ ai ðxÞeib ^ a2 ðxÞ ¼
00 piffiffi ^ a ðxÞeib ðxÞl0 2 i
þ
ðxÞl0
;
0 p1ffiffi ^ a ðxÞeib ðxÞ2d ; 2 s
ð6Þ
where the propagation constants of the fiber link d and the fiber coiling l0 are denoted as b0 and b00 , respectively. Due to the Heisenberg uncertainty relation, these operators obey the canonical communication relations: ^ aj ðxj Þ; ^ ak ðxk Þ ¼ 0; h i ^ aþ aþ k ðxk Þ ¼ 0; j ðxj Þ; ^ ð7Þ ^aj ðxj Þ; ^ aþ ðx Þ ¼ d dðx x Þ; k jk j k k j; k ¼ 1; 2; s; i: According to the Mandel formula for detection (Hong et al., 1987), the coincidence rate at the photodetectors is given as follows: Z T Z T þ þ PC ¼ dt1 dt2 hWjE ð8Þ 1 ðt 1 ÞE 2 ðt 2 ÞE2 ðt 2 ÞE 1 ðt 1 ÞjWi; 0
0
where T is the integration time window of the detectors. Omitting the irrelevant normalization constants, the positive- and negative-frequency field operators at detector j are defined by R Eþ dxj ^ aj ðxj Þeixj tj ; j ðt j Þ ¼ R ð9Þ þixj tj E dxj ^ ; forj ¼ 1; 2: aþ j ðxj Þe j ðt j Þ ¼ Substituting (9) and (6) into (8) and taking the limit T ! 1 without loss of generality, one obtains Z P C / dx1 Z dx2 hWj^ aþ aþ a2 ðx2 Þ^ a1 ðx1 ÞjWi: ð10Þ 1 ðx1 Þ^ 2 ðx2 Þ^ Using Eq. (1)–(5) and neglecting the dispersion terms higher than second order dispersion, we evaluate the integrand as below
From Eq. (12), it is apparent that the GVD terms have no contribution to the final HOM interferometric measurement. In the simple case of a Gaussian spectrum |f(x0 )|2 with variance Dx2, Eq. (12) becomes 2 ðb00 l b0 2dÞ2 1 0 1
P C ¼ 1 e2Dx
ð13Þ
It follows from Eq. (13) that, the zero coincidence is obtained at b0 12d = b00 1l0 with a dip width 1/(2Dx). Since the adjustable fiber coiling is in the laboratory, the time delay in it (sl0 ¼ b001 l0 ) can be measured and controlled accurately. Under the assumption that Dsl0 ¼ 0, the delay in the fiber link is then determined by sd = b0 1d = b00 1l0/2 with an accuracy Dsd = 1/(4Dx), which is shown independent of the group-velocity-dispersion (GVD) effect inherent in the glass fiber. Accordingly, given the group velocity parameter b0 1 of the fiber link, its length will be estimated as d = b00 1l0/(2b0 1) with an uncertainty Dd = 1/(4Dxb0 1). If the fiber link and the fiber coiling are both normal singlemode fibers, the group refractive indices will be approximately the same, i.e. b0 1 = b00 1, therefore d l0/2. As described in Section 2, the time offset between clocks A and B t0 is given by t0 = (tB tA) b0 1d. To measure t0, a series of registration time difference tB(i)tA(i) must be measured first, where tA(i) is the ith SPDC photon pair creation time recorded by clock A while tB(i) the ith signal photon arrival time recorded by clock B after traveling through the fiber link. It is mathematically expressed by the following second-order coherence function (Valencia et al., 2004) 2
þ Gð2Þ ðtB tA Þ ¼ jh0jEþ s ðd; t B ÞE i ð0; t A ÞjWij :
ð14Þ
The electric field operators of the signal and idler phoþ tons Eþ s ðd; t B Þ; Ei ð0; t A Þ are given by
hWj^aþ aþ a2 ðx2 Þ^ a1 ðx1 ÞjWi 1 ðx1 Þ^ 2 ðx2 Þ^ ¼ jh0j^ a1 ðx1 Þ^ a2 ðx2 ÞjWij2 2 00 0 0 00 ¼ 12 h0j^ ai ðx1 Þ^ as ðx2 Þeib ðx1 Þl0 þib ðx2 Þ2d ^ as ðx1 Þ^ ai ðx2 Þeib ðx1 Þ2dþib ðx2 Þl0 jWi " # 2 xp xp xp xp
xp ei½b000 þb001 ðx1 2 Þþ12b002 ðx1 2 Þ2 l0 þi½b00 þb01 ðx2 2 Þþ12b02 ðx2 2 Þ2 2d 1 ¼ 2 dðxp x1 x2 Þf 2 x1 xp xp 2 xp xp 2 0 0 0 00 00 00 1 1 i½b0 þb1 ðx1 2 Þþ2b2 ðx1 2 Þ 2dþi½b0 þb1 ðx2 2 Þþ2b2 ðx2 2 Þ l0 e h 00 0 i xp xp 2 xp 0 00 00 0 0 1 00 x 2 ¼ j 12 dðxp x1 x2 Þf ð 2p x1 Þeiðb0 l0 þb0 2dÞþi2ðb2 l0 þb2 2dÞðx1 2 Þ eiðb1 l0 b1 2dÞðx1 2 Þ eiðb1 2db1 l0 Þðx1 2 Þ j h i xp 00 0 x 2 2 ¼ 12 jdðxp x1 x2 Þj jf ð 2p x1 Þj 1 Re eiðb1 l0 b1 2dÞ2ð 2 x1 Þ :
We substitute (11) into (10) and drop the overall numerical factors, the final coincidence rate is yielded Z P C / dx0 jf ðx0 Þj2 f1 cos½2x0 ðb001 l0 b01 2dÞg: ð12Þ
Eþ s ðd; t B Þ
¼
Eþ i ð0; t A Þ ¼
Z Z
ð11Þ
0
dx^as ðxÞeixðtB tB0 Þ eib ðxÞd ; dx^ai ðxÞeixðtA tA0 Þ ;
ð15Þ
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where tA0 and tB0 denote the times at which Clock A and B start their clocks, t0 = tB0 tA0 is thus the time offset between these clocks. Substituting (15)–(23) and neglecting the dispersion terms higher than the second-order dispersion term, one gets Z 2 2 x0 0 0 Gð2Þ ðtB tA Þ ¼ dx0 f ðx0 Þei 2 b2 d eix s Z 2 2 x0 0 0 ¼ dx0 f ðx0 Þei 2 b2 d eix s ; ð16Þ where s = (tB tA) t0 b01 d. When the far-field condition is satisfied, which is reasonable in most application cases, the second-order correlation function is approximated to: 2 2 1 2 0s 2 s 2Dx ðb dÞ ð2Þ 0 2 G ðtB tA Þ f x ¼ 0 e : ð17Þ b2 d One can see that, it has the same shape as the Fourier transform of the SPDC spectrum, only broadened by a factor of Dx2b0 2d. Since d is well measured by d l0/2 based on the HOM setup, and b02 is normally specified by manufacturer of the fiber, the measurement of tB tA can reach, in principle, the same order of resolution as the natural width of G(2) for SPDC (Valencia et al., 2004) DðtB tA Þ 1=Dx:
ð18Þ
The accuracy of the time difference between clock A and B t0 is then given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 17 Dt0 ¼ D2 ðtB tA Þ þ D2 ðb01 dÞ ¼ : ð19Þ 16Dx2 It follows from Eq. (19) that, the accuracy of the scheme is dependent only on the bandwidth Dx of the twin beam. Suppose that the bandwidth is Dx 1013 Hz, the accuracy is given by Dt0 0.1 ps. However, the propagation delay in the fiber coiling is fluctuating, i.e., Dsl0 – 0. Among various factors, the uncertainty of the propagation delay in the fiber coiling is mainly caused by the fiber temperature variation and can be expressed as Dsl0 ¼ DT l0 ðb001 l0 Þ ¼ l0
@b001 @l0 DT l0 þ b001 DT l0 : @T @T
ð20Þ
where DT l0 denotes the temperature variation of the fiber coiling, the first right-hand-side term concerns the temperature dependence of the group velocity, and the second one the fiber thermal expansion. The two terms describe the thermal sensitivity of the fiber coiling propagation delay. Such fluctuation contributes an error to the determination of the delay in the fiber link Dsd sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2 1 1 ð21Þ Dsd ¼ þ DT l0 ðb001 l0 Þ : 4Dx 2
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Likewise, the measurement of tB tA is also affected by the temperature variation of the fiber link between clocks A and B, which can be given by s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 DðtB tA Þ ¼ þ Dx DT d ðb02 dÞ Dx s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 1 2 ð22Þ þ ½Dk DT d ðD0 dÞ ; ¼ Dx where DTd denotes the temperature variation of the fiber link, Dk is the corresponding wavelentgh bandwidth of the biphoton spectrum. Go back to Eq. (19), the error in determinating the time offset of clock A and B is then given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dt0 ¼ D2 ðtB tA Þ þ D2 sd sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 00 17 b 1 l0 2 2 þ DT l ð23Þ þ ½Dk DT d ðD0 dÞ : 2 0 16Dx 2 Suppose that the twin beam is centered at 1.55 lm, the bandwidth of Dx 1013 Hz corresponds to Dk 13 nm. According to Ref. (Sliwczynski et al., 2010), the thermal sensitivity of the fiber propagation delay in 1.55 lm transmission window is around 37 ps/(K km). Suppose that the fiber coiling length in our scheme is l0 = 20 km, and its temperature variation DT l0 is controlled within 1 mK, the uncertainty of the propagation delay in the fiber coiling will be Dsl0 0:74 ps . Substituting it to Eq. (21), one gets Dsd 0.37 ps. In single-mode fibers, the thermal sensitivity of the GVD parameter can be taken as 1.5 103 ps/(K km nm) (Sliwczynski et al., 2010). For the fiber link with the length of d l0/2 10 km and the temperature fluctuation of 2 K, the determination error of tB tA caused by the temperature fluctuation is Dk DT d ðD0 dÞ 0:39 ps. Adding this to Eq. (22), the total accuracy of the measurement of tB tA then gives D(tB tA) 0.4 ps. After including the temperature variation induced fluctuations, the error of the time offset of clock A and B, which is departed at a fiber link distance of about 10 km, is then Dt0 0.54 ps. Such error determines the synchronization accuracy between the two clocks. 4. Conclusion In conclusion, we have demonstrated a high-accuracy fiber-based quantum clock synchronization scheme. In this scheme, two distant clocks, which are linked by an unknown fiber, can be accurately synchronized with the help of a frequency anti-correlated entangled biphoton source, quantum correlation measurement setups, and a reference fiber coiling, which is placed in the laboratory with its length capable of being precisely controlled and measured. In ideal case, the achievable synchronization accuracy Dt0 is only dependent on the bandwidth of the biphoton spectrum. So the higher the bandwidth of the
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biphoton spectrum is, the more accurate the synchronization will be. The accuracy is unaffected either by the GVD effect or by the fiber propagation length. In reality however, due to the inevitable fiber temperature variations, the accuracy will be limited by the fiber propagation lengths. Based on our scheme, utilizing a biphoton source with a bandwidth of 13 nm to synchronize the two clocks at a fiber distance of 10 km, 0.54 ps accuracy can be achieved, which provides a highly promising application in comparing two atomic clocks within a distance of a few to tens of kilometers with an accuracy below a picosecond. Acknowledgments This work was supported by the NSFC 11174282, NSFC Y133ZK1101, the key project fund of the Chinese Academy of Sciences for the “Western light” Talent Cultivation Plan 180 (2011). The authors thank Professor Longsheng Ma for valuable discussions. References Agrawal, G.P. Nonlinear Fiber Optics, third ed Academic Press, San Diego, 2001. Bahder, T.B., Golding, W.M., Clock synchronization based on secondorder coherence of entangled photons. In: Proceeding 7th International Conference on Quantum, Communication, July 2004. Bauch, A., Piester, D., Blanzano, B., Koudelka, O., Kroon, E., Dierikx, E., Whibberley, P., Achkar, J., Rovera, D., Lorini, L., Cordara, F., Schlunegger, C., Results of the 2008 TWSTFT calibration of seven european stations. In: Proceedings European Frequency and Time Forum – IEEE Frequency Control Symposium Joint Conference, Besancßon, France, pp. 1209–1215, 2009. Cacciapuoti, L., Salomon, C. Space clocks and fundamental tests: the ACES experiment. Eur. Phys. J. Special Topics 172, 57–68, 2009. Calhoun, M., Huang, S., Tjoelker, R.L. Stable photonic links for frequency and time transfer in the deep-space network and antenna arrays. Proc. IEEE 95, 1931–1946, 2007. Czubla, A., Konopka, J., Gornik, M., Adamowicz, W., Strus, J., Pawszak, T., Romsicki, J., Lipinski, M., Krehlik, P., Sliwczynski, L., Wolczko, A. Comparison of precise time transfer with usage of multi-channel GPS CV receivers and optical fibers over distances of about 3 kilometers. In: Proceedings 38th Annual Precise Time and Time Interval (PTTI) Systems and Applications Meeting, Reston, Virginia, USA, pp. 337–345, 2006. Einstein, A. Zur Elektrodynamik bewegter Korper. Ann. Phys. (Leipzig) 17, 891–921, 1905. Fridelance, P., Samain, P.E., Veillet, C. T2L2-time transfer by laser link: a new optical time transfer generation. Exp. Astron. 7, 191–207, 1997.
Fridelance, P., Veillet, C. Operation and data analysis in the LASSO experiment. Metrologia 32, 27–33, 1995. Giovannetti, V., Lloyd, S., Maccone, L. Quantum-enhanced positioning and clock synchronization. Nature 412, 417–419, 2001a. Giovannetti, V., Lloyd, S., Maccone, L. Clock synchronization with dispersion cancellation. Phys. Rev. Lett. 87, 117902(4pp), 2001b. Giovannetti, V., Lloyd, S., Maccone, L. Quantum cryptographic ranging. J. Opt. B: Quantum Semiclass. Opt. 4, 413–414, 2002. Giovannetti, V., Lloyd, S., Maccone, L., Shapiro, J.H., Wong, F.N.C. Conveyor-belt clock synchronization. Phys. Rev. A 70, 043808(8pp), 2004. Giovannetti, V., Maccone, L., Shapiro, J.H., Wong, F.N.C. Extended phase-matching conditions for improved entanglement generation. Phys. Rev. A 66, 043813(9pp), 2002. Guillemot, P.H., Samain, E., Vrancken, P., Exertier, P., Leon, S. Time transfer by laser link – T2L2: an opportunity to calibrate rf links. In: Proceedings of the PTTI, pp 95–106, 2008. Hofmann-Wellenhof, B., Lichtenegger, H., Collins, J. Global positioning system theory and practice. Springer-Verlag, New York, 1993. Hong, C.K., Ou, Z.Y., Mandel, L. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett. 59, 2044–2046, 1987. Kaplan, E.D. Understanding GPS: Principles and Applications Mobile Communications Series. Artech House, Boston, 1996. Kartner, F.X., Kim, J., Cox, J., Chen, J., Nejadmalayeri, A.H. Femtosecond precision timing distribution for accelerators and light sources. In: Proceedings IEEE International Frequency Control Symposium, Newport Beach, CA, USA, pp. 564–568, 2010. Kirchner, D. Two-way satellite time and frequency transfer (TWSTFT): principle, implementation, and current performance. Review of Radio Sciences. Oxford University Press, pp. 27–44, 1996. Louisell, W.H., Yariv, A., Siegman, A.E. Quantum fluctuations and noise in parametric processes. I. Phys. Rev. 124, 1646–1654, 1961. Mollow, B.R. Photon correlations in the parametric frequency splitting of light. Phys. Rev. A 8, 2684–2694, 1973. Parkinson, B.W., Spilker, J.J. Global Positioning System: Theory and Applications. Vols. I and II, Progress in Astronautics and Aeronautics, vols. 163 and 164 Amer. Inst. Aero. stro, Washington, D.C., 1996. Rubin, M.H., Klyshko, D.N., Shih, Y.H., Sergienko, A.V. Theory of twophoton entanglement in type-II optical parametric down-conversion. Phys. Rev. A 50, 5122–5133, 1994. Sliwczynski, L., Krehlik, P., Lipinski, M. Optical fibers in time and frequency transfer. Meas. Sci. Technol. 21, 075302(11pp), 2010. Steinberg, A.M., Kwiat, P.G., Chiao, R.Y. Dispersion cancellation in a measurement of the single-photon propagation velocity in glass. Phys. Rev. Lett. 68, 2421–2424, 1992a. Steinberg, A.M., Kwiat, P.G., Chiao, R.Y. Dispersion cancellation and high-resolution time measurements in a fourth-order optical interferometer. Phys. Rev. A 45, 6659–6665, 1992b. Valencia, A., Scarcelli, G., Shih, Y. Distant clock synchronization using entangled photon pairs. Appl. Phys. Lett. 85, 2635–2657, 2004. Vrancken, P. Characterization of T2L2 (time transfer by laser link) on the Jason 2 ocean altimetry satellite and micrometric laser ranging. Thesis, Universite´ de Nice – Sophia Antipolis, 2008.
Advances in Space Research 50 (2012) 1489–1494 www.elsevier.com/locate/asr
Dispersion-free quantum clock synchronization via fiber link Fei-yan Hou a, Rui-fang Dong a,⇑, Run-ai Quan a, Yu Zhang a,b, Yun Bai a,b, Tao Liu a, Shou-gang Zhang a, Tong-yi Zhang c a
Key Laboratory of Time and Frequency Primary Standards, National Time Service Center, Chinese Academy of Sciences, Xian 710600, China b Graduate University of Chinese Academy of Sciences, Beijing 100039, China c State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xian 710119, China Received 24 February 2012; received in revised form 24 July 2012; accepted 26 July 2012 Available online 4 August 2012
Abstract We proposed a fiber-based quantum clock synchronization protocol by employing the dispersion cancellation feature of the frequency anti-correlated entangled source under the quantum interference measurement. It is shown that, the accuracy of the synchronization is mainly dependent on the bandwidth of the frequency entangled biphoton spectrum and on the temperature variation induced fluctuations in both the reference fiber coiling and the fiber link. With this proposal, synchronization between two clocks at a distance of ten kilometers can be implemented with an accuracy below a picosecond. Ó 2012 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Frequency anti-correlated entanglement; Dispersion-free; Quantum clock synchronization via fiber link
1. Introduction High-accuracy synchronization of spatially separated atomic clocks plays an important role in fundamental physics and in a wide range of applications such as communications, message encryption, navigation, geolocation and homeland security. Einstein synchronization is the most applicable method for clock synchronization which is implemented by an exchange of electromagnetic pulses between two spatially separated clocks and measuring their time of arrival (Einstein, 1905). Based on this method, atomic clocks are compared by means of the satellite-based Global Positioning System (GPS) (Parkinson and Spilker 1996; Kaplan 1996; Hofmann-Wellenhof et al., 1993), and by employing telecommunication satellites for twoway satellite time and frequency transfer (TWSTFT) (Kirchner, 1999; Bauch et al., 2009). The time scales can
⇑ Corresponding author.
E-mail address: [email protected] (R.-f. Dong).
be synchronized with an uncertainty up to 1nanosecond. However, such accuracy is far from enough to satisfy the growing requirement for comparison of the new generation of high-precision atomic clocks, meanwhile limits many other prospective applications, such as coherent detection of electromagnetic signals and fundamental physics principle tests. In a recent future, the ACES mission (Atomic Clock Ensemble in Space) based on an elaborated manipulation of radiofrequency signals will demonstrate an improved accuracy as low as 100 ps (Cacciapuoti and Salomon, 2009). To further improve the synchronization accuracy, the time transfer by laser link (T2L2) was proposed and applied in the real satellite-based synchronization system (Fridelance and Veillet, 1995; Fridelance et al., 1997; Guillemot et al., 2008; Vrancken, 2008). With a better control of light pulses than radio wave propagation, the recent T2L2 experiment gives the best clock synchronization accuracy within 50 ps (Guillemot et al., 2008; Vrancken, 2008), though it has a strong dependence on the weather condition. Owing to the low loss, strong anti-jamming ability, high transfer security, and wider transmission
0273-1177/$36.00 Ó 2012 COSPAR. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.asr.2012.07.029
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bandwidth of optical fiber, time transfer through optical fibers has demonstrated a further reduction of uncertainty (Czubla et al., 2006; Sliwczynski et al., 2010). Therefore, synchronization via optical fibers becomes a promising candidate as a part of the infrastructure connecting the ground station setups during forthcoming T2L2 and ACES experiments, supporting deep-space network and antenna arrays in astronomy (Calhoun et al., 2007) or in accelerators for particle physics (Kartner et al., 2010). One of the most important characteristics among the above mentioned methods is the uncertainty evaluation of atomic clocks and time scales. Upon all kinds of noises contributing to the achievable accuracy, the noise resulting from the time-of-arrival measurement is the fundamental noise and is classically limited by the available power and bandwidth, which is the so called shot noise limit. For example, according to the settings of the T2L2 instrument on board the Jason-2 space vehicle, this limit contributes an accuracy of about 3.5 ps over an integration time of 100 s. Quantum clock synchronization is a new technology which utilizes quantum sources and quantum measurement technique in the clock synchronization system to break through the shot noise limit on the classical system, finally reaches the fundamental limit for quantum mechanics, that is, the Heisenberg limit. Since it was proposed in the beginning of this century (Giovannetti et al., 2001a), quantum clock synchronization has attracted much research interests (Giovannetti et al., 2002; Giovannetti et al., 2001b; Giovannetti et al., 2004; Valencia et al., 2004; Bahder and Golding, 2004). According to recent investigations, quantum clock synchronization, as a highly promising technology to improve the classical synchronization accuracy dramatically, will not only be capable of quantum encrypted (Giovannetti et al., 2002), but also have the ability of cancelling propagation noise induced by dispersion in the optical path (Giovannetti et al., 2001b; Giovannetti et al., 2004). As light pulses will be broadened by the group velocity dispersion (GVD) after passing through dispersive medium, dispersion is one of the main restrictions to improve the accuracy of the fiber-link based clock synchronization systems. In this paper we propose a fiber-based quantum clock synchronization scheme to eliminate the timing error due to the GVD in the fiber, based on the quantum feature that the dispersion experienced by the frequency anticorrelated entangled source can be cancelled by the quantum interference measurement in the frame of reference of a Hong-Ou-Mandel (HOM) interferometer (Bahder and Golding, 2004; Hong et al., 1987; Steinberg et al., 1992a,b). Through a thorough analysis, we show that the final synchronization accuracy is jointly determined by the bandwidth of the frequency entangled biphoton spectrum as well as the temperature variation induced fluctuations in both the reference fiber coiling and the fiber link. The rest of the paper is organized as follows. Section 2 is a brief description of the scheme. Section 3 gives the theoretical derivation. Section 4 is the conclusion.
2. Scheme description The dispersion-free fiber-link based quantum clock synchronization scheme is sketched in Fig. 1. Positions of clock A and clock B are linked by a fiber with an unknown length d. To synchronize clock A and clock B, the following procedures are implemented. At clock A position, frequency anti-correlated photon pairs are generated by spontaneous parametric down-conversion (SPDC) in a nonlinear crystal (Louisell et al., 1961; Mollow, 1973; Rubin et al., 1994). In the SPDC process, a photon from an intense cw pump beam is occasionally annihilated due to the nonlinear interaction inside the crystal and two photons with lower frequencies, named as signal and idler, are created. The signal and idler photons are then departed spatially and disseminated through different fiber paths. The signal is traveled from A to B and then reflected back through the fiber link, while the idler is held at A and sent through the reference fiber coiling with an adjustable length of l0. After traveling through different fiber paths, the signal and idler photons are interfered at a 50/50 beam splitter. The two outputs of the beam splitter are subsequently detected by two single-photon counters D1 and D2. The photon count outputs are then sent into a high-speed AND-logic circuit for coincidence measurement. Such measurement is the so-called Hong-Ou-Mandel (HOM) interferometric measurement (Hong et al., 1987), which is also named as second-order quantum interference measurement. The length of the reference fiber coiling l0 is adjusted until the interferometer is balanced, which corresponds to the minimum coincidence. Once the HOM interferometer is balanced, the time delay in the fiber coiling l0 will be equal to the round-trip time delay in the fiber link 2d , that is, b0 12d = b00 1l0, where b0 1 and b00 1 denote the inverse group velocities in the fiber link and the adjustable fiber coiling, respectively. The achievable accuracy of such balance is then determined by the resolution of the HOM interferometer, which is shown limited by the spectral bandwidth of the entangled photons and unaffected by group velocity dispersion in the fiber (for details see Section 3). Since the adjustable fiber coiling l0 is in the laboratory, standard techniques can be applied to determine the delay in the fiber coiling accurately and, hence, the delay in the fiber link. Maintaining the balance of the HOM interferometer, clocks A and B are used to record the SPDC photons creation times at position A {tA(i)}, and the arrival times of the signal photons at position B {tB(i)} after traveling through the fiber link, where i = 1. . .N denotes a series of N photon pairs. The time data are brought together through a classical communication channel for comparison, and the registration time difference tB tA is obtained by maximum “coincidences” of the photon registration time records. When the path length d is known, the measurement resolution is determined by the natural width of G(2) for SPDC (Valencia et al., 2004), which is given by the Fourier transform of the
F.-y. Hou et al. / Advances in Space Research 50 (2012) 1489–1494
A
B
Signal
Frequency entanglement source Idler
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BS D1
D2
C Classical Channel Fig. 1. Sketch of the fiber-link quantum clock synchronization scheme. Positions of clock A and clock B is linked by a fiber path with an unknown length d. The Signal and Idler are a pair of frequency anti-correlated entangled beams. The length l0 of the fiber coiling is adjustable and measurable. After passing through the different fiber paths, the Signal and Idler are interfered on a 50/50 beam splitter. D1 and D2 are two single photon detectors for counting the two output photons of the beam splitter. The C box represents the logic unit for coincidence detection. Clock A records the SPDC photon pair creation time and clock B the time when the Signal photon arrive at B after traveling through the fiber link. The registration time tA(i) and tB(i) are brought together via a classical channel to measure tB tA.
biphoton spectrum and typically on the order of a few femtoseconds to hundreds of femtoseconds. Assume the real time difference between clock A and B is t0 , it is then estimated by t0 = (tB tA) – b0 1d (Valencia et al., 2004). Informed this time offset, the clocks are then synchronized accordingly. 3. Theoretical analysis To investigate the achievable accuracy of the above fiber-linked clocks synchronization scheme, the theoretical analysis is presented as follows. Consider a cw pump at the frequency of xp and the degenerate phase matching such that the center frequencies of the SPDC generated signal and idler both equal half the pump frequency. This system produces a biphoton source of the form Z x x p p 0 þ 0 ^ jwi ¼ dx0 f ðx0 Þ^ þ x x aþ a j0i; ð1Þ s i 2 2 ^þ where ^ aþ s ðxs Þ and a i ðxi Þ are the creation operators for the signal and idler, respectively; j0i is the vacuum state. |f(x0 )|2 is the frequency spectrum of the biphoton state. As a signal photon at frequency xp/2 + x0 is accompanied by an idler photon at frequency xp/2 x0 , the signal and idler are frequency anti-correlated entangled. The signal and idler photons are then separated and traveled through the fiber link and the adjustable reference fiber coiling, respectively. Due to the frequency dependence of the refractive index n(x) in optical fibers, chromatic dispersion is induced during propagation of the signal and idler photons. This effect is described by the mode-propagation constant b(x), which can be expanded as Taylor series about the frequency x0 = xp/2 (Agrawal, 2001)
bðxÞ ¼
nðxÞx c
¼ b0 þ b1 ðx x0 Þ þ
1 b ðx x0 Þ2 þ ; 2! 2
ð2Þ
where c is the velocity of light in vacuum, bm is defined as m d b bm ¼ with m ¼ 1; 2; 3; . . . ð3Þ dxm x¼x0 The first order term b1 can be interpreted as the inverse of the group velocity tg, the speed at which the pulse envelop travels (Agrawal, 2001): 1 dn 1 dn ng 1 nþx nk b1 ¼ ð4Þ ¼ ¼ ¼ ; c dx c dk tg c where k is the wavelength corresponding to x, and ng is the group refractive index. The second order term describes the frequency dependence of the propagation speed (Agrawal, 2001): 1 dn d 2n x d 2n k3 d 2 n b2 ¼ 2 þx 2 c dx dx c dx2 2pc2 dk2 ¼
k2 D; 2pc
ð5Þ
which represents the dispersion of the group velocity and is responsible for pulse broadening or distortion. This phenomenon is known as group velocity dispersion (GVD), and D ¼ 2pc b is the GVD parameter. b1 and b2 are nork2 2 mally specified by manufacturer of the fiber. After passing through the fiber link d and the fiber coiling l0 respectively, the signal and idler photons are then interfered on a 50/50 beam splitter, as shown in Fig. 1. The annihilation operators for the two outputs 1 and 2 of the beam splitter are related to those for the signal and idler by the following formula:
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00
^ a1 ðxÞ ¼ piffiffi2 ^ as ðxÞeib ðxÞ2d þ p1ffiffi2 ^ ai ðxÞeib ^ a2 ðxÞ ¼
00 piffiffi ^ a ðxÞeib ðxÞl0 2 i
þ
ðxÞl0
;
0 p1ffiffi ^ a ðxÞeib ðxÞ2d ; 2 s
ð6Þ
where the propagation constants of the fiber link d and the fiber coiling l0 are denoted as b0 and b00 , respectively. Due to the Heisenberg uncertainty relation, these operators obey the canonical communication relations: ^ aj ðxj Þ; ^ ak ðxk Þ ¼ 0; h i ^ aþ aþ k ðxk Þ ¼ 0; j ðxj Þ; ^ ð7Þ ^aj ðxj Þ; ^ aþ ðx Þ ¼ d dðx x Þ; k jk j k k j; k ¼ 1; 2; s; i: According to the Mandel formula for detection (Hong et al., 1987), the coincidence rate at the photodetectors is given as follows: Z T Z T þ þ PC ¼ dt1 dt2 hWjE ð8Þ 1 ðt 1 ÞE 2 ðt 2 ÞE2 ðt 2 ÞE 1 ðt 1 ÞjWi; 0
0
where T is the integration time window of the detectors. Omitting the irrelevant normalization constants, the positive- and negative-frequency field operators at detector j are defined by R Eþ dxj ^ aj ðxj Þeixj tj ; j ðt j Þ ¼ R ð9Þ þixj tj E dxj ^ ; forj ¼ 1; 2: aþ j ðxj Þe j ðt j Þ ¼ Substituting (9) and (6) into (8) and taking the limit T ! 1 without loss of generality, one obtains Z P C / dx1 Z dx2 hWj^ aþ aþ a2 ðx2 Þ^ a1 ðx1 ÞjWi: ð10Þ 1 ðx1 Þ^ 2 ðx2 Þ^ Using Eq. (1)–(5) and neglecting the dispersion terms higher than second order dispersion, we evaluate the integrand as below
From Eq. (12), it is apparent that the GVD terms have no contribution to the final HOM interferometric measurement. In the simple case of a Gaussian spectrum |f(x0 )|2 with variance Dx2, Eq. (12) becomes 2 ðb00 l b0 2dÞ2 1 0 1
P C ¼ 1 e2Dx
ð13Þ
It follows from Eq. (13) that, the zero coincidence is obtained at b0 12d = b00 1l0 with a dip width 1/(2Dx). Since the adjustable fiber coiling is in the laboratory, the time delay in it (sl0 ¼ b001 l0 ) can be measured and controlled accurately. Under the assumption that Dsl0 ¼ 0, the delay in the fiber link is then determined by sd = b0 1d = b00 1l0/2 with an accuracy Dsd = 1/(4Dx), which is shown independent of the group-velocity-dispersion (GVD) effect inherent in the glass fiber. Accordingly, given the group velocity parameter b0 1 of the fiber link, its length will be estimated as d = b00 1l0/(2b0 1) with an uncertainty Dd = 1/(4Dxb0 1). If the fiber link and the fiber coiling are both normal singlemode fibers, the group refractive indices will be approximately the same, i.e. b0 1 = b00 1, therefore d l0/2. As described in Section 2, the time offset between clocks A and B t0 is given by t0 = (tB tA) b0 1d. To measure t0, a series of registration time difference tB(i)tA(i) must be measured first, where tA(i) is the ith SPDC photon pair creation time recorded by clock A while tB(i) the ith signal photon arrival time recorded by clock B after traveling through the fiber link. It is mathematically expressed by the following second-order coherence function (Valencia et al., 2004) 2
þ Gð2Þ ðtB tA Þ ¼ jh0jEþ s ðd; t B ÞE i ð0; t A ÞjWij :
ð14Þ
The electric field operators of the signal and idler phoþ tons Eþ s ðd; t B Þ; Ei ð0; t A Þ are given by
hWj^aþ aþ a2 ðx2 Þ^ a1 ðx1 ÞjWi 1 ðx1 Þ^ 2 ðx2 Þ^ ¼ jh0j^ a1 ðx1 Þ^ a2 ðx2 ÞjWij2 2 00 0 0 00 ¼ 12 h0j^ ai ðx1 Þ^ as ðx2 Þeib ðx1 Þl0 þib ðx2 Þ2d ^ as ðx1 Þ^ ai ðx2 Þeib ðx1 Þ2dþib ðx2 Þl0 jWi " # 2 xp xp xp xp
xp ei½b000 þb001 ðx1 2 Þþ12b002 ðx1 2 Þ2 l0 þi½b00 þb01 ðx2 2 Þþ12b02 ðx2 2 Þ2 2d 1 ¼ 2 dðxp x1 x2 Þf 2 x1 xp xp 2 xp xp 2 0 0 0 00 00 00 1 1 i½b0 þb1 ðx1 2 Þþ2b2 ðx1 2 Þ 2dþi½b0 þb1 ðx2 2 Þþ2b2 ðx2 2 Þ l0 e h 00 0 i xp xp 2 xp 0 00 00 0 0 1 00 x 2 ¼ j 12 dðxp x1 x2 Þf ð 2p x1 Þeiðb0 l0 þb0 2dÞþi2ðb2 l0 þb2 2dÞðx1 2 Þ eiðb1 l0 b1 2dÞðx1 2 Þ eiðb1 2db1 l0 Þðx1 2 Þ j h i xp 00 0 x 2 2 ¼ 12 jdðxp x1 x2 Þj jf ð 2p x1 Þj 1 Re eiðb1 l0 b1 2dÞ2ð 2 x1 Þ :
We substitute (11) into (10) and drop the overall numerical factors, the final coincidence rate is yielded Z P C / dx0 jf ðx0 Þj2 f1 cos½2x0 ðb001 l0 b01 2dÞg: ð12Þ
Eþ s ðd; t B Þ
¼
Eþ i ð0; t A Þ ¼
Z Z
ð11Þ
0
dx^as ðxÞeixðtB tB0 Þ eib ðxÞd ; dx^ai ðxÞeixðtA tA0 Þ ;
ð15Þ
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where tA0 and tB0 denote the times at which Clock A and B start their clocks, t0 = tB0 tA0 is thus the time offset between these clocks. Substituting (15)–(23) and neglecting the dispersion terms higher than the second-order dispersion term, one gets Z 2 2 x0 0 0 Gð2Þ ðtB tA Þ ¼ dx0 f ðx0 Þei 2 b2 d eix s Z 2 2 x0 0 0 ¼ dx0 f ðx0 Þei 2 b2 d eix s ; ð16Þ where s = (tB tA) t0 b01 d. When the far-field condition is satisfied, which is reasonable in most application cases, the second-order correlation function is approximated to: 2 2 1 2 0s 2 s 2Dx ðb dÞ ð2Þ 0 2 G ðtB tA Þ f x ¼ 0 e : ð17Þ b2 d One can see that, it has the same shape as the Fourier transform of the SPDC spectrum, only broadened by a factor of Dx2b0 2d. Since d is well measured by d l0/2 based on the HOM setup, and b02 is normally specified by manufacturer of the fiber, the measurement of tB tA can reach, in principle, the same order of resolution as the natural width of G(2) for SPDC (Valencia et al., 2004) DðtB tA Þ 1=Dx:
ð18Þ
The accuracy of the time difference between clock A and B t0 is then given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 17 Dt0 ¼ D2 ðtB tA Þ þ D2 ðb01 dÞ ¼ : ð19Þ 16Dx2 It follows from Eq. (19) that, the accuracy of the scheme is dependent only on the bandwidth Dx of the twin beam. Suppose that the bandwidth is Dx 1013 Hz, the accuracy is given by Dt0 0.1 ps. However, the propagation delay in the fiber coiling is fluctuating, i.e., Dsl0 – 0. Among various factors, the uncertainty of the propagation delay in the fiber coiling is mainly caused by the fiber temperature variation and can be expressed as Dsl0 ¼ DT l0 ðb001 l0 Þ ¼ l0
@b001 @l0 DT l0 þ b001 DT l0 : @T @T
ð20Þ
where DT l0 denotes the temperature variation of the fiber coiling, the first right-hand-side term concerns the temperature dependence of the group velocity, and the second one the fiber thermal expansion. The two terms describe the thermal sensitivity of the fiber coiling propagation delay. Such fluctuation contributes an error to the determination of the delay in the fiber link Dsd sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2 1 1 ð21Þ Dsd ¼ þ DT l0 ðb001 l0 Þ : 4Dx 2
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Likewise, the measurement of tB tA is also affected by the temperature variation of the fiber link between clocks A and B, which can be given by s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1 DðtB tA Þ ¼ þ Dx DT d ðb02 dÞ Dx s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 1 2 ð22Þ þ ½Dk DT d ðD0 dÞ ; ¼ Dx where DTd denotes the temperature variation of the fiber link, Dk is the corresponding wavelentgh bandwidth of the biphoton spectrum. Go back to Eq. (19), the error in determinating the time offset of clock A and B is then given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dt0 ¼ D2 ðtB tA Þ þ D2 sd sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 00 17 b 1 l0 2 2 þ DT l ð23Þ þ ½Dk DT d ðD0 dÞ : 2 0 16Dx 2 Suppose that the twin beam is centered at 1.55 lm, the bandwidth of Dx 1013 Hz corresponds to Dk 13 nm. According to Ref. (Sliwczynski et al., 2010), the thermal sensitivity of the fiber propagation delay in 1.55 lm transmission window is around 37 ps/(K km). Suppose that the fiber coiling length in our scheme is l0 = 20 km, and its temperature variation DT l0 is controlled within 1 mK, the uncertainty of the propagation delay in the fiber coiling will be Dsl0 0:74 ps . Substituting it to Eq. (21), one gets Dsd 0.37 ps. In single-mode fibers, the thermal sensitivity of the GVD parameter can be taken as 1.5 103 ps/(K km nm) (Sliwczynski et al., 2010). For the fiber link with the length of d l0/2 10 km and the temperature fluctuation of 2 K, the determination error of tB tA caused by the temperature fluctuation is Dk DT d ðD0 dÞ 0:39 ps. Adding this to Eq. (22), the total accuracy of the measurement of tB tA then gives D(tB tA) 0.4 ps. After including the temperature variation induced fluctuations, the error of the time offset of clock A and B, which is departed at a fiber link distance of about 10 km, is then Dt0 0.54 ps. Such error determines the synchronization accuracy between the two clocks. 4. Conclusion In conclusion, we have demonstrated a high-accuracy fiber-based quantum clock synchronization scheme. In this scheme, two distant clocks, which are linked by an unknown fiber, can be accurately synchronized with the help of a frequency anti-correlated entangled biphoton source, quantum correlation measurement setups, and a reference fiber coiling, which is placed in the laboratory with its length capable of being precisely controlled and measured. In ideal case, the achievable synchronization accuracy Dt0 is only dependent on the bandwidth of the biphoton spectrum. So the higher the bandwidth of the
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biphoton spectrum is, the more accurate the synchronization will be. The accuracy is unaffected either by the GVD effect or by the fiber propagation length. In reality however, due to the inevitable fiber temperature variations, the accuracy will be limited by the fiber propagation lengths. Based on our scheme, utilizing a biphoton source with a bandwidth of 13 nm to synchronize the two clocks at a fiber distance of 10 km, 0.54 ps accuracy can be achieved, which provides a highly promising application in comparing two atomic clocks within a distance of a few to tens of kilometers with an accuracy below a picosecond. Acknowledgments This work was supported by the NSFC 11174282, NSFC Y133ZK1101, the key project fund of the Chinese Academy of Sciences for the “Western light” Talent Cultivation Plan 180 (2011). The authors thank Professor Longsheng Ma for valuable discussions. References Agrawal, G.P. Nonlinear Fiber Optics, third ed Academic Press, San Diego, 2001. Bahder, T.B., Golding, W.M., Clock synchronization based on secondorder coherence of entangled photons. In: Proceeding 7th International Conference on Quantum, Communication, July 2004. Bauch, A., Piester, D., Blanzano, B., Koudelka, O., Kroon, E., Dierikx, E., Whibberley, P., Achkar, J., Rovera, D., Lorini, L., Cordara, F., Schlunegger, C., Results of the 2008 TWSTFT calibration of seven european stations. In: Proceedings European Frequency and Time Forum – IEEE Frequency Control Symposium Joint Conference, Besancßon, France, pp. 1209–1215, 2009. Cacciapuoti, L., Salomon, C. Space clocks and fundamental tests: the ACES experiment. Eur. Phys. J. Special Topics 172, 57–68, 2009. Calhoun, M., Huang, S., Tjoelker, R.L. Stable photonic links for frequency and time transfer in the deep-space network and antenna arrays. Proc. IEEE 95, 1931–1946, 2007. Czubla, A., Konopka, J., Gornik, M., Adamowicz, W., Strus, J., Pawszak, T., Romsicki, J., Lipinski, M., Krehlik, P., Sliwczynski, L., Wolczko, A. Comparison of precise time transfer with usage of multi-channel GPS CV receivers and optical fibers over distances of about 3 kilometers. In: Proceedings 38th Annual Precise Time and Time Interval (PTTI) Systems and Applications Meeting, Reston, Virginia, USA, pp. 337–345, 2006. Einstein, A. Zur Elektrodynamik bewegter Korper. Ann. Phys. (Leipzig) 17, 891–921, 1905. Fridelance, P., Samain, P.E., Veillet, C. T2L2-time transfer by laser link: a new optical time transfer generation. Exp. Astron. 7, 191–207, 1997.
Fridelance, P., Veillet, C. Operation and data analysis in the LASSO experiment. Metrologia 32, 27–33, 1995. Giovannetti, V., Lloyd, S., Maccone, L. Quantum-enhanced positioning and clock synchronization. Nature 412, 417–419, 2001a. Giovannetti, V., Lloyd, S., Maccone, L. Clock synchronization with dispersion cancellation. Phys. Rev. Lett. 87, 117902(4pp), 2001b. Giovannetti, V., Lloyd, S., Maccone, L. Quantum cryptographic ranging. J. Opt. B: Quantum Semiclass. Opt. 4, 413–414, 2002. Giovannetti, V., Lloyd, S., Maccone, L., Shapiro, J.H., Wong, F.N.C. Conveyor-belt clock synchronization. Phys. Rev. A 70, 043808(8pp), 2004. Giovannetti, V., Maccone, L., Shapiro, J.H., Wong, F.N.C. Extended phase-matching conditions for improved entanglement generation. Phys. Rev. A 66, 043813(9pp), 2002. Guillemot, P.H., Samain, E., Vrancken, P., Exertier, P., Leon, S. Time transfer by laser link – T2L2: an opportunity to calibrate rf links. In: Proceedings of the PTTI, pp 95–106, 2008. Hofmann-Wellenhof, B., Lichtenegger, H., Collins, J. Global positioning system theory and practice. Springer-Verlag, New York, 1993. Hong, C.K., Ou, Z.Y., Mandel, L. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett. 59, 2044–2046, 1987. Kaplan, E.D. Understanding GPS: Principles and Applications Mobile Communications Series. Artech House, Boston, 1996. Kartner, F.X., Kim, J., Cox, J., Chen, J., Nejadmalayeri, A.H. Femtosecond precision timing distribution for accelerators and light sources. In: Proceedings IEEE International Frequency Control Symposium, Newport Beach, CA, USA, pp. 564–568, 2010. Kirchner, D. Two-way satellite time and frequency transfer (TWSTFT): principle, implementation, and current performance. Review of Radio Sciences. Oxford University Press, pp. 27–44, 1996. Louisell, W.H., Yariv, A., Siegman, A.E. Quantum fluctuations and noise in parametric processes. I. Phys. Rev. 124, 1646–1654, 1961. Mollow, B.R. Photon correlations in the parametric frequency splitting of light. Phys. Rev. A 8, 2684–2694, 1973. Parkinson, B.W., Spilker, J.J. Global Positioning System: Theory and Applications. Vols. I and II, Progress in Astronautics and Aeronautics, vols. 163 and 164 Amer. Inst. Aero. stro, Washington, D.C., 1996. Rubin, M.H., Klyshko, D.N., Shih, Y.H., Sergienko, A.V. Theory of twophoton entanglement in type-II optical parametric down-conversion. Phys. Rev. A 50, 5122–5133, 1994. Sliwczynski, L., Krehlik, P., Lipinski, M. Optical fibers in time and frequency transfer. Meas. Sci. Technol. 21, 075302(11pp), 2010. Steinberg, A.M., Kwiat, P.G., Chiao, R.Y. Dispersion cancellation in a measurement of the single-photon propagation velocity in glass. Phys. Rev. Lett. 68, 2421–2424, 1992a. Steinberg, A.M., Kwiat, P.G., Chiao, R.Y. Dispersion cancellation and high-resolution time measurements in a fourth-order optical interferometer. Phys. Rev. A 45, 6659–6665, 1992b. Valencia, A., Scarcelli, G., Shih, Y. Distant clock synchronization using entangled photon pairs. Appl. Phys. Lett. 85, 2635–2657, 2004. Vrancken, P. Characterization of T2L2 (time transfer by laser link) on the Jason 2 ocean altimetry satellite and micrometric laser ranging. Thesis, Universite´ de Nice – Sophia Antipolis, 2008.