System and method for clock synchronization and position determination using entangled photon pairs

A system and method for clock synchronization and position determination using entangled photon pairs is provided. The present invention relies on the measurement of the second order correlation function of entangled states. Photons from an entangled photon source travel one-way to the clocks to be synchronized. By analyzing photon registration time histories generated at each clock location, the entangled states allow for high accuracy clock synchronization as well as high accuracy position determination.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to clock synchronization and position determination and, more specifically, to a system and method for accurate one way clock synchronization and position determination using entangled photon pairs.

2. Background of the Related Art

Accurate timing and positioning metrological measurements are important for both fundamental research and practical applications. In particular, distant clock synchronization has attracted a great deal of attention in recent years due to its essential role in the Global Positioning system (GPS) and telecommunications.

Modern clocks have been improved to such a level, that the resolution and accuracy of the comparison techniques have become the limiting factors to determine their relative rates and synchronization. There are two standard methods for synchronizing two distant clocks: the classic Einstein protocol and the Eddington slow transportation method. Both methods have certain limitations and difficulties in high accuracy nonlocal synchronization in which relativistic effects, such as the rotating disk problem, have to be taken into consideration. The Einstein protocol is a two-way method and, hence, it requires (1) an accurate knowledge of the one-way speed of light that, until now, has not been measured conclusively on rotating reference systems and (2) the light propagation path to be the same in each direction. The Eddington transportation method relies on the physical movement of a clock, therefore, this method is not practical for space applications.

SUMMARY OF THE INVENTION

Therefore, an object of the present invention is to provide high accuracy one-way synchronization of clocks using entangled photon pairs.

Another object of the present invention is to provide high accuracy position determination using entangled photon pairs.

To achieve the at least above objects, in whole or in part, there is provided a system for clock synchronization and position determination, including an entangled photon source for generating entangled photon pairs and associated quantum mechanical probability amplitudes of directing one of the photons in each pair to a first clock positioned at a first location and the other photon in the pair to a second clock positioned at a second location, and a processor for synchronizing the first and second clocks and/or determining an unknown distance using information on arrival times of the photons at the first and second clocks.

To achieve at least the above objects, in whole or in part, there is further provided a system for clock synchronization and position determination, including an entangled photon source for generating entangled photon pairs and associated quantum mechanical probability amplitudes of directing one of the photons in each pair to a first clock positioned at a first location and the other photon in the pair to a second clock positioned at a second location, a first detector at the first location for detecting one of the photons in each pair, a first event timer in communication with the first clock and the first detector for determining arrival times of the photons at the first detector and generating a first detection event history, a second detector at the first location for detecting the other photons in each pair, a second event timer in communication with the second clock and the second detector for determining arrival times of the photons at the second detector and generating a second detection event history, and a processor for synchronizing the first and second clocks and/or determining an unknown distance using the first and second detection event histories.

To achieve at least the above objects, in whole or in part, there is further provided a method for determining a synchronization state between a first clock at a first location and a second clock at a second location, including generating a plurality of entangled photon pairs and associated quantum mechanical probability amplitudes of directing one of the photons in each pair to a first clock positioned at a first location and the other photon in the pair to a second clock positioned at a second location, generating a first detection event history based on arrival times of the photons at the first location, generating a second detection event history based on arrival times of the photons at the second location, and determining if the first and second clocks are synchronized based on a comparison of the first and second detection event histories.

To achieve at least the above objects, in whole or in part, there is further provided a method of synchronizing a first clock at a first location and a second clock at a second location, including generating a plurality of entangled photon pairs and associated quantum mechanical probability amplitudes of directing one of the photons in each pair to a first clock positioned at a first location and the other photon in the pair to a second clock positioned at a second location, detecting one of the photons from a first photon pair with wavelength λs(“photon1A”) at the first location and detecting the other photon from the first photon pair with wavelength λi(“photon2A”) at the second location, determining a first registration time difference based on arrival times of photons1A and2A at the first and second locations, respectively, detecting a first photon from a second photon pair with wavelength λs(“photon1B”) at the second location and the other photon from the second photon pair with wavelength λi(“photon2B”) at the first location, determining a second registration time difference based on arrival times of photons1B and2B at the second and first locations, respectively, subtracting the second registration time difference from the first registration time difference to yield an intermediate result, determining a time offset based on the intermediate result, and synchronizing the first and second clocks using the time offset.

To achieve at least the above objects, in whole or in part, there is further provided a method for determining a distance between a first location and a second location, wherein a first clock is located at the second location and a second clock is located at a third location, including generating a plurality of entangled photon pairs at the first location and associated quantum mechanical probability amplitudes of directing one of the photons in each pair to a first clock positioned at a second location and the other photon in the pair to a second clock positioned at a third location, detecting a first photon from a first photon pair with wavelength λs(“photon1A”) at the second location and detecting the other photon from the first photon pair with wavelength λi(“photon2A”) at the third location, determining a first registration time difference based on arrival times of the photons1A and2A at the second and third locations, respectively, detecting a first photon from a second photon pair with wavelength λs(“photon1B”) at the third location and the other photon from the second photon pair with wavelength λi(“photon2B”) at the second location, determining a second registration time difference based on arrival times of photons1B and2B at the third and second locations, respectively, subtracting the second registration time difference from the first registration time difference to yield an intermediate result, determining a time offset based on the intermediate result, and determining the distance between the first location and the second location based on the intermediate result.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

The system and method of the present invention relies on the measurement of the second order correlation function of entangled states. In a preferred embodiment, the entangled photon pairs produced in a continuous wave (CW) pumped spontaneous parametric down conversion (SPDC) are utilized for clock synchronization and position determination.

Generally, the process of SPDC involves sending a pump laser beam into a nonlinear material, such as a non-centrosymmetric crystal. Occasionally, the nonlinear interaction inside the crystal leads to the annihilation of a high frequency pump photon and the creation of two lower frequency photons named as signal and idler. The creation time of either the signal photon or the idler photon is unknown. However, if the signal photon is detected at a certain time, the detection time of the idler photon can only happen at a unique precise time. In a performed experiment, which will be discussed in more detail below, both the signal photon and the idler photon are in the form of continuous wave, i.e., Δt=∞. Nevertheless, the time correlation measurement of the signal/idler at distance of 3 km has shown uncertainty in the order of a picosecond.

According to quantum field theory, the probability of having a joint photo-detection even at space-time points (r1, t1) and (r2, t2) is proportional to the second-order correlation function of the fields as follows:
G(2)(r1,t1;r2,t2)=E(−)(r1,t1)E(−)(r2,t2)E(+)(r2,t2)E(+)(r1,t1)×E(+)(r1,t1),  (1)
where E(−)and E(+)are the negative-frequency and the positive-frequency field operators of the detection events at space-time points (r1, t1) and (r2, t2). For the two-photon entangled state of SPDC, G(2)(r1, t1;r2, t2) can be written as the modulus square of a two-photon effective wavefunction, or biphoton:
G(2)(r1,t1;r2,t2)=|0|E(+)(r2,r2)E(+)(r1,t1)|Ψ|2≡|ψ(r1,t1;r2,t2)|2(2)
where |0stands for the vacuum, and |Ψis the state of the signal/idler photon pair.

The two-photon effective wavefunction is calculated to be:

Ψ⁡(r1,t1;r2,t2)=ⅇ-ⅈ⁡(ωs0⁢r11+ωi0⁢r2)⁢Fr1-r2⁢{f⁡(Ω)}(3)
where Fr1−r2{ƒ(Ω)} is the Fourier transform of the spectrum amplitude function ƒ(Ω), rj=tj−τj/uj, j=1,2, and ujis the group velocity at frequencies ωs0and ωi0along the optical paths1and2, respectively ωs0and ωi0are the control frequencies of the signal/idler radiation field.

The G(2)(r1,t1;r2,t2) function for the two-photon entangled state of SPDC is thus
G(2)(r1,t1;r2,t2)=|Fr1−r2{ƒ(Ω)}|2.  (4)
This function, depending on τ1−τ2, is independent of the chosen reference coordinates—it is a Lorentz invariant. The spectrum amplitude function of the SPDC, ƒ(Ω), provides all the information about the spectrum and the correlation properties of the signal/idler pair. In the collinear case, for type-II and nondegenerate type-I SPDC, the spectral function is calculated as ƒ(Ω)˜sinc(DLΩ/2), where L is the length of the crystal and

D=1us-1ui
is the inverse group velocity difference for the signal and idler. For an 8 mm LBO crystal pumped at 458 nm (type-II), the estimated width of G(2)(t1−t2) is about 800 femtoseconds. For collinear degenerate type-I SPDC, the spectral function is ƒ(Ω)˜sinc(D″LΩ2/2), where D″ is the second derivative of the dispersion function of the nonlinear material. In this case, the width of G(2)(t1−t2) is about 30 femtoseconds for the same size LBO crystal. Typical values for the natural width of G(2)for SPDC are, then, on the order of a few femtoseconds to hundreds of femtoseconds. If r1and r2are well controlled, the measurement of t1−t2can reach, in principle, the same order of resolution, making SPDC particularly suitable for implementing protocols for timing and positioning measurements with ultra high accuracy.

FIG. 1is a schematic diagram of a clock synchronization system, in accordance with the present invention. In the example shown, two clocks, clock1and clock2, are located at separate locations. Specifically, clock1is located at the first location110, and clock2is located at a second location120. An entangled photon source130generates entangled photon pairs and the quantum mechanical probability amplitude of directing one of the photons in the pair to clock1and the other photon in the pair to clock2.

A photon counting detector D1and an event timer140are used to detect photons at location110. Similarly, a photon counting detector D2and an event timer150are used to detect photons at location120. The photon registration times of the detectors, t1and t2, are recorded by two event timers140and150, whose time bases are provided by clock1and clock2, respectively. The individual detection event histories160and170are sent to a processor190, preferably through classical communication channels180for comparison. The processor190can be located at a location separate from locations110and120, or can be located at either location110or120. If clock1and clock2are synchronized, the joint detection of the entangled photon pair obtained by matching the detection event history records160and170will show maximum “coincidences”. If the clocks lose their synchronization, one has to rematch the detection event history records160and170to achieve maximum coincidences by shifting one of them by a certain amount. The amount that one of the records needs to be shifted with respect to the other corresponds to how much the two clocks have lost their synchronization. The clocks can be adjusted and kept synchronized accordingly.

The processor190can be a general purpose computer. However, it can also be a special purpose computer, programmed microprocessor or microcontroller and peripheral integrated circuit elements, ASICs or other integrated circuits, hardwired electronic or logic circuits such as discrete element circuits, programmable logic devices such as FPGA, PLD, PLA or PAL or the like. In general, any device on which a finite state machine capable of executing code can be used to implement the processor190.

Communications channels180may be, include or interface to any one or more of, for instance, the Internet, an intranet, a PAN (Personal Area Network), a LAN (Local Area Network), a WAN (Wide Area Network) or a MAN (Metropolitan Area Network), a storage area network (SAN), a frame relay connection, an Advanced Intelligent Network (AIN) connection, a synchronous optical network (SONET) connection, a digital T1, T3, E1 or E3 line, Digital Data Service (DDS) connection, DSL (Digital Subscriber Line) connection, an Ethernet connection, an ISDN (Integrated Services Digital Network) line, a dial-up port such as a V.90, V.34bis analog modem connection, a cable modem, and ATM (Asynchronous Transfer Mode) connection, or an FDDI (Fiber Distributed Data Interface) or CDDI (Copper Distributed Data Interface) connection. Communications channels180may furthermore be, include or interface to any one or more of a WAP (Wireless Application Protocol) link, a GPRS (General Packet Radio Service) link, a GSM (Global System for Mobile Communication) link, CDMA (Code Division Multiple Access) or TDMA (Time Division Multiple Access) link such as a cellular phone channel, a GPS (Global Positioning System) link, CDPD (Cellular Digital Packet Data), a RIM (Research in Motion, Limited) duplex paging type device, a Bluetooth radio link, or an IEEE 802.11-based radio frequency link. Communications channels180may yet further be, include or interface to any one or more of an RS-232 serial connection, an IEEE-1394 (Firewire) connection, a Fibre Channel connection, an IrDA (infrared) port, a SCSI (Small Computer Systems Interface) connection, a USB (Universal Serial Bus) connection or other wired or wireless, digital or analog interface or connection.

FIG. 2is a flowchart of a method for maintaining at least two clocks synchronized after they have been initially synchronized (a method for initially synchronizing the at least two clocks will be described in connection withFIG. 3below). The method starts at step200, where a plurality of entangled photon pairs, along with the quantum mechanical probability amplitude of directing one of the photons in the pair to clock1and the other photon in the pair to clock2, are generated using the source of entangled photons130. The method then moves to step210, where one of the photons in a photon pair (termed “photon1”) is detected at the first detector D1and the other photon in the pair (termed the “photon2”) is detected at the second detector D2. It should be appreciated that it does not matter which detector receives photon1and which detector receives photon2. Thus, the first detector D1could receive photon2and the second detector D2could receive photon1.

At step220, the photon arrival times at each detector are recorded by respective event timers140and150to yield respective detection event histories160and170. Next, at step230, the respective detection event histories160and170are compared. Then, at step240, it is determined whether the clocks are synchronized based on the comparison of the respective detection event histories160and170. If the clocks are synchronized, the method returns to step200. If the clocks are not synchronized, the method proceeds to step250, where the clocks are synchronized based on a difference between respective detection event histories160and170. This is preferably done by shifting the detection event timing history of one of the clocks with respect to the other by an amount sufficient to achieve maximum coincidences between the respective detection event histories160and170. The method then returns to step200.

FIG. 3is a flowchart of a method for initially synchronizing the two clocks shown inFIG. 1. The method starts at step300, at which entangled photon pairs, along with the quantum mechanical probability amplitude of directing one of the photons in a pair to clock1and the other photon in a pair to clock2, are created using the entangled photon source130. The method then proceeds to step310, at which one of the photons in a pair, termed “photon1A” (with wavelength λs) is detected at detector D1and other photon of the pair, termed the “photon2A” (with wavelength λi) is detected at detector D2.

t1-t2=r1us+t0-r2ui(5)
where t0is the time offset of the non-synchronized clocks. Then, at step330, photon1B (with wavelength λs) from a second photon pair is detected at D2and photon2B from the second photon pair (with wavelength λi) is detected at D1. It should be appreciated that, due to the quantum mechanical probability amplitudes of directing one of the photons in a pair to clock1and the other photon in a pair to clock2, the second photon pair detected at clocks1and2is not necessarily the photon pair that is generated immediately after the previously detected photon pair. The registration time difference, t′1−t′2, of D1and D2is the calculated at step340as:

t1′-t2′=r1us+t0-r2us(6)
At step350, the registration time differences calculated at step330is subtracted from the registration time difference calculated at step320to yield:
Δt_=(t1−t2)−(t′1−t′2)=D(r1+r2)  (7)
where

D=1us-1ui,
Δt_ is obtained from direct measurements, and r2is known. In some cases, D is known or independently measurable, therefore the distance between location110and the entangled photon source130, r1, is predictable through the measurements of Δt_. In other cases, r1may be given or independently measurable, so, the value of D can be calibrated in the above procedure with ultra-high accuracy.

In both cases, the time offset to is estimated at step350by substituting into either Eq. (5) or Eq.(6). The accuracy of the estimated time offset to is of the same order as the accuracy of the measurement t1−t2. At step370, clock1and clock2are synchronized using the estimated time offset t0. The measurement can be repeated for different frequencies and different values of r2. Thus, even in the case in which both D and r1are unknown, measurements of Δt_ with different known values of r2allow the evaluation of D and r1simultaneously.

As discussed above, at least one of the distances r1or r2must be known in order to synchronize the clocks. However, the method of the present invention can also be used to determine the sum of two unknown distances (r1or r2) or one of the unknown distances, r1or r2, if the other is known using the method shown inFIG. 4.

Steps400-450of the method ofFIG. 4are identical to steps300-350of the method ofFIG. 3, and thus they will not be described again. At step460, the unknown distance is calculated using the result from step450. As discussed above, the unknown distance could be the sum of two unknown distances (r1or r2) or one of the unknown distances, r1or r2, if the other is known. If desired, one could also optionally correct the distance measurement for fluctuations in the atmospheric index of refraction using optional step470. Any known techniques for making such correction, such as multi-color ranging techniques, can be used for correcting the distance measurement for fluctuations in the atmospheric index of refraction.

An experimental demonstration was performed in the case in which r1and r2were known. In the experiment, long optical fibers of known lengths were used to simulate long distances to a remote location. The experimental setup is shown inFIG. 5. A single frequency Artlaser line500at a wavelength of 457.9 nm was used to pump an 8 mm LBO crystal510for type-II SPDC. The signal/idler radiations (centered at ˜901 nm and at ˜931 nm, respectively) were separated from the pump laser beam500by using filtering devices520and530. Filter device520is preferably a a high reflectivity laser cavity mirror to block the pump beam, and filtering device530is preferably a band-cut filter to further block the remaining pump beam. The orthogonally polarized signal/idler pair was split by means of a polarizing beam splitter550. Before the beam splitter550, a half-wave plate540was positioned in order to perform the two measurements described previously. When the waveplate540is at 0°, the signal is transmitted to D1and the idler reflected to D2. When the waveplate540is at 45°, the idler is transmitted to D1and the signal is reflected to D2.

In both measurements, the signal and idler radiation were fed into two 1.5 km-long commercial optical fibers560and570optimized for single-mode operation at 1300 nm. The signal/idler pair was then detected by two-single photon counting detectors D1and D2. After a large number of signal/idler pair measurements, a histogram of the number of counts against t1−t2(the resolution of the fast-timing electronics is 3 ps) can be obtained. This distribution function corresponds to the G(2)(t1−t2) function previously described.

FIG. 6is a plot of the experimental results. The left distribution function600corresponds to the case of signal-D1and idler-D2(half-wave plate540at 0°), while the right distribution function610corresponds to the case of idler-D1and signal-D2(half-wave plate540at 45°). The presence of multiple peaks on each individual distribution function is a consequence of intermodal dispersion in the optical fibers560and570, which is a known effect in fiber optics.

The calculated width of the effective two-photon wavefunction from a 8 mm type-II LBO SPDC, without the long optical fibers, is about 800 fs. The measured width of G(2)(t1−t2), with the fibers, is around 750 ps. There are two contributions for the broadening of the G(2)function: (i) dispersion in the optical fiber, which may be compensated nonlocally (the compensation is not included in this proof-of-principle experiment); and (ii) the time jitter of the photodetectors D1and D2. The behavior of the biphoton in dispersive medium has been previously studied. Using two fibers of 1.5 km length, the far-field zone condition is satisfied. Therefore, G(2)(t1−t2) is expected to take the shape of the spectrum function of the type-II SPDC |ƒ(Ω)| with a full width at half maximum of 600 ps.

FIG. 7is a plot showing the central peak of the experimental data (for the case of the half-wave plate540at 45° ) compared with the theoretical expectation when the broadening contributions of (i) and (ii) above are taken into consideration. The fitting parameters k″sand k″iof the signal/idler radiations, 2.76×10−28and 2.96×10−28s2/cm, respectively, are in agreement with the values specified by the manufacturer of the optical fiber.

By measuring the displacement of the central peak when the half-wave plate540is rotated from 0° to 45° (Δt=5432±1 ps), and knowing the length of the fibers560and570, the experimental value for D, using Eq. (7), was found to be 1799.9±0.4 ps/km, in agreement with the parameters of the fibers560and570. Substituting the estimated value of D into either Eq. (5) or Eq. (6), the time offset is measured to be t0=40369±1 ps, which has the same order of accuracy of the t1−t2measurement.