Doppler shift and ambiguity velocity caused by relative motion in quantum-enhanced measurement

Yanghe Shen; Luping Xu; Hua Zhang; Peng Yang

doi:10.1364/OE.23.018445

1. Introduction

Distance ranging is widely used in the fields of satellite formation flying and large-scale manufacturing. In distance measuring systems, there are three critical parameters: precision, ambiguity range, and update rare (i.e., the time taken to acquire the data for a given precision). In terms of traditional techniques which are based on Maxwell’s equations, the first two parameters are generally contradictory. For instance, using shorter-wavelength signals, laser interferometer tends to obtain a better precision, however, at the cost of an insufficient ambiguity range [1].

Multi-wavelength interferometry combines measurements at several optical wavelengths, and can increase the ambiguity range [2, 3], but this technique is still insufficient for measuring a large distance. Recently, femtosecond optical frequency combs have been developed, which can offer a maximum ambiguity range of tens of kilometers [4–6]; however, a complex system and the susceptibility to systematic errors (i.e., spurious reflections and environment noise) are common defects of this scheme [7].

Employing some ‘quantum trick’ such as frequency entanglement, quantum-enhanced measurement can beat the conventional bounds to the accuracy of measurements [8–10]. Giovannetti et al. demonstrated that using M frequency entangled pulses could theoretically increase the ranging accuracy by a factor of M^1/2 compared to using M classical pulses with the same bandwidth. Thus, a frequency-entangled-based ranging (FEB) scheme has been proposed [11, 12], which is generally based on the second-order coherence (SOC) of frequency entangled photons. In fact, when M = 2 in Ref [9, 10], the joint probability for the two detectors to detect the photons pair is just proportional to the SOC function, which corresponds to the principle of the FEB scheme. Besides an ultrahigh ranging precision, the FEB scheme has advantages of high immunity to systematic errors and no ambiguity range. Additionally, it is possible to implement a quantum cryptographic scheme that prevents an eavesdropper from obtaining the information of distance [13]. Considering all of the factors above, this novel scheme may have a promising future in the applications of distance measurement.

Traditional ranging techniques (for example, pulsed radar systems) directly measure the time-of-flight of signals, while the FEB scheme measures the SOC of the photons pair and requires an accumulated time interval [t₀, t₀ + T_a] spent by the coincidence measurement, (t₀ is the time when the detectors initially record the TOAs of the photons pair), which results in a slower update rate; it means that only when t = t₀ + T_a, can we obtain the information of distance at time t = t₀. The previous works [9–12] paid attention to a model of motionless targets for the FEB scheme. However, most practical situations involve the relative motion between targets and the measuring platform, and the distance we want to measure can vary during the accumulated time interval [t₀, t₀ + T_a]; because the SOC of the photons pair contains all of the information of distance during [t₀, t₀ + T_a], the distance information we obtained is no longer the distance at time t = t₀. Additionally, Doppler shift due to the relative motion may have an influence on the performance of the FEB scheme. Thus, in this work, we theoretically analyze the major puzzles due to the effect of relative motion, and propose a technique to overcome them.

2. Scheme description

As shown in Fig. 1, Alice located at the measuring platform wants to measure the distance (i.e., d₀) from Bob using the FEB scheme. Here Alice and Bob are no longer the consenting parties in a quantum cryptography experiment, Bob is passive and he merely reflects the photons to provide a delay from which the range can be determined. The procedure is as follows: a bunch of continuous-wave (CW) laser is focused on a BBO crystal through a Glan prism, and a pair of frequency entangled photons (i.e., idler photon and signal photon) is then generated, which is known as the spontaneous parametric down-conversion (SPDC). Once generated, the idler photons propagate directly towards detector D1 while signal photons travel through a round trip between Alice and Bob, and are then collected by detector D2. The times of arrival (TOAs) of the signal and idler photons are recorded by the two detectors, and a coincidence measurement is then performed on the time difference of arrival (TDOA) τ of the two photons with a coincidence counts rate (i.e., P_c) proportional to the SOC of the signal-idler photon pair (i.e., G(τ)). After an accumulated time interval T_a, we gains the accumulated profile n(τ) (a distribution histogram of τ), which corresponds to G(τ) [11].

Fig. 1 Schematic diagram of the FEB scheme

Download Full Size | PDF

Since G(τ) is a single-peak function, one can find the maximum coincidence count at a single value (i.e., τ_max). Assuming the time taken for the idler and signal photons to be travelled are t₁ and t₂, respectively, then we can obtain that:

τ_{max} = t_{2} - t_{1} = τ |_{\max [c (τ)]} = τ |_{\max [G (τ)]}

where τ|_max[_G₍_τ_)] denotes the value of τ that makes G(τ) equal its maximum value.

Then, the difference in signal and idler photon paths (i.e., l_dif) can be written as follows:

l_{dif} = 2 d_{0} - l_{reference} = c \times τ |_{\max [G (τ)]}

where l_reference is the idler photon path on-board the platform which can be measured in advance. Thus, the distance d₀ between Alice and Bob can be determined as follows:

d_{0} = \frac{1}{2} (l_{reference} + c \times τ |_{\max [G (τ)]})

The precision of the FEB scheme is determined by the full-width at half-maximum (FWHM) of G(τ), which depends on the resolution of the coincidence measurement. Using two-photon absorption (TPA) in a GaAs photon-counting module [14], the FWHM of G(τ) can reach 50 fs. Therefore, the precision of ranging is on the order of 7.5µm.

In this section, the target Bob in Fig. 1 is motionless; therefore, the accumulated profile n(τ) is stable, and τ_max can be utilized to measure the distance d. When Bob starts to move, another situation will happen, and details are presented in Section 3.

3 Major puzzles caused by the relative motion

3.1 Effect of Doppler shift

In a scenario of relative motion, for simplicity, we assume Bob moves at a constant speed of v with respect to Alice. Specifying a certain time t during the accumulated time interval [t₀, t₀ + T_a], the distance d = d₀ – v (t - t₀), and the difference in the light paths is varying during the time interval.

The SPDC satisfies the condition of degenerate phase matching so that the center frequency of the signal-idler photon pair equals a half of the pump frequency (i.e., ω_p); then, the state of signal-idler photon pair can be written as follows:

| ψ 〉 = \int d ω^{'} f (ω^{'}) {| ω_{p} / 2 + ω^{'} 〉}_{s} {| ω_{p} / 2 - ω^{'} 〉}_{i}

where f (ω^’ ) is the spectral amplitude of the signal-idler radiation and f (ω^’ ) = f (- ω^’ ); the indices s and i denote the signal and idler photons, respectively.

The signal-idler photon pair then travels a different light paths in a non-dispersive medium and is detected by the two detectors. Doppler shift unquestionably exists in the scenario of relative motion, therefore, the energy of the reflected signal photon is red or blue shifted. As Bob moves towards to Alice, the frequency shift of the reflected signal photon (i.e., Δω₂) can be expressed as follows:

Δ ω_{2} = [ω_{2} (1 + β) / (1 - β)] - ω_{2} = 2 ω_{2} β / (1 - β)

where ω₂ denotes the frequency of the signal photon once generated by the SPDC; β is caused by the Doppler shift, and β = v/c. If the frequency of reflected signal photon detuned beyond the spectral response of detector D1 (i.e., the coincidence rate drops to the dark count rate or solar background count rate of the detector), the relative velocity would reach its upper limit. Then, in the following, we will estimate the maximum of the frequency shift. It is known that the sum of the frequencies of the idler and signal photons equals the pump frequency ω₀, then, the frequency of reflected photon should be less than the pump frequency ω₀; thus, Eq. (5) can be rewritten as follows:

Δ ω_{2} < 2 ω_{0} β / (1 - β) = 2 ω_{0} / [(1 / β) - 1]

We select a detector which can offer everything for single photon detection from 350 – 1000 nm [15], namely, with a pass band 1.885 × 10¹⁵ – 5.386 × 10¹⁵ Hz. The speed of the currently fastest aircraft can reach Mach 20 (i.e., 20 times the speed of sound), and we select a CW pump at a wavelength of 363.8 nm which is the same to that used in the simulation, then the frequency shift can be calculated as 2.3489 × 10¹¹ Hz; thus, it is concluded that even with the fastest aircraft, the frequency of reflected photon cannot detune beyond the spectral response of the detector, and we should not be concerned about it.

Additionally, due to the effect of Doppler shift, the annihilation operator of the reflected signal photon can be affected [16]. As the coincidence counts rate is related to the annihilation operators, this may result in an influence on the distance ranging. Based on Ref [16], the annihilation operators (i.e., a₁ (ω₁) and a₂ (ω₂)) at the detectors 1 and 2, respectively, can be written as follows:

\begin{array}{l} a_{1} (ω_{1}) = a_{i} (ω_{1}) \exp (i ω_{1} l / c) \\ a_{2} (ω_{2}) = χ^{1 / 2} \times a_{s} (χ ω_{2}) \times \exp {- i ω_{2} [2 β (- d / c) / (1 - β) - 2 d / c]} \\ = χ^{1 / 2} a_{s} (χ ω_{2}) \exp {i ω_{2} [2 d / (1 - β) c]} \end{array}

where a_i (ω₁) and a_s (ω₂) denote the annihilation operators for the idler and signal photons at the crystal, respectively; c is the speed of light; l and 2d denote the lengths of the idler and signal light paths, respectively; χ is associated with the Doppler shift introduced by the motion of Bob and χ = (1 + β)/(1-β); β is identical to that used in Eq. (5).

Then, the electromagnetic field operators at detectors 1 and 2 can be written as follows:

\begin{array}{l} E_{j}^{(-)} (t_{j}) = \int d ω_{j} a_{j}^{+} (ω_{j}) \exp (i ω_{j} t_{j}) \\ E_{j}^{(+)} (t_{j}) = \int d ω_{j} a_{j} (ω_{j}) \exp (- i ω_{j} t_{j}), for j = 1, 2 \end{array}

where t_j denotes the time when the signal-idler photon pair arrive at the j-th detector; and a⁺_j (ω_j) denotes the creation operator for idler (signal) photons at the detector 1 (2). According to the Mandel formula for detection [17], the coincidence rate is written as follows:

P_{c} = \int_{0}^{T} d t_{1} \int_{0}^{T} d t_{2} 〈 ψ | E_{1}^{(-)} (t_{1}) E_{2}^{(-)} (t_{2}) E_{2}^{(+)} (t_{2}) E_{1}^{(+)} (t_{1}) | ψ 〉

where T is the coincidence gate window. Equation (7) can be further represented by the following form [11, 18]:

\begin{array}{l} P_{c} (t_{1}, t_{2}) \propto G (t_{1}, t_{2}) \\ = {| 〈 0 | E_{1}^{+} (t_{1}) E_{2}^{+} (t_{2}) | ψ 〉 |}^{2} \end{array}

Combining Eqs. (4) through (8), Eq. (10) can be derived to be:

\begin{array}{l} P_{c} (t_{1}, t_{2}) \propto G (t_{1}, t_{2}) \\ = {| \begin{array}{l} \int d ω_{1} \int d ω_{2} 〈 0 | a_{i} (ω_{1}) \times \exp (i ω_{1} l / c) \times \exp (- i ω_{1} t_{1}) \times χ^{1 / 2} \\ \times a_{s} (χ ω_{2}) \times \exp {i ω_{2} [2 d / (1 - β) c]} \times \exp (- i ω_{2} t_{2}) | ψ 〉 \end{array} |}^{2} \\ = χ {| \begin{array}{l} \int d ω_{1} \int d ω_{2} 〈 0 | a_{i} (ω_{1}) a_{s} (χ ω_{2}) | ψ 〉 \\ \times \exp {i ω_{1} l / c + i ω_{2} [2 d / (1 - β) c]} \times \exp [- i (ω_{1} t_{1} + ω_{2} t_{2})] \end{array} |}^{2} \end{array}

And

\begin{array}{l} 〈 0 | a_{i} (ω_{1}) a_{s} (χ ω_{2}) | ψ 〉 \\ = \int d ω^{'} f (ω^{'}) \times {}_{s}{〈 χ ω_{2} |} {}_{i}{〈 ω_{1} |} {| ω_{p} / 2 + ω^{'} 〉}_{s} {| ω_{p} / 2 - ω^{'} 〉}_{i} \\ = \int d ω^{'} f (ω^{'}) \times δ [χ ω_{2} - (ω_{p} / 2 + ω^{'})] δ [ω^{'} - (ω_{p} / 2 - ω_{1})] \\ = f (ω_{p} / 2 - ω_{1}) δ (ω_{1} + χ ω_{2} - ω_{p}) \end{array}

Substituting Eq. (12) into Eq. (11), and defining ω = ω₂ –ω_p /2, we can obtain:

\begin{array}{l} P_{c} (t_{1}, t_{2}) \propto {| 〈 0 | E_{i}^{+} (t_{1}) E_{s}^{+} (t_{2}) | ψ 〉 |}^{2} \\ = χ {| \begin{array}{l} \int d ω_{1} \int d ω_{2} f (ω_{p} / 2 - ω_{1}) δ (ω_{1} + χ ω_{2} - ω_{p}) \\ \times \exp {i ω_{1} l / c + i ω_{2} [2 d / (1 - β) c]} \times \exp [- i (ω_{1} t_{1} + ω_{2} t_{2}) \end{array} |}^{2} \\ = χ {| \begin{array}{l} \int d ω f (ω) \times \exp (- i ω {t_{2} - χ t_{1} - [2 d / (1 - β) c - χ l / c]}) \\ \times \exp {i ω_{p} / 2 \times [2 d / (1 - β) c - χ l / c + 2 (l / c - t_{1}) - (t_{2} - χ t_{1})]} \end{array} |}^{2} \\ = χ {| \int d ω f (ω) \exp (- i ω {t_{2} - χ t_{1} - [2 d - (1 + β) l] / ((1 - β) c)}) |}^{2} \end{array}

Equation (13) shows the coincidence rate depends on t₂ - χt₁ instead of TDOA (i.e., τ = t₂ - t₁); thus, Doppler shift can invalidate the result of the coincidence measurement of the FEB scheme during condition of rapid relative motion.

3.2 Ambiguity velocity

When relative motion is rapid, the FEB scheme is useless to measure the distance. In this section, we evaluate the performance of the FEB scheme under condition of slow relative motion where the effect of Doppler shift can be ignored; then, a₂ (ω₂) in Eq. (7) is changed into the following form:

\begin{matrix} a_{2} (ω_{2}) = a_{s} (ω_{2}) \exp [i ω_{2} (2 d / c)] & β \to 0, χ \to 1 \end{matrix}

Then, G(t₁,t₂) in Eq. (11) is rewritten to be:

G (t_{1}, t_{2}) = {| \begin{array}{l} \int d ω_{1} \int d ω_{2} 〈 0 | a_{i} (ω_{1}) a_{s} (ω_{2}) | ψ 〉 \times \exp [i ω_{1} l / c + i ω_{2} (2 d / c)] \\ \times \exp (i ω_{1} t_{1} + i ω_{2} t_{2}) \end{array} |}^{2}

And Eq. (12) can be rewritten as follows:

〈 0 | {\hat{a}}_{i} (ω_{1}) {\hat{a}}_{s} (ω_{2}) | ψ 〉 = f (ω_{p} / 2 - ω_{1}) δ (ω_{1} + ω_{2} - ω_{p})

Substituting Eq. (16) into Eq. (15), and considering the distance d = d₀ - v(t - t₀), we derive the final form of P_c:

P_{c} (τ, t) \propto {| \int d (ω) f (ω) \exp (- i ω {τ - [2 d_{0} - 2 v (t - t_{0}) - l] / c}) |}^{2}

where τ = t₂ – t₁.

Equation (17) shows the current coincidence counts rate (i.e., P_c) depends on t₂ - t₁, thus the FEB scheme is valid during condition of slow relative motion; however, a new puzzle arises: P_c is time-variant during [t₀, t₀ + T_a]. As the value of τ (i.e., τ_max), which makes P_c equal its maximum value, is related to the difference in the signal and idler light paths (i.e., l_dif (t)); then τ_max is shifted along the axis of TDOA during [t₀, t₀ + T_a], and the accumulated profile n(τ) in Section 2 is distorted after a time interval T_a, which causes a lower ranging accuracy.

Thus, we try to find a situation where the accumulated profile is ‘slightly distorted’. The ‘slightly distorted’ means that the distance d is able to be measured with an accuracy reaching the same order as that achieved by a non-distorted profile. Note that the essential principle of the FEB scheme is measuring the difference in the light path (i.e., l_dif); thus, if the distance variation of l_dif during the time interval T_a was less than the ranging accuracy, the shift of τ_max along the axis of TDOA would be very slight and the ‘slightly distorted’ profile could be obtained:

Δ l_{dif} = l_{dif} (t_{0}) - l_{dif} (t_{0} + T_{a}) \leq ξ \times p_{d 0}

where p_d₀ denotes the ranging accuracy of the FEB scheme in a motionless scenario (see Section 2); ξ (ξ ≤ 1) denotes the scale factor which represents the degree of the “slightly distortion” (a smaller value of ξ yields a more slightly distorted profile); Δl_dif describes the distance variation of l_dif during the interval [t₀, t₀ + T_a]; l_dif (t₀) and l_dif (t₀ + T_a) denote the difference in the light paths at time t₀ and t₀ + T_a, respectively. It is evident that:

\begin{array}{l} l_{dif} (t_{0}) = 2 d_{0} - l_{reference} \\ l_{dif} (t_{0} + T_{a}) = 2 (d_{0} - v \times T_{a}) - l_{reference} \end{array}

where d₀ is the same as that used in Eq. (3); v is the relative velocity that Bob moves towards Alice.

Substituting Eq. (19) into Eq. (18), we can easily derive that:

v \leq \frac{ξ \times p_{d 0}}{2 T_{a}}

Thus, the ambiguity velocity v_ambiguity can be described as:

\begin{matrix} v_{ambiguity} = \max (v) = \frac{ξ \times p_{d 0}}{2 T_{a}} & p_{d 0} \geq p_{limit} \end{matrix}

where p_limit denotes the limit of the ranging precision that can generally reach the order of µm, restricted by the resolution of the coincidence measurement; If v exceeds v_ambiguity, the accumulated profile will be distorted, which causes a lower range accuracy.

Since the ranging precision of FEB scheme depends on the half of the FWHM of the accumulated profile, we can obtain:

p_{d 0} = c (τ |_{\max [G (τ)]} - τ |_{\max [G (τ)] / 2})

where

τ |_{\max [G (τ)]}

is the same as that used in Eq. (2);

τ |_{\max [G (τ)] / 2}

is the value of τ which makes G(τ) obtain a half of the maximum value.

Substituting Eq. (22) into Eq. (21), the final form of Eq. (21) is:

\begin{matrix} v_{ambiguity} = \frac{ξ c (τ |_{\max [G (τ)]} - τ |_{\max [G (τ)] / 2})}{2 T_{a}} & p_{d 0} \geq p_{limit} \end{matrix}

3.3 Effect of background noise on the FEB scheme

As the idler light path is not exposed to background light, we merely consider the effect of jamming noise that has the same bandwidth on the signal light path; based on Refs [7, 12], the coincidence count rate can be written as follows:

P_{c} (t_{1}, t_{2}) \propto 〈 E_{i}^{(-)} (t_{1}) [E_{s}^{(-)} (t_{2}) + E_{n}^{(-)} (t_{2})] [E_{s}^{(+)} (t_{2}) + E_{n}^{(+)} (t_{2})] E_{i}^{(+)} (t_{1}) 〉

where subscripts i, s, and n represent the idler photons, the signal photons and the jamming noise, respectively. Because the jamming noise is independent of the idler and signal photons, the cross terms can be ignored [7, 12]; then, Eq. (24) can be written further as follows:

\begin{array}{l} P_{c} (t_{1}, t_{2}) \propto 〈 E_{i}^{(-)} (t_{1}) E_{s}^{(-)} (t_{2}) E_{s}^{(+)} (t_{2}) E_{i}^{(+)} (t_{1}) 〉 \\ + 〈 [E_{i}^{(-)} (t_{1}) E_{i}^{(+)} (t_{1}) 〉 〈 E_{n}^{(-)} (t_{2}) E_{n}^{(+)} (t_{2}) 〉 \\ = G (t_{1}, t_{2}) + 〈 E_{i} 〉 〈 E_{n} 〉 \end{array}

where G(t₁, t₂) is just the second-order Glauber correlation function of the signal-idler photon pair; the extra term 〈E_i〉〈E_n〉 is caused by the environment noise. If the magnitude of the random variations of this extra term approached the magnitude of the signal peak, the measuring range would become unreliable. Based on Ref [19], a reasonable definition of signal to noise ratio (SNR) is dividing the expected total coincidence count (i.e., R_s) by the standard deviation of the total counts of two coincidence channels. Ref [14]. shows that both the coincidence photon counts and the background noise counts (i.e., b) obey the Poissonian distribution, then, the standard deviation of the total counts of two coincidence channels can be expressed as follow:

σ = {(R_{s} T_{a} + 2 b T_{a})}^{1 / 2}

Then, SNR is written as follows:

S N R = R_{s} T_{a} / {(R_{s} T_{a} + 2 b T_{a})}^{1 / 2}

Generally, the SNR should be no less than 1 so that we can clearly distinguish the peak position of the accumulated profile. Xiao et al. had conducted an experiment that an incandescent lamp is used to irradiate the signal light path in order to investigate the noise with arbitrary wavelength [12]. The measurement result still has a good performance and the FWHM of the accumulated profile is unchanged, even in an environment that has a noise level 12.84 dB above the signal photons. Additionally, another method can also be used: the extra term 〈E_i〉〈E_n〉 can be estimated through measuring the average power of the noise, then it can be eliminated through careful data calibration [7]; thus, the environment noise cannot affect the coincidence measurement. By the way, we derived the expression of the accumulated time T_a which determines the update rate through Eq. (27):

T_{a} = S N R^{2} (R_{s} + 2 b) / R_{s}^{2}

Because SNR denotes the distinguishability of the peak position of the accumulated profile, it is evident that T_a is a function of both the photon pair flux and the accuracy to which the range must be determined.

3.4 Simulation and analysis

3.4.1 Distortion of the profile in a scenario of relative motion

Valencia et al. [11] and Xiao et al. [12] have already performed a proof-of-principle experiment on the FEB scheme in a motionless scenario. The signal-idler photon pair were produced from two 0.8-mm-thick type-II BBO crystals pumped by an Ar laser at a wave-length of 363.8 nm. The TDOAs of photons were measured by a coincidence measurement system with a resolution of 4 ps; the coincidence count rate which also contains the dark counts of the two detectors was [12]:

p (τ) = {n_{0} \sin^{2} [ω (τ - μ)]} / {[ω (τ - μ)]}^{2}

where n₀, µ, ω denote the total number of the detected signal-idler photon pair, the displacement of the center peak, and the parameter related to the FWHM of the SOC, respectively. After an accumulated time of 10s, the accumulated profile n(τ) corresponded to p(τ).

In this simulation, the distance between Alice and Bob at t₀ (i.e., d₀) was set to 502.5 m, and the reference photon path on-board the platform (i.e., l_reference) was set to 5 m. We used the same entangled photon source and coincidence measurement system as above [12] (i.e., n₀ = 5820, ω = 1.85 $\times$ 10⁻³, the accumulated time interval T_a = 10 s, and the FWHM of G(τ) equals 1504.38 ps); thus, the coincidence count rate could be derived using Eq. (29):

\begin{array}{l} p_{c} (τ) = {n_{0} \sin^{2} [ω (τ - μ)]} / [^{ω (τ - μ)] 2} \\ = n_{0} \sin^{2} {ω [τ - (2 d_{0} - l_{reference}) / c]} / {^{ω [τ - (2 d_{0} - l_{reference}) / c]} 2} \end{array}

The probability of having a joint photo-detection event at two detectors is proportional to the coincidence count rate [11]. Keep this idea in mind, then, we used the normalized p_c(τ) as the probability function of the joint-detection of the signal-idler photon pair; after an accumulated time (T_a = 10 s), a histogram of the TDOA of the signal-idler photon pair was obtained and was plotted in Fig. 2, which is the accumulated profile in a motionless scenario. The natural value of the width of this profile is generally in the order of hundreds of femtoseconds (i.e., 10⁻¹³ second), which is determined by the bandwidth of the down-converted photons [20]; while in this work, the measured value of this width is in the order of nanosecond, which is restricted by the detector resolving time.

Fig. 2 Accumulated profile in a motionless scenario

Download Full Size | PDF

On the other hand, combining Eqs. (17) and (30), the coincidence count rate in the scenario of relative motion (i.e., p_motion(τ,t)) could be derived as follows:

\begin{array}{l} p_{motion} (τ, t) = n_{0} \sin^{2} {ω τ - ω [2 d_{0} - 2 v (t - t_{0}) - l_{reference}] / c} \\ / {^{ω τ - ω [2 d_{0} - 2 v (t - t_{0}) - l_{reference}] / c} 2} \end{array}

Equation (31) shows that the probability function of the joint-detection of the signal-idler photon pair during condition of relative motion varies with time. Then, we divided the time interval [0, 10 s] into a great amount of small time bins, in which the corresponding probability functions are different. Selecting three values of the velocity of Bob (i.e., 0 m/s, 0.1 m/s and 1 m/s) and using the same accumulated time (T_a = 10 s), we obtained three accumulated profiles (histograms of the TDOA) in the Fig. 3.

Fig. 3 Accumulated profile in a scenario of relative motion

Download Full Size | PDF

Through the observations shown in Fig. 3, we can conclude: 1) compared with the black curve in the motionless scenario, both of the red and blue curves (correspond to values of 0.1 m/s and 1 m/s, respectively) are distorted; 2) a more rapidly motion of Bob yields a greater broadening of the FWHM of the accumulated profile, and it is thus difficult for us to measure the distance, which coincides with the analysis in Section 3.2; 3) although Bob moves towards Alice, the total number of signal-idler photon pair remains unchanged, which indicates the areas of three curves should be equal; therefore, the peaks of the distorted curve have dropped with the broadening of the FWHM of the distorted profiles, see Fig. 3.

3.4.2 Ambiguity velocity

The scale factor ξ in this simulation was set to 1, which means we loosen the degree of “slightly distortion” to the greatest extent; then, the ambiguity velocity was calculated to be 1.128 × 10⁻² m/s, using Eq. (23) with the same parameters (i.e., FWHM = 1504.38 ps, T_a = 10 s) used in Section 3.4.1.

As shown in Fig. 4, the red curve represents the accumulated profile when Bob moves at the ambiguity velocity, and the shape of this profile remains almost unchanged; however, the shape of the blue curve is distorted seriously when Bob moves at a speed approximately ten times faster than the ambiguity velocity.

Fig. 4 Accumulated profile in a scenario of relative motion

Download Full Size | PDF

Additionally, the TDOA shift of the peak of the distorted profile with respect to that of the non-distorted profile inspires us to analyze Eq. (18) at another point of view. Then, we can change Eq. (18) into another form (ξ = 1 in this section):

\frac{Δ l_{dif}}{c} = \frac{l_{dif} (t_{0}) - l_{dif} (t_{0} + T_{a})}{c} \leq \frac{p_{d 0}}{c}

Substituting Eq. (2), (19) and (22) into Eq. (32), we can obtain:

\begin{array}{l} \frac{l_{dif} (t_{0}) - l_{dif} (t_{0} + T_{a})}{c} = \frac{l_{dif} - l_{dif} (t_{0} + T_{a})}{c} \\ = τ |_{\max [G (τ)]} - τ_{reference} = \frac{2 v T_{a}}{c} \leq τ |_{\max [G (τ)]} - τ |_{\max [G (τ)] / 2} \end{array}

where l_dif is the same as that used in Eq. (2); τ|_max[G(τ)] and τ|_max[G(τ)]/2 represent the abscissas of the peak (i.e., O_peak in Fig. 4) and the half-maximum point (i.e., O_half in Fig. 4) of the non-distorted profile, respectively; τ_reference is the abscissa of a reference point (i.e., A and B in Fig. 4) of the distorted profile. Then, the two equations are easily derived from Eq. (33):

τ_{reference} = τ |_{\max [G (τ)]} - 2 v T_{a} / c

τ_{reference} \geq τ |_{\max [G (τ)] / 2}

The reference point is used to judge whether the distorted profile is ‘slightly distorted’: if the displacement of τ_reference with respect to τ|_max[G(τ)] is less than a half of the FWHM of the non-distorted profile (i.e., τ_reference lies on the right of τ|_max[G(τ)]/2), the accumulated profile is ‘slightly distorted’.

While Bob moves at the ambiguity velocity, the reference point A on the red distorted profile in Fig. 4 has the same abscissa as the half-maximum point O_half, thus, it can be described as ‘slightly distorted’. However, when Bob moves at a speed exceeding the ambiguity velocity, the reference point B lies on the left of O_half. Thus, the blue profile is distorted seriously. The results of the simulation coincide with Eqs. (34) and (35); as the two equations are derived from Eq. (18), on the contrary, the results simultaneously verify the correctness of Eq. (18).

4. Doppler shift and ambiguity velocity cancellation

If the relative speed between Alice and Bob is slower than the ambiguity velocity (i.e., v_ambiguity), the FEB scheme can be used to measure the distance d. Though we loosen the degree of “slightly distortion” to the greatest extent in Section 3.4.2, the value of the ambiguity velocity is still so small that it severely restricts the applications of the FEB scheme. Thus it is urgent for us to remove the effect of the ambiguity velocity and Doppler shift.

As mentioned in Section 3, Bob moves towards Alice at a speed of v; based on Ref [16], the motion of Bob performs the same role as a time-varying delay δl_B(t), see Fig. 5. Consider the case of linear time dependence, namely:

δ l_{B} (t) = - v (t - t_{0})

where t₀ denotes the initial time which is the same as that used in Section 3.

Fig. 5 Schematic diagram of the improved FEB scheme

Download Full Size | PDF

The motion of Bob simply exists in the light path of the signal photon, thus, we could attempt to introduce another time-varying delay δl_A(t) in the signal photon path to cancel the effect of δl_B(t). See Fig. 5, the time-varying delay δl_A(t) is placed in front of the detector 2, which can be represented as:

δ l_{A} (t) = v_{add} (t - t_{0})

where v_add denotes the time-varying rate of δl_A(t).

Then, let us analyze the evolution of the annihilation operator for the signal photon. Once produced by the photon source, this operator is denoted by a_s(ω₂); one find that a_s(ω₂) can be evolved into a_Bob(ω₂) at Bob’s position (at a distance d₁ from Alice) and then into a^’₂(ω₂) at the detector 2’s position (at a distance d₂ from Bob). As Bob is far from Alice, for simplicity, we assume δl_A(t) and detector 2 are at the same position, i.e., d₁ = d₂ = d₀ at the initial time t₀. Therefore, we can obtain:

\begin{array}{l} a_{Bob} (ω_{2}) = χ^{1 / 2} \times a_{s} (χ ω_{2}) \times \exp {- i ω_{2} [2 β (- d_{1} / c) / (1 - β) - d_{1} / c]} \\ = χ^{1 / 2} a_{s} (χ ω_{2}) \exp [i ω_{2} χ (d_{1} / c)] \end{array}

\begin{array}{l} a_{2}^{'} (ω_{2}) = χ_{add}^{- 1 / 2} \times a_{Bob} (ω_{2} / χ_{add}) \\ \times \exp {- i ω_{2} [- 2 β_{add} (- d_{2} / c) / (1 + β_{add}) - (d_{2} / c)]} \\ = (^{χ / χ_{add}) 1 / 2} a_{s} (ω_{2} χ / χ_{add}) \times \exp {i ω_{2} [(χ d_{1} + d_{2}) / c χ_{add}]} \\ = (^{χ / χ_{add}) 1 / 2} a_{s} (ω_{2} χ / χ_{add}) \\ \times \exp {i ω_{2} [(χ + 1) d_{0} + χ δ l_{B} (t) + δ l_{A} (t)] / c χ_{add}} \end{array}

where χ_add = (1 + β_add) / (1-β_add), and β_add = v_add / c.

To eliminate the effect of relative motion, the annihilation operator a^’₂(ω₂) should be independent of the time t, namely:

δ l_{A} (t) = v_{add} (t - t_{0}) = - χ δ l_{B} (t) = χ v (t - t_{0})

Equation (40) shows the time-varying rate of δl_A(t) (i.e., v_add) should equal χv, and χ_add can be represented as follows:

\begin{array}{l} χ_{add} = (1 + β_{add}) / (1 - β_{add}) = (1 + χ v / c) / (1 - χ v / c) \\ = {c + [v (1 + β) / (1 - β)]} / {c - [v (1 + β) / (1 - β)]} \\ = [c (c - v) + v (c + v)] / [c (c - v) - v (c + v)] \\ = (c^{2} + v^{2}) / (^{c - v) 2} = (1 + β^{2}) / (^{1 - β) 2} \end{array}

Substituting Eq. (40) into Eq. (39) and we can obtain:

\begin{array}{l} a_{2}^{'} (ω_{2}) = (^{χ / χ_{add}) 1 / 2} a_{s} (ω_{2} χ / χ_{add}) \times \exp {i ω_{2} [(χ + 1) d_{0}] / c χ_{add}} \\ = (^{γ) 1 / 2} a_{s} (γ ω_{2}) \times \exp [i ω_{2} (μ d_{0} / c)] \end{array}

where µ = (χ + 1) /χ_add, and γ = χ /χ_add, which can be further written based on Eq. (41):

\begin{array}{l} γ = χ / χ_{add} = [(1 + β) / (1 - β)] / [(1 + β^{2}) / (^{1 - β) 2}]} \\ = (1 - β^{2}) / (1 + β^{2}) \end{array}

The subsequent procedure of deriving the coincidence rate P_c corresponds to that of Eqs. (7-12); thus, the details are omitted here, and we can obtain:

\begin{array}{l} P_{c} (t_{1}, t_{2}) \propto {| \int d (ω) f (ω) \exp {- i ω [t_{2} - γ t_{1} - (μ d_{0} - l) / c]} |}^{2} \\ = {| \int d (ω) f (ω) \exp {- i ω [t_{2} - t_{1} + (1 - γ) t_{1} - (μ d_{0} - l) / c]} |}^{2} \end{array}

Through introducing a time-varying delay δl_A(t), t₂ - γt₁ becomes a quadratic term related to β, compared with t₂ - χt₁ in Eq. (13) that is a first-order term related to β. The speed of the currently fastest aircraft can reach 20 times the speed of sound, then the maximum of 1-χ can reach an order of 10⁻⁵ while that of 1-γ reaches an order of 10⁻¹⁰. Although P_c in Eq. (44) is still dependent on t₂ - γt₁ at first sight, due to the extremely small value of 1-γ, P_c can in practice be written as a function of TDOA (i.e., τ):

P_{c} (τ) \propto {| \int d (ω) f (ω) \exp {- i ω [τ - (μ d_{0} - l) / c]} |}^{2}

Finally, the coincidence rate P_c is independent of the time t, and a stable accumulated profile n(τ) of the coincidence measurement can be obtained after a time interval T_a. Thus, we can determine the distance d₀ between Alice and Bob at the time t₀:

d_{0} = \frac{1}{μ} {l + c \times τ | \max [G (τ)]}

Equation (46) coincides with Eq. (3) in Section 2, which indicates that the effect of relative motion on the FEB scheme has been removed. In summary, a time-varying delay δl_A(t) with the time-varying rate of χv can be introduced to solve the two major puzzles.

5 Conclusions

In this study, we have analyzed the effect of relative motion on the FEB scheme. Two major puzzles have arisen, namely, Doppler shift and ambiguity velocity. When the effect of the Doppler shift is obvious, the SOC of signal-idler photon pair would not depend on the DTOA of photon pairs; thus, the FEB scheme is invalid to measure the distance. When the relative motion is relatively slow, the effect of Doppler shift can be ignored; however, if the relative speed between Alice and Bob exceeds the ambiguity velocity, the accumulated profile will be distorted, which causes a lower ranging accuracy. The ambiguity velocity was calculated as 1.128 cm/s in Section 3.4.2, and this small value would severely restrict the applications of the FEB scheme.

Through the theoretical analysis, introducing a time-varying delay δl_A(t), which is placed in front of the detector 2, is able to solve the two major puzzles; the time-varying rate of δl_A(t) is directly related to the relative velocity v, which can be determined by the use of time-tagged photons in time correlated singe photon counting: if the absolute time t₀ is recorded for the detection of the coincident pairs, then the order in which the counts are entered into the histograms in Figs. 3 and 4 is known; this allows the relative velocity to be determined from an initial guess, since the TDOA can be corrected to account for the relative velocity; then, the correct relative velocity is the one which gives the highest and shortest (i.e. sharpest) peak. Our work has extended the application of the FEB scheme from the static scenario to the motion one, and hopefully, will give some guidance on the practical designs of quantum ranging sensor for measuring the moving target.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 61172138, No. 61401340), the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2013JQ8040), the Open Research Fund of The Academy of Satellite Application (Grant No. 2014_CXJJ-DH_12), the Shandong Aerospace Innovation Fund (2013JJ04), the Open Fund of Key Laboratory of Precision Navigation and Timing Technology, National Time Service Center, CAS(NO. 2014PNTT07)

References and links

1. K. N. Joo and S. W. Kim, “Absolute distance measurement by dispersive interferometry using a femtosecond pulse laser,” Opt. Express 14(13), 5954–5960 (2006). [CrossRef] [PubMed]

2. H. Wu, F. Zhang, S. Cao, S. Xing, and X. Qu, “Absolute distance measurement by intensity detection using a mode-locked femtosecond pulse laser,” Opt. Express 22(9), 10380–10397 (2014). [CrossRef] [PubMed]

3. H. J. Yang, J. Deibel, S. Nyberg, and K. Riles, “High-precision absolute distance and vibration measurement with frequency scanned interferometry,” Appl. Opt. 44(19), 3937–3944 (2005). [CrossRef] [PubMed]

4. N. Schuhler, Y. Salvadé, S. Lévêque, R. Dändliker, and R. Holzwarth, “Frequency-comb-referenced two-wavelength source for absolute distance measurement,” Opt. Lett. 31(21), 3101–3103 (2006). [CrossRef] [PubMed]

5. Y. Salvadé, N. Schuhler, S. Lévêque, and S. Le Floch, “High-accuracy absolute distance measurement using frequency comb referenced multiwavelength source,” Appl. Opt. 47(14), 2715–2720 (2008). [CrossRef] [PubMed]

6. I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009). [CrossRef]

7. J. Zhu, X. Chen, P. Huang, and G. Zeng, “Thermal-light-based ranging using second-order coherence,” Appl. Opt. 51(20), 4885–4890 (2012). [CrossRef] [PubMed]

8. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004). [CrossRef] [PubMed]

9. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning and clock synchronization,” Nature 412(6845), 417–419 (2001). [CrossRef] [PubMed]

10. V. Giovannetti, S. Lloyd, and L. Maccone, “Positioning and clock synchronization through entanglement,” Phys. Rev. A 65(2), 022309 (2002). [CrossRef]

11. A. Valencia, G. Scarcelli, and Y. Shih, “Distant clock synchronization using entangled photon pairs,” Appl. Phys. Lett. 85(13), 2655–2657 (2004). [CrossRef]

12. J. J. Xiao, C. Fang, X. C. Han, J. K. Zhao, and G. H. Zeng, “Distance ranging based on quantum entanglement,” Chin. Phys. Lett. 30(10), 100301 (2013). [CrossRef]

13. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum cryptographic ranging,” J. Opt. B Quantum Semiclassical Opt. 4(4), S413–S414 (2002). [CrossRef]

14. F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009). [CrossRef]

15. Laser. Componets, “Single photon counting module count® blue series,” (Laser Componets, 2015), http://www.lasercomponents.com/fileadmin/user_upload/home/Datasheets/lcp/count-blue-series.pdf.

16. V. Giovannetti, S. Lloyd, L. Maccone, and F. N. C. Wong, “Clock synchronization with dispersion cancellation,” Phys. Rev. Lett. 87(11), 117902 (2001). [CrossRef] [PubMed]

17. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987). [CrossRef] [PubMed]

18. F. y. Hou, R. Dong, R. Quan, Y. Zhang, Y. Bai, T. Liu, S. Zhang, and T. y. Zhang, “Dispersion-free quantum clock synchronization via fiber link,” Adv. Space Res. 50(11), 1489–1494 (2012). [CrossRef]

19. J. G. Walker, S. F. Seward, J. G. Rarity, and P. R. Tapster, “Range measurement photon by photon,” Quantum Opt. 1(1), 75–82 (1989). [CrossRef]

20. S. Friberg, C. K. Hong, and L. Mandel, “Measurement of Time Delays in the Parametric Production of Photon Pairs,” Phys. Rev. Lett. 54(18), 2011–2013 (1985). [CrossRef] [PubMed]

Doppler shift and ambiguity velocity caused by relative motion in quantum-enhanced measurement

Abstract

1. Introduction

2. Scheme description

3 Major puzzles caused by the relative motion

3.1 Effect of Doppler shift

3.2 Ambiguity velocity

3.3 Effect of background noise on the FEB scheme

3.4 Simulation and analysis

3.4.1 Distortion of the profile in a scenario of relative motion

3.4.2 Ambiguity velocity

4. Doppler shift and ambiguity velocity cancellation

5 Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Equations (46)

Optics Express