Coherent Two-Photon Atmospheric Lidar Based on Up-Conversion Quantum Erasure

Sep. 07 , 2024photonics1

Abstract

The Hong-Ou-Mandel (HOM) effect, also called two-photon interference, which is pivotal in quantum information processing, serves as the foundation for our coherent two-photon wind detection method, achieving time-frequency discrimination. Our method capitalizes on the complete erasure of phase and color distinctions with up-conversion detectors, allowing for the observation of HOM interference among independently sourced photons without necessitating delay tuning or active synchronization. The integration of quantum erasure enhances the bandwidth in both space-time and frequency domains, paving the way for ultrafast, continuous velocity detection up to tens of km/s. The demonstration uses just 14% pulse energy of previous levels, enabling a wind detection range up to 16 km, marking a quantum leap in remote sensing capabilities.

Click to copy section linkSection link copied!

Since its discovery in 1987, the Hong-Ou-Mandel (HOM) interference has been a cornerstone in distinguishing quantum phenomena from classical physics, (1) heralding a new era in quantum exploration. (2) Beyond its foundational role in precise time measurement (3) and quantum state analysis, (4,5) HOM interference has become integral in an array of quantum information processing applications, (6) ranging from quantum computing (7) to quantum-optical coherence tomography. (8) Its exploration extends across various quantum systems like plasmons, (9) phonons, (10) and atoms, (11) underlining its versatility. Additionally, HOM interference between independent weak coherent sources have been demonstrated (12−14) and played a key role toward practical implementation of the measurement-device-independent quantum key distribution (MDI-QKD) protocol. (15,16) Central to understanding HOM interference is the principle of indistinguishability in all degrees of freedom (d.o.f.), including spatial mode, polarization, color, and phase, a concept derived from Bohr’s complementarity principle. (17) The advent of quantum erasure has further expanded the horizons, enabling interference between independent particles by effectively’erasing’ their distinguishing features. (18) This principle has led to breakthroughs like observing interference between celestial bodies and Earth-based quantum systems. (19,20) In the realm of practical applications, color erasure stands out, extending interference capabilities across different wavelengths and adding a new dimension to quantum networks. (21) Taking advantage of the very low background noise and the high quantum efficiency of up-conversion single-photon detector (UCSPD) in the infrared band, the distance of two hard targets beyond the diffraction limit was measured by color erasure. (22) Despite its successful implementation in hard target detection, leveraging its potential for atmospheric detection has been challenging due to the significantly smaller backscattering cross-section and higher speeds of particle targets. The backscattering cross section of the atmosphere is 7 orders of magnitude smaller than that of hard targets, so there is neither atmosphere detection nor time-frequency discrimination. Enthalpy probes and laser scattering methods (23,24) are commonly used high-speed particle detection methods. Although the laser method is nonintrusive measurement and has higher precision and spatial resolution, accurate distributed and continuous time-frequency detection of high-speed particles is still challenging due to their high Doppler shift and weak signal. Here, we demonstrate a coherent two-photon atmospheric lidar based on up-conversion quantum erasure, paving the way for ultrafast, continuous velocity detection up to tens of km/s. The details are given in the following sections.

Click to copy section linkSection link copied!

In HOM interference, being indistinguishable is necessary, as mentioned above. It is intuitive to consider the overlap or “bunching” between two photons at the beam splitter. However, many studies have suggested that indistinguishability is not strictly necessary at beam splitters (BS) but at detectors. (25−27) The key is to detect the photons in such a way that the distinguishing information is erased. (28) Therefore, for erasure of color d.o.f., a third frequency is introduced to mask the distinguishability. Specifically, we perform frequency up-conversion for both photons γ_{1},γ_{2} with a large frequency difference to “fool” the detector so that it is unable to determine which source the converted photons γ_{3} came from. In this experiment, the core device for color erasure is an up-conversion periodically poled lithium niobate (PPLN) waveguide with a variable central wavelength, which can upconvert the frequency of an incoming photon without changing its nonclassical properties. The process can be described more intuitively by the following formulas. Let *f*_{1(2)} denotes the frequency of photon γ_{1(2)}, and let ${f}_{3}^{1(2)}$ denote the frequency of γ_{3} converted from γ_{1(2)}. Consider the initial state
where |Ω⟩ is the vacuum state and ${a}_{{\gamma}_{1(2)}}^{\u2020}$ are the creation operators of γ_{1(2)} in some fixed spatiotemporal mode. Suppose we have two PPLNs and prepare the coherent state of the pump photons *γ*_{p1(p2)} with frequency$\mathrm{\Delta}{f}_{31(32)}={f}_{3}^{1(2)}-{f}_{1(2)}$
where $|n{\u27e9}_{{\gamma}_{p1(p2)}}$ is the number state in which *n γ*_{p1(p2)} photons are in an incoming mode. Then, the input states of the two PPLNs are described by

$$|{\psi}_{0}\u27e9=({\alpha}_{1}{a}_{{\gamma}_{1}}^{\u2020}+{\alpha}_{2}{a}_{{\gamma}_{2}}^{\u2020})|\mathrm{\Omega}\u27e9$$

(1)

$${|\beta ,coh.\u27e9}_{{\gamma}_{p1(p2)}}={e}^{-{|\beta |}^{2}/2}\sum _{n=0}^{\infty}\frac{{\beta}^{n}}{\sqrt{n!}}{|n\u27e9}_{{\gamma}_{p1(p2)}}$$

(2)

$${|1\u27e9}_{{\gamma}_{1}}\otimes {|\beta ,coh.\u27e9}_{{\gamma}_{p1}},{|1\u27e9}_{{\gamma}_{2}}\otimes {|\beta ,coh.\u27e9}_{{\gamma}_{p2}}$$

(3)

When the average number of photons ${|\beta |}^{2}{=}_{{\gamma}_{p1(p2)}}{\u27e8\beta ,coh.|\hat{n}|\beta ,coh.\u27e9}_{{\gamma}_{p1(p2)}}$is sufficiently large, the input states evolve with the Hamiltonian
where ξ_{31(32)} represents the conversion efficiency and ?_{31(32)} represents the phase of the pump light. The Hamiltonians cause the γ_{1},γ_{2} entangled with the pump photons of large average number |β|^{2}, which allows us to lose track of the loss or gain of single photons. (29) Then, the converted pho tons are combined by a 50–50 beam splitter. Finally, applying *f*_{3} filters, which is equivalent to projecting onto ${a}_{{\gamma}_{3}}^{\u2020}|\mathrm{\Omega}\u27e9$, and the resulting state can be described by
where *τ* is the traveling time of the photons through the PPLN. In the above equation, *τ,* ξ, and ? are constant and controllable. If tuning ξ_{31}*τ* = *π*/2, ξ_{31}*τ* = 2*π,* ξ_{31}*τ* = 2*π, ?*_{31} = ?_{32} = 0, the preceding process accomplishes the following transformation:

$${H}_{31}=i{\xi}_{31}({e}^{i{?}_{31}}{a}_{{\gamma}_{1}}{a}_{{\gamma}_{3}}^{\u2020}-{e}^{-i{?}_{31}}{a}_{{\gamma}_{1}}^{\u2020}{a}_{{\gamma}_{3}})$$

(4)

$${H}_{32}=i{\xi}_{32}({e}^{i{?}_{32}}{a}_{{\gamma}_{2}}{a}_{{\gamma}_{3}}^{\u2020}-{e}^{-i{?}_{32}}{a}_{{\gamma}_{2}}^{\u2020}{a}_{{\gamma}_{3}})$$

(5)

$$\frac{1}{2}({\alpha}_{1}{e}^{i{?}_{31}}\mathrm{sin}({\zeta}_{31}\tau )+{\alpha}_{2}{e}^{i{?}_{32}}\mathrm{cos}({\zeta}_{32}\tau )){a}_{{\gamma}_{3}}^{\u2020}|\mathrm{\Omega}\u27e9$$

(6)

$$({\alpha}_{1}{a}_{{\gamma}_{1}}^{\u2020}+{\alpha}_{2}{a}_{{\gamma}_{2}}^{\u2020})|\mathrm{\Omega}\u27e9\to \frac{1}{2}({\alpha}_{1}+{\alpha}_{2}){a}_{{\gamma}_{3}}^{\u2020}|\mathrm{\Omega}\u27e9$$

(7)

Note that, ${f}_{3}^{1}$ and ${f}_{3}^{2}$ are not exactly identical in most cases, but they are indistinguishable to the detector as long as their difference is less than the bandwidth of the detector. Furthermore, the color erasure of photons with a larger frequency difference can be achieved by tuning the wavelengths of the two pumps.

The above process achieves the erasure of color d.o.f. For the polarization d.o.f., the PPLN is polarization sensitive, making it inevitable that all photons passing through the waveguide will have the same polarization state. Last but not least, for the phase d.o.f., in hard target detection, the phases of the two photons are correlated, and control of the time delay will affect the visibility of the interference. So, it is necessary to tune their level of indistinguishability. (30−32) For independent sources, strict synchronization is required. (33) In this study, our purpose is to use high-order quantum erasure to achieve more compact, large bandwidth, and single-photon time-frequency discrimination. Fortunately, the phase of backscattering photons from atmosphere is completely random, (34) which inherently implies the indistinguishability of the phase, eliminating the need for tuning delays or active synchronization.

To verify the high-order quantum erasure theory and demonstrate performance improvements, we conducted two experiments.

An ultrafast speed particle will generate a large Doppler frequency, which is calculated as 2*v*/*λ*, where *v* is the moving speed and *λ* is the wavelength of the detection laser. When λ = 1.5 μm, a speed of 10 km/s corresponds to a Doppler frequency of 12.9 GHz. In order to simulate the detection of ultrafast speed particles, we use two lasers to generate a tunable lager frequency difference as the Doppler frequency. A setup diagram of the experiment is shown in Figure 1b. A fixed-wavelength continuous-wave (CW) laser with a central wavelength of 1550.1230 nm and a tunable-wavelength CW laser with a tuning range of 1549.6000 to 1550.5700 nm are prepared to serve as two input photons with different colors. The two lasers are totally independent of phase locking. The input photons from the two lasers are combined with pump light and focus on the PPLN1 and PPLN2 waveguides for frequency conversion. The number of input photons is adjusted by variable attenuators (VAs). The acceptance wavelengths of the two PPLNs are tuned by a temperature controller. After the PPLNs, we used 863 nm band-pass filters to filter out the unwanted photons. The filtered photons are then combined by a 50–50 BS and guided to two silicon single photon detectors (SSPDs). The arrival time of detected photons is recorded by two time-to-digital converters (TDC). To obtain the interference pattern oscillating at ${f}_{HOM}=|{f}_{3}^{1}-{f}_{3}^{2}|$, we calculate the second-order quantum mechanical correlation function *G*^{(2)}(τ). The time window is set to 100 ps. Then, based on the Fourier transform of *G*^{(2)}(τ) and the wavelength difference of the two pump lasers, we can achieve our goal of obtaining the actual frequency difference between γ_{1} and γ_{2}.

During the experiment, the pump laser of PPLN1 is 1950.3091 nm, while the tunable pump wavelength of PPLN2 is 1950.0828 nm. The interference of the photons with a greater frequency difference of 17.1396–17.8385 GHz. Figure 2 shows the experimentally measured wavelength of the tunable laser against the set value. The deviation is less than 30 MHz, which is smaller than the systematic error caused by the limited measurement accuracy (60 MHz) of the High Finesse wavelength meter, as shown in Figure 2a. The frequency difference of ∼17 GHz demonstrated in our experiment corresponds to a speed of ∼13 km/s, which will need ultrafast data sampling rate and huge storage space in traditional measuring method. Benefiting from the up-conversion detection, the interference fringes are on the order of MHz. Thus, a sample rate of MHz is sufficient to record the GHz signal, which is with optically compressed data below the Nyquist sampling theorem. It can be expected that when the wavelength difference of the pump laser is further increased, the QEHOM method has the potential to realize long-term continuous observation of targets moving at speeds of hundreds of km/s.

A free space wind detection experiment is demonstrated to show the weak signal performance of the QEHOM. Wind sensing is of great significance for weather warning, meteorological model analysis, aviation safety, energy structure, deep space exploration, and various other applications. (35) There are two fundamental methods for determining wind velocity: direct detection Lidar (DDL) and coherent Doppler Lidar (CDL). DDL, which uses molecules and atoms as tracers, can achieve longer distances and higher altitude detection. It requires high-powered lasers and high-precision frequency discriminators. CDL uses aerosols as tracers and detects Mie scattering signals and is thus greatly influenced by aerosol content. As of 2012, the longest measurable range for CDL, achieved with high-output power laser amplifiers, extends beyond 30 km. (36) For all-fiber systems, ONERA realized a measurable range of 16 km based on the parallel connection of multiple fiber amplifiers. (37) Here, the QEHOM provides a brand-new method for high quantum efficiency wind measurement, achieving a long detection range at low laser power for an all-fiber system.

Figure 1c shows the setup of the QEHOM for the wind speed measurement. A CW laser generates linearly polarized light with a central wavelength of 1548.5 nm, which is divided into a reference beam and transmitted beam by a beam splitter. The transmitted beam is chopped into a pulse train by an acousto-optic modulator (AOM). A frequency shift of 80 MHz is carried out in the 100 ns laser pulse. After amplification by an erbium-doped fiber amplifier (EDFA), the pulses are transmitted to the atmosphere through a fiber circulator. Then, the backscattering signal is received by a coaxial telescope. The reference beam, which is attenuated to the order of a single photon by an electro-optic modulator (EOM), and the echo photons enter the PPLN1 and PPLN2 waveguides for frequency up-conversion. The pump wavelength of both waveguides is 1950.3 nm. After frequency conversion and filtering, reference and echo photons are both converted to 863 nm and then incident into the BS. Photons emitted from BS are then detected by SSPDs, and the arrival time of photons is recorded by TDCs. To validate the results of wind measurement, a traditional CDL is constructed simultaneously for comparison. We use two 50:50 BS to separate half of the backscattering signal and reference beam for the conventional coherent module. The beat signal is converted to an intermediate frequency (IF) signal by a balanced detector (BD). The IF signal will be performed with the FFT by ADC and the digital signal processor (DSP) for Doppler shift retrieval.

A continuous observation from 12:24 to 12:54 on March 23 was carried out, and the wind velocity measurement results based on QEHOM and CDL were compared. The spatial and temporal resolutions in this experiment were 15 min and 20 s, respectively. Figure 3a shows the single-detector counting rate of each range gate, which is roughly on the order of 1 × 10^{3}. Figure 3b shows the coincidence counts at different ranges of a typical time slice. The frequency of the oscillation pattern is ∼80 MHz. The amplitude of the envelope gradually decreases as the distance increases because of the attenuation of the SNR (Signal-to-Noise Ratio). Figure 3c–e shows the comparison between the wind velocity measurement results of the two methods. More quantitatively, we analyzed the residual distribution between them based on *y* = *y*_{0} + *A* exp(−(*v*_{HOM} *– v*_{CDL} *– u*)^{2})/(2*w*^{2})), where *u* and *w* represent the average and variance of their residual, and *v*_{QEHOM} and *v*_{CDL} denote the wind velocity retrieved from our scheme and the CDL, respectively. We obtained *u* = 0.13506 ± 0.0024 and *w* = 0.31947 ± 0.0026. This shows that the result of QEHOM has good consistency with that of CDL.

The imbalance of BS will result in reduced visibility. (16) Additionally, quantum efficiency and noise are related to the number of incoming photons for nonideal detectors due to the existence of shot noise and thermal noise. To optimize the measurement performance and to get close to the shot-noise limit, (38) we optimized the relative intensity of the two beams by revising the number of reference photons. For the conventional CDL, if the shot noise and thermal noise brought by the local oscillator are considered, the carrier-to-noise ratio (CNR) can be expressed as (39)
where Θ is the sensitivity of the detector, *η*_{h} is the heterodyne efficiency, *P*_{sd} is the signal power, *P*_{LO} is the local oscillator power, α is the saturation response factor, *B* is the bandwidth of the detector, and *R*_{L} is the load resistance, all of which are constant. It can be seen from this formula that the local oscillator power is not always higher, the better, and the gain of the SNR is nonlinear. Considering the coherent two-photon interference, there will be an optimal ratio *k* = *N*_{ref}/*N*_{Sig} corresponding to the peak of SNR, where *N*_{ref},*N*_{Sig} is the counts per second (cps) of reference and signal photons. The number of reference and backscattering photons needs to have the same “shape”, or in other words, to keep the same ratio in any range gate. We realized the optimal one based on EOM and AWG (33622A Keysight). First, *N*_{Sig}(*n*) is obtained, where *n* is the number of range gates. Then, *N*_{ref}(*i,n*) is updated every second, where *i* is the number of recursions. The relative deviation between *N*_{ref}(*i,n*) and *N*_{Sig}(*n*) is calculated as follows:

$$CNR=\frac{2{\mathrm{\Theta}}^{2}{\eta}_{h}{P}_{sd}{(1-2\alpha {P}_{LO})}^{2}}{4KTB/{R}_{L}+2e\mathrm{\Theta}{P}_{LO}(1-\alpha {P}_{LO})B}$$

(8)

$$\begin{array}{l}\delta (i,n)=\frac{{N}_{ref}(i,n)-{N}_{ref}(0,n)}{{N}_{ref}(0,n)},\\ i=1,2,3,...,{N}_{ref}(0,n)={N}_{Sig}(n)\end{array}$$

(9)

When δ(*i,n*) is less than the set value, calculate the waveform of the modulated wave loaded into the EOM through the AWG

$$\begin{array}{l}{N}_{ref}(i+1,n)=({N}_{ref}(i,n)-{N}_{ref}(0,n))/st\\ +{N}_{ref}(i,n),\\ st=\{\begin{array}{l}3,\delta (i,n)\ge 0.2\\ 10,0.1<\delta (i,n)<0.2\end{array}\end{array}$$

(10)

We adopt the variable step method to obtain a faster correction speed and a better correction effect. The next step in the formula is the empirical value obtained from the experiment. When δ(*i,n*) is less than 0.1, the recursive correction terminates.

Based on the above method, the SNR corresponding to different *k* was analyzed, and we obtained the best results at *k =* 1.7. With this optimal ratio, we demonstrate a measurement range of 10 km @ 1 s with pulse energy of 70 μJ, and 16 km @ 2 min, which is only 14% of ONERA, (37) as shown in Figure 4a. Figure 4b shows the raw visibility without any subtraction of the background. The visibility is limited to 0.5 (40) and the decrease of visibility beyond 3 km is mainly due to the increase of ASE noise. (41)

Furthermore, we made a comparison between QEHOM and the CDL with the longest detection range mentioned above, as shown in Table 1. We define Efficiency = Measurable Range/Pulse Energy, and it can be seen that the efficiency achieved by the QEHOM is 1 order of magnitude higher than that of the CDL. This mainly benefits from the lower noise and higher quantum efficiency of SPD compared with that of BD.

In addition to the huge performance improvement above, our scheme also has another significant advantage. Our measurement is at the single photon level, which consists of counts of photon events. These counting statistics are described by the Poisson distribution. (42) In contrast, the shot noise and thermal noise are Gaussian processes, which means that their distributions are random and relatively independent. Thus, the SNR can continue to increase as the accumulation time increases. The SNR versus accumulated times is shown in Figure 4c, and the result is approximately linear. Based on this advantage, QEHOM has the potential to achieve high-performance in weak signal measurements. Due to the absence of a standard single-photon source or other nonclassical light sources such as squeezed light in the experiment, the standard quantum limit could not be overcome in the experiment. Nonetheless, the incorporation of a single-photon source and detectors with higher quantum efficiency holds the potential for achieving higher quantum gain and further enhancing QEHOM’s capacity for weak signal measurement.

Click to copy section linkSection link copied!

On the basis of color erasure and further phase erasure techniques, we have achieved higher order quantum erasure detection and applied this method to wind detection. This method can achieve the quantum limit noise of single-photon detectors and offers advantages such as high quantum efficiency, large detection bandwidth, and multiple wavelength applicability. Benefiting from the quantum erasure detection, an optical compressed sampling method is used, and a sample rate of MHz is sufficient to record 17 GHz bandwidth signals (corresponding to a moving speed of about 13 km/s), solving the problem of high sampling rates and large data storages for continuous detection of ultrahigh-speed targets. The long-range wind detection shows that the QEHOM has the potential to achieve high-performance in weak signal measurements. Above all, optical frequency can be retrieved without frequency discriminator device in QEHOM, which is a brand-new detection method combining the advantages of direct detection and coherent detection. As the experimental results, the “Efficiency” is 1 order of magnitude higher than traditional detection methods. QEHOM has achieved fiber integration and compactness; it has potential applications in the future for continuous measurement of ultrahigh-speed moving targets, such as explosions, plasma, and celestial motion.

In this experiment, the visibility is inherently limited to a maximum of 0.5 (43) due to the absence of nonclassical light sources. Additionally, the visibility decreases further with increasing distance, which is attributed to the presence of ASE noise. Despite the fact that the results presented in this paper do not surpass the classical limit of 0.5, the quantum effects observed in this study are unequivocal. Prior studies have substantiated that nonclassical quantum interference effects can manifest even when visibility is below 0.5. (44,45) It demonstrates that visibility can convey the statistical characteristics of a light source; however, it is not an adequate criterion for determining the presence of quantum effects. In our work, we obtained HOM interference fringes by performing a series of quantum erasure operations (color erasure, phase erasure, etc.) to render the two particles quantum-mechanically indistinguishable, followed by the calculation of the second-order quantum mechanical correlation function *G*^{(2)}(τ). For any given system, whether involving a single-photon source or a weak coherent source, these are merely different quantum state inputs to the same quantum operator. Consequently, the quantum erasure operations themselves provide evidence of the quantum effects inherent in QEHOM. In addition, we have also made a detailed theoretical derivation on color erasure, which is the most critical part of quantum erasure in QEHOM, and obtained matching experimental results. These provide compelling evidence of the quantum effects intrinsic to QEHOM. In future research, we are planning to replace the current light source with a single-photon source to improve interference visibility and further optimize QEHOM’s capability for weak signals measurement. Besides, in some circumstances, where the intensity of the echo signal exceeds the saturation count of the single-photon detector, QEHOM may not effectively demonstrate its advantages. It may be beneficial to use traditional coherent detection methods for regions with strong near-field signals and to employ QEHOM for regions with weak, single-photon-level signals in the far field. By integration of these detection techniques, full-range, continuous, and ultralong-distance atmospheric parameter detection can be achieved. Finally, the current experimental results are constrained by the low quantum efficiency of the single-photon detectors (∼10%). If detectors with higher quantum efficiency, such as superconducting detectors, are employed, it is anticipated that the signal-to-noise ratio (SNR) and the detection range of the QEHOM will be significantly enhanced.