Introduction

Single-photon spectroscopy (SPS) can provide high-resolution information in the optical frequency domain for extremely weak fields with photon-level flux (picowatt level), playing a crucial role in extreme light field detection and significantly expanding the detection limit of humans in cutting-edge fields such as astronomy, physics, biomedical science, and environmental science1,2,3,4. In photon-scarce scenarios where the light field is severely attenuated or strong light is not permitted, a high-performance single-photon spectrometer is urgently demanded for astronomical galaxy observation, dark matter detection, trace molecule telemetry, extreme wavelength spectral analysis, single molecule precision analysis and trace characterization of phototoxicity or photosensitive matter5,6,7,8,9,10. However, the detector used for single-photon spectral measurements is a single-pixel binary device without spectral resolution capability, which outputs discrete “1” (high level) and “0” (low level) signals to indicate the presence or absence of photons for photon counting11. This binary nature poses significant challenges for reconstructing the “high-dimension” spectral information from the “low-dimension” photon-counting signal in SPS.

Traditionally, SPS often utilizes single-photon detection devices to continuously collect photon signals and accumulate energy distribution dispersed in time or spatial domain to compensate for the binary nature of single-photon detection, enabling spectral measurements from statistic histogram of photon counts. However, most dispersive single-photon spectrometers can only achieve sub-nanometer spectral resolutions with a slow-speed (a few minutes up to hours)12,13,14,15. In 2024, B. Sun et al., reported a time-stretching spectroscopy for high-speed measurements of photon-level (~300 pW) spectral information, utilizing a single-photon detector to collect the photon distribution after time-domain dispersion16. They have recorded supercontinuum spectra in the region of 2.4–4.2 μm with a resolution of 0.5 cm⁻¹ (15 GHz) in a sampling time of 30–1000 seconds aided by the up-conversion technique. The Fourier-transform single-photon spectrometer uses an interferometer to continuously perform time-domain delayed scanning of weak light, and reconstruct the interferogram through the time-domain photon count distribution, enabling nanometer-level spectral resolutions with typical measurement times of several minutes17. Its performance also suffers from optical alignment, thermal and mechanical instability especially in long-distance, high-precision scanning measurements18. In summary, it has been a significantly long-standing challenge to achieve either high-speed or high-resolution spectral measurements in photon-deficient scenarios.

The emerging dual-comb spectroscopy (DCS) offers a unique solution for high-speed, high-resolution and high-precision broadband spectral measurements, which utilizes two optical frequency combs (OFCs) with slightly different repetition rates to perform optical self-scanning without any moving mechanical stages and acquire time-domain interferograms at a single high-speed photodetector19,20. DCS has brought tremendous technological innovations to traditional spectroscopic fields in the past twenty years, e.g., cavity-enhanced dual-comb spectroscopy (CE-DCS)21, multidimensional coherent spectroscopy (MDCS)22, and coherent anti-Stokes Raman spectroscopy (CARS)23, but mostly in photon-rich scenarios. In 2024, N. Picqué and colleagues proposed a photon-counting DCS by repeatedly counting discrete photons at different time points to reconstruct the dual-comb interferogram, which pushed the spectral resolution down to an astonishing level of 200–500 MHz (~ pm) under extremely weak light (90 pW) conditions24,25. However, extensive recordings of “low-dimensional” discrete single-photon signals are required to statistically reconstruct continuous time/frequency-domain signals, leading to an hour-long measurement time. To date, high-speed high-resolution spectral measurements have not been accomplished yet under extremely weak light-field conditions.

In this paper, we propose single-photon dual-comb ghost imaging spectroscopy (DC-GIS), which directly reconstructs high-resolution spectral information from photon-counting signals within milliseconds. Ghost imaging is a nonlocal imaging technique that produces object shape by exploiting the correlation of intensity fluctuations between light fields of the reference arm and the measurement arm, neither of which could independently reconstruct the object information26. This approach significantly reduces the performance requirements for detectors in the measurement arm and has rapidly evolved into a cornerstone methodology for single-pixel imaging and single-photon imaging27,28. In recent years, ghost imaging has rapidly evolved from a spatial-domain imaging technique to encompass temporal and spectral domains, addressing the growing demands for dynamic imaging of ultrafast waveforms and high-sensitivity spectroscopic detection29,30. Our single-photon DC-GIS utilizes a dual-comb source with mode-resolved spectra modulated by an orthogonal matrix and recovers the high-resolution spectrum through ghost imaging. In DC-GIS, we have realized high-precision matrix coding with an extinction ratio better than −20 dB for the dual-comb spectrum with a frequency interval of 125 MHz and a linewidth of 12.5 kHz. Under the condition of photon depletion down to 210 fW (5 × 105 photons per second with a detection efficiency of 30%), the fingerprint absorption spectrum of acetylene molecules is directly obtained through photon counts in tens of milliseconds, with a spectral resolution of 125 MHz (~ 1 pm) and a single spectral range of 8 GHz. Benefiting from the frequency agility of the single electro-optic dual-comb system, the broadband absorption spectrum covering 1525–1540 nm was obtained through central wavelength tuning. Besides, single-photon DC-GIS has allowed ultra-long distance (125 km) fiber sensing of phase-shift fiber Bragg grating (PS-FBG) with only femtowatt-level power per comb line. Our scheme brings the high-speed and high-resolution spectroscopy to the photon-scarce regime with optical power down to femtowatt, providing a significant foundation for high-sensitivity trace molecule telemetry, long-distance fiber sensing and fingerprint spectral characterization of phototoxicity or photosensitive molecules.

Results

Basic principle

Figure 1a illustrates the basic concept of the single-photon spectroscopy based on DC-GIS. The electric fields \({\sum}_{{{\rm{n}}}}{A}_{{{\rm{n}}}}\cos \left[\left({\Delta f}_{0}+n{\Delta f}_{{{\rm{r}}}}\right)t\right]\) of multi-heterodyne dual-comb signal \({E}_{{{\rm{DCS}}}}\left(t\right)\) can be written in a vector form

$${E}_{{{\rm{DCS}}}}\left(t\right)={{{\bf{A}}}}_{1\times {{\rm{N}}}}\times {{{\bf{F}}}}_{{{\rm{N}}}\times 1}=\left[\begin{array}{ccc}{A}_{1} & {A}_{2} & \begin{array}{ccc}{A}_{3} & \cdots & {A}_{{{\rm{N}}}}\end{array}\end{array}\right]\times \left[\begin{array}{c}\cos \left(1\times {\Delta f}_{{{\rm{r}}}}+{\Delta f}_{0}\right)t\\ \cos \left(2\times {\Delta f}_{{{\rm{r}}}}+{\Delta f}_{0}\right)t\\ \begin{array}{c}\cos \left(3\times {\Delta f}_{{{\rm{r}}}}+{\Delta f}_{0}\right)t\\ \vdots \\ \cos \left(N\times {\Delta f}_{{{\rm{r}}}}+{\Delta f}_{0}\right)t\end{array}\end{array}\right]$$
(1)

where \(N\) is the number of frequency teeth, the frequency amplitude vector \({{{\bf{A}}}}_{1\times {{\rm{N}}}}=1\) is used to simplify formula derivation and \({{{\bf{F}}}}_{{{\rm{N}}}\times 1}\) is the frequency vector, \({\Delta f}_{{{\rm{r}}}}\) and \({\Delta f}_{0}\) denote the difference of repetition rates and the frequency difference between the first tooth of two optical combs, respectively. In DC-GIS, the amplitude \({{{\bf{A}}}}_{1\times {{\rm{N}}}}\) of frequency teeth is encoded by a sampling matrix \({{{\bf{H}}}}_{{{\rm{M}}}\times {{\rm{N}}}}\), into the dual-comb matrix for the ghost-imaging sampling. The dual-comb matrix \({{{\bf{E}}}}_{{{\rm{M}}}\times 1}\) can be represented as

$${{{\bf{E}}}}_{{{\rm{M}}}\times 1}={{{\bf{H}}}}_{{{\rm{M}}}\times {{\rm{N}}}}{{\boldsymbol{\times }}}{{{\bf{F}}}}_{{{\rm{N}}}\times 1}=\left[\begin{array}{cc}\begin{array}{cc}{H}_{1,1} & {H}_{1,2}\\ {H}_{2,1} & {H}_{2,2}\end{array} & \begin{array}{cc}\ldots & {H}_{1,{{\rm{N}}}}\\ \ldots & {H}_{2,{{\rm{N}}}}\end{array}\\ \begin{array}{cc}{H}_{3,1} & {H}_{3,2}\\ \vdots & \vdots \\ {H}_{{{\rm{M}}},1} & {H}_{{{\rm{M}}},2}\end{array} & \begin{array}{cc}\ldots & {H}_{3,{{\rm{N}}}}\\ \ldots & \vdots \\ \ldots & {H}_{{{\rm{M}}},{{\rm{N}}}}\end{array}\end{array}\right]\times \left[\begin{array}{c}\cos \left(1\times {\Delta f}_{{{\rm{r}}}}+{\Delta f}_{0}\right)t\\ \cos \left(2\times {\Delta f}_{{{\rm{r}}}}+{\Delta f}_{0}\right)t\\ \begin{array}{c}\cos \left(3\times {\Delta f}_{{{\rm{r}}}}+{\Delta f}_{0}\right)t\\ \vdots \\ \cos \left(N\times {\Delta f}_{{{\rm{r}}}}+{\Delta f}_{0}\right)t\end{array}\end{array}\right]$$
(2)

where \({H}_{{{\rm{i}}},{{\rm{j}}}}\) (\(i=1,\cdots,{M;j}=1,\cdots,N\)) is the element of the sampling matrix \({{{\bf{H}}}}_{{{\rm{M}}}\times {{\rm{N}}}}\). According to the principle of optical asynchronous sampling in DCS31, the duration \(\Delta T\) of each dual-comb spectrum is equal to the period of the dual-comb time-domain interferogram, \(\Delta T=1/\Delta {f}_{{{\rm{r}}}}\). Similarly, the resonance absorption of the sample molecules can be written as a one-dimensional vector to operate the frequency vector of the dual-comb system,

$${R}_{N\times 1}=\left[\begin{array}{c}{e}^{-\frac{{\alpha }_{1}L}{2}}\\ {e}^{-\frac{{\alpha }_{2}L}{2}}\\ \begin{array}{c}{e}^{-\frac{{\alpha }_{3}L}{2}}\\ \vdots \\ {e}^{-\frac{{\alpha }_{{{\rm{N}}}}L}{2}}\end{array}\end{array}\right]$$
(3)

where \({\alpha }_{{{\rm{i}}}}\) is the absorption coefficient of the sample at different frequency \(i\times {\Delta f}_{{{\rm{r}}}}+{\Delta f}_{0}\) (\(i=1,\cdots,N\)). The absorption signal after the interaction between the dual-comb matrix and the sample molecules can be expressed as

$${{{\rm{E}}}}_{{{\rm{M}}}\times 1}^{{\prime} }={{{\rm{H}}}}_{{{\rm{M}}}\times {{\rm{N}}}}\times \left({{{\rm{R}}}}_{{{\rm{N}}}\times 1}\cdot {{{\rm{F}}}}_{{{\rm{N}}}\times 1}\right)$$
(4)

where \({E}_{{{\rm{i}}},1}^{{\prime} }={\sum}_{{{\rm{n}}}=1}^{{{\rm{N}}}}{H}_{{{\rm{i}}},{{\rm{n}}}}{R}_{{{\rm{n}}},1}\cos (n{\Delta f}_{{{\rm{r}}}}+{\Delta f}_{0})t\). Furthermore, the photon-counting signal \({{{\bf{P}}}}_{{{\rm{M}}}\times 1}\) detected by a single-photon avalanche diode (SPAD) is proportional to the time-domain integration of the absorption signal,

$${{{\bf{P}}}}_{{{\rm{M}}}\times 1}=\left[\begin{array}{c}{P}_{1}\\ {P}_{2}\\ \begin{array}{c}{P}_{3}\\ \vdots \\ {P}_{{{\rm{M}}}}\end{array}\end{array}\right]\propto {\int }_{{t}_{0}}^{{t}_{0}+\Delta T}{\left|{{{\bf{E}}}}_{{{\rm{M}}}\times 1}^{{\prime} }\right|}^{2}{dt}$$
(5)

where \(\Delta T\) is the time-domain duration of DCS. The reference spectra \({{{\bf{S}}}}_{{{\rm{M}}}\times 1}\) can be obtained by performing Fourier transform on the vector of the sampling signal \({\left|{{{\bf{E}}}}_{{{\rm{M}}}\times 1}\right|}^{2}\),

$${{{\bf{S}}}}_{{{\rm{M}}}\times 1}\left(\omega \right){\mathfrak{=}}{\mathfrak{F}}\left[{\left|{{{\bf{E}}}}_{{{\rm{M}}}\times 1}\right|}^{2}\right]\propto \left[\begin{array}{cc}\begin{array}{cc}{H}_{1,1}^{2} & {H}_{1,2}^{2}\\ {H}_{2,1}^{2} & {H}_{2,2}^{2}\end{array} & \begin{array}{cc}\ldots & {H}_{1,{{\rm{N}}}}^{2}\\ \ldots & {H}_{2,{{\rm{N}}}}^{2}\end{array}\\ \begin{array}{cc}{H}_{3,1}^{2} & {H}_{3,2}^{2}\\ \vdots & \vdots \\ {H}_{{{\rm{M}}},1}^{2} & {H}_{{{\rm{M}}},2}^{2}\end{array} & \begin{array}{cc}\ldots & {H}_{3,{{\rm{N}}}}^{2}\\ \ldots & \vdots \\ \ldots & {H}_{{{\rm{M}}},{{\rm{N}}}}^{2}\end{array}\end{array}\right]\times \left[\begin{array}{c}{{\rm{\delta }}}\left[\omega -\left(1\times {\Delta f}_{{{\rm{r}}}}+{\Delta f}_{0}\right)\right]\\ {{\rm{\delta }}}\left[\omega -\left(2\times {\Delta f}_{{{\rm{r}}}}+{\Delta f}_{0}\right)\right]\\ \begin{array}{c}{{\rm{\delta }}}\left[\omega -\left(3\times {\Delta f}_{{{\rm{r}}}}+{\Delta f}_{0}\right)\right]\\ \vdots \\ {{\rm{\delta }}}\left[\omega -\left(N\times {\Delta f}_{{{\rm{r}}}}+{\Delta f}_{0}\right)\right]\end{array}\end{array}\right]$$
(6)

Here the unit impulse \(\delta\)-function denotes the ideal frequency tooth. Based on \(M\) patterns of dual-comb spectra in ghost imaging spectroscopy32,33,34, the dual-comb response spectrum \(R(\omega )\) can be reconstructed from the photon-counting signal by Sun et al., Sun et al. 35,36

$$R\left(\omega \right)=\frac{1}{M}{{{\bf{P}}}}_{{{\rm{M}}}\times 1}^{{{\rm{T}}}}\times {{{\bf{S}}}}_{{{\rm{M}}}\times 1}\left(\omega \right)=\frac{1}{M} \sum _{m}^{M} \sum _{n}^{N}{P}_{{{\rm{m}}}}{H}_{{{\rm{m}}},{{\rm{n}}}}^{2}{{\rm{\delta }}}\left[\omega -\left(n{\Delta f}_{{{\rm{r}}}}+{\Delta f}_{0}\right)\right]$$
(7)

where the superscript \(T\) represents the transpose operation of a matrix. When using a structured Hadamard matrix in DC-GIS, at least \(N\) different dual-comb patterns (\(M=N\)) need to be sampled to reconstruct the spectral information of the sample with a spectral resolution of dual-comb system37. By leveraging the sub-Nyquist sampling advantage of compressed sensing, the dual-comb patterns required for interrogating the sample can be significantly reduced compared to the number of comb teeth (\(M < N\))38,39. As demonstrated in Supplementary note 5, single-photon DC-GIS based on compressed sensing can achieve a compression ratio down to 18%, allowing further improvement on the measurement speed compared to the current ghost-imaging algorithm. Thus, the measuring end in DC-GIS only needs to perceive the power information of the transmitted light for reconstructing the high-resolution spectrum, without using time or spatial domain dispersive elements and high-bandwidth detectors, which offers a routine platform for precision spectroscopy using underperforming detectors in extreme scenarios.

Fig. 1: Basic principles and experimental setup of the single-photon DC-GIS scheme.
figure 1

a Sketch of the basic principles of single-photon DC-GIS. The dual-comb signal with a time-domain duration of \(\Delta T\) is composed of a series of evenly spaced teeth in the radio frequency, which can be converted into a dual-comb spectrum by performing a Fourier transform (FT) operation. A series of mode-resolved spectral patterns form a dual-comb matrix to sequentially interrogate sample molecules. Then, a single-photon avalanche diode (SPAD) is used to detect the discrete photon signal and count the photon number of different dual-comb patterns after sample absorption. The response spectrum of the sample can be directly reconstructed from the dual-comb matrix measured in the reference arm and the photon-counting signal by ghost-imaging calculation. b Experimental setup of single-photon DC-GIS. The output of the seed continuous-wave (CW) laser is modulated by a single electro-optic modulator (EOM) to generate a dual-comb time-domain interference signal, which can directly generate a dual-comb source. An arbitrary waveform generator (AWG), which outputs different dual-comb waveforms, is used to drive the EOM generating different dual-comb spectral patterns. A tunable optical filter (TOF) is employed to avoid aliasing. The signal is split into two beams: one is directly incident on a fast photodetector (PD) as the reference signal, while the other is detected by a SPAD after passing through a variable attenuator (VA) and the sample. The data streams from the PD and the SPAD are digitized and recorded synchronously by a data-acquisition card (AlazarTech, ATS9350) for DC-GIS reconstruction.

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Experimental setup

For the proof-of-concept demonstration of single-photon DC-GIS, this article introduces a dual-comb system based on a single electro-optic modulator (EOM)40,41, where the intensities of all teeth have been encoded by the binary matrix shown in Fig. 2a. As shown in Fig. 1(b), a seed continuous-wave (CW) laser passes through an EOM driven by an arbitrary waveform generator (AWG). Different dual-comb waveforms are generated by the AWG, driving the EOM to modulate the CW intensity and produce a dual-comb matrix. (See more details in Supplementary Information note 1). In this paper, the electrical temporal waveform driving the EOM consists of 128 sets of dual-comb composite signals, which are shown in Fig. S2a. The working wavelength of the CW laser is tunable from 1480 to 1640 nm with 20-mW output power. The line spacings are chosen as \({f}_{{{\rm{r}}},1}=125\) MHz and \({f}_{{{\rm{r}}},2}=125.2\) MHz, respectively, and the spectral bandwidth is 8 GHz, containing a total of 64 spectral lines. The offset frequencies are \({f}_{0,1}=7\) GHz and \({f}_{0,2}=7.03\) GHz to create enough spectral gap for filtering out a certain sideband and the residual CW, which is detailed in Fig. S3. The series of waveforms are played by AWG with a sampling rate of 32 GSa/s and a vertical resolution of 8 bits. The DC bias voltage of EOM is locked at the point of the minimum transmitted power with about 4 μW. The optical power after filtered by a tunable optical filter (TOF) is ~0.4 μW, and 99% of the total power is directly incident on a fast photodetector (PD) as the reference signal, while the residual is detected by a SPAD after passing through a variable attenuator (VA) and the sample in the gas cell under test. The reference arm is set to record the real-time matrix through high-speed and high-resolution DCS for ghost-imaging calculation. The sample filled with gas-phase acetylene has an absorption path length of 40 cm in low pressure. The delay between the reference signal and the photon-counting signal (not shown in Fig. 2a) is compensated by adjusting the PD position to synchronize the reference and photon-counting signals. The reference signal and the photon-counting signal are digitized and recorded synchronously by a data-acquisition card for ghost-imaging calculation.

Fig. 2: Spectral encoding of the dual-comb system and the mode-resolved spectrum from single-photon DC-GIS.
figure 2

a Encoding matrix used for DC-GIS. b Dual-comb matrix after encoding by the matrix from a. c First dual-comb mode-resolved pattern. d Zoom-in the single comb with 12.5 kHz linewidth from c. e Fourteen dual-comb mode-resolved pattern; grey lines are “missing” teeth encoded with “0”. f Zoom-in spectrum of the comb encoded with “0” from e, the extinction ratio is better than −20 dB. g Photon-counting signal with a measurement duration of 1.92 seconds, and the counting time for each data point is 15 ms. h Mode-resolved spectrum reconstructed by ghost imaging based on the photon-counting signal from g; the blue line represents the simulated absorption lines from HITRAN database; the inset is the background spectrum without the sample. The optical frequency \({f}_{{{\rm{c}}}}\) is 196.484 THz in these figures.

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Figure 2a depicts the 128 × 64 binary encoding matrix used for DC-GIS in this work, where the matrix elements only include 0 and 1. The intensities of dual-comb lines are then encoded using each row array in sequence, and the dual-comb signal is detected by a InGaAs photodetector (Thorlabs, APD430C/M) to obtain the current dual-comb spectral pattern. After fast Fourier transform (FFT) processing, a series of normalized spectra are shown as a matrix in Fig. 2b. Due to the EOM’s nonlinearity and the limited vertical resolution of AWG, the intensity of the dual-comb lines cannot be fully encoded in a binary manner, leading to the mismatch between the encoding matrix (Fig. 2a) and the interrogation dual-comb matrix (Fig. 2b) generated by the EOM42. Besides, the power of high-frequency lines is relatively low, which attributes to the limited bandwidth of the used EOM ( ~ 20 GHz, 3 dB). As shown in Fig. S7 in Supplementary Information note 2, direct reconstruction with the computer-generated encoding matrix cannot recover the absorption spectrum due to the difference between the computer-generated encoding matrix and the dual-comb matrix34. Thus, the reference arm, enabling real-time measurements of the interrogation matrix through high-speed and high-resolution DCS, is fundamentally important for the reconstruction of high-quality single-photon spectrum based on ghost-imaging calculation19,22. Figure 2c and 2e show the first and fourteenth pattern of mode-resolved dual-comb spectra respectively, where the time length of each data stream is 0.05 s. The spectra are composed of discrete combs with a frequency spacing of 125 MHz and a linewidth of 12.5 kHz, as shown in the zoom-in Fig. 2d. The average SNR reaches 650 in a measurement time of 0.05 s43. Figure 2f shows a mode-resolved line with intensity encoded as “0”. The intensity contrast between the dual-comb lines encoded by “0” element and “1” element has been measured to be better than −20 dB.

Figure 2h shows the mode-resolved absorption spectrum reconstructed from the photon-counting signal (as shown as Fig. 2g) by DC-GIS. In this measurement, every spectral pattern of dual-comb system lasted for 0.015 s. After passing through a 40 cm optical path, 10 mbar pressure, and 30% concentration of acetylene molecules, the optical signal was attenuated to 2.5 × 106 photons/s (~1 pW) and collected by a SPAD with a detection efficiency of 30%. The photon-counting signal is shown in Fig. 2g. From Eq. (7), the P(1) line of \({\upsilon }_{1}+{\upsilon }_{3}\) vibrational band and the R(10) line of \({\upsilon }_{1}+{\upsilon }_{3}+{({\upsilon }_{4}+{\upsilon }_{5})}_{+}^{0}-{({\upsilon }_{4}+{\upsilon }_{5})}_{+}^{0}\) hot band were reconstructed with the mode-resolved teeth, as shown in Fig. 2h. The result has been normalized by the background spectrum shown in the inset.

High-speed and high-resolution single-photon spectral measurements by DC-GIS

The AWG loops the playback of encoded time-domain waveforms as shown in Figure. S2a to achieve encoded dual-comb spectra. A fast photodetector was used to receive the radiofrequency dual-comb interferograms. Figure 3a shows the time-domain detector signal filtered by a 78 MHz low-pass filter within a time window of 12.8 ms, including about 20 loops. 128 interferograms in a loop are depicted in Fig. 3b, which are separated by gray dashed lines. The zoom-in picture in Fig. 3c depicts the single dual-comb interferogram with the duration of 5 μs (\(1/\Delta {f}_{r}\)), where the down-converted radiofrequency signal in time domain exhibits typical interferogram with the centerburst.

Fig. 3: Reconstructed single-photon spectra using DC-GIS.
figure 3

a Continuous multi-heterodyne signal within 12.8 ms including multiple interferograms. b Time-domain dual-comb interferograms corresponding to different spectral encodings with a total duration of 0.64 ms; gray dashed lines are used to distinguish different interferogram periods. c Dual-comb interferogram with a duration of 5 μs. d Photon-counting signals for 0.64-ms, 6.4-ms, 64-ms and 640-ms measurement times. The zoom-in inset is the photon-counting signal with the 640-ms acquisition. e Comparison between normalized absorption signature of the gas-phase acetylene obtained by DC-GIS for 640 ms photon-counting time (red dots) and the simulated result from HITRAN database (blue line). The areas colored in pink, gray, yellow, and black represent the standard deviation intervals in photon-counting times of 0.64 ms, 6.4 ms, 64 ms and 640 ms, respectively. f Broadband dual-comb ghost imaging spectrum (red line) covering 1525–1540 nm. The blue line is the simulated spectrum from HITRAN for comparison.

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The sample cell was filled with gas-phase C2H2 at a concentration of 30% in nitrogen buffer gas at a total pressure of 10 mbar. Under extremely weak light conditions with a photon-counting rate of 5 × 106 photons/s (about 2.13 pW), a SPAD was used to count the number of photons containing acetylene absorption information within each dual-comb interferogram period. Figure 3d compares the number of photons in different measurement windows of 0.64 ms (black line), 6.4 ms (red line), 64 ms (blue line), 640 ms (green line). Within a single-loop time of 0.64 ms, an average of only 25 photons are counted per interferogram period. The low photon counts in short measurement time constrain the overall signal-to-noise ratio (SNR, proportional to the square root of photon counts) of the reconstructed signal25. Therefore, increasing the photon-counting time to accumulate photon counts is an effective way to improve the SNR of the reconstructed spectrum. The blue curve (64 ms) and the green curve (640 ms) exhibit the same trend as shown in the inset, differing in the number of photons counted. The encoded dual-comb spectra recorded by the photodetector and the absorbed photon-counting signal were correlated to reconstruct the absorption spectrum by Eq. (7). Figure 3e compares the reconstructed spectra from photon counting of 0.64-ms, 6.4-ms, 64-ms, 640-ms windows with HITRAN simulated result. Red dots depict the absorption lines P(1) \({\upsilon }_{1}+{\upsilon }_{3}\) and R(10) \({\upsilon }_{1}+{\upsilon }_{3}+{({\upsilon }_{4}+{\upsilon }_{5})}_{+}^{0}-{({\upsilon }_{4}+{\upsilon }_{5})}_{+}^{0}\) of C2H2 reconstructed from an 640-ms acquisition, and the areas colored in pink, gray, yellow, and black represent the standard deviation intervals for different photon-counting durations of 0.64 ms, 6.4 ms, 64 ms, 640 ms, respectively. The standard deviation interval of 0.64 ms duration is significant with a maximum exceeding 50%, because the correlation between photon counting in statistics and optical power is accompanied by significant noise, as shown in Fig. 3d. The grey area for photon-counting duration of 6.4 ms achieves a residual error of ~30%, while the blue (64 ms) and brown (640 ms) area show excellent agreement between the reconstructed spectral line and the simulated line (black line) with residual errors of 15% and 5%, respectively. The SNR of the photon-counting signal increases with the measurement time, and the SNR of the reconstructed absorption spectrum has also been significantly improved.

Benefiting from the frequency agility of the electro-optic dual-comb system44, the broadband transition spectrum can be obtained through central wavelength tuning of the CW laser. Figure 3f shows the stitching spectrum (red line) from 1525 to 1540 nm with a spectral resolution of 125 MHz, covering the P-branch spectral lines of the acetylene vib-rotational band (\({\upsilon }_{1}+{\upsilon }_{3}\)), where the total power is as low as 2.13 pW and the measurement time of the single unstitching spectrum is 1 s. The broadband spectrum reconstructed by DC-GIS has excellent agreement with the simulation result of the HITRAN database45, as shown by the blue line in Fig. 3f. In Supplementary Information note 3, we present a practical scheme for broadband spectral measurements of single-photon DC-GIS, which we are currently working on.

Signal-to-noise ratio (SNR) characteristics of single-photon DC-GIS

Under photon-scarce conditions, it is important to obtain the high-speed and high-resolution spectrum with a sufficient SNR that is defined as \(\bar{s}/\sigma\), where \(\bar{s}\) represents the average intensity of the teeth in the reconstructed results, and \(\sigma\) is the standard deviation of the noise in those results43,46. We have thus investigated the influence of the light intensity (photon counts per second) and the measurement time on the SNR of the ghost-imaging spectrum to quantitatively compare these single-photon DC-GIS spectra. Multiple sets of data streams with a duration of 100 seconds were collected and evaluated, with optical powers of 210 fW, 420 fW, 1.27 pW, 2.13 pW, and 2.96 pW (corresponding to photon count rates of 5 × 105, 1 × 106, 3 × 106, 5 × 106, and 7 × 106 photons/s), respectively. Figure 4 compares the relationship between the SNR and the measurement time under different light-intensity conditions. For a photon-counting rate of 5 × 105, the SNR of the single-photon spectrum containing 64 comb teeth can reach 203 with a measurement time of 5 s.

Fig. 4: Signal-to-noise ratio (SNR) in single-photon DC-GIS.
figure 4

SNR as a function of the measurement time \(T\) under different photon count rates of 5 × 105 photons/s, 1 × 106 photons/s, 3 × 106 photons/s, 5 × 106 photons/s and 7 × 106 photons/s (pink dots, red dots, blue dots, green dots and orange dots). The black line is the linear fitting curve with a slope of 1/2. The insets are the fitting curves with the experimental SNR curves of the photon counts of 1 × 106 and 5 × 105 photons/s, respectively.

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As shown in Fig. 4, the blue-, green-, and orange-dot curves of the SNR scales with the square root of the measurement time, where the same trends with the measurement time up to 100 s. Moreover, the SNR of single-photon DC-GIS exhibits independence from the photon count rate at photon levels above 3 × 106 photons/s. At photon levels of 1 × 106 and 5 × 105 photons/s, the SNR curves deviate significantly within a short measurement time, but still show the same trend as the measurement time increases. The experimental SNR of single-photon DC-GIS exhibits independence from the photon-counting rate, which is different from the SNR of SPS based on DCS scaling as the square root of the photon-counting rate25. This phenomenon is because that the SNR of the retrieved photon-level spectrum is not only determined by the SNR of photon-counting signal, but also the SNR of the reference signal.

According to the discussion in Supplementary Information note 2, the SNR equation for single photon DC-GIS is given by,

$${SNR}\propto \frac{T}{\sqrt{\left(\frac{{\alpha }_{{{\rm{DCS}}}}{N}^{2}}{0.64}+\frac{1}{{n}_{{{\rm{pr}}}}}\right)T+\frac{\beta {\alpha }_{{{\rm{DCS}}}}{N}^{2}}{0.64{n}_{{{\rm{pr}}}}}}}$$
(8)

where \(T\) is the measurement time, \(N\) is the number of frequency teeth, \({n}_{{{\rm{pr}}}}\) is the photon-counting rate of the SPAD, \({\alpha }_{{{\rm{DCS}}}}\) is the noise coefficient of DCS43 and \(\beta\) is the noise coupling factor between DCS and the photon-counting signal. Within a shorter measurement time (\(T\ll \frac{\beta {\alpha }_{{{\rm{DCS}}}}{N}^{2}}{\beta {\alpha }_{{{\rm{DCS}}}}{{n}_{{{\rm{pr}}}}N}^{2}+0.64}\)), the SNR of single-photon DC-GIS increases linearly with the measurement time and the square root of the photon-counting rate. As the measurement time increases to satisfy \(T\gg \frac{\beta {\alpha }_{{{\rm{DCS}}}}{N}^{2}}{\beta {\alpha }_{{{\rm{DCS}}}}{n}_{{{\rm{pr}}}}{N}^{2}+0.64}\), the SNR is proportional to the square root of the measurement time \({T}^{1/2}\). The two insets in Fig. 4 compare the fitting curves with the experimental SNR curves of the photon counts of 1 × 106 and 5 × 105 photons/s, respectively, where the consistency between the experimental and simulation results validates the theory. When the photon-counting rate is high enough to satisfy \({n}_{{{\rm{pr}}}}\gg \frac{{\alpha }_{{{\rm{DCS}}}}{N}^{2}T}{0.64T+\beta {\alpha }_{{{\rm{DCS}}}}{N}^{2}}\), the long-term SNR of single-photon DC-GIS is independent from the photon-counting rate. More details can be found in the discussion of Figure. S5b in Supplementary Information note 2. In addition, the relationship of \({SNR}\propto {T}^{1/2}\), as shown by the black fitting curve in Fig. 4, is maintained within the 100-s acquisition time, proving that the system outperforms the mutual coherence time up to 100 seconds47.

Over 100-km fiber sensing based on single-photon DC-GIS

Single-photon DC-GIS can perform high-resolution and high-speed spectral analysis on extremely weak light signals, which can reduce the optical power required for spectral sensing technology, and is expected to greatly improve the detection distance of spectral sensing. In this work, we have completed the long-distance spectral sensing with extremely weak light based on single-photon DC-GIS. As shown in Fig. 5a, a SPAD enables long-distance spectral sensing by detecting the photon-counting signal from the sensor after transmission through long fibers with substantial attenuation.

Fig. 5: Long-distance spectral strain sensing based on phase-shifted fiber Bragg grating (PS-FBG).
figure 5

a Concept of long-distance fiber sensing using single-photon DC-GIS. b Experimental configuration. c Periodic trigger signal for avoiding the backscattering in long-distance fiber. d Reference signal recorded by DCS. e Photon-counting signal of the SPAD with a counting time of 5 μs. The photons detected within 0–1.25 ms are caused by the limited isolation (~45 dB) of the circulator and the backscattering signals, while the photons within 1.25–1.89 ms are the reflected signals of PS-FBG. f Reflected spectra of PS-FBG reconstructed by single-photon DC-GIS under different strain conditions; the solid-dot curve represents the linear relationship between the frequency shift and the applied axial strain with a slope of −4.04 με/GHz.

Full size image

The experimental setup is shown in Fig. 5b, where the signal from the spectral-encoded dual-comb source passed a fiber circulator to interrogate a phase-shifted fiber Bragg grating (PS-FBG) after 125 km communication fiber with a loss coefficient of 0.15 dB/km. The reflected photons from PS-FBG are detected and counted by a SPAD after passing through the circulator. In order to avoid the influence of backscattering signals during long-distance fiber optic propagation, this work uses a periodic interrogated spectral-encoded dual-comb light to separate photon signals from PS-FBG and backscattering signals in time domain48,49. A complete detection cycle is the sum of the time delay (1.25 ms) caused by the long fiber and the duration (0.64 ms) of the dual-comb matrix50, where the trigger signal is shown in Fig. 5c. The reference matrix detected by DCS is shown in Fig. 5d, which corresponds to the non-zero triggering limit (0.64 ms) in the time domain. The SPAD receives backscattering signals within zero trigger signal time and reflected signals from PS-FBG within non-zero trigger signal time, as shown in Fig. 5e. The sensing distance can be further improved by increasing the zero-trigger time.

With a DCS power of 20 nW, the total photon count received by the SPAD is about 4 × 106 photons/s, while the photon count originating from the PS-FBG is only 0.5 × 106 photons/s. The PS-FBG to be tested is fixed on a translation stage, and axial strain is applied to the PS-FBG by moving the stage. The high-resolution reflection spectrum of PS-FBG can be directly obtained from the photon-counting signal and the dual-comb matrix measured in the reference arm through the ghost imaging correlation. The axial strain from the translation stage movement causes uniform deformation of the PS-FBG, and high-resolution spectra shown in Fig. 5f are used to achieve strain sensing of the PS-FBG. By performing ghost-imaging calculations on photon counts within 1 s, the PS-FBG spectra under different axial strains are obtained with a SNR of up to 90, a full width at half maximum (FWHM) of about 4 pm, and a spectral resolution of about 1 pm. With the increment of strain at a step of 6.2 με, the peak of the reflection spectrum shifts uniformly towards lower frequency with a step size of about 1.5 GHz51,52. The linear relationship between the frequency shift and the applied axial strain is fitted with a slope of −4.04 με/GHz, corresponding to a sensitivity of 0.5 με for a 125-MHz spectral resolution of the dual-comb source. This experiment shows 125 km spectral sensing for extremely weak light without any relay, and its sensing distance can be further enhanced by increasing the detection power and the duration of the zero-trigger signal, which is expected to play a more significant role in the field of ultra-long distance fiber sensing.

Furthermore, a single-photon fiber sensing experiment has also been performed with single-comb GIS, detailed in Supplementary note 3. The CW laser is modulated by an EOM driven by the time-waveform (Fig. S8b) from an AWG, to generate a series of single-comb spectral patterns. As direct acquisition of the single-comb matrix using a reference arm is challenging, single-comb GIS utilizes the computer-generated AWG matrix as a reference to reconstruct the PS-FBG spectrum from photon-counting signals. However, significant discrepancies were observed between the PS-FBG reflection spectrum reconstructed via single-comb GIS and that obtained through tunable CW laser scanning. This error arises because nonlinearity in the EOM causes the actual single-comb matrix in the measurement arm to deviate from the computer-generated AWG matrix, degrading reconstruction quality significantly. The comparison results between PS-FBG spectra (Fig. S8c and S8d) further confirm that, in photon-deficient scenarios, single-photon DC-GIS offers superior accuracy for spectral sensing through high-precision DCS measurements on the actual spectral matrix in the reference arm.

Discussion

In this study, we proposed and experimentally demonstrated the concept of single-photon DC-GIS, which explores a distinctive avenue for single photon spectral measurements with a high resolution of 125 MHz (about 1 pm) and a short measurement time down to milliseconds. Its potential applications are successfully demonstrated through high-resolution spectral analysis of acetylene fingerprint spectra and interrogation of a fiber sensor at 125 km, both performed under photon-scarcity conditions. In Table S2, the performance parameters of single-photon DC-GIS were compared with other SPS techniques. Single-photon DC-GIS can directly reconstruct the photon-level precise spectrum from photon-counting signals through a matrix-based dual-comb source and ghost-imaging calculations. In addition, this scheme is not limited to extremely weak light field applications at the photon level, but is also suitable for other applications where detector performance is limited, such as low-speed detectors with large light sensing surfaces, power meters, etc. In the present demonstration, a single EO dual-comb system is used to generate the dual-comb spectral matrix and excite the molecular response at the photon level within a narrow spectral range. This scheme can be extended to dual-comb systems based on fiber OFCs to achieve photon-level broadband precision spectroscopy. We further propose a experimental configuration consisting of the broadband fiber OFCs, a two-dimensional disperser53,54,55 and digital micromirror device (DMD) to obtain matrix-based broadband comb source for broadband single-photon DC-GIS. See Supplementary Information note 3 for more details. It is expected to achieve encoding control of tens of thousands of mode-resolved teeth covering several terahertz spectral regions, which will meet the application requirements of broadband precision spectral detection in photon deficient scenarios. This approach can be further adapted for a wider range of OFC sources, such as fiber OFCs19, broadband electro-optic combs20 and a single-cavity dual-comb system56,57, for high-sensitivity massively parallel sensing of trace multi-molecules and single-molecule spectral analysis with extremely weak light. Single-photon DC-GIS holds potential for extension into the mid- and far-infrared regions, which currently lack efficient single-photon detectors58. By leveraging nonlinear frequency conversion techniques, mid- or far-infrared spectra can be upconverted to the near-infrared band for photon-counting detection. The mid-infrared temporal ghost imaging based on frequency up-conversion has been experimentally demonstrated in recent reports, which confirms the viability of implementing mid-infrared ghost imaging59,60. Further experimental validation of mid-infrared single-photon DC-GIS necessitates addressing practical challenges, such as calibrating spectral matrix distortions in mid-infrared dual-comb systems arising from frequency conversion processes. Besides, this scheme has the potential to further integrate with emerging compressed sensing spectroscopy61,62 (see Supplementary note 5 for references) and computational spectroscopy63,64, providing a powerful way for intelligent computing to empower precision spectroscopy applications.

In conclusion, single-photon DC-GIS offers a host of powerful features, such as high resolution, high precision and rapid measurements under extremely weak light conditions, which generates unprecedented opportunities for the cutting-edge research of photon-scarce scenarios, e.g., extremely long-range sensing, single-molecule analysis, high-sensitivity trace molecule telemetry, and fingerprint spectral characterization of phototoxicity or photosensitive molecules.

Methods

Dual-comb matrix based on a single electro-optic modulator

An electro-optic modulator (EOM, IM with a bandwidth of 40 GHz, KG-AM-15-40G-PPFA Conquer) is used to modulate the intensity of a tunable continuous-wave (CW, 20 mW, TSL-770, Santec Corporation) laser in the communication band as the dual-comb waveform in time domain, which is driven by an arbitrary waveform generator (AWG, Keysight M8195A). The output voltage of the AWG is set as 1 V, and the actual peak-to-peak voltage of the electrical waveform is about 0.7 V. The driving signal is calculated using MATLAB and loaded into the AWG in.mat format for continuous playback. The driving electrical signal \({E}_{{{\rm{DCS}}}}\left(t\right)\) of AWG is defined by the dual-comb time-domain waveform of the corresponding spectral pattern,

$${E}_{{{\rm{DCS}}}}\left(t\right)=\sum_{i} \sum_{n}{A}_{{{\rm{n}}},{{\rm{i}}}}\cos \left[2\pi \left(n{f}_{{{\rm{r}}},{{\rm{i}}}}+{f}_{0,{{\rm{i}}}}\right)t+{\phi }_{{{\rm{n}}},{{\rm{i}}}}\right]$$
(9)

where \(i=1,\,2\) and \(n\) are the indices of the combs and the OFC’s lines, \({A}_{{{\rm{n}}},{{\rm{i}}}}\) and \({\phi }_{{{\rm{n}}},{{\rm{i}}}}\) are the amplitude and phase of the corresponding wave, \({f}_{{{\rm{r}}},{{\rm{i}}}}\) and \({f}_{0,{{\rm{i}}}}\) are the repetition rate and the frequency of the first tooth of the \(i\)-th comb. The values of \({A}_{{{\rm{n}}},1}={A}_{{{\rm{n}}},2}\) are sequentially the elements of the row array of the matrix \({{{\bf{H}}}}_{2{{\rm{N}}}\times {{\rm{N}}}}\), which is transformed from an \(N\times N\) Hadamard matrix \({{\bf{H}}}\),

$${{{\bf{H}}}}_{2{{\rm{N}}}\times {{\rm{N}}}}=\left[\frac{{{\bf{H}}}+1}{2};1-\frac{{{\bf{H}}}+1}{2}\right]$$
(10)

where \(N\) is the number of frequency teeth. The phase relationship between OFCs’ lines is defined by \({\phi }_{{{\rm{n}}},1}={\phi }_{{{\rm{n}}},2}=\tan {n}^{2}\), which results in the low peak-to-average power ratio (PAPR) in time domain to obtain the high energy per comb line. The dual-comb temporal waveform with intensity-encoded teeth is obtained by superposing cosine functions modulated by the row array of the matrix \({{{\bf{H}}}}_{2{{\rm{N}}}\times {{\rm{N}}}}\).

Data acquisition and processing

In this study, the electric signals from a fast PD and a SPAD are digitized and recorded synchronously with a vertical resolution of 12 bits and a sampling rate of 125 MHz. The dual-comb matrix, serving as the reference signal, can be obtained through fast Fourier transform. Furthermore, a ghost-imaging reconstruction program has been developed to reconstruct the signal spectrum from the photon-counting signal. All of the above data processing was implemented using MATLAB software.