1. Introduction

A new paradigm of quantum optical measurement technique using quantum entangled photon pairs has recently emerged, which promises a new platform for spectroscopy and microscopy surpassing classical limits [1–7]. Early studies of optical measurements utilizing various quantum correlations of a pair of photons were based on the correlated detection of two-photon interferences with nonlinear interferometers [8] or entangled input sources [9–11], and they demonstrated unprecedented enhancements in measurement sensitivity [6,7,12–14].

Only recently, techniques requiring one-photon interference measurements only [15], such as quantum spectroscopy and imaging using path-entangled photon pairs, have been proposed and demonstrated [16–25]. They represent a crucial step towards surpassing the limits of two-photon interference measurements. Without employing heralded detection schemes, the quantum coherence of a single party (signal) in the pairs of path-entangled photons (e.g., signal and idler) generated by dual spontaneous parametric down conversion (SPDC) [26,27] was shown to be inducible by erasing the which-path information of its conjugate (idler) photons [15,28]. The induced coherence of signal photons from two different pathways critically depends on the degree of indistinguishability of the conjugate idler photons [15,23,24,29,30] so that the visibility of the interference fringe of signal field is modulated according to their amplitude and phase differences, i.e., the absorption and dispersion, respectively, of the conjugate idler fields. Therefore, quantum spectroscopy using one-photon interference measurements enables one to study the structure and dynamics of molecular systems by means of detecting photons conjugate with material-probing or material-interacting photons in an unheralded manner.

The dual SPDC-based QSUP method with non-degenerate signal and idler photon pairs and without coincident detection of both signal and idler fields overcomes the limited availability of high-sensitivity detectors in certain frequency regions of probe fields interacting with material of interest [9,10,31]. For example, a linear IR spectroscopy experiment is made possible by detecting the interference fringe visibility of the visible photons that are quantum mechanically entangled with the IR photons directly interacting with vibrationally resonant molecules [20,21]. In addition, dual SPDC-entangled photon pairs have been applied to quantum information processing [32,33] and fundamental physics studies of quantum entanglement [34]. Although those studies demonstrated the great potential of the method for overcoming the quantum projection noise problem that significantly limits measurement sensitivity [8], the experimental possibility of high-resolution QSUP has not yet been explored, mainly because of the inevitably broad bandwidths of SPDC-generated signal and idler spectra [20,21].

In this report, we demonstrate both ultra-high resolution (sub-Doppler-broadened linewidth) and remote-measurement QSUP method using induced one-photon coherence detection, which involves a dual stimulated parametric down-conversion (StPDC) scheme [35–38]. We show that the frequency resolution is not limited by the PDC-generated signal or idler spectral bandwidth but by the linewidth of an injected seed beam. We specifically use a Fabry-Pérot (FP) cavity as an optical sample because it has an extremely narrow transparent frequency window with a width typically many orders of magnitude smaller than that of the rovibrational spectra of gaseous molecules at low pressure. In addition, it is shown that our QSUP based on frequency-comb single photon interferometry [37] has both an improved signal-to-noise ratio (SNR) compared to a single StPDC spectroscopy and the capability of remote-measurement quantum spectroscopy with undetected photons.

To emphasize the essential difference between the previous dual SPDC or StPDC schemes and the present approach, let us begin by describing the quantum states of generated photons. When the signal and idler modes are initially vacuum states, the pair of signal and idler photons generated by each individual PDC is in a photon number-entangled two-photon state. Indistinguishability between two idler fields from a dual SPDC or StPDC process was shown to be induced via erasing the which-path information of the idler fields through aligning them collinearly and then overlapping them spectrally and spatially, regardless of the presence of the injected coherent seed beam at an idler frequency [36,38]. Because the spectral bandwidths of the generated signal and idler photons are broad due to intrinsic phase-matching conditions, the frequency resolution of such QSUP with induced coherence geometry must be limited by dispersive optics for spectral resolution, e.g., grating. Although the broadband QSUP utilizing two SPDC crystals has been demonstrated recently [20,21], a QSUP with sufficiently high frequency-resolution that should be of critical use for measuring the absorption spectra of atomic and molecular vapor samples with narrow bandwidths has not been reported. Here, we demonstrate that the QSUP with dual StPDC as a high frequency-resolution Doppler-free quantum spectroscopy is experimentally feasible. Furthermore, in contrast to the dual SPDC scheme, the present dual StPDC scheme does not require the two idler beams generated by the corresponding nonlinear crystals to be perfectly aligned or spatially overlapped at all. In the previous dual SPDC-based QSUP, an optical sample under spectroscopic investigation should be located in between the two crystals or two SPDC processes. In contrast, our dual StPDC-based QSUP allows the optical sample to be placed anywhere in space as long as its distance from the interferometric detection system is shorter than the coherence length of the seed laser. In the present work, we also demonstrate such remote measurements with undetected photons, where the light-matter interactions and the photon detections are performed in spectrally and spatially different regions.

2. Main results

2.1. Schematics of QSUP

In Fig. 1, presents a schematic diagram of our QSUP setup, which consists of four modules: (a) dual StPDC single-photon interferometry, (b) signal beam interference detection unit, (c) radiation source, and (d) matter-field interaction unit. The two nonlinear crystals (periodically poled lithium niobate, PPLN) are pumped by a frequency-comb with a central wavelength of 530 nm, which is produced by frequency doubling a 1060 nm frequency-comb laser. The dual PDC’s are stimulated by an ultra-narrow linewidth (< 1.0 Hz) continuous wave (CW) laser with an idler wavelength of 1542.384 nm, which results in the generation of a path-entangled non-degenerate signal (807 nm) and idler photons. Here, the single photon interferometry in Fig. 1(a) differs from conventional Mach-Zehnder interferometry, because the latter has only one optical field input, whereas ours has two radiation sources (PPLNs) producing coherent signal photons under interference measurement. Here, the role of the seed beam in our QSUP scheme is essentially twofold: (i) stimulating the generation of frequency-entangled idler-signal photons with narrow linewidth and high emission rate and (ii) making two idler fields from two PDC crystals indistinguishable in their nonlocal photon numbers. The PDC efficiency is set extremely low so that the resulting signal field at the output port of the interferometry can be considered to be in a path-entangled single-photon state. Interestingly, one photon coherence of the dual StPDC-generated signal field at the output port is induced by the erasure of idler path information due to the indistinguishability of idler field photon statistics in the presence of coherent seed beams rather than due to the perfect alignment of the two idler beams [37].

 

Fig. 1 Schematic representation of QSUP experimental setup. Yb fiber optical frequency-comb laser (λ_f = 1060 nm) is used for second-harmonic generation (SHG) of optical frequency-comb at λ_p = 530 nm (a). The SHG field is then split into two paths by a polarizing beam splitter (PBS) to pump the two spatially separate but identical PPLN crystals. The parametric down-conversion processes, PDC1 and PDC2, are stimulated by injected seed beams from a CW laser at 1542 nm with a linewidth of <1 Hz (c). Each StPDC produces a stream of path-entangled pairs of signal and idler photons. In the radiation source module, a 50:50 beam splitter (BS) is used to divide the coherent CW light into two paths. The lower path CW light interacts with an FP cavity whose cavity length is 7.5 mm and free spectral range is 10 GHz (d), which is a model optical sample. In (a), a variable neutral density (VND) filter is used to adjust or attenuate the CW beam intensity at the PDC1 crystal, to enhance the measured visibility. The signal beam is separated from the collinearly propagating idler beam using a dichroic mirror (DM) placed after each periodically poled lithium niobate (PPLN) crystal. Only the signal beams from the two PPLN crystals are combined by a 50:50 BS and the one-photon interference of the band-pass-filtered (BPF) signal field is recorded by spectrometer and EMCCD (b). The phase modulation is introduced by periodically changing the difference in the two pump (530 nm) pathlengths in (a). Here, the FP cavity with a finesse of 135 exhibits a resonant peak with a linewidth of 74 MHz.

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The signal photons generated from the two PPLNs are combined at a beam splitter (BS) and allowed to interfere with each other to produce fringe patterns. To carry out the quantum spectroscopy experiment, the coherent CW (~1542 nm) light is split into two paths by a 50:50 BS as illustrated in Fig. 1(c). The upper CW beam stimulates PDC1, whereas the lower CW beam is coupled to a long optical fiber and allowed to interact with the Fabry-Pérot (FP) cavity that serves as a frequency-selective light transmitter, i.e., an optical sample of narrow linewidth (~74 MHz). Note that the location of the spectroscopic probing module (Fig. 1(d)) can be an arbitrarily far distance from the single-photon interferometry laboratory, where the upper distance limit is the coherence length of the ultra-narrow CW laser in either optical fiber or free space. The coherent (CW) seed photons directly interacting with the model optical sample, the FP cavity, do not reach the detector in Fig. 1(b) and the coherence nature of the StPDC-generated idler photons is transferred to their conjugate signal photons, which carry information on the property of the optical sample, i.e., the FP cavity transmission spectrum.

When the two nonlinear crystals are identical, the quantum state in the interferometry in Fig. 1(a), in the low StPDC efficiency regime, is approximately given as

2.2. Experimental QSUP for FP cavity

To extract the transmission spectrum from the measured single-photon interference of signal field, either the seed laser frequency ν_i or the FP cavity resonance frequency ν_FP can be tuned or scanned, where the detuning frequency Δν ≡ ν_FP -ν_i needs to cover a sufficiently broad range (>2ν_FSR) for frequency calibration. Here, ν_FSR is the frequency of the cavity free spectral range, which is 10 GHz in this case. Experimentally, the detuning frequency Δν is scanned in time either by tuning the coherent seed laser frequency or by changing the FP cavity length, with the scanning speed of Δν set much slower than the modulation frequency Δφ_p (t) for signal field interferogram measurements.

To change the FP cavity resonance frequency, a Piezo-electric transducer (PZT) is used to control the cavity length at a frequency scan speed of 1.27 GHz/s. In Fig. 2(a), the single-photon counting rate of the signal field is plotted with respect to the scanned detuning frequency Δν (t) = ν_FP (t)-ν_i (t), and clearly shows two resonant interference peaks. The average R_s value of 1.1 × 10⁶ photons per 10 ms integration time corresponds to that of the signal field of the upper path only. We carried out fitting analyses of the experimentally measured interference patterns with the theoretical Eq. (2). Then, the transmission spectrum was normalized to the peak value. The normalized transmission spectrum with respect to the detuning frequency is shown in Fig. 2(b), which can be quantitatively fitted to an Airy function. To confirm that the measured spectrum extracted from the single-photon interferometry is the FP cavity transmission spectrum, we carried out a direct measurement of the FP cavity transmission spectrum with a conventional near-IR photodiode (1.54 μm). The two spectra are found to be in excellent agreement with each other. Furthermore, noting that the ability of our QSUP for monitoring cavity frequency change (1.27 GHz/s) with respect to time, it is believed that this QSUP setup can be used to study relatively slow photo-induced chemical or biological reactions or even the conformational changes of reactive molecules in condensed phases that result in frequency shifts in time. Unlike the conventional spectroscopic method, the QSUP measurement of reaction dynamics utilizes the quantum entanglement between the signal and idler fields, where the information on the dynamic molecular transition frequency shifts in the idler frequency region is instantaneously transferred to the entangled signal field and subject to interferometric detection in a completely different spectral region.

 

Fig. 2 One-photon interference and FP cavity transmission spectrum. (a), Experimentally measured single photon counting rate of signal field in Fig. 1(b) is plotted with respect to the detuning frequency, which is changed by scanning the FP cavity length with PZT. Here, the voltage applied to the PZT attached to one of the cavity mirrors is also shown. Each scan time is 10 s. By assuming that the frequency was adjusted linearly with the voltage scan time, the frequency was calibrated with a time interval between the two resonant peaks (10 GHz). In a, the coherent CW seed laser frequency is fixed. The pump pathlength difference (Δx_p) is modulated by a 0.5 Hz triangle wave between 0 to 6 μm (11 oscillations per second), to obtain interference fringes. The SNR (the ratio of peak amplitude to standard deviation of noise) is about 20. (b) The measured (normalized) transmission (blue square) is plotted with respect to the detuning frequency. In this figure, open black circles are the transmission obtained by directly measuring the transmitted 1540 nm laser with a near-IR photodiode detector. The red solid line in (b) is the fitted Airy function that is known to describe the transmission spectrum of the FP resonator, i.e., |T_FP (ν_FP ,ν_i )|² = |T₀|² /{1 + (2F/ π)² sin² [δ/2]}(with δ = 2πν_i /ν_FSR, T₀ is the characteristic constant depending on each FP cavity, ν_FSR = c /4L for a confocal cavity, and L the cavity length. The resonant transmission peaks appear at δ = 2πq (q = integer), i.e., ν_i = ν_FP = qν_FSR, with a peak width Δν = ν_FSR /F and a Finesse$F=\pi \sqrt{R}/(1-R)$ with R being the mirror reflectance of the FP cavity. Two resonant peaks appear in (b) and the linewidth of each peak is estimated to be 74 MHz. The blue squares represent the QSUP results (averages over 5 independent measurements), which are extracted from the signal field interference fringe analyses, where the EMCCD detection window is ± 5 nm around 807 nm. (c) One-photon interference of signal photons is modulated by idler transmission. Here, the FP cavity length (resonance frequency) is fixed, but the CW laser frequency is scanned in the frequency window of ± 150 MHz around the FP cavity resonance frequency with scan time 2 s by applying FM voltage to WGM micro-resonator of the laser. The frequency is calibrated by the full width at half maximum (FWHM) to 74 MHz based on the assumption of a linear relation between the FM voltage scan time and frequency shift. The applied FM voltage scan rate is 10 V/s and the frequency-voltage relationship is 15 MHz/V. The pump pathlength difference (Δx_p) is modulated by a 0.5 Hz triangle wave between 0 to 6 μm (11 oscillations per second). (d) The retrieved QSUP spectrum (blue squares) of the FP cavity. The black dots represent transmission data obtained by directly measuring the transmitted idler photon intensity with a near-IR photodiode and the solid red line is a fitted Airy function. The sign of the detuning frequency refers to the opposite sign to the detuning around the resonant frequency.

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Next, the FP cavity resonance frequency (cavity length) is fixed and the coherent seed laser frequency, Δν (t) = ν_FP (t) - ν_i (t), is scanned in time to measure the cavity spectrum, where the frequency is scanned at 150 MHz/s across a frequency window ± 150 MHz around the FP cavity resonance frequency as shown in Fig. 2(c). Since the free spectral range of our cavity is 10 GHz we did not observe another resonant peak due to a limitation of the continuous frequency tuning range of the seed laser (see Section 3). The single-photon counting rate of the signal field with respect to the detuning frequency clearly shows the frequency dependence of the FP cavity transmission. Fitting Eq. (2) to the experimental results in Fig. 2(c) provided the transmission spectrum of the FP cavity shown in Fig. 2(d). Again, for the sake of comparison, we directly measured the cavity spectrum with the near-IR photodiode and found that the agreement is quantitative. Furthermore, the fact that the experimental spectrum can be fit to an Airy function of the FP cavity suggests that the present QSUP technique is reliable and phase-stable during the interferometric measurement time of 2 s. This is especially important because both the phase matching condition and the initial seed beam intensity fluctuate during each frequency-scan of the seed laser. However, it should be noted that the interference fringe patterns are not strongly affected by such deleterious fluctuations because they contribute to both StPDC processes simultaneously causing the corresponding noise patterns to be canceled out when the interference measurements are performed on the dual StPDC-generated signal field. Such a noise cancellation effect does not apply to single StPDC-type quantum spectroscopy. To experimentally confirm that our single-photon interferometry based on the dual StPDC scheme shown in Fig. 1 provides significantly improved SNR data in QSUP measurements when compared to the single path measurement approach using just one nonlinear crystal, we additionally carried out a series of experiments with one seed laser, one PDC crystal, and a non-interferometric detector at 807 nm (see Appendix B). In fact, the comparison clearly shows that the SNR of the present approach is better than that of the single StPDC quantum spectroscopy.

2.3. Visibility optimization

The fringe visibilities at the peaks in Figs. 2(a) and 2(c) are found to be 0.18 which is apparently much smaller than unity. Such low visibility is due to the following reasons: (i) the transmission coefficient of the FP cavity at resonance is as small as 0.01, (ii) the signal beams do not overlap perfectly at the detector, (iii) undesired signal photons are generated due to noncollinear phase mismatching, and (iv) we use very weak seed beams for StPDCs. The measured visibility without the FP cavity is approximately 0.8, which is also smaller than unity mainly because of the imperfectness in the spatial overlap of the two signal beams. In the presence of the FP cavity, the unbalance between the idler beam intensities at the two PDC crystals makes the fringe visibility of signal fields very small. In experiments, we thus needed to have an additional control variable for enhancing the visibility SNR, which led us to introduce a variable neutral density (VND) filter on the upper path of the seed beam. This VND was used to attenuate the intensity of the upper seed beam to ultimately make the intensities of the seed beams at the two crystals equivalent to each other, which enhances the fringe visibility of signal fields and enables us to perform higher SNR measurements.

The experimentally measured visibility is plotted with respect to |T_VND |² in Fig. 3, where the seed beam intensity is completely determined by the transmission coefficient T_FP(ν_FP) of the FP cavity at its resonance frequency ν_FP. Here, note that the intensity of the seed beam on the upper path is modulated by |T_VND|². For simplicity, we first consider the case of balanced pump beam intensity. The fitting analysis with visibility expression obtained by treating the coherent seed beam-FP cavity interaction quantum mechanically shows that the pump beam is balanced with intensity ratio I₂ /I₁ = 1.3, the average photon number of the upper seed beam is 40, and that of the lower seed beam at the FP cavity resonance frequency is approximately 0.1 (see Section 3). When the pump beam intensity ratio I₂ /I₁ is 5.7 as shown in the inset in Fig. 3, the overall visibility values are larger than those in the case of I₂ /I₁ = 1.3, because the unbalance in the two seed beam intensities is compensated by yet another unbalance in the two pump beam intensities. Again, these experimental results are consistent with those obtained by independently and separately measuring the pump powers and by using the conversion efficiencies of the SPDC and StPDC processes.

 

Fig. 3 Fringe visibility at the FP cavity resonance frequency versus the intensity ratio of the two seed beams, where the latter is controlled by adjusting the transmissivity of variable neutral density (VND) filter. The intensity of the upper seed beam in Fig. 1(a) is modulated by |T_VND|², whereas that of the lower seed beam by the frequency-dependent |T_FP|². Here, the experimentally measured visibility (blue square) is at the FP cavity resonance frequency and it is plotted with respect to |T_VND |² when the pump intensity ratio I₂/I₁ is close to unity. The inset in this figure shows the sample plot when I₂/I₁ = 5.7. The solid red line is a fitted curve with the theoretical equation obtained from quantum mechanical descriptions of the coherent seed beam-cavity interaction and the single-photon interferometry, $V=2\sqrt{I_2/I_1}\left|T_{VND}{T}_{FP}\right|\left|{\alpha }_1{\alpha }_2\right|{[{\left|T_{VND}\right|}^2{\left|{\alpha }_1\right|}^2+1+(I_2/I_1)({\left|T_{FP}\right|}^2{\left|{\alpha }_2\right|}^2+1)]}^{-1}$. Here, the experimental parameters, such as the average photon numbers of the seed (in an idler mode) beams at the PDC crystals and the degree of intensity unbalance of the pump beam, are measured independently. The dashed line corresponds to the visibility obtained from a classical mechanical description with the same parameters for the pump beam intensities. The error bars represent the standard deviation estimated from ten consecutive, independent measurements.

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In Fig. 3, our experimental data are compared with the visibility (dashed line) obtained by treating the seed laser as a classical field. That is, by using the theoretical expression $V_{CM}=2\sqrt{I_2/I_1}\left|T_{FP}{T}_{VND}\right|\left|{\alpha }_1{\alpha }_2\right|{(|T_{VND}{|}^2{\left|{\alpha }_1\right|}^2+(I_2/I_1)|T_{FP}{|}^2{\left|{\alpha }_2\right|}^2)}^{-1}$(see Appendix A). The deviation between the experimental data and the classical theory clearly indicates that the coherent seed beam should be treated as a quantum mechanical field and that the classical description of the present single-photon interferometry is not valid for the interpretation of our experimental results because the signal photon states in the interferometry are in the quantum regime due to very weak coherent seed beam intensity, i.e., |α|²<100. To understand the quantum nature of this modified Mach-Zehnder interferometry featuring two embedded StPDC crystals, it should be noted that, in a very low conversion efficiency regime, the two signal fields from the OFC-pumped PPLNs become indistinguishable and can be described as a superposition state like $c_1{|1,0〉}_{s_1{s}_2}+c_2{|0,1〉}_{s_1{s}_2}$because the two single photon added coherent states ${\widehat{a}}_{i_1}^{†}{|\alpha 〉}_{i_1}\otimes {|T_{FP}{}_{}\alpha 〉}_{i_2}$ and ${|\alpha 〉}_{i_1}\otimes {\widehat{a}}_{i_2}^{†}{|T_{FP}{}_{}\alpha 〉}_{i_2}$ be indistinguishable when |α|≫1 for |α₁| = |α₁| = |α| [37](see Eq. (1)).

3. Experimental details

3.1. Radiation source

In the experimental setup of Fig. 1, the pump beam with a center wavelength of 530 nm is prepared from the second-harmonic generation by using LBO crystal from an optical frequency comb (OFC) laser (Menlo Systems) centered at 1060 nm with a spectral bandwidth of 46 nm. The frequency comb pump beam has long-term phase coherence where the repetition rate and carrier-offset phase of the frequency comb laser are actively stabilized at 250 MHz and 20 MHz, respectively, with reference to our GPS disciplined Rb atomic clock. The spectral width of the pump beam at 530 nm is approximately 3 nm, which is narrower than that of the fundamental (1060 nm) beam due to the noncritical phase-matching bandwidth of the LBO crystal (Newlight Photonics, type I, 3 × 5 × 5 mm³) at the phase matching temperature of 143 °C.

3.2. Parametric down-conversion

The pump beam with a power of 6 mW is divided into two beams with variable intensity ratio by using a half wave plate (HWP) and a polarizing beam splitter (PBS). The pump beam is focused to a periodically poled lithium niobate (PPLN) crystal (HCP, type-0, 15 × 7.9 × 0.5 mm³, grating width of 7.3 μm) by using the achromatic lens with a focal length of 100 mm. The signal and idler beams generated from each parametric down-conversion are collimated with the same dichroic lens. In the case of spontaneous parametric down-conversion (SPDC), a pair of photons consisting of a signal photon with wavelength (${\lambda }_s$) at around 807 nm and an idler photon with wavelength (${\lambda }_i$) at around 1542 nm are generated and their linewidths as broad as 20 nm at a noncritical phase-matching temperature around 120.5 °C. The signal and idler beams propagate collinearly with the same polarization. Since the SPDC efficiency is extremely low (approximately one out of 10 billion pump photons is converted into a pair of signal and idler photons), multi-photon generation processes can be safely ignored. Each pair of signal and idler photons from a single SPDC are in a path-entangled state and show just two-photon coherence property not one-photon coherence property. Thus, SPDC is not useful for high-frequency resolution QSUP. To enhance the PDC rate, a monochromatic laser with a sub Hz linewidth centered at 1542.384 nm (OEwaves, OE403) that overlaps with the emission spectrum of idler photons from the SPDC crystal is injected along the paths of the two pump beams by using two dichroic beam splitters (DM).

3.3. Single photon interferometry

As increasing the average photon numbers of the two seed beams, the indistinguishability of idler photons generated by the two PDC crystals increases accordingly. As a result, the quantum coherence between the signal photons on the two paths increases, which manifests the complementary relation between visibility and distinguishability of quantum objects [37]. To overlap the spectral distributions of thus generated signal fields from the two StPDC crystals, we controlled the phase-matching temperatures of the two nonlinear crystals to be 121.5 °C and 120.5 °C for PDC1 and PDC2, respectively. After each StPDC, the transmitted pump and seed beams and the generated idler beam are all reflected off by the dichroic mirror (DM). Only the signal photons with wavelengths centered at around 807.2 nm are transmitted through the DM and combined at a beam splitter (BS). Finally, the induced interference fringes are measured with employing a single-photon-sensitive two-dimensional EMCCD camera (Andor Technology) attached to the output port of a spectrometer (Shamrock SR−303i−B, Andor Technology, Grating 600 lines/mm at 500 nm blaze). The exposure time is 10 ms. Interference fringes are recorded with respect to relative phase (path) differences of the pump, seed, or signal beams propagating along the two different paths. Here, the path length differences, $\Delta x_p$, $\Delta x_{seed}$, and $\Delta x_s$, associated with the corresponding phase differences as $\Delta {\varphi }_m=2\pi \Delta x_m/{\lambda }_m$ (for m = p, s, and seed), are controlled by three Piezo-electric transducers (PZTs) as illustrated in Fig. 1. To ensure identical signal beam conditions for the generation of high-contrast interference fringe patterns, we optimized the overlaps of the two signal beams in spectral, spatial, and temporal domains. The spectrum of each signal beam is tuned by adjusting the phase-matching temperature of PPLN crystal according to the Sellmeier equation and the visibility is maximized when they are exactly matched within the spectrometer’s resolution of 0.1 nm.

3.4. Spectroscopy of Fabry-Pérot cavity

Febry-Pérot cavity is used for an optical sample, which is placed on one of the two idler beam paths. The degree of path-indistinguishability of thus generated idler beams is modulated due to the intensity imbalance of the two injected seed beams. This results in modulation of the degree of induced one-photon interference of quantum-entangled signal photons. The confocal FP cavity has a free spectral range (FSR) of 10 GHz and a full-width at half-maximum (FWHM) of 74 MHz (Thorlabs, SA210-12B). The modulated one-photon signal interference pattern due to the change of FP cavity length is measured, where the FP cavity length is changed so that it cavity resonance frequency change covers more than two FSRs within the scanning time of 10 s while the input laser frequency is fixed. Ignoring nonlinearity in translation motion and hysteresis in repeatability of the PZT, we found that the frequency scan rate is about 1.27 GHz /s.

Independently, the same FP transmission spectrum can be measured by scanning the seed beam frequency. For a fixed FP cavity, the seed beam frequency is scanned from −150 MHz to + 150 MHz around the FP cavity resonance, where the whole scan time is 2 s. Here, the experimental relationship between the applied FM (Frequency Modulation) voltage and seed beam frequency shift is 15 MHz/V so that the frequency scan rate is 15 MHz/s. In this case, we were not able to scan the seed beam frequency to cover at least two FP cavity resonant peaks, which is mainly due to the limitation of the continuous frequency tuning range of the seed laser based on the whispering gallery mode (WGM) micro-resonator (OE Waves). The interference pattern is further phase-modulated by a fast triangle modulation of the pump path length with 0.5 Hz so that almost 11 cycles appeared in one second across the cavity resonance. We obtained integrated single photon counts for 10 ms exposure time (~0.2 π step size) for signal photons incident to the whole spectrometer arrays at the center wavelength of 807.2 nm to increase the SNR.

3.5. Remote-measurement QSUP

Since the length of the seed beam path (lower, path 2) on which the FP cavity is located is longer by 1.7 m than another one (upper, path 1), the spatial profiles (e.g., beam waist and divergence) of the two seed beams at the two nonlinear crystals could be slightly different. This causes difficulties in matching the spatial profiles (beam sizes) and the photon numbers per unit area of the two generated signal beams at the position of the beam combining BS. This critically results in reduced visibility (< 1). In fact, if the two seed beam path lengths are adjusted to be the same and if only the central part of the signal beams are selected to make them perfectly overlap in space, one can observe almost unit visibility for unit transmission as experimentally demonstrated recently [37]. To overcome this technical difficulty, we intentionally adjusted the pump beam intensity ratio between the lower path to the upper path to be 5.7. However, still the measured fringe visibility in the presence of the confocal FP cavity is 0.18 at the FP cavity resonance frequency. We could further maximize the visibility by matching the average photon numbers of the seed beams at the two crystals by means of controlling the transmission coefficient, T_VND, of variable neutral density (VND) filter.

3.6. Visibility measurement and analysis

To achieve maximum visibility measurements for high signal-to-noise ratios, we needed to estimate the average photon numbers, $|{\alpha }_1{|}^2$ and $|T_{FP}{\alpha }_2{|}^2$, of the seed beams in the upper path and the lower path separately, when T_VND = 1. This can be achieved by directly comparing the conversion efficiency of the SPDC with that of the StPDC with the seed beam as demonstrated in [37]. We found that they are about 40 and 0.1, respectively. To confirm that these estimated values are accurate, we obtained the average photon numbers by fitting our experimentally measured visibility data to the quantum mechanical expression for the visibility function, i.e., $V\propto 2\left|T_{VND}{\alpha }_1{T}_{FP}{\alpha }_2\right|\sqrt{I_2/I_1}/[{\left|T_{VND}{\alpha }_1\right|}^2+1+(I_2/I_1)({\left|T_{FP}{\alpha }_2\right|}^2+1)]$ versus $T_{VND}$ [37]. The fitting analyses provide that $|{\alpha }_1{|}^2\approx 40$ and $|T_{FP}{\alpha }_2{|}^2\approx 0.1$, respectively, where the pump intensity ratio $I_2/I_1$ is either 1.3 (balanced) or 5.7 (unbalanced). Thus obtained average photon numbers are in quantitative agreement with those obtained by using the experimental comparison of the conversion efficiency of the SPDC without any seed beam and that of the StPDC in the presence of the seed beam.

The measured visibility reaches a maximum value when the two generated signal beam intensities are the same. This is the case when the two terms in the denominator of the visibility equation above are identical to each other, i.e., $I_1(|T_{VND}{|}^2|{\alpha }_1{|}^2+1)=I_2(|T_{FP}{\alpha }_2{|}^2+1)$. Indeed, as can be seen in Fig. 3, the visibility is found to reach its maximum value when the transmission coefficient of the VND is $|T_{VND}{|}^2=[(I_2/I_1)(|T_{FP}{\alpha }_2{|}^2+1)-1]/|{\alpha }_1{|}^2\approx 0.06$for $I_2/I_1=1.3$. In this case of the maximum visibility, the experimentally measured single photon counting rate is $R_s~3.6\times {10}^4/10\text{ms}$ . If the coherent seed laser is treated as classical radiation, the maximum visibility is predicted to be observed when $I_1|T_{VND}{\alpha }_1{|}^2=I_2|T_{FP}{\alpha }_2{|}^2$. If we use the estimated average photon numbers and the pump beam intensity ratio determined independently, the maximum visibility is predicted to be at $|T_{VND}{|}^2=(I_2/I_1)(|T_{FP}{\alpha }_2{|}^2/|{\alpha }_1{|}^2)\approx 0.003$ for $I_2/I_1=1.3$, which clearly differs from our experimental result in Fig. 3. This indicates that our QSUP is in a quantum regime.

4. Conclusion

Here, as a proof-of-principle experiment, we show that a high-resolution spectroscopic measurement of a sample with an optical resonance at 1542 nm can be achieved by detecting the one-photon interference of an 807 nm field in a quantum mechanical single-photon state. Since our spectroscopic measurement is based on scanning either an optical sample (FP cavity resonance) or coherent seed laser frequency, the frequency resolution is intrinsically limited by the laser linewidth (< 1 Hz). This is in stark contrast with the single or dual SPDC (not StPDC) based QSUP schemes because the optical sample interacts with the spectrally broad idler beam, which inevitably requires a prism or diffraction grating for frequency resolution.

An interesting feature of the present dual StPDC single-photon interferometric spectroscopy is that the data acquisition time is significantly shorter than other single-photon interferometric approaches using just one PDC crystal because the single-photon counting rate increases with the average photon number of the injected seed beam. Since the detection is still performed at the single-photon level, the present QSUP technique requires neither high PDC efficiency nor cavity enhancement of the field-matter interaction. In summary, we experimentally demonstrated a dual StPDC based QSUP without any heralded detection, which would be used (i) to measure natural linewidths of atoms or molecules in the gas phase because of its high frequency-resolution at the sub-Doppler broadening limit, (ii) to perform remote-measurement of molecular or optical samples located far away from the interferometric detection module, and (iii) to measure optical spectra to a higher SNR and over a shorter data acquisition time than single PDC based quantum spectroscopy. Thus, we anticipate that the present QSUP technique can be useful in the fields of quantum optical measurement and even quantum state engineering [37]. Our QSUP based on a quantum state manipulation, which overcomes the limitations of available detectors at the wavelength of interacting photons, is expected to be widely applicable to not only extending conventional time and/or frequency-resolved spectroscopy with undetected photons but also quantum information processing and metrology.

Appendix A Classical spectroscopy with undetected photons

In the quantum description of StPDC, we treat both the signal and idler beams as quantized fields so that the Hamiltonian operator is given by

(3)$$H_{QM}=i\hslash g{E}_p{a}_s^{+}{a}_i^{+}-i\hslash g^{*}{E}_p^{*}{a}_s{a}_i,$$

where g is the PDC coupling coefficient and E_p is the complex ampliude of the pump field. Note that the Hamiltonian in Eq. (3) is semiclassical because the pump field is treated as a classical field. In fact, the time-evolution operator described by this Hamilonian creates a two-mode squeezed vacuum state [35]. In the main text, we presented the photon counting rate of the signal beams at the detector and the visibility of the interference pattern produced by the signal fields.

If the coherent seed beam is treated as a classical field, the Hamiltonian becomes

(4)$$H_{CM}=i\hslash g{E}_p{E}_i{a}_s^{+}-i\hslash g^{*}{E}_p^{*}{E}_i^{*}{a}_s,$$

where E_i is the complex amplitude of the seed field. It should be noted that the subscripts ‘QM’ and ‘CM’ in Eqs. (3) and (4) indicate that the coherent seed beam in an idler mode is treated quantum mechanically and classical mechanically, respectively. The time-evolution operator described by the Hamiltonian in Eq. (4) can be viewed as the displacement operator that creates the coherent state from the vacuum state of the signal field. Here, for the sake of simplicity, we consider a single mode for both the signal and idler fields. With Eq. (4), the coherent state of the generated signal field after each nonlinear interaction with PDC crystal can be written as $|{\psi }_j(t)〉=e^{{\beta }_j{}^{*}{a}_{s_j}^{+}-{\beta }_j{a}_{s_j}}|0〉=|{\beta }_j〉$, where ${\beta }_j=-i{g}^{*}t{E}_{p_j}^{*}{E}_{i_j}^{*}$ (j = 1,2). When one of the seed beams undergoes an attenuation by T_FP, we have $|{\psi }_t(t)〉={|{\beta }_1〉}_{s_1}\otimes {|T_{FP}{\beta }_2〉}_{s_2}$. Then, the photon counting rate of the signal beams at the detector is given by

(5)$$\begin{array}{c}{R}_s=〈{\psi }_t(t)|(a_{s_1}^{†}+e^{-i\Delta {\phi }_t}{a}_{s_2}^{†})(a_{s_1}^{}+e^{i\Delta {\phi }_t}{a}_{s_2}^{})|{\psi }_t(t)〉\\ ={\left|{\beta }_1\right|}^2+{\left|T_{FP}{\beta }_2\right|}^2+2\left|{\beta }_1{\beta }_2{T}_{FP}\right|\cos \Delta {\phi }_t\\ \propto I_1{\left|{\alpha }_1\right|}^2+I_2{\left|T_{FP}{\alpha }_2\right|}^2+2\sqrt{I_1{I}_2}\left|{\alpha }_1{\alpha }_2{T}_{FP}\right|\cos \Delta {\phi }_t,\end{array}$$

where ${\alpha }_j$(j = 1,2) is the complex amplitude of the injected coherent CW beam on the path j. If the VND filter is placed in the path 1 as illustrated in Fig. 1, the complex amplitude is replaced by ${\alpha }_1\to T_{VND}{\alpha }_1$. From Eq. (5), one can find that the visibility in this classical limit is given by

(6)$$V_{CM}=2\sqrt{I_2/I_1}\left|T_{FP}{T}_{VND}\right|\left|{\alpha }_1{\alpha }_2\right|{(|T_{VND}{|}^2{\left|{\alpha }_1\right|}^2+(I_2/I_1)|T_{FP}{|}^2{\left|{\alpha }_2\right|}^2)}^{-1}.$$

When $I_1=I_2$and $\left|{\alpha }_1\right|=\left|{\alpha }_2\right|=\left|\alpha \right|$, the visibility in Eq. (6) becomes $V_{CM}=2\left|T_{FP}\right|/(1+{\left|T_{FP}\right|}^2)$. In fact, this is identical to the visibility obtained by using the quantum mechanical description of the coherent seed beam in the limit of large α, i.e., $V_{CM}=\underset{\left|\alpha \right|\to \infty }{\lim }{V}_{QM}$. In the present work, we experimentally showed that such a classical description of single photon interferometry in Fig. 1 is not valid for interpreting our experimental results because the signal photon states in the interferometry are indeed in quantum regime with very weak coherent seed beam intensity.

Appendix B Comparison between conventional spectroscopy and our QSUP

In order to clarify how our QSUP scheme is differentiated from the traditional spectroscopic techniques, we obtained experimental results with employing the conventional spectroscopy method and directly compare them with our QSUP results. The schematic representations of conventional spectrometry and our QSUP are shown in Fig. 4. First, the spectrum of FP cavity is measured directly by using the detector D1 in Fig. 4(a), where the CW laser has a center wavelength of 1542 nm. Second, the change of CW laser intensity due to the presence of FP cavity is indirectly measured with detector D2, where this detector is sensitive to the signal beam photons at a center wavelength of 807 nm illustrated in Fig. 4(b). Note that the measured photons are signal photons generated from one nonlinear crystal (PDC) but the photons interacting with the FP cavity are CW laser photons at an idler mode frequency. This experimental scheme in Fig. 4(b), which can be referred to as single StPDC QSUP, was considered and used in the previous frequency upconversion experiments by other groups [39–41]. In Fig. 4(c), our frequency comb single-photon interferometry setup is schematically depicted, where the change of the upper CW beam intensity due to its interaction with the FP cavity is measured indirectly with the detector D3. Note that the measurement involves a single-photon interference phenomenon. To obtain the spectrum, we scanned either the FP cavity length (resonant frequency) for a fixed CW laser frequency as shown in the bottom panel of Fig. 5 or the laser frequency of the seed beam for a fixed FP cavity length as shown in the bottom panel of Fig. 6. Figure 2 in the main text and Figs. 5 and 6 in this appendix clearly demonstrate that the pure transmittance spectrum of any optical sample (FP cavity in the present case) in idler wavelength region can be successfully measured from the spectral interferograms. Although the spectra obtained with employing the conventional (direct) spectroscopic method (see those in the top panels of Figs. 5 and 6) exhibit high SNR’s, because of the fortunate availability of highly sensitive NIR detector in this wavelength (1542 nm) range. However, this will not be the case when there is no high-sensitivity detector available in the frequency range of interacting seed (idler) photons. In this regard, our experimental results as shown in the bottom panels of Figs. 5 and 6 unambiguously demonstrate the capability of our QSUP for quantum optical measurements with a visible or near-infrared (NIR) detector, regardless of the wavelength range of the radiation directly interacting with materials or molecules of interest in condensed phases. For example, our scheme will enable one to obtain the far-IR spectra of molecules in condensed phases with a highly sensitive visible-NIR detector (800 nm), whereas the optical measurement of the far-IR absorption spectrum would be severely limited by detectors that have typically very low quantum efficiency.

 

Fig. 4 Schematic diagrams representing conventional spectroscopy, single StPDC QSUP, and our dual StPDC QSUP. (a) The transmission spectrum of the FP cavity can be directly measured with detector D1 at 1524 nm. (b) Single StPDC QSUP can be used to indirectly measure the transmission spectrum, where the quantum entangled signal beam at a center wavelength of 807 nm is measured with detector D2. This is the ordinary frequency conversion setup with one nonlinear crystal (PDC). (c) Our dual StPDC QSUP uses two nonlinear crystals and the one-photon interference of thus generated signal fields is detected with D3 at around 807 nm.

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Fig. 5 Experimentally measured transmission spectra of the FP cavity with the FP cavity length scan. The transmission spectrum of the FP cavity is obtained by tuning the resonance frequency of FP cavity, which is achieved by scanning the cavity length for 10 s. The scan rate is 1.27 GHz/s. The spectrum in the top panel is the transmission intensity (arbitrary unit) of injected seed beam, which is measured with NIR (1542 nm) photodiode (D1) in Fig. 4(a). That in the middle panel shows the frequency converted signal photons modulated by the idler beam transmission, where EMCCD (D2) in Fig. 4(b) is used. The spectrum in the bottom panel is one-photon interference fringe of signal photons modulated by idler beam transmission, which is detected by EMCCD (D3) in Fig. 4(c). To obtain the scan time-dependent signals in the middle and bottom panels, the single photon counting rates are measured for 10 ms (exposure time) and the detection wavelength window is 807.2 nm ± 0.1 nm. In this FP cavity length scanning mode, SNR improvement in our QSUP setup is significant compared to the conventional single path technique using just one nonlinear crystal.

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Fig. 6 Experimentally measured transmission spectra of the FP cavity with a seed beam frequency scan. To measure the transmission spectrum of the FP cavity, we scan the seed laser frequency with a fixed FP cavity length (resonance frequency). The scan time is 2 s and the seed beam frequency scan rate is 150 MHz/s. The spectrum in the top panel corresponds to the transmission intensity (arbitrary unit) of the injected seed beam, where NIR photodiode (D1) in Fig. 4(a) is used. That in the middle panel is single-photon counting rates with respect to the seed beam frequency scan time (or equally seed beam frequency), where the used detector is EMCCD (D2) in Fig. 4(b). The spectrum in the bottom panel corresponds to the one-photon interference fringe of signal photons modulated by the seed beam transmission, where EMCCD (D3) in Fig. 4(c) is the detector. Here, the single photon counting rates are measured for 10 ms (exposure time) and the wavelength window of the EMCCD is 807.2 nm ± 0.05 nm.

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In addition, we have experimentally shown that our high frequency-resolution QSUP technique has almost 4 times higher SNR than that based on detection with a single StPDC crystal as shown in the single StPDC QSUP (the middle panels in Figs. 5 and 6) and those with our QSUP in the high resolution laser scan mode shown in the bottom panels in the same figures. This enhanced SNR of our interferometric QSUP can be understood by noting that the spectral distortions of beams during their propagations through each nonlinear crystal, intrinsic fluctuation of the laser beam intensity, and other common noises on the two paths are canceled with each other in the present QSUP interferometry involving two identical nonlinear crystals. To further demonstrate such enhancement effects in the measured SNR’s, we collected and measured the signal photons at the center wavelength of 807.2 nm by using a narrow band pass filter with bandwidth of ±0.05 nm, which gives a factor of 60 smaller counting rate compared to the case when all the incident signal photons are counted as shown in the main text. The spectrum with QSUP in Fig. 5 (bottom panel) indicates that the SNR (the ratio of peak amplitude to standard deviation of noise) is about 4, which is a significant enhancement compared to the spectrum with one nonlinear crystal shown in Fig. 5 (middle) where the SNR cannot even be defined because the signal is under noise variance. The same enhancement in the SNR of our QSUP interferometry setup can be seen in the high resolution laser scan mode too by comparing Fig. 6 (middle panel) and Fig. 6 (bottom panel).

Finally, it should be noted that the present QSUP using dual StPDC scheme has practically better points than that using dual SPDC scheme. First, the spectral resolution is not limited by the SPDC process but dictated by the seed laser linewidth. Note that the spectra of SPDC-generated idler and signal beams are inherently broad. Consequently, the spectral resolution is strictly determined by the quality of grating. It would be extremely difficult or even impossible to make any grating for frequency resolution beyond the sub-Doppler broadening limit. Second, the light-matter coupling at each nonlinear crystal in the dual StPDC scheme is much stronger than that with a dual SPDC scheme because it is driven by an injected coherent light. Third, the overall data acquisition time including frequency scanning time for measuring the absorption spectrum can be shortened by increasing the average photon number of seed beams. However, still our method requires a time-consuming frequency tuning of seed laser, which potentially limits any further shortening of the data acquisition time. These novel aspects of our QSUP that is capable of high frequency resolution and remote-measurement will allow one to apply this technique to study chemical and biological reactions and processes with photons, where there is no sensitive detector available.

Funding

Institute for Basic Science (IBS-R023-D1).

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