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Abstract
Ultrafast quantum optics with time-frequency entangled photons is at the forefront of progress towards future quantum technologies. However, to unravel the time domain structure of entangled photons and exploit fully their rich dimensionality, a single-photon detector with sub-picosecond temporal resolution is required. Here, we present ultrafast single-photon detection using an optical Kerr gate composed of a photonic crystal fiber (PCF) placed inside a Sagnac interferometer. A near-rectangle temporal waveform of a heralded single-photon generated via spontaneous parametric down-conversion is measured with temporal resolution as high as 224 ± 9 fs. The large nonlinearity and long effective interaction length of the PCF enables maximum detection efficiency to be achieved with only 30.5 mW gating pulse average power, demonstrating an order-of-magnitude improvement compared to optical gating with sum-frequency generation. Also, we discuss the trade-off relationship between detection efficiency and temporal resolution.
Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. Introduction
Single-photon detection with precise arrival timing information is of vital importance in a broad range of applications such as high accuracy, high-resolution three-dimensional depth imaging [1–3] and measurement of the dynamics of ultrashort lifetime fluorescence in physical and biological samples [4,5]. Additionally, recent progress of quantum optical experiments has directly clarified the time-frequency behavior of entangled photon pairs, requiring treatment using a two-dimensional Fourier transform [6–8]. This higher-order Fourier optical treatment brings us new manipulation techniques of the quantum characteristics of entangled photons in a femtosecond range [7]. Moreover, a recent approach combining a quantum pulse gate, the sum frequency generation using spectrally engineered gate pulses, with quantum estimation theory provides a high-resolution multiparameter measurement of two optical pulses in close temporal proximity [9]. To further develop such ultrafast quantum optics fields, a single-photon detector with sub-picosecond temporal resolution is needed to elucidate the time domain characteristics of the entangled-photon wave packet in detail. Recently, cutting-edge technology in a superconducting nanowire single-photon detector (SNSPD) has achieved sub-3 picosecond temporal resolution [10]. However, it is still not enough to capture the time-domain distribution of a single-photon wave packet in detail.
On the other hand, ultrafast optical gating techniques can efficiently achieve sub-picosecond temporal resolution. For example, single-photon detection with a temporal resolution of several hundred femtoseconds has been demonstrated by using optical gating with sum-frequency generation in a second-order nonlinear optical effect [6–8,11,12]. However, the sum-frequency generation method has an intrinsic drawback: the phase-matching condition restricts the available spectral bandwidth of the gating pulse, limiting the temporal resolution. In contrast, an optical Kerr gate utilizing cross-phase modulation (XPM) due to third-order nonlinearity may offer higher temporal resolution because of the relaxation of phase matching requirements and commensurate increase in bandwidth of the gating pulse. The optical Kerr gate has been widely deployed in classical optics for ultrafast spectroscopy and imaging [13–17], but its use at the single-photon level is still challenging [18,19]. Conventional Kerr gating utilizing solid or gaseous materials requires a higher peak power of the gating pulse, which usually requires the use of a low-repetition rate laser and is unable to offer efficient detection of photons for quantum optical experiments. However, optical fiber can induce the Kerr effect at relatively low peak powers, and hence its use has been reported for frequency conversion [20] and switching of photons [21]. We note that in Ref. [21] the authors exploit the temporal walkoff between the pump pulse and single-photon wave packet in conventional single-mode fiber; this achieves a uniform phase shift for efficient optical switching but renders their setup unsuitable for single-photon detection with high temporal resolution.
Our work is devoted to developing the optical Kerr gate for ultrafast time-resolved measurement of a single-photon wave packet. Our scheme utilizes a polarization-maintaining photonic crystal fiber (PCF) as the Kerr medium, placed inside a Sagnac interferometer. The PCF is designed so that the group velocity difference between the gating pulse and the signal photon is close to zero, providing a long effective interaction length and maximizing the temporal resolution. We successfully measured the temporal profiles of a heralded single-photon wave packet generated via spontaneous parametric down-conversion (SPDC) with a resolution of 224 ± 9 fs.
2. Experimental setup
Figure 1 illustrates our experimental setup for the time-resolved measurement of heralded single-photon wave packets with a Sagnac interferometer. We used a mode-locked titanium sapphire (Ti:S) laser, operating at a center wavelength of 792 nm, pulse duration of 220 fs (FWHM, full width at half-maximum), and a repetition rate of 76 MHz, to pump both the SPDC process and the optical Kerr effect. The laser pulses from the Ti:S laser were divided by a polarizing beam splitter (PBS). A half-wave plate (HWP) controls the splitting ratio at the PBS. The horizontally polarized pulse was sent to a periodically poled KTiOPO4 (PPKTP) crystal with a crystal length of L = 30 mm and a poling period of Λ = 46.1 µm for the generation of photon pairs via type-II group-velocity-matched SPDC [22–24]. We optimized the waist size of the pump beam to be 800 µm at the middle of the PPKTP crystal to maximize the collection efficiency of the photons into the PCF. The maximum collection efficiency of the SPDC photons into the PCF is about 28%, which was estimated in a different experimental setup. Down-converted photon pairs with orthogonal polarizations were generated around a wavelength of 1584 nm. The second PBS (PBS2) separated the photon pairs; each vertically polarized photon functioned as a herald with the corresponding horizontally polarized photon as a signal photon to be input to the optical Kerr gate. The herald was coupled into single-mode fiber and detected by a superconducting nanowire single-photon detector, labeled SNSPD2.
Fig. 1. Schematic diagram of the experimental setup. HWP, half-wave plate; PBS, polarizing beam splitter; M, Mirror; NPBS, non-polarizing beam splitter; PCF, photonic crystal fiber; QWP, quarter-wave plate; DM, dichroic mirror; BT, beam trap; LPF, long-pass filter; BPF, band-pass filter; FC, fiber coupler; SMF, single-mode fiber; SNSPD, superconducting nanowire single-photon detector.
The PCF was fabricated from pure silica raw material with a microstructure formed by stacking capillaries into a triangular array and drawing to fiber. An electron micrograph of the cleaved PCF end face can be seen in Fig. 2(a). The pitch of the holes in the cladding was 3.1 microns with a hole-diameter-to-pitch ratio of approximately 0.25. During the fiber draw selective pressurization was used to increase the diameter of two holes adjacent to the core by a factor of 1.7, thus inducing birefringence in the PCF. The structural parameters were chosen to match group velocities at the wavelengths of 792 nm and 1584 nm while providing polarization-maintaining guidance. The group delay per unit length of both polarization axes was measured by spectrally-resolved white-light interferometry. Group delay on one fiber axis is plotted in Fig. 2(b) and the associated group velocity dispersion calculated from the polynomial fit to the group delay is shown in panel (c). The difference in group delay between 792 nm and 1584 nm on this axis is 48 fs/m as retrieved from the fit. Note that on this scale the group delay of the second fiber axis would appear largely overlapped with that of the first as the difference in group delay between the two axes is 96 fs/m and 230 fs/m at 792 nm and 1584 nm respectively.
Fig. 2. (a) Scanning electron micrograph of cleaved PCF end face. (b) Group delay per unit length relative to 792 nm on one polarization axis. Points are data measured in a 494 mm length of PCF with errors smaller than the markers, and the line is a polynomial fit. Vertical lines indicate wavelengths of 792 nm and 1584 nm. (c) Group velocity dispersion calculated from the fit to the data in (b).
The signal photon was sent into the optical Kerr gate composed of a length of the PCF shown in Fig. 2(a) within a Sagnac interferometer. After being separated by a non-polarizing beam splitter (NPBS), the signal photon propagated along either the clockwise (CW) or the counterclockwise (CCW) path and was coupled into the 28-cm-long PCF, the Kerr medium, from either end. The PCF is advantageous for large nonlinearity due to the large core-cladding refractive index difference and small effective core area. Combined with the group-velocity matching condition, our PCF enables highly efficient optical gating even for single-photon level light. The reflected beam from PBS1, working as a gating pulse, was sent to a delay system composed of PBS3, a quarter-wave plate (QWP), and a retroreflector set on a mechanical moving stage with a resolution of 1 µm per step. The gating pulse from PBS3 was combined into the CCW path of the Sagnac interferometer by a dichroic mirror (DM1) to induce an optical Kerr effect. The coupling efficiency of the gating pulse into the PCF is about 45%. Thanks to the group-velocity matching at the two wavelengths of 792 nm and 1584 nm, the gating pulse and the signal photon can be propagated together over the entire length of the PCF, resulting in efficient XPM. After passing through the PCF, the gating pulse is excluded from the Sagnac interferometer by the second dichroic mirror (DM2).
The signal photon passing through the PCF is output to either the input port or the output port of the Sagnac interferometer, depending on the relative phase difference of the propagated photons in the CW and the CCW paths. In this experiment, the relative phase difference corresponds to the nonlinear phase shift (ϕXPM) produced by the XPM in the PCF, which is proportional to the intensity of the Kerr gating pulse (Igate) and the interaction length (Leff): ϕXPM ∝ Igate × Leff. Therefore, the probability that the signal photon is distributed to the output port is described as $1 - \textrm{cos}{\phi _{\textrm{XPM}}}$. Blocking the gating pulse leads to zero relative phase difference, and all the signal photons are returned to the SPDC source. In contrast, the photons would be sent to a superconducting nanowire single-photon detector (SNSPD1) when achieving the relative phase value π by an optical Kerr effect. In this experiment, the duration of the gating pulse is sufficiently shorter than the temporal width of the signal photon wave packet. Therefore, by sweeping the time delay of the gating pulse relative to the signal photon wave packet, we can perform the time-resolved measurements of the temporal shape of the signal photon wave packet. Here we note that the Sagnac configuration allows us to retain the phase stability in the measurement period without any active stabilization, due to the common optical path in the CW and CCW propagations. The signal photon output from the Sagnac interferometer was coupled into a single-mode fiber after passing through long-pass and band-pass filters which limit the transmission wavelength range to 1550 to 1600 nm, then detected by the SNSPD1. The coupling efficiency of the PCF output photons to the SMF was estimated at up to 54%. The single-photon detection efficiency of our SNSPD is ∼70% with the dark count rate of ∼100 counts/sec. We measured the coincidence counts of the signal photon and the herald while sweeping the time delay of the gating pulse to measure the single-photon wave packet.
3. Results and discussion
Figure 3 shows the temporal waveform of the heralded single-photon generated via SPDC, measured at an average SPDC pump power of 130 mW and a Kerr gating pulse average power of 30.2 mW. According to the photon-pair production rate reported in Ref. [23], the number of photon pairs generated at the average SPDC pump power of 130 mW (corresponding to 1.71 nJ pulse energy) is less than 0.1 pairs/pulse. We collected the coincidence counts while changing the time delay of the Kerr gating pulse with a step of 67 fs; the coincidence window was 3 ns and counts were accumulated for 10 seconds. The solid line is the fitting curve composed of the complementary error functions, which are represented by
Fig. 3. The measured temporal waveform of the heralded single-photon generated via SPDC. Average SPDC pump power was set to 130 mW, and the Kerr gating pulse average power was set to 30.2 mW. The solid line is the fitting curve using the complementary error functions.
Next, we measured the temporal waveform of the single-photon wave packet at different gating pulse average powers from 5.0 mW to 36.3 mW. Figure 4 shows the results with the gating pulse average power of 5.0, 9.8, 15.3, 20.2, and 30.2 mW. The height of the temporal waveform varied with the gating pulse power, indicating the dependence on gating pulse power of the nonlinear phase shift and therefore the time-resolved single-photon detection efficiency through the Sagnac interferometer. As mentioned above, the output efficiency of the Sagnac interferometer depends on the relative phase difference between the CW and CCW paths and is expressed as $1 - \textrm{cos}{\phi _{\textrm{XPM}}}$. Therefore, to maximize the detection efficiency of the phase-modulated signal photon, the Kerr gating pulse power should be optimized so that the phase shift approaches π. Figure 5 shows the gating pulse power dependence of the coincidence counts averaged in the time range of −1 to 1 ps. The solid line is the fitting curve as a function of the gating pulse average power (pgate) represented by
where y1 is the background count and D is the amplitude of the cosine curve and E is a conversion coefficient that converts the gating pulse average power to radians. We obtained E = 0.103 rad/mW as the nonlinear phase shift per 1 mW gating pulse average power from the fitting result, which implies the high energy efficiency of our system. Note that since the gating pulse has a Gaussian temporal profile, the signal photons were given a non-constant phase shift, depending on its temporal position relative to the gating pulse. Therefore, we can evaluate only the effective phase shift averaged over the temporal profile of the gating pulse, but not exactly specify the amount of phase shift of each signal photon. We found that the detection efficiency is maximized at an average gating power of 30.5 mW, which is one order of magnitude lower than the typical gating power in the sum-frequency generation method [6,7]. This outstanding feature of the optical Kerr gate method allows us to use lower-power ultrashort pulsed laser sources to perform higher resolution time-resolved measurement more efficiently and at a low cost.Fig. 4. Temporal waveforms of the single-photon wave packet measured at the Kerr gating pulse average power of 5.0, 9.8, 15.3, 20.2, and 30.2 mW from lower to upper. The solid lines are the fitting curves using the complementary error functions.
Fig. 5. Kerr gating pulse power dependence of the coincidence count averaged in the time range of −1 to 1 ps. The solid line is the fitting curve using Eq. (2).
Finally, we evaluated the temporal resolution of our ultrafast single-photon detection system. The theoretical envelope of the single-photon wave packet in the time domain is provided by the convolution of a rectangular function, determined by the phase-matching condition, and the temporal profile of the SPDC pump pulse. Thus, we estimated the temporal resolution by fitting the rising edge of the detected temporal waveform to the complementary error function. The standard deviation of the rise time (σerf) contains the information about the temporal resolution of the system (σres) and the Gaussian SPDC pump pulse (σpump), and one can estimate the temporal resolution of the system from the relationship σerf2 = σres2 + σpump2. Figure 6 shows the estimated temporal resolution of the system at the different gating pulse powers. Here, we plotted the system temporal resolution as the full width at half maximum of the detection timing distribution ($2\sqrt {\textrm{ln}2} {\sigma _{\textrm{res}}}$). We repeatedly measured the rising edge of the temporal waveform five times at each power and the averaged values of the resolution with their standard error are plotted in Fig. 6. We find that the system temporal resolution is gradually degraded when the Kerr gating pulse average power is increased above 20 mW, an effect that might be attributable to self-phase modulation (SPM) of the Kerr gating pulse. As the gating pulse propagates in the normal dispersion regime of the PCF, new frequency components that are generated by SPM result in more rapid spreading of the gating pulse in the time domain and hence a reduction in the temporal resolution. Note that the dispersion length of the PCF is approximately 1.7 m at 792 nm, thus in the low-power limit, the gating pulses are not significantly stretched as they pass through a 28 cm length of the PCF. Consequently, ultrafast single-photon detection systems using the optical Kerr effect are likely to have an intrinsic trade-off between the detection efficiency and the temporal resolution. We note that propagating the gating pulse as an optical soliton in the anomalous dispersion regime might allow the SPM of the gating pulse to be compensated. In this situation, optimum detection efficiency could be achieved alongside maximum temporal resolution if the peak power required for a π phase shift was coincident with the pulse energy of a fundamental soliton.
Fig. 6. Kerr gating pulse power dependence of the estimated temporal resolution of the system. The error bars show the standard errors.
4. Conclusion
We have developed an ultrafast single-photon detection system using optical Kerr gating and demonstrated sub-picosecond temporal resolution in the measurement of heralded single-photon wave packets generated via SPDC. Our system has successfully captured the near-rectangle temporal waveform of the SPDC photons. The system temporal resolution of 368 fs was achieved at 31.6 mW of Kerr gating pulse average power where the nonlinear phase shift is close to π, and near-maximum gating efficiency is achieved. This gating pulse power is one order of magnitude lower than that at which upconversion detection was previously demonstrated using the sum frequency generation method [6,7]. Moreover, by slightly reducing the Kerr gating pulse average power to 20.0 mW, we achieved the highest temporal resolution demonstrated in single-photon detection. Therefore, we can conclude that the optical Kerr gate with the PCF can efficiently work even for a single-photon wave packet and be very useful for detecting the single-photon with sub-picosecond temporal resolution.
Our next challenge is the polarization multiplexing of our system presented here, allowing us to expand into a two-photon detection system. A two-photon detector with high temporal resolution is indispensable for precision measurement of the joint temporal intensity distribution of entangled photon wave packets. In addition, such an ultrafast two-photon detection system would enable active control of the time-frequency characteristics of entangled photons in the time domain. For instance, temporal filtering of an entangled photon wave packet might enable us to manipulate its spectral characteristics via a Fourier transform. We anticipate our system will help unravel the quantum nature of time-frequency entangled photons and enable their control in the femtosecond domain.
Funding
Ministry of Education, Culture, Sports, Science and Technology (JPMXS0118069242); Japan Society for the Promotion of Science (JP18H05245); National Natural Science Foundation of China (11704290, 12074299, 91836102); Engineering and Physical Sciences Research Council UK National Quantum Technology Hub in Quantum Computing and Simulation (EP/T001062/1).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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