We report the generation of biphotons, with a temporal full width at the half maximum (FWHM) of 13.4 ± 0.3 µs and a spectral FWHM of 50 ± 1 kHz, via the process of spontaneous four-wave mixing with laser-cooled atoms. The temporal width is the longest, and the spectral linewidth is the narrowest to date. This is also the first biphoton result that obtains a linewidth below 100 kHz, reaching a new milestone. The very long biphoton wave packet has a signal-to-background ratio of 3.4, which violates the Cauchy–Schwarz inequality for classical light by 4.8 folds. Furthermore, we demonstrated a highly tunable-linewidth biphoton source and showed that while the biphoton source’s temporal and spectral width were controllably varied by about 24 folds, its generation rate only changed by less than 15%. A spectral brightness or generation rate per pump power per linewidth of 1.2× 106 pairs/(s mW MHz) was achieved at the temporal width of 13.4 µs. The above results were made possible by the low decoherence rate and high optical depth of the experimental system, as well as a novel scheme of classical fields’ and biphotons’ propagation directions in the experiment. This work has demonstrated a high-efficiency ultranarrow-linewidth biphoton source and has made substantial advancements in quantum technology utilizing heralded single photons.

Sources of biphotons or pairs of time-correlated single photons are widely utilized in quantum information research and applications. While one of the paired photons is detected to trigger, or start, a quantum operation, the other will be used in the operation as a heralded single photon or, if carrying a wave function or quantum state, a heralded qubit. Biphotons are produced mainly via two kinds of schemes: spontaneous parametric down conversion (SPDC) and spontaneous four-wave mixing (SFWM). A SPDC biphoton source is commonly made of a nonlinear crystal placed inside an optical cavity.1–10 A SFWM biphoton source generally consists of a cold atom cloud,11–17 a hot atomic vapor,18–28 or an integrated photonics device with a built-in micro-resonator or waveguide.29–37 Both the SPDC biphoton source and the micro-resonator/waveguide SFWM biphoton source can achieve a high generation rate.1–10,29–37 On the other hand, the atomic SFWM biphoton source, which utilizes the effect of electromagnetically induced transparency (EIT), is able to accomplish a long correlation time between the pair of single photons, i.e., a long temporal width as well as a narrow spectral linewidth of the biphoton wave packet.11–16,18–24

A quantum operation using biphotons of a narrower linewidth or, equivalently, a longer temporal width can achieve better efficiency. For example, a narrower linewidth of input photons can result in a higher storage efficiency (SE) for quantum memory,38–44 a greater success rate for quantum phase gates,45–52 and a larger conversion efficiency for quantum frequency converters,53–56 which utilize resonant or quasi-resonant transition schemes. As another example, while the entanglement swapping is a key process in the quantum repeater protocol for long-distance quantum communication,57–61 the deterministic entanglement swapping with the fidelity of 79% has been demonstrated with ion qubits,62 which operated at a narrow-linewidth transition. In addition, the EIT effect has recently been observed in superconducting qubits or artificial atoms driven by narrow-linewidth microwaves.63–65 One can foresee that narrow-linewidth photonic qubits, which are converted to and from microwave coherences based on the EIT effect,38–44 will be used in a quantum network that links superconducting-qubit-based quantum computers.66–68 

Five groups have reported the sources of biphotons with sub-MHz linewidths. Utilizing nonlinear crystals in the cavity-assisted SPDC process, Rambach et al. showed that their biphotons had a linewidth of 430 kHz,4 and Liu et al. demonstrated a biphoton source with a linewidth of 265 kHz.10 Both sources produced multi-mode biphotons, and the frequency modes of the biphotons spanned several hundreds of MHz. Although SPDC biphoton sources can generate single-mode biphotons, currently all of these single-mode biphotons have linewidths larger than 1 MHz, e.g., Refs. 5–7. Using laser-cooled atoms in the EIT-based SFWM process, Han et al. achieved a biphoton source with a linewidth of 380 kHz,12 and Zhao et al. reported that their biphotons had a linewidth of 430 kHz11 or 250 kHz.13 Using an atomic vapor heated to 38 °C in the EIT-based SFWM process, Hsu et al. generated biphotons with a linewidth of 320 kHz.22,24 Among all the biphoton sources of integrated photonics devices, the narrowest linewidth was 92 MHz.37 There has been no report on the biphoton source with a linewidth below 100 kHz until now. Please see Table I for the list of all biphoton sources with linewidths below 1 MHz.

TABLE I.

Biphoton sources with spectral profiles of the full width at the half maximum (FWHM) below 1 MHz.

ProcessMediumTypeaTemporal FWHM (μs)Spectral FWHM (kHz)gs,as(2)(0)bReferences
SPDC Nonlinear crystal MM 0.33c 430d 5.2 4  
SPDC Nonlinear crystal MM 0.83c 265d 3.9 10  
SFWM Cold atom cloud SM 1.7 430 11 11  
SFWM Cold atom cloud SM 2.1 380 6.8 12  
SFWM Cold atom cloud SM 2.9 250 6.1 13  
SFWM Cold atom cloud SM 13.4 50 4.4 This work 
SFWM Hot atomic vapor SM 0.66 320 6.4 22 and 24  
ProcessMediumTypeaTemporal FWHM (μs)Spectral FWHM (kHz)gs,as(2)(0)bReferences
SPDC Nonlinear crystal MM 0.33c 430d 5.2 4  
SPDC Nonlinear crystal MM 0.83c 265d 3.9 10  
SFWM Cold atom cloud SM 1.7 430 11 11  
SFWM Cold atom cloud SM 2.1 380 6.8 12  
SFWM Cold atom cloud SM 2.9 250 6.1 13  
SFWM Cold atom cloud SM 13.4 50 4.4 This work 
SFWM Hot atomic vapor SM 0.66 320 6.4 22 and 24  
a

MM denotes multimode, and SM denotes single mode.

b

Cross-correlation function between the Stokes and anti-Stokes photons estimated from the SBR of the biphoton wave packet.

c

The FWHM of the envelope formed by all peaks in a temporally comb-like structure.

d

The spectral FWHM of the envelope.

Here, we report a cold-atom SFWM source of biphotons with a tunable temporal width. A novel scheme for the arrangement of classical fields’ and biphotons’ propagation directions was used in the experiment. The scheme not only maintains a nearly phase-mismatch-free condition but also effectively reduces classical light’s leakages to the single-photon counting modules (SPCMs). Consequently, while maintaining non-classicality, the biphotons had a temporal width as long as 13.4 µs or a spectral linewidth as narrow as 50 kHz. A large optical depth (OD) as well as a negligible decoherence rate in the experimental system enabled the propagation delay time of the EIT effect to dominate the temporal profile of the biphoton wave packet. Furthermore, we were able to tune the temporal or, equivalently, spectral width by about 24 folds, and at the same time, the change in the generation rate of the biphoton source was less than 15%. A generation rate per pump power per linewidth of 1.2 × 106 pairs/(s mW MHz) was achieved at the temporal width of 13.4 µs. In quantum operations, this biphoton source can meet any harsh requirements of linewidth.

We carried out the experiment in cold 87Rb atoms, which were produced by a magneto-optical trap (MOT). Details of the cold atoms and the MOT can be found in Ref. 54. Before each measurement of biphotons, we momentarily performed the dark-MOT and optically pumped all population to the single Zeeman state of |5S1/2, F = 1, m = 1⟩, i.e., the state |1⟩ in Fig. 1(a). The processes of the dark-MOT and the optical pumping are very similar to those described in our previous studies of Refs. 69 and 38. At the end of the processes, the optical depth (OD) of the system was 110 ± 5 throughout this work.

FIG. 1.

(a) Relevant energy levels of 87Rb atoms and the transition diagram in the experiment. Ground states |1⟩ and |2⟩ and excited states |3⟩ and |4⟩ are the Zeeman states of |5S1/2, F = 1, m = 1⟩, |5S1/2, F = 2, m = 1⟩, |5P3/2, F = 2, m = 2⟩, and |5P1/2, F = 2, m = 0⟩, respectively. The pump field was blue-detuned with a detuning Δp of 200 MHz. The signal (probe) photons are also called the Stokes (anti-Stokes) photons. The frequency difference between |1⟩ and |2⟩ is 6.8 GHz, corresponding to a wavelength difference of merely 0.014 nm. (b) Scheme of the experimental setup. The angular separation between the pump and coupling propagation directions was ∼1°. The 795-nm signal photons and the 780-nm coupling field (the 780-nm probe photons and the 795-nm pump field) propagated in the same direction. PMF: polarization-maintained optical fiber, M: mirror, PBS: polarizing beam splitter, QWP: quarter-wave plate, BF780 (BF795): 780 nm (795 nm) bandpass filter, PF: polarization filter, SMF: single-mode optical fiber, EF: etalon filter, and SPCM: single-photon counting module.

FIG. 1.

(a) Relevant energy levels of 87Rb atoms and the transition diagram in the experiment. Ground states |1⟩ and |2⟩ and excited states |3⟩ and |4⟩ are the Zeeman states of |5S1/2, F = 1, m = 1⟩, |5S1/2, F = 2, m = 1⟩, |5P3/2, F = 2, m = 2⟩, and |5P1/2, F = 2, m = 0⟩, respectively. The pump field was blue-detuned with a detuning Δp of 200 MHz. The signal (probe) photons are also called the Stokes (anti-Stokes) photons. The frequency difference between |1⟩ and |2⟩ is 6.8 GHz, corresponding to a wavelength difference of merely 0.014 nm. (b) Scheme of the experimental setup. The angular separation between the pump and coupling propagation directions was ∼1°. The 795-nm signal photons and the 780-nm coupling field (the 780-nm probe photons and the 795-nm pump field) propagated in the same direction. PMF: polarization-maintained optical fiber, M: mirror, PBS: polarizing beam splitter, QWP: quarter-wave plate, BF780 (BF795): 780 nm (795 nm) bandpass filter, PF: polarization filter, SMF: single-mode optical fiber, EF: etalon filter, and SPCM: single-photon counting module.

Close modal

The transition diagram of the spontaneous four-wave mixing (SFWM) process is shown in Fig. 1(a). Since all population was optically pumped to |1⟩, only the four Zeeman states specified in the caption were relevant to the experiment. The pump field had the σ− polarization and drove the transition from |1⟩ to |4⟩ with a blue detuning of 200 MHz. The coupling field had the σ+ polarization and drove the transition from |2⟩ to |3⟩ resonantly. In the SFWM process, a pair of signal and probe photons was emitted by the transitions of |4⟩ → |2⟩ and |3⟩ → |1⟩ and had the polarizations of σ and σ+, respectively. The frequencies of the pump field, signal photon, coupling field, and probe photon maintain the four-photon resonance. Since the energy level of |1⟩ is lower than that of |2⟩, the signal and probe photons are also called the Stokes and anti-Stokes photons. The pump field and Stokes photons formed the Raman transition scheme, and their wavelengths were about 795 nm. The coupling field and anti-Stokes photons formed the electromagnetically induced transparency (EIT) transition scheme, and their wavelengths were about 780 nm.

The sketch of the experimental setup is depicted in Fig. 1(b). We merged the pump and coupling fields with a polarizing beam splitter (PBS). The two fields propagated in the directions with a small angle separation of about 1°. After the PBS, a quarter-wave plate made the pump and coupling fields become σ− and σ+ polarized. Their beam profiles completely overlapped with the cigar-shaped atom cloud.54 The pump and coupling fields had the same e−2 full widths of 2.0 mm. In this work, we set the pump power to 56 μW and varied the coupling power from 60 μW to 1.5 mW. The Rabi frequency of the pump field was estimated according to the Gaussian beam width and the transition dipole matrix element of |5S1/2, F = 1, m = 1⟩ → |5P1/2, F = 1, m = 0⟩. The peak intensity of the 56 μW pump beam corresponds to the Rabi frequency of 0.32Γ. The determination of the coupling Rabi frequency is illustrated in the next paragraph.

We experimentally determined the quoted parameters of coupling Rabi frequency (Ωc), optical depth or OD (α), and decoherence rate (γ).54,70 First, the EIT spectrum of a weak probe field was measured, and the separation distance between two transmission minima, i.e., the Autler–Townes splitting, determined Ωc. In the spectrum measurement, the OD was intentionally reduced such that the two minima can be clearly observed. The Rabi frequency of the 60 μW (or 1.5 mW) coupling beam corresponded to Ωc of 0.4Γ (or 2.0Γ). Knowing the value of Ωc, we next measured the delay time of a short probe pulse to determine α used in the experiment. Finally, we used the values of Ωc and α as well as measured the peak transmission of a long probe pulse to determine γ. This input probe pulse was long enough that its output is not affected by the EIT bandwidth. Once Ωc, α, and γ were determined, we further compared the short-pulse data with the theoretical predictions. The good agreement between the experimental data and theoretical predictions demonstrated that the experimentally determined parameters of Ωc, α, and γ are convincing.

The counter-propagation scheme was commonly used in the cold-atom SFWM biphoton sources.11–16 In the scheme, the pump field and signal photons propagate in one direction, and the coupling field and probe photons propagate in the opposite direction. However, there existed a significant phase mismatch in the counter-propagation scheme. Here, we utilized a novel scheme to make the coupling and pump fields and the Stokes and anti-Stokes photons all propagate nearly in the same direction. In the all-copropagation scheme, we intentionally collected Stokes photons along the coupling propagation direction and anti-Stokes photons along the pump propagation direction. This scheme enabled us to effectively reduce the contributions of the coupling and pump fields to the SPCMs. Details of the reduction will be described in the next two paragraphs, and it attenuated the coupling and pump fields by 147 and 157 dB, respectively. In addition, the scheme resulted in a nearly phase-mismatch-free condition, i.e., L|(kpks+kckas)ẑ| 0.23 rad, where L is the medium length, kp, kc, ks, and kas are the wave vectors of the pump and coupling fields and the Stokes and anti-Stokes photons, and ẑ is the unit vector of the Stokes or anti-Stokes propagation direction. This phase mismatch of 0.23 rad is a very manageable downside since it makes a negligible reduction of the biphoton’s generation rate.

Since the anti-Stokes (Stokes) photons and the pump (coupling) field propagated nearly in the same direction, we installed a polarization filter and a 780-nm bandpass filter (two 795-nm bandpass filters) to prevent the pump (coupling) field from entering the single-mode optical fiber (SMF), collecting the anti-Stokes (Stokes) photons. Each of the polarization filters is the combination of a quarter-wave plate, a half-wave plate, and a polarizing beamsplitter, which reduced the pump (coupling) power by 49 dB (49 dB). The bandpass filter(s) added an additional attenuation of 38 dB (40 dB). After each SMF, we further used two etalons in series (an etalon) to decrease the pump (coupling) leakage by about 70 dB (58 dB). The total extinction ratio of 157 dB (147 dB) reduced the pump (coupling) power of 56-μW (1.5-mW) to merely zero (12) photons/s.

Due to the anti-Stokes (Stokes) photons and the coupling (pump) field having very similar wavelengths and the same polarization, the above-mentioned polarization and bandpass filters cannot block the leakage of the coupling (pump) field into the anti-Stokes (Stokes) SMF. Fortunately, the angular separation of 1° between the anti-Stokes and coupling (between the Stokes and pump) propagation directions had already reduced the leakage significantly. After the SMFs, the two etalons in series (the etalon) further decreased the coupling (pump) leakage by about 60 dB (60 dB). The total extinction ratio of 130 dB (118 dB), including the attenuations due to the angle separation and etalon(s), reduced the coupling (pump) power of 1.5 mW (56 μW) to about 600 (350) photons/s. It should be noted that the anti-Stokes (Stokes) SPCM (Excelitas SPCM-AQRH-13-FC) has a quantum efficiency of 0.55 (0.56) and a dark count rate of 220 (140) counts/s.

The two etalons in series for the anti-Stokes photons (the etalon for the Stokes photons) had a net linewidth of 46 MHz (80 MHz) and a peak transmission of around 18% (30%). Taking into account the attenuation due to optics, SMF’s coupling ratio, two etalons’ total transmission, and SPCM’s quantum efficiency, the overall collection efficiency of the anti-Stokes (Stokes) photons was 7.7% ± 2% (13% ± 1%). Inside the atomic cloud, the Stokes photon propagated at light speed in vacuum, and the anti-Stokes photon was slow light due to the EIT effect. The Stokes SPCM detected a photon first and triggered a time tagger (Fast ComTech MCS6A5T8). After some delay time, the anti-Stokes SPCM detected another photon, which was recorded as a coincidence count by the time tagger.

The time correlation function between the anti-Stokes and Stokes photons, i.e., the biphoton wave packet, is given by71 

G(2)(τ)=dδ2πeiδτkasksL2χ(δ)sincksL4ξ(δ)ei(ksL/4)ξ(δ)2,
(1)

where τ is the delay time of detecting an anti-Stokes photon upon a Stokes photon’s trigger, δ is the two-photon detuning between the Stokes photon and pump field (or −δ is that between the anti-Stokes photon and coupling field), kas and ks are the wave vectors of the two photons, L is the medium length, χ(δ) is the cross-susceptibility of the anti-Stokes photon induced by the Stokes photon, and ξ(δ) is the self-susceptibility of the anti-Stokes photon. The formulas for the cross-susceptibility and self-susceptibility are shown in the following:

kasksL2χ(δ)=αΓ4ΩpΔp+iΓ/2ΩcΩc24(δ+iγ)(δ+iΓ/2),
(2)
ksL4ξ(δ)=αΓ2δ+iγΩc24(δ+iγ)(δ+iΓ/2),
(3)

where α represents the OD of the entire atoms, Ωp and Ωc are the Rabi frequencies of the pump and coupling fields, Γ = 2π × 6 MHz is the spontaneous decay rate of the excited state, Δp is the detuning of the pump field, and γ is the dephasing rate of the ground-state coherence, i.e., the decoherence rate. Since the difference between the spontaneous decay rates of the excited states |3⟩ and |4⟩ is merely about 5% in our case, we neglect the difference and set the two rates to Γ.

We systematically measured the biphoton wave packets or two-photon time correlation functions by varying the coupling powers. The representative data taken at the highest and lowest coupling powers of 1.5 mW and 63 μW are shown by the circles in Figs. 2(a) and 2(b), respectively. Each inset is the result of 324 000 measurements, and each measurement had a time window of 240 µs. Since we switched off (and on) the MOT during (and after) each measurement, the duty cycle of the biphoton generation is 0.8%. The width of the time bin is 6.4 ns in (a) or 51.2 ns in (b). Considering the overall collection efficiencies of the Stokes and anti-Stokes photons, the generation rates of this biphoton source at the coupling powers of 1.5 mW and 63 μW were 3460 ± 140 and 3340 ± 40 pairs/s, respectively.

FIG. 2.

Representative biphoton wave packets. Circles connected with black lines are the data of two-photon coincidence counts. Red lines are the theoretical predictions of G(2)(τ) in Eq. (1). Green lines are the results of the four-point moving average on the data points. In each inset, squares represent the discrete Fourier transform of the data, and the blue line is the Fourier transform of the red line. (a) The coupling power was 1.5 mW, and the pump power and detuning were 56 μW and +200 MHz. Red line corresponds to (α, Ωc, γ) = (115, 2.1Γ, 4.0 × 10−3Γ) and (Ωp, Δp) = (0.32Γ, 33.3Γ). The biphoton wave packet had a temporal full width at the half maximum (FWHM) of 0.57 µs, a spectral FWHM of 1.20 MHz, and a signal-to-background ratio (SBR) of 40. (b) The coupling power was reduced to 63 μW, while the pump power and detuning were the same. The red line corresponds to (α, Ωc, γ) = (110, 0.42Γ, 3 × 10−4Γ) and (Ωp, Δp) being the same. The biphoton wave packet had a temporal FWHM of 13.4 µs, a spectral FWHM of 50 kHz, and an SBR of 3.4.

FIG. 2.

Representative biphoton wave packets. Circles connected with black lines are the data of two-photon coincidence counts. Red lines are the theoretical predictions of G(2)(τ) in Eq. (1). Green lines are the results of the four-point moving average on the data points. In each inset, squares represent the discrete Fourier transform of the data, and the blue line is the Fourier transform of the red line. (a) The coupling power was 1.5 mW, and the pump power and detuning were 56 μW and +200 MHz. Red line corresponds to (α, Ωc, γ) = (115, 2.1Γ, 4.0 × 10−3Γ) and (Ωp, Δp) = (0.32Γ, 33.3Γ). The biphoton wave packet had a temporal full width at the half maximum (FWHM) of 0.57 µs, a spectral FWHM of 1.20 MHz, and a signal-to-background ratio (SBR) of 40. (b) The coupling power was reduced to 63 μW, while the pump power and detuning were the same. The red line corresponds to (α, Ωc, γ) = (110, 0.42Γ, 3 × 10−4Γ) and (Ωp, Δp) being the same. The biphoton wave packet had a temporal FWHM of 13.4 µs, a spectral FWHM of 50 kHz, and an SBR of 3.4.

Close modal

The red lines in Fig. 2 are calculated from the two-photon time correlation function, i.e., G(2)(τ) in Eq. (1). During the biphoton-generation time window of 240 µs, the OD of the system gradually decayed to about 90% of its initial value. We take the OD’s decay into account, and all the quoted values of the OD are the average value within this 240 µs. Since the OD varied merely 10% and the variation was nearly linear during the measurement, we just use its average value in the evaluation of G(2)(τ). In the calculation, α and γ are set to the values, which are determined by the classical-light data of slow light, and the values of Ωp and Δp are also experimentally measured or determined. The value of Ωc is adjustable to get the best match of the biphoton’s temporal width. It should be noted that α, Ωc, and γ affect the profile of the biphoton wave packet, but Ωp and Δp do not. In Figs. 2(a) and 2(b), Ωc = 2.1Γ and 0.42Γ, respectively, while their values determined by the Autler–Townes splitting in the EIT spectrum are Ωc = 2.0Γ and 0.40Γ. The parameters of α, γ, Ωp, and Δp (or Ωc) used in G(2)(τ) were measured (or verified) by the methods without using th e biphoton data. Hence, the red lines are regarded as theoretical predictions.

The coupling field also drove the far-detuned transition of |5S1/2, F = 2, m = 1⟩ → |5P3/2, F = 3, m = 2⟩, and the transition caused the photon switching effect.72,73 Consequently, the decoherence rate (γ) was approximately linear to the coupling Rabi frequency square (Ωc2),38,42,74 i.e.,

γ=γ0+aΩc2Γ2,
(4)

where γ0 is the intrinsic decoherence rate of the experimental system, and a is the proportionality due to the photon switching effect. The value of γ0 was 2 × 10−4Γ or 2π × 1.2 kHz. The uncertainty plus day-to-day fluctuation of the decoherence rate is the larger one of ±1 × 10−4Γ and ±20%. Please see the  Appendix for the determination method of the decoherence rate.

The green lines in Fig. 2 are the results of the four-point moving average, which reduces the fluctuation of the data points. Since the temporal full width at the half maximum (FWHM) of 0.57 (or 13.4) μs as determined by the red line is much longer than the time-bin width of 6.4 (or 51.2) ns, the moving average affects the profile of the biphoton wave packet very little. The consistency between the experimental data (the green lines) and the theoretical predictions is satisfactory.

We performed the discrete Fourier transform (DFT) on the biphoton wave packets and the Fourier transform (FT) on the theoretical predictions. The baseline or background count of the data was removed before the DFT or FT. The DFT procedure is illustrated as follows: First, the baseline was subtracted from the temporal data. Next, we took the absolute values to ensure the original data points below the baseline were positive. Then, the square root was applied, and the negative sign was added only to the data points originally below the baseline. Please note that the fluctuation of the baseline still remained in the data. Finally, we performed the DFT to obtain the data points in the frequency domain, and they were squared to represent the spectrum. The representative spectra are shown in the insets of Fig. 2.

The temporal width or spectral linewidth of the biphoton wave packet is mainly determined by the propagation delay time in the experimental system of a large OD and a negligible decoherence rate. This is exactly demonstrated by Fig. 3, showing that the FWHM of the biphoton’s temporal profile and the corresponding spectral FWHM as functions of Ωc2. Under a large OD, the propagation delay time, τd(=αΓ/Ωc2), is much larger than the inverse of the EIT bandwidth, τb (=1/ΔωEIT or αΓ/Ωc2). Under a negligible decoherence rate, the coherence time, τc (=1/γ), is far greater than τd. Using the condition of τcτdτb, one can show that the biphoton’s spectral FWHM, Δω/2π, is given by12,71

Δω2π0.88τd=0.88ΓαΩcΓ2.
(5)

In Fig. 3, the blue circles are the experimental data of Δω/2π vs (Ωc/Γ)2, and the blue line is the best fit of a linear function with zero interception. The initial OD during the biphoton-generation time window, i.e., α0, was 110 ± 5. Using α = 110 ± 5 in Eq. (5), we obtain Δω/2π = 300 ± 10 kHz ×(Ωc/Γ)2. The slope of the best fit in Fig. 3 is 280 kHz. Therefore, the experimental result is very close to the theoretical expectation of a nearly ideal biphoton source with a negligible decoherence rate.

FIG. 3.

Temporal FWHM (red circles) and spectral FWHM (blue circles) of the biphoton wave packet as functions of Ωc2. In the measurement, α = 110 ± 5. The blue line is the best fit of a linear function with zero interception, which determines Δω/2π (the spectral FWHM) = 280 kHz ×(Ωc/Γ)2. In theory, i.e., Eq. (5), it is expected that Δω/2π = 300 kHz ×(Ωc/Γ)2 at α = 110.

FIG. 3.

Temporal FWHM (red circles) and spectral FWHM (blue circles) of the biphoton wave packet as functions of Ωc2. In the measurement, α = 110 ± 5. The blue line is the best fit of a linear function with zero interception, which determines Δω/2π (the spectral FWHM) = 280 kHz ×(Ωc/Γ)2. In theory, i.e., Eq. (5), it is expected that Δω/2π = 300 kHz ×(Ωc/Γ)2 at α = 110.

Close modal

A cold-atom SFWM biphoton source has a large tunability of linewidth as demonstrated by Fig. 3. Its continuously tuning range of frequency is around the order of magnitude of the natural linewidth of the excited state driven by the coupling field, which is about 6 MHz in the present work. This is the drawback of a cold-atom biphoton source. Nevertheless, the frequency tuning range can be significantly increased by the Doppler broadening effect in a hot-atom SFWM biphoton source. In Refs. 22 and 24, Rb atomic vapor was heated to temperatures between 38 and 65 °C and used to produce biphotons of a linewidth as narrow as 290 kHz. Since the same SFWM generation process was utilized in those works and also in this work, a hot-atom biphoton source can have a linewidth below 100 kHz in the near future, while its frequency tuning range will be kept around 600 MHz with Rb atoms or larger with other species of lighter atoms.

The longest temporal or narrowest spectral FWHM in Fig. 3 is 13.4 ± 0.3 µs or 50 ± 1 kHz as demonstrated by Fig. 2(b). This is the best record to date and also the first result of the biphoton linewidth below 100 kHz. The very long biphoton had a signal-to-background ratio (SBR) of 3.4, showing that the cross-correlation function between the Stokes and anti-Stokes photons, gs,as(2)(0), is 4.4.13 Both the auto-correlation function of the Stokes photons, gs,s(2)(0), and that of the anti-Stokes photons, gas,as(2)(0), are approximately equal to 2.24,75 With gs,as(2)(0)=4.4 and gs,s(2)(0)=gas,as(2)(0)=2, the Cauchy–Schwarz inequality for classical light is violated by 4.8 folds, which clearly demonstrates the biphoton’s non-classicality. Furthermore, by increasing the coupling power, we were able to tune the temporal or spectral width to 0.57 ± 0.02 µs or 1200 ± 40 kHz as demonstrated by Fig. 2(a). While the temporal or spectral width of the biphoton was shortened or enlarged by about 24 folds, its SBR was significantly enhanced to 40.

It is the natural consequence in the biphoton generation that a broader (narrower) biphoton linewidth results in a larger (smaller) value of gs,as(2)(0). This is demonstrated by Figs. 2(a) and 2(b), i.e., gs,as(2)(0) = 41 at a linewidth of 1.2 MHz and gs,as(2)(0) = 4.4 at a linewidth of 50 kHz. As presented in Table I, narrow-linewidth biphoton sources typically have low values of gs,as(2)(0). Since the accidental photons are usually incompatible with quantum operations, a larger value of gs,as(2) is more desirable. On the other hand, due to the incompatibility of accidental photons, one might be able to utilize quantum memory to enhance gs,as(2) of narrow-linewidth biphotons and, at the same time, to store heralded single photons for further usages.

We compared the generation rates of the biphoton source operating at different coupling powers or equivalently producing various temporal widths. Under the condition of a large OD and a negligible decoherence rate, i.e., Ωc22ΔωEITΓ and Ωc22γΓ, the generation rate is approximately proportional to

dτG(2)(τ)αΓ2πΩp24Δp2+Γ2expαγΓΩc2.
(6)

As long as the value of αγΓ/Ωc2 is small, varying the coupling power or tuning the biphoton temporal width changes the generation rate a little. This is exactly the case in our experiment. Although we changed the biphoton temporal width by about 24 folds, the generation rate varied in the range of 3300–3800 pairs/s or fluctuated merely about ±7% as shown by Fig. 4. That is we generated either a longer biphoton pulse with a smaller amplitude or a shorter biphoton pulse with a larger amplitude, while the area below the pulse or the pulse energy was approximately the same.76 Only a negligible decoherence rate in the EIT system can make it happen.

FIG. 4.

Brightness, i.e., the generation rate per pump power (red circles), and spectral brightness, i.e., the brightness per linewidth (blue circles), as functions of Ωc2. The pump power was kept at 56 μW, and the generation rates of different coupling Rabi frequencies varied in the range of 3300–3800 pairs/s. During the entire measurement, the value of γ was (340)×104Γ corresponding to Ωc=(0.42.0)Γ, and that of α was between 105 and 115. Considering the uncertainty or fluctuation in γ of ±20% and that in α of ±5, the gray area represents the predictions based on Eq. (6). Blue line is the best fit of the function y = b/x, where b is the fitting parameter.

FIG. 4.

Brightness, i.e., the generation rate per pump power (red circles), and spectral brightness, i.e., the brightness per linewidth (blue circles), as functions of Ωc2. The pump power was kept at 56 μW, and the generation rates of different coupling Rabi frequencies varied in the range of 3300–3800 pairs/s. During the entire measurement, the value of γ was (340)×104Γ corresponding to Ωc=(0.42.0)Γ, and that of α was between 105 and 115. Considering the uncertainty or fluctuation in γ of ±20% and that in α of ±5, the gray area represents the predictions based on Eq. (6). Blue line is the best fit of the function y = b/x, where b is the fitting parameter.

Close modal

The brightness is defined as the generation rate per pump power, and the spectral brightness is defined as the brightness per linewidth. Figure 4 shows the brightness and the spectral brightness as functions of Ωc2. During the measurement, the pump power was kept the same, and thus, the behavior of the brightness is essentially the same as that of the generation rate. We considered the fluctuations of the OD and the decoherence rate and utilized the predictions of Eq. (6) to plot the gray area. The consistency between the data and the predictions is satisfactory. Furthermore, since the brightness varied a little against Ωc2 and the biphoton spectral width is linearly proportional to Ωc2, the spectral brightness or generation rate per pump power per spectral linewidth approximately depends on 1/Ωc2. Such dependence is clearly seen in Fig. 4. In this work, the highest spectral brightness is 1.2 × 106 pairs/(s mW MHz).

The generation rate of the biphoton source can be enhanced by increasing the pump power. In Ref. 24, the authors show that the generation rate of a SFWM hot-atom biphoton source is linearly proportional to the pump power and is further enhanced by the optical depth. This SFWM biphoton source had a generation rate of 3.7× 105 pairs/s, resulting in a generation rate per linewidth of 3.8 × 105 (pairs/s)/MHz, which is the highest record to date among all kinds of biphoton sources including SPDC and micro-cavity ones. To our knowledge, a SFWM biphoton source can achieve a generation rate higher than 106 pairs/s, but its generation rate per linewidth cannot exceed 106 pairs/s/MHz due to a larger generation rate per linewidth, resulting in a smaller SBR or equivalently gs,as(2).24 

To verify the biphoton linewidth obtained in Fig. 2(b), we further measured the FWM spectrum with classical light under a very similar experimental condition. In the presence of the coupling and pump fields, we applied the signal field at the input and measured the FWM-generated probe field at the output. Figure 5 is the FWM spectrum, showing the normalized power of the generated probe field as a function of the frequency of the signal field. The red line in the figure is the theoretical prediction calculated from the biphoton spectrum of |χ(δ) sinc[(ksL/4)ξ(δ)] exp[i(ksL/4)ξ(δ)]|2 used in Eq. (1), where δ represents the signal detuning or frequency. In the calculation, we used the same values of α, Ωc, and γ as those shown in the caption of Fig. 2(b). It should be noted that the spectral profile is very insensitive to χ(δ). The experimental data are in good agreement with the theoretical prediction. The FWHM of the measured spectrum is 47 kHz, which confirms the biphoton linewidth of 50 kHz shown in the inset of Fig. 2(b).

FIG. 5.

The FWM spectrum measured with classical light at the experimental condition is very similar to that in Fig. 2(b). Circles are the experimental data. The black line is the B-spline curve connecting the data points and has a FWHM of 47 kHz. The red line is the theoretical prediction calculated with the same parameters shown in the caption of Fig. 2(b).

FIG. 5.

The FWM spectrum measured with classical light at the experimental condition is very similar to that in Fig. 2(b). Circles are the experimental data. The black line is the B-spline curve connecting the data points and has a FWHM of 47 kHz. The red line is the theoretical prediction calculated with the same parameters shown in the caption of Fig. 2(b).

Close modal

The biphoton linewidth here means the linewidth, i.e., the reciprocal of the coherence time, of the two-photon wave function, and it does not mean the linewidth or frequency fluctuation of the probe photons. The coherence time of the two-photon wave function can be determined by the Franson interferometer.77–80 In this work, the coherence time of the biphoton generation process was much longer than the biphoton temporal width, which enabled us to perform the Fourier transform of the biphoton wave packet to determine the biphoton linewidth.

In the SFWM process, the coupling field and probe photons form the two-photon transition scheme of the EIT effect. The frequency difference between the probe photons and the coupling field in the two-photon transition is called the two-photon frequency. During the biphoton generation, the EIT effect makes the two-photon frequency locked to the frequency difference between the two ground states.76,81,82 The uncertainty or fluctuation of the two-photon frequency manifests the biphoton linewidth. In this work, the coupling field frequency fluctuated around 1 MHz as indicated by the frequency stabilization scheme, and the probe photon frequency did likewise. On the other hand, the biphoton linewidth ensured the two-photon frequency of the probe photons and coupling field to fluctuate within (or have a linewidth of) 50 kHz as shown by the inset of Fig. 2(b) and also by Fig. 5.

The linewidth of the two-photon frequency, i.e., the biphoton linewidth, is key to the applications, such as photonic quantum memory and quantum frequency converter. We use the following example to explain why the uncertainty or fluctuation of the two-photon frequency, but not the linewidth or frequency fluctuation of the heralded single photons, is an important parameter in EIT-based photonic quantum memories. In a SFWM biphoton source like the one in this work, the frequencies of the coupling field and generated probe photons fluctuate around 1 MHz, and their difference or two-photon frequency maintains constant with uncertainty or fluctuation of 50 kHz, which is given by the biphoton linewidth. One can utilize the coupling laser of the biphoton source to stabilize the frequency of the coupling laser of an EIT-based quantum memory (called the QM coupling) via the injection lock or other frequency-lock schemes. Consequently, the heralded probe photons generated from the biphoton source and the QM coupling laser maintain a fixed frequency difference, i.e., their two-photon frequency is kept around the two-photon resonance of the EIT-based memory, with an uncertainty or fluctuation of 50 kHz. This 50 kHz of the biphoton linewidth, but not the probe frequency fluctuation’s 1 MHz, affects the storage efficiency of the memory.38,42 That is to say, photonic quantum memories based on the EIT transition, Raman transition, or similar two-photon transitions concern the spectrum of the two-photon frequency between the stored photons and the control field, i.e., the QM coupling field in the EIT case.

The cold atoms used in the experiment had a temperature of 300 ± 50 μK and were set by the laser-cooling mechanism of the magneto-optical trap. The short-term and day-to-day fluctuations of the atom temperature did not degrade the performance of the biphoton source. Suppose the optical depth of the system is fixed. As long as the Doppler width of the two-photon frequency caused by the atomic thermal motion is negligible as compared with the biphoton linewidth,70 the performance of the biphoton source will not be degraded by the atom temperature. Considering the angular separation between the coupling field and probe photons (also the pump field and signal photons) propagation directions of about 1° and a biphoton linewidth of 50 kHz, the atom temperature can, in theory, be as high as a few Kelvins.83,84

In conclusion, we were able to produce the narrow-linewidth biphoton with a temporal FWHM of 13.4 µs, corresponding to a spectral FWHM of 50 kHz, owing to a large optical depth and a negligible decoherence rate in the experimental system. As for a wide range of the Rabi frequency or power of the coupling field used in this work, the large optical depth maintained the criterion that the propagation delay time is far greater than the reciprocal of the EIT bandwidth, and the negligible decoherence rate preserved the criterion that the coherence time is much longer than the propagation delay time. Consequently, not only the temporal profile of the biphoton wave packet was predominately determined by the propagation delay time but also we tuned the temporal or spectral width by about 24 folds and, at the same time, kept the generation rate approximately the same. This work has demonstrated a high-efficiency ultranarrow-linewidth biphoton source and achieved the milestone of an ultralong biphoton wave packet.

This work was supported by the National Science and Technology Council of Taiwan under Grant Nos. 109-2639-M-007-002-ASP and 110-2639-M-007-001-ASP.

The authors have no conflicts to disclose.

Yu-Sheng Wang: Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Validation (lead); Writing – review & editing (supporting). Kai-Bo Li: Data curation (supporting); Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Validation (supporting); Writing – review & editing (supporting). Chao-Feng Chang: Data curation (supporting); Investigation (supporting); Methodology (supporting); Writing – review & editing (supporting). Tan-Wen Lin: Data curation (supporting); Investigation (supporting); Methodology (supporting); Writing – review & editing (supporting). Jian-Qing Li: Investigation (supporting); Methodology (supporting). Shih-Si Hsiao: Investigation (supporting); Methodology (supporting); Validation (supporting); Writing – review & editing (supporting). Jia-Mou Chen: Methodology (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Yi-Hua Lai: Methodology (supporting); Writing – review & editing (supporting). Ying-Cheng Chen: Conceptualization (supporting); Funding acquisition (supporting); Validation (supporting); Writing – review & editing (supporting). Yong-Fan Chen: Conceptualization (supporting); Funding acquisition (supporting); Validation (supporting); Writing – review & editing (supporting). Chih-Sung Chuu: Conceptualization (supporting); Funding acquisition (supporting); Validation (supporting); Writing – review & editing (supporting). Ite A. Yu: Conceptualization (lead); Funding acquisition (lead); Project administration (lead); Supervision (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The determination method of the decoherence rate in the EIT system is described as follows: We determined the decoherence rate (γ) from the measurements of slow light with a classical-light probe field. According to the EIT theory,85,86 the output transmission and propagation delay time of slow light are given by exp(2αγ/Ωc2) and αΓ/Ωc2. We first determined the value of Ωc directly from the Autler–Townes splitting in the EIT spectrum. Then, at a given Ωc, the propagation delay time of a Gaussian probe pulse gave the value of α. Using the values of Ωc and α, we finally measured the probe output transmission to determine the value of γ. Please refer to Figs. 3(a)–3(c) in Ref. 70 for the examples of data in the above-mentioned measurements. The value of γ depends on Ωc2 as shown in Eq. (4), which is the consequence of the photon switching effect;38,42 for examples, in this work, γ = 3 × 10−4Γ (or 2π × 1.8 kHz) at Ωc = 0.42Γ, and γ = 4 × 10−3Γ (or 2π × 24 kHz) at Ωc = 2.1Γ. We extrapolated the data points of γ vs Ωc2 to obtain the intrinsic decoherence rate (γ0), i.e., γ at Ωc → 0.

In the EIT system, as long as the probe frequency is locked to the coupling frequency and their difference maintains the two-photon resonance, the coupling frequency fluctuation or linewidth makes a negligible contribution to the decoherence rate; for example, the EIT-based coherent optical memory with high storage efficiency (SE) has been experimentally demonstrated in Refs. 38 and 42. The SE is sensitive to the decoherence rat,e and a larger decoherence rate results in a lower SE, as demonstrated by the two blue lines in Fig. 2(b) of Ref. 38. In the works of the two references, the injection-lock scheme was utilized to lock the probe frequency to the coupling frequency with a stable frequency offset. The frequency offset was set to the frequency difference between the two ground states, which is the condition of the two-photon resonance. Although the fluctuation of the coupling (also the probe due to the frequency lock) frequency was around 1 MHz in each of the works, the probe pulse well fitted the memory, which had a decoherence rate of less than 2π × 4 kHz, and the frequency fluctuation did not degrade the SE. The example demonstrates that the coupling (also the probe) frequency fluctuation or linewidth can play a negligible role in the decoherence rate of the EIT system.

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