1. Introduction

Photons exhibit the wave-particle duality, i.e., interference versus anti-bunching, viz. anti-correlation on a beam splitter. In the early days of quantum mechanics, interference experiments replicating Young^’s two-slit one with faint light were considered to provide alternative but decisive evidence for the duality: Spatially-localized photon detection events, reflecting the corpuscular nature of light, add up on a screen at a distance with an alternating intensity contrast due to interference fringes of the photon field, i.e., light wave. Such a notion has been challenged over time with a semi-classical argument invoked thereby [1, 2]. The contention is that the spatial quantization in the detector material rather than the quantization of photon fields could explain the observation equally well. Behind this is the physical limitation that a coherent state cannot be reduced to a single-photon state simply by attenuation. In this context, Grangier et al. were the first to demonstrate the legitimate single-photon interference, in which they used true single photons created in cascade fluorescence from calcium ions [3]. Nowadays, the significance of such single-photon interferences is more relevant than ever in the many and expanding fields of quantum information processing [4] though largely restricted to the space-momentum domain.

Interference fringes in the time domain or transient fringes are also known as heterodyne beats. Albeit a close relativistic analogue, they are not readily comparable from the experimental viewpoint with those in the space-momentum domain, e.g. Young^’s two-slit experiment, which is essentially homodyne detection. Heterodyning, building on the first-order correlation between photon fields with different frequencies, has become a technique of practical value for use in precision metrology [5], biometrics [6], laser line-width analysis [7], and fluid velocimetry [8]. Surprising enough, the heterodyning at a single photon level remains to be challenged.

In analogy to the interference fringes in the space-momentum domain, an intuitive explanation for the heterodyne beats of single photons is such that (i) sporadic photon detection events randomly occur along the time line and (ii) many such events collected over a large ensemble add up to trace out the heterodyne beats developing on a virtual capture frame. To accomplish this, one must defeat the incoherence and the decoherence, or interference fringes are eventually averaged out. Importantly, the relative phase between the signal and the reference photon fields must be maintained at all times, which excludes a local oscillator unlike the existing heterodyning. In addition, one must secure a virtual capture frame onto which dynamic photon detection events are mapped point-by-point to reconstruct the otherwise underdeveloped beat patterns.

Here, we attempt to capture transient interference fringes using a stream of true single photons. Self-heterodyne beats (SHBs) are obtained by letting the nonclassical light field associated with a single photon interfere with itself [9]. The significance of capturing single-photon SHBs is twofold: First, it settles an as yet unqualified premise that the visibility of the first-order interference is independent of the photon statistics unlike the higher-order ones in the time domain as in the space-momentum domain. Second, it marks a great step forward in enabling such applications that can benefit from handling interfering photons in the time domain. Specifically, those requiring ultimately low light level and/or building on the nonclassical photon statistics are the ones. As to the former, single-photon interferometry acquires an almost ideal immunity to DC noise [10] while interferometric biometrics is applicable to highly light-sensitive targets. For the latter, quantum metrology based on the nonclassical nature of single photons, such as phase super-resolution [11] and quantum optical coherence tomography [12], should make most of the single-photon heterodyning technology. More importantly, the potential of the single-photon heterodyning shall be manifest in the frequency-based quantum information processing [13–17], in which a photon superposition state encoded with frequency plays a central role.

2. Photon heterodyning

Figure 1 schematically shows the experimental setup. It is essentially a Hanbury-Brown and Twiss two-photon interferometer (HBT-TPI) [18] with a built-in Mach-Zehnder interferometer (MZI). An acousto-optic modulator (AOM) at the entrance port of the MZI replaces the non-polarizing beam splitter in standard MZI. The first-order diffracted beam out from the AOM consists of an up-converted stream of single photons (That they are indeed “single” photons will be proven later). Therefore the MZI with an AOM can generate a frequency superposition state

 

Fig. 1 (a) Frequency-labeled Mach-Zehnder interferometer (MZI) to observe single-photon self-heterodyne beats (SHBs); AOM, acousto-optic modulator; LP, long-pass filter; HWP, half-wave plate; NPBS, non-polarizing beam splitter; M, mirror; SPCM, single-photon counting module. (b)(c) Hanbury-Brown and Twiss-type two-photon interferometers (HBT-TPIs) for checking the anti-bunching status of single photons: (b) emanating from a type-II BBO (β-barium borate), and (c) undergoing 80-MHz up-conversion through the AOM. (d) HBT-TPI built with an AOM as NPBS for observing cross-frequency anti-bunching.

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The frequency conversion of light in an AOM is based on the emission/absorption of phonons, so a concern may be raised about the path(mode)-phonon entanglement. This is because posterior knowledge of a single phonon emission/absorption can compromise the indistinguishability of the photon streams in the MZI arms [19]. However, AOMs cause no such problems as relevant phonons are in a coherent state. Therefore, the photon state exiting one of the ports of the beam splitter of the MZI reduces to a frequency superposition state (leaving only the relevant modes)

An SHB experiment is arguably a time analogue of the Young’s two-slit counterpart in many respects as already mentioned. However, what makes it unique is that there is no preset tangible screen to map photons onto unlike the space domain. With this in mind, we design an experiment in the following way. First, there is one photon at most per unit spatio-temporal area on average. Next, we prepare the otherwise sparse time-series data I(t) being T-periodic, which facilitates the observation of transient interference fringes. In practice, unless T is known a priori, it must be found heuristically by an exhaustive search or by the Fourier transform: One has to split I(t), fold it back by the modulus T′, and add it up over an ensemble so that t = t′ + kT′ (k ∈ Z) where the relative coordinate t′ is uniquely determined in the interval [0, T′) (Fig. 2(a)). If T′ = T (= 2π/Δω), i.e., the true period, all this reduces to a point-by-point mapping of singlephoton detection events onto the dynamic capture frame of a fixed width T (See inset), tracing out the anticipated semi-normalized SHB profile, $I\left(t\right)=1+\sin \left(\frac{2\pi t}{T}\right)$. After modulo-T′ folding 2n + 1 times, we obtain

 

Fig. 2 (a) Schematic modulo-T′ folding of time-series data, I(t). Single photon detection events are mapped point-by-point onto the virtual capture frame of a fixed width T′. (b) Visibility V_n (θ) (Eq. (6)) for modulo-T′ folding of n-times as a function of θ = 2π T′/T where T is the true period. Only commensurate foldings (θ = 2mπ; m ∈ Z) provide V_n (θ) = 1. (Inset) Cumulative photon counts over a large ensemble reconstruct the otherwise underdeveloped profile of the self-heterodyne beats.

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Heralded single photons were generated by spontaneous parametric down-conversion (SPDC). A 3-mm-thick type-II BBO (β-barium borate) was pumped by a 405-nm cw diode laser. Beam-like degenerate 810-nm SPDC photons emanating from the crystal were input-coupled to single mode fibers [20]. The 10-nm bandwidth of our photon source (≫Δω) compromises the monochromaticity, but it suffices for the SHB observation so long as the path-length difference in the MZI is small. The second-order correlation function of the photon field E at a null delay, $g^{\left(2\right)}\left(0\right)=〈E_{\mathrm{T}}^{*}\left(t\right)E_{\mathrm{R}}^{*}\left(t\right)E_R\left(t\right)E_T\left(t\right)〉/〈E_{\mathrm{T}}^{*}\left(t\right)E_T\left(t\right)〉〈E_{\mathrm{R}}^{*}\left(t\right)E_R\left(t\right)〉$ where the subscripts denote the arms of the HBTI, was measured to assess the anti-bunching statistics, g⁽²⁾(0) = 0. Table 1 summarizes the singles counts and the heralded coincidence counts obtained for the setup in Fig. 1(b). N_T and N_R are the singles counts at the two output ports (T, R) of the HBTI, respectively, while N_G is the singles count in the heralding arm (G). Thus the coincident counts N_GT and N_GR are regarded as the useful singles counts. N_GTR is the triple coincidence count for evaluating g⁽²⁾. Compared are the results under different gating conditions. While N_G and N_T(N_R) are in excess of 10⁶, N_GTR remains small, 9–17, depending on the gating conditions. We find that [21]

Table 1. Singles count and heralded coincidence counts of SPDC photons receiving different anti-bunching tests. The left three columns correspond to normal HBT-TPI of Fig. 1(b). Note that multimode fibers were used for Δt = 7.3 and 18.8 ns while single mode fibers were used for Δt = 27.3 ns for collecting photons. The right two columns show the results for 80-MHz up-conversion through the AOM (Fig. 1 (c)) and cross-frequency anti-bunching (Fig. 1 (d)). Note that g⁽²⁾(0) = N_GTRN_G/N_GTN_GR.

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We further checked the statistics of the single photons undergoing the frequency conversion. An AOM driven by an 80-MHz rectangular pulse train was inserted in front of the HBT-TPI as shown in Fig. 1(c). The singles and the heralded coincidence counts of the SPDC photons passing through the AOM are shown in Table 1. Apparently, the photon anti-bunching is confirmed, g⁽²⁾(0) ≈ 0.04, even after the up-conversion. We also examined cross-frequency anti-bunching, g⁽²⁾(0) ≈ 0.04, which occurs when the detector R replaces the beam stop in Fig. 1(c), which is illustrated in Fig. 1(d). Based on these findings, we assert that the single-photon properties remain unchanged even after the frequency conversion, which opens up the possibility of manipulating photons in the frequency domain without compromising their nonclassical properties.

Figure 3 shows the cumulative triple-coincidence counts taken for 30 s. A pulse generator was used to drive the AOM at 80 MHz, the output of which was frequency-divided at 20 MHz to synchronously trigger the time interval analyzer. This resulted in the time-series data folded back every 50 ns, so they are projected onto a virtual capture frame of as much width. The clear single-photon SHBs, viz. temporal interference fringes, are captured with visibility V=0.91 due to true single photons. The V values improved as the measurement time decreased. As the time evolution of the single photon statistics did not show such degradation with time, this indicates that the instability of the MZI is responsible for a visibility loss for prolonged data accumulation.

 

Fig. 3 Single-photon self-heterodyne beats (SHBs). Coincident counts were accumulated for 30 s without dark count correction. The solid line is the least-squares with visibility V = 0.91. Inset: the Poincaré sphere representation of Eq. (2). The digital heterodyning at a particular position (time) allows one to determine the azimuth angle ϕ_m ∈ [0, 2π).

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The ability to capture a digitized SHB profile is of significance in light of phase measurement: A small displacement d in the upper arm of the MZI in Fig. 1(a) adds a spectral phase (SP) d(ω +Δω)/c (c is the speed of light) so that the interference fringes are uniformly shifted in time. Conversely, the SP is known by measuring the beats. So is d. More precisely, for a pure state like |ψ(φ)〉 of Eq.(2) (the equator of the Poincaré sphere in Fig. 3) or classical mixed states (on the equatorial plane), the photon detection probability p at a designated time t′ in our virtual capture frame is proportional to the expectation value of the projection onto a state with the corresponding SP, ϕ_m. As such, the phase information is available by detecting individual photons. For example, p for a state ρ at the arrowed position in Fig. 3 is proportional to Tr[ρ|ψ(π)〉〈ψ(π)|], and fewer counts there indicate high purity of the output state with ϕ = 0. Such an SHB-based phase measurement will find possible applications in quantum metrology.

3. Conclusions

We investigated the SHB of true single photons from the quantum mechanical point of view. The SHBs with visibility up to 91 % were found to develop as the time-series photon-counting data were added up at a repetition period given by the inverse of the detuning Δω. We found that the frequency conversion with an AOM compromises neither the nonclassical nature of single photons nor a quantum mechanical superposition of the single-photon states. At this point, it is fair to claim that SHBs are caused by the interference of a photon with itself. The digital heterodyning, i.e, SHB at a single photon level, likely boosts the photon manipulation capability.