1. Introduction

Two-photon interference, in which indistinguishable photons are incident on the two input ports of a balanced beam splitter and always emerge from the same output port, is a canonical behavior in the quantum mechanical treatment of light. First demonstrated by Hong, Ou and Mandel [1], this nonclassical effect reflects the bosonic nature of light and is at the heart of many quantum information protocols including boson sampling [2–4], linear optical quantum computing [5], and quantum networks that utilize quantum repeaters [6]. Suppression of coincidence events between both outputs of the beam splitter, quantified by the visibility in Hong-Ou-Mandel interference (HOMI), depends critically on the photons being indistinguishable, that is, occupying the same state, and not being in a mixture of states, that is, occupying a pure state [7,8]. A means to determine the state of single-photon sources is thus essential to diagnosing and engineering quantum light sources and matching single-photon emitters for optimal interference.

When only a single photon is excited in the quantized electromagnetic field, then the only additional information needed to fully describe the state of the field is the mode occupied by the single photon. In this way, one can define the wave function of a single photon as being given by the mode that it occupies [9–12]. The mode functions are solutions to the Maxwell equations and describe the polarization, transverse-spatial, and spectral-temporal degrees of freedom for a beam of light [9,11]. Here we focus on the spectral-temporal degree of freedom, which describes the pulse mode occupied by a single photon and has gained significant interest recently for quantum information applications [13,14].

Characterization of single-photon frequency-time structure has been demonstrated by interference with a known reference field [15–17] and more recently in a self-referencing interferometer [18,19]. The self-referencing method has shown reconstruction of single-mode, pure-state photon sources, but has yet to achieve characterization of partially coherent (partially mixed) single-photon states, which would require scanning the frequency shift in the spectral-interferometer used in these demonstrations. The methods based upon interference with known reference pulses, which utilize either strong classical fields [15,16] or weak single-photon-level fields [17] to measure overlap between the unknown single-photon source and the reference field mode, have shown reconstruction of mixed-state single-photon sources. However, these approaches require the reference field to be scanned across multiple modes to extract the unknown single-photon state. This inevitably introduces experimental challenges related to the accuracy of the reference mode, stability of the experimental setup and time required for data acquisition.

Here we propose and demonstrate a method to determine the pulse-mode state of a single-photon source by interference with a single reference photon that eliminates the need to scan. This is accomplished by interfering the photons on a balanced beam splitter and performing frequency-resolved coincidence measurements at the output, or frequency-resolved HOMI (FR-HOMI). The resultant FR-HOMI coincidence events as a function of measured frequencies give a joint-spectral interference pattern that depends upon the time delay between the two input photon pulses and their mode overlap. Previous work has demonstrated the feasibility of FR-HOMI between independent sources [20] and its sensitivity to time-frequency mode distinguishability [21]. Our work builds on this, using Fourier analysis of FR-HOMI data to directly determine the state of the unknown source, which may be pure or partially mixed. The only requirement on the reference photon is that it must occupy a single pulse mode that has spectral support spanning that of the unknown single-photon source. By performing the joint multi-mode frequency-resolved coincidence detection, this method utilizes only one experimental configuration that does not require scanning of the reference pulse, making it fast and resilient to experimental fluctuations.

2. Single-photon state and frequency-resolved HOMI

The state of the electromagnetic field when a single photon occupies the pulse mode represented by the complex-valued function $\tilde {\psi }(\omega )$ can be expressed by $|1\rangle_{\psi } = \hat {a}^\dagger _\psi |\mbox {vac}\rangle$, where $|\mbox {vac}\rangle$ is the vacuum state of the field and

To determine the density matrix elements in the frequency basis, $\rho (\omega ,\omega ')$, we employ Hong-Ou-Mandel interference (HOMI) between the unknown single-photon source described by Eq. (2) and a known reference single-photon source occupying a single mode $\tilde {\chi }(\omega )$. When an unknown single-mode single-photon source is interfered with a calibrated single-photon source on a balanced beam splitter, no coincidence detection events should be registered between the outputs for the case in which the reference photon occupies the same mode as the unknown source - this is the hallmark of HOMI. It has been experimentally demonstrated that by recording the coincidence count rates while scanning the mode structure of the reference pulse, the density matrix elements of the unknown source can be determined [17]. In this case the reference was derived from a well-characterized, highly-attenuated laser pulse. Although this technique can characterize the single-photon density matrix elements, it requires a two-dimensional scan of the reference signal, leads to long acquisition times and requires stable experimental conditions over the whole acquisition. Here we employ spectrally-resolved detection at the output of the HOMI (as depicted in Fig. 1), which significantly reduces the experimental complexity and data acquisition time required to determine the spectral density matrix elements.

 

Fig. 1. General depiction of spectrally-resolved HOMI wherein the spectral coincidences between an unknown single photon described by a density matrix $\hat {\rho }$ and a reference photon in a pure state $|1\rangle_\chi$ are retrieved in the form of a two-dimension interferogram $\tilde {S}(\omega _c,\omega _d)$.

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To see how the FR-HOMI coincidence measurements between the reference and unknown single photons enables the extraction of the single-photon density matrix, consider a train of single-photon reference pulses at input $b$ of the balanced beam splitter occupying a single mode, $\tilde {\chi }(\omega )$, as depicted in Fig. 1. The input-output beamsplitter transformation reads:

 

Fig. 2. (a) Experimental scheme for pulse characterisation by spectrally-resolved Hong-Ou-Mandel interference using the idler photon as a reference. Note that the final beam splitter and the spectrometers are fibre-coupled. TDC: Time-to-digital converter. PD: photodiode. DM: dichroic mirror. DG: diffraction grating. (P)BS: (Polarising) beam splitter. (b) Example 2-dimensional Fourier transform of two-photon spectral interferogram $S(\omega _c,\omega _d)$ (inset), giving the pseudo-temporal distribution $\bar {S}(T_c,T_d)$ in which the three peaks corresponding to the one unmodulated and two oscillatory terms are evident. The red contour represent the supergaussian filters at 10% and the following dashed contour represent respectively 50 and 95% of their maximum value of 1.

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Where it cannot be assumed that the test pulse is in a pure state, this method generalizes straightforwardly to the mixed-state formalism and allows full extraction of the spectral density matrix $\rho (\omega _1,\omega _2)$ without further measurements. To show this, we represent the state of the single photon by the following density matrix:

Note that in any cases, since the final stage of the extraction is division by the spectral amplitude of the reference, it is necessary that the reference pulse completely covers the spectral support of the test signal. Spectral gaps in the reference photon will lead to the inability to characterise the phase of the signal at those frequencies.

3. Experimental scheme

Our experimental scheme is depicted in Fig. 2(a). A commercial Ti:Sa femtosecond oscillator (Spectra-Physics Tsunami) delivers pulses with a bandwidth-limited duration of $100$ fs full width at half maximum (FWHM) at $12.5$ ns intervals, corresponding to a spectrum centered at $830$ nm and a bandwidth of $10$ nm FWHM with a comb structure with a spacing of $80$ MHz. A beam pickoff at the output of the laser directs some light to a fast photodiode, which generates an electronic time-reference signal or ‘clock’ that is used throughout the experiment. Spectral detection is achieved with time-resolved single photon spectrometers (TRSPS) [23], which consists of a dispersive element mapping frequency to time followed by time resolved detection. The specifications of the TRSPS such as resolution and spectral range may be found in [23].

A pump beam is prepared by second harmonic generation in a $1$ mm-long bismuth borate (BiBO) crystal, generating a $3$ nm FWHM spectrum at $415$ nm. Heralded single photons are generated by collinear, type-II spontaneous parametric down conversion in an $8$ mm-long potassium dihydrogen phosphate (KDP) crystal [7]. The orthogonally-polarised idler and signal fields have, respectively, $12$ nm and $3$ nm FWHM bandwidths and degenerate 830 nm centre wavelengths. The two beams are separated at a polarizing beam splitter. To achieve the experimental goals, both photons need to be recombined with a given temporal delay and their resulting interference pattern spectrally resolved. We use as a reference the broadband idler photon produced by the down-conversion whilst the narrowband signal photon is treated as the unknown state. Due to the strong correlations in photon number between the signal and idler arms, this approach greatly increases the probability of desired coincidences relative to the background two-photon events that would arise if the reference pulse were derived from an attenuated coherent state. However, this solution has the drawback of requiring the independent characterisation of the idler photon. For the purposes of this demonstration, the full spectral modefunction of the idler photon was determined by a direct reconstruction using the electro-optical shearing interferometer detailed in [18,19]. However, if full characterization of the reference photon is not possible, the scope of the measurement recovers the relative spectral phase between the two single-photon pulses on the region where the spectral amplitude of both photons overlaps, which is still of considerable interest for mode-matching applications.

To test that our characterization scheme can perform arbitrary mode reconstruction, we prepare ensembles of test pulses in a range of states by directing the signal photon to a fiber-coupled pulse shaper capable of performing arbitrary spectral phase operations, as well as spectral amplitude carving [24]. This approach is sufficient to show if the setup can reconstruct arbitrary modes, since the spectrum and intrinsic phase of the SPDC source was already known. Any extracted phase thus results from what is imprinted by the pulse shaper, which allows comparison of the experimental results with the theoretical expectation. The shaper consists of a standard $4$-f line built with a $2000$ lines/mm diffraction grating (Spectrogon) and $200$ mm focal length cylindrical lens (Thorlabs) with a $2$D phase mask (Hamamatsu SLM, $1272 \times 1024$ pixel mask) and is capable of achieving arbitrary spectral phase shaping with a resolution of $0.015$ nm/pixel and near-uniform spectral intensity transmission of $\approx 60$%. Losses are equally distributed between the diffraction grating efficiency and the insertion losses in fiber coupling at the output of the device.

To ensure high-visibility two-photon spectral interference at the output of the beam splitter requires both the spatial and polarization modes of the input photons to be matched. The former was accomplished by implementing the interferometer in polarization-maintaining single-mode fiber, using an evanescently coupled fiber beam splitter to close the interferometer. This ensured any photon events registering at the outputs correspond only to photons propagating in the fundamental spatial mode of the fiber. Polarization matching was achieved using half- and quarter-wave plates and verified using an auxiliary fiber-coupled polarizer.

The value of $\tau$ must be large enough for the terms in Eq. (10) to be distinguishable in the pseudo-temporal domain, but small enough that the spectral fringes can be resolved by the TRSPSs. $\tau$ must also be known precisely if the delay of the signal at the input is to be determined by the measurement. A value in the range of several picoseconds satisfies these constraints, allowing the interference fringes to be clearly distinguished without loss of contrast from the spectral point-spread function of the TRSPS. The two paths were matched using fast single-photon detectors whilst fine tuning was achieved by scanning the linear spectral phase between the two paths with the pulse shaper over a 30 ps window. An interference filter of 3 nm FWHM bandwidth was inserted in the broadband idler beam to maximize the spectral mode overlap. The delay that minimised the distinguishability of the two photons, and hence the number of coincidences, was taken to correspond to the point where $\tau =0$. The delay line was then moved to the setting where $\tau =5.5$ ps, the interference filter was removed, and the outputs of the interferometer routed into the two TRSPS. This delay was chosen to provide a sufficient number of high-visibility fringes in the joint spectrogram to reconstruct high order spectral phase given the high-resolution of the TRSPS (see [18]).

The signal photon is passed through the pulse shaper, which also served to fine-tune the relative delay $\tau$ between the two arms by writing linear spectral phase onto the signal photon. The photon paths are then recombined at a beam splitter, after which they are directed into time-of-flight spectrometers and undergo spectrally resolved coincident detection with a resolution of $0.05$ nm [23]. The single-photon pulses were prepared in a series of modes approximating the first five Hermite-Gauss modes [13] by applying a $\pi$-phase shift across the points in the spectrum corresponding to the nodes of those states. This allowed a nontrivial and near-orthogonal basis of modes to be prepared without the need for spectral amplitude carving (which introduces losses). Moreover, such phase jumps are challenging to extract using reference-free mode reconstruction methods, even using classical light [25]. These prepared modes were measured using a broadband single-photon spectrometer and are shown in Figs. 3(a)–(e).

 

Fig. 3. (a-e) Theoretical test modes obtained by pulse shaping. (a) is the signal photon spectrum directly measured, whilst the remaining have been applied the $\pi$-phase jumps according to the setup of the pulse shaper. The $y$ axis is the spectral amplitude in arbitrary units; (f) mode mixing ratios $P_i$ for mixed states A (dots) and B (square). The mode number refers to the Hermite Gauss basis. The boundaries of the phase flips were added in post processing by making use of an artefact of pulse shaping whereby diffraction caused by a sharp change in the orientation of the nematic crystals on the SLM mask results in a narrow dip in spectral intensity. These points mark unambiguously the location of the phase shifts.

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The interferograms were obtained by binning roughly 70,000 spectral coincidences events in a 2D histogram with the resolution of the TRSPS. The acquisition time, on the order of one hour at a coincidence rate of 30 Hz, was chosen mostly to ensure a high fidelity in the reconstruction and good statistics. We have observed that as little as 5,000 events were sufficient to successfully reconstruct our test modes in a matter of minutes, at the expense of higher uncertainties due to reduced signal to noise ratio .

4. Results

The signal photon acquires a quadratic component of the spectral phase in propagation through 2 m of coupling fibre (approximately 90,000 fs$^2$). In the case of the zero-order mode, this extra contribution was measured by the experiment. The pulse shaper then applied an additional component to the spectral phase to compensate for this, yielding a resultant flat spectral phase. Note that this value is consistent with the one measured in [18,19]. In the subsequent cases, the measurement was taken with this quadratic spectral phase component compensated.

The interferograms $S(\omega _c,\omega _d)$ are shown in the bottom row of Fig. 4 for all prepared test pure and mixed states (described in the following). Fourier filtration was used to remove high frequency noise of the detection (see Fig. 2(b)). Isolating only the sideband, the density matrices were extracted using the procedure described above. The modulus of these, $|\rho (\omega _c,\omega _d)|$ as well as the phase $\phi (\omega _c,\omega _d)=\mbox {Arg}\{\rho (\omega _c,\omega _d)\}$ are presented in Fig. 4 (rows b and d). The signal-to-noise ratio (SNR) for the amplitude arises primarily from Poisson count statistics in the detection, while SNR of the phase will depend on the spectral fringe visibility as well. For clarity, we chose to plot the Cosine of $\phi$ for both theoretical and experimental cases. This allows for better distinction between the multiple phase shifts, since any continuous phase is non-existent (see plot 0 in row d).

 

Fig. 4. Top (a-b): modulus of the theoretical and experimental density matrices for the five test pure states (0-4) and the two mixed states (A-B). Middle (c-d): argument of the density matrices similar to the two top rows. Below row (d): overlap between the theoretical and experimental matrices. Bottom (e): low-pass filtered interferograms.

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A phase difference of approximately $\pi$ was measured between the regions of alternating phase, demonstrating the capability of this technique to characterise spectral phase. The locations of the boundaries of these regions are consistent with the positions written on the phase mask. Lastly, a mixed state was simulated by flickering the pulse shaper alternately between modes 0,1 and 2 (mixed state A) and 1,2 and 3 (mixed state B). This has the effect of simulating a mixed state described by a density operator comprising of a mixture of these three pure states. The mixing coefficients were monitored at the end of the measurement, and are show in Fig. 3(f)). Interferograms for these two mixed states are also shown in Fig. 4 (bottom row), in which overlayed interference patterns from the different contributions are visible. The modulus and phase of the density matrices is shown in Fig. 4 (row b and d). The suppression of the off-diagonal elements of the density matrix is clear, as the state begins to approximate a spectrally incoherent mixture for state A, while that effect is less visible for state B.

Since the relative spectral phase between the signal and idler path is known to be flat (apart from the linear phase responsible for delay), we can construct theoretical density matrices $\rho _\textrm {th}$ from the measured spectra depicted in Fig. 3 to verify the fidelity of the reconstruction. We compute the overlap between the theoretical and experimental matrices with a Hilbert-Schmidt inner product. The values are shown at the bottom of row d in Fig. 4. We achieve high fidelity in the reconstruction of lower order pure state, although the highest order reconstruction is poorer. The main reason for this decrease in fidelity is ringing in the Fourier tranform due to sharp variations in both amplitude and phase, which is smoothed out by the filtering, thus reducing overlap with the theoretical states. We also find that the mixed states show good overlap with theory. The biggest difference arises again from the phase which is even more structured in this case. Note that utilizing finite bandwidth filters necessarily results in a smoother variation of both the amplitude and the phase of the reconstructed mode. This ensures that the recovered phase is free from any artifacts that arise from the photon-counting interferograms that manifest as high frequency noise on the spectrum. Nevertheless, in the limit of high rate coincidence-detection events resulting in a smooth interferogram, a larger bandwidth filter could be utilized to reduce blurring of the reconstructed density matrices. Finally, the speed of the reconstruction was mostly limited by the poor detection efficiency (on the order of 10%) and the binning of the interferograms at the highest resolution allowed by the TRSPSs. This can nevertheless be vastly improved by using high speed, high efficiency detectors such as superconducting nanowires and by further reducing the resolution of the acquisition, depending on the optical delay imprinted on the reference path and on the structure to be resolved.

5. Conclusion

In this work we proposed and demonstrated an externally-referenced technique to reconstruct the full spectral density matrix of an unknown single photon. The method has the advantages that it uses resolved measurements and hence does not require repeated reconfiguration in order to perform a reconstruction. Furthermore, there is no need to make the assumption of state purity: the full spectral density matrix of the photon is directly extracted from the interference pattern. Moreover, this method is potentially applicable to temporally mixed states, where each pure state has a different delay, since the information is contained along the diagonal in the pseudo-temporal Fourier domain. This approach may be of particular use when one photon from a well-characterized photon pair source is used to probe some unknown process and the other retained, in which case tomography on the state of the probe using this method can provide useful information about its evolution under the unknown process. The mode resolved HOMI approach can also be generalized for determining spatial or polarization state.

Future work on this technique may revisit the strategy of using a reference other than the twin photon from the same downconversion source. This strategy becomes practical when the spectrally-resolved detection efficiency is sufficiently high, and hence with advances in detector technology may become a feasible measurement. Alternatively, a reference with sub-Poissonian photon number statistics could be used to suppress the two-photon term. Ideally, the reference would be another broadband single photon, perhaps from a well-characterised heralded source.