All-fiber source and sorter for multimode correlated photons

Photons occupying multiple spatial modes hold a great promise for implementing high-dimensional quantum communication. We use spontaneous four-wave mixing to generate multimode photon pairs in a few-mode fiber. We show the photons are correlated in the fiber mode basis using an all-fiber mode sorter. Our demonstration offers an essential building block for realizing high-dimensional quantum protocols based on standard, commercially available fibers, in an all-fiber configuration.

prove that the photons are correlated in the guided mode basis, by mapping the modes the photons occupy to their arrival times at the end of a 1-km-long fiber. The 1-km fiber acts as an all-fiber in-line mode sorter, in contrast to bulk free-space mode sorters that are typically used for measuring correlations between transverse modes 15,[34][35][36] . Our in-line mode sorting configuration allows us to measure the two-dimensional (2D) histogram of the arrival times of the photons, which reveals that the photons occupy three guided modes of the fiber. By analyzing the histogram we achieve the twophoton modal decomposition and verify the spatial correlations of photon pairs generated in the multimode fiber.

Multimode correlated photons source
Our source is based on coupling Ti:Sapphire mode-locked pulses (pulse duration 140 fs, wavelength λ pump = 695 nm) into a fewmode fiber as shown in Fig. 1. In SFWM, two pump photons are spontaneously annihilated, and two photons called signal and idler are generated in two spectral channels (λ s = 542 nm, λ i = 970 nm). Each spectral channel is composed of many different spatial modes. The photons occupy the guided modes of the fiber, which can be approximated by the linearly polarized (LP) modes of a weakly guiding optical fiber. The state of the photons is determined by the phase-matching conditions and can be written as: where subscripts s (i) mark the mode of the signal (idler) photon and the coefficients α, β are determined by the nonlinear overlap integral (see Supplementary Equation 3). The term LP 01 j i s LP 02 j i i is not present in the quantum state as the mode LP 02 is not guided in our fiber for the wavelength of the idler photon. The extension of this scheme to higher dimensions and other spectral bands is presented in Supplementary Note 1.
The photon pairs are generated mostly in the first few tens of centimeters of the fiber, after which the peak power of the pump pulse is too weak for SFWM due to its temporal spreading (see Supplementary Note 2 for more information). To quantify the efficiency of the pair generation we use a 20 cm section of SMF-28 to measure the coincidence detection rate as a function of the pump average power, exhibiting a quadratic scaling as expected for a four-wave mixing process (Fig. 1b). The coincidence to the accidental ratio we obtain for a pump average power of 10 mW is 850 (see Supplementary Note 3 for more details). In principle, to improve the coincidence rate we could use higher pump powers. Increasing the pump power, however, will also increase parasitic Raman scattering. In our system, Raman scattering hardly adds noise since it is temporally separated from the generated photon pairs. However, the pump power is limited since the photon counts due to Raman scattering exceed the maximal count rate of our detectors (≈5 Mhz). This limitation can be circumvented by using superconducting nanowire detectors with an order of magnitude higher maximal count rates (≈50 Mhz), or by using inline fiber Bragg gratings to filter the pump light before the sorter so that the pump will not generate Raman scattering along the 1km-fiber.

Multimode photons sorter
Next, we use a 1 km section of the same fiber, which serves as a photon pairs source and as a mode sorter of the fiber's guided modes. Due to modal Group Delay Dispersion (GDD), the arrival times of the photons at the end of the fiber depend on their modal distribution and their spectral channel, as depicted in Fig. 1. We can therefore map the arrival times of the photons to their modal decomposition, up to modal degeneracy in symmetric fiber cores. Although this sorting scheme is quite common in classical optics 37 , it was only recently demonstrated at the single-photon level for weak coherent pulses 38 . Here we use the same principle for entangled photons. In our setup, the temporal resolution is limited by the jitter of the avalanche photodiodes which is 400 ps. Since the GDD of our fiber is on the scale of 1 ns/km, a 1-km-long fiber is sufficient to temporally separate the modes.

Two-photon modal distribution measurement
To investigate the modal distribution of the two-photon state, we use the mode-to-time mapping and study the temporal two-photon probability P(T s , T i ) that describes the probability to detect a signal photon at time T s and an idler photon at time T i . To this end, we plot the two-dimensional histogram of the arrival times after compensating for chromatic dispersion (Fig. 2a). Two correlation peaks are observed, corresponding to the delay between either LP 02 j i s and LP 01 j i i or between LP 11 j i s and LP 11 j i i . Clearly, the two-photon probability is not-separable, indicating that photons are correlated in the modal basis. To quantify the correlation of the two photons we post select two arrival times for the signal ðT  Fig. 1 An all-fiber multimode source and mode sorter for photon pairs correlated in the fiber modes. a Ultrashort pulses of 140fs (λ pump = 695 nm) are coupled into a 1-km-long fiber. Pump photons are spontaneously annihilated and pairs of signal and idler photons are generated at two different spectral channels (λ s = 542 nm, λ i = 970 nm). At these wavelengths, the fiber (SMF-28) supports a few modes, where the modal distribution of the photon pairs is determined by the phase-matching condition of the fiber. After the first few tens of centimeters, the temporal spread of the pump pulse prevents SFWM. In the next 1-km of the fiber, the different modes are separated due to modal dispersion (inset). Higher spatial modes arrive after lower spatial modes, and shorter wavelengths arrive after longer wavelengths. At the output of the fiber, the signal and idler photons are spectrally separated by a dichroic mirror (DM), filtered by a bandpass filter (BPF), and their arrival times are registered using two single-photon detectors and a time-to-digital converter (TDC). An electronic delay of 70ns is introduced to the idler detector to compensate for the chromatic delay between the signal and idler photons. b Experimentally measured coincidence rate as a function of the pump average power for a 20-cm-long fiber, exhibiting a quadratic scaling. i À μ Ti Þ=ðσ Ts σ Ti Þ where μ Ts ; μ Ti are the mean arrival times of the signal and idler photons and σ Ts ; σ Ti are their standard deviations. We obtain PCC = 0.51 ± 0.012, which indicates a strong correlation. The main source of correlation degradation in our system is the 400 ps jitter of the detectors, which causes circular smearing of the histogram peaks. Another source of decorrelation is the uncertainty in the creation times of the pairs, which results in a diagonal spread of about ≈200 ps that hardly effects the PCC between the chosen arrival times. In principle, the intermodal coupling can also add decorrelation, however, the PCC is sensitive only to mode mixing that occurs in the first few tens of centimeters of the fiber because the arrival times of photons which experience mode coupling after a longer distance will be different from the post-selected times ðT

Modal group delay simulation
To show that the measured delays between the signal and idler photons match the expected delays for an SMF-28 fiber, we numerically calculated its modal group delays. We solve the scalar wave equation for an SMF-28 fiber, with a 4.2 um core radius, corecladding index difference of Δ = 0.33%, and a step-index profile with a typical dip shape. The modal delay of LP 11 , LP 02 modes, relative to the fundamental mode is presented in Fig. 3. We chose the fundamental mode as a reference to cancel the chromatic dispersion. At the signal's wavelength, the delay between the LP 02 and LP 11 is ΔT s = 1 ns. At the idler's wavelength, the delay of LP 01 and LP 11 is ΔT i = 0.5 ns. These delays are in agreement with the temporal correlations found experimentally, supporting the modeto-time mapping scheme (Fig. 1b).

DISCUSSION
In conclusion, we have demonstrated generation and sorting of correlated photon pairs occupying high-order modes of a commercially available fiber. The all-fiber configuration opens the door for implementing high-dimensional photonic quantum bits in fiber-based applications. For example, the mode-to-time mapping can potentially solve the challenge of scaling the number of required detectors with the number of fiber modes, an outstanding challenge in conventional mode sorters. Towards this end, it is necessary to improve the temporal resolution of the system, for example by using superconducting nanowire singlephoton detectors with jitter times as low as a few picoseconds, and faster electronics. It will allow sorting more transverse modes and using shorter fibers for the temporal mode sorter, which in turn will decrease the background noise caused by fluorescence and parasitic nonlinear processes in the fiber.
In order to manipulate the photons coherently and apply projective measurements in two mutually unbiased bases one can use multi-plane light converters (MPLC) 34,35,40,41 . We note that by combining the all-fiber temporal sorter with an all-fiber wavefront modulator that we recently developed 42 , it would be possible to demonstrate an all-fiber sorter on a mutually unbiased basis, opening the door for all-fiber quantum communication protocols with high-dimensional quantum bits.
Addressing these challenges will allow exploring applications of the all-fiber source and sorter. For example, using an in-line multimode fiber beam splitter one could split the photon pairs and route each photon to a different remote user. Such configuration is relevant for device-independent quantum key distribution, where an untrusted user (Charlie) distributes entangled photon pairs to Alice and Bob, who generate a secure key based on Bell measurements 43 . A more immediate application of the all-fiber source is quantum communication protocols that rely on sending both photons to the same target. Examples include quantum dense coding 44,45 , high capacity quantum key distribution 46,47 , and direct quantum communication 48 .

Experimental setup
An optical fiber (SMF-28) is pumped by a Ti:Sapphire laser (Coherent Chameleon Ultra II, 680-1060 nm, 140 fs duration, 80 MHz repetition rate). Before coupling to the fiber, the laser was filtered using a bandpass filter (Thorlabs FB700-40). The signal and idler photons were separated using a dichroic mirror (DM) with an edge at 925 nm (Semrock FF925-Di01). In each arm, the pump beam was blocked using spectral filters. In the signal arm, we employed a short-pass filter (Semrock BSP01-633R), and a bandpass filter (Semrock FF01-540). In the Idler arm, we employed a longpass filter (Semrock BLP01-808R) and a bandpass filter (Semrock LL01-976). The signal and idler photons were coupled into two optical fibers (SMF-28) and detected using avalanche photodetectors (Excelitas SPCM-AQ4C), with a quantum efficiency of 50% for the signal photons and 15% for the idler photons. The arrival times of the photons were registered using a time-todigital converter (Swabian Time Tagger 20).

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.  Fig. 3 Numerical computation of the modal delays of the LP 11 (blue curve) and the LP 02 (red curve) modes. The delays are presented relative to the fundamental mode LP 01 , to compensate for the chromatic delay. For the signal photon at λ s = 542 nm, the delay between the LP 02 and LP 11 modes is ΔT ðthÞ s ¼ 1 ns=km, in agreement with the experimentally measured delays presented in Fig. 2c. For the idler photon at λ i = 970 nm, the delay between the LP 11 and LP 01 modes is ΔT ðthÞ i ¼ 0:5 ns=km, in agreement with measured delays reported in Fig. 2b.