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Abstract
Hyper-entanglement between two or more photonic degrees of freedom (DOF) can enhance and enable new quantum protocols by allowing each DOF to perform the task it is optimally suited for. Here we demonstrate the generation of photon pairs hyper-entangled between pulse modes and frequency bins. The pulse modes are generated via parametric downconversion in a domain-engineered crystal and subsequently entangled to two frequency bins via a spectral mapping technique. The resulting hyper-entangled state is characterized and verified via measurement of its joint spectral intensity and non-classical two-photon interference patterns from which we infer its spectral phase. The protocol combines the robustness to loss, intrinsic high dimensionality and compatibility with standard fiber-optic networks of the energy-time DOF with the ability of hyper-entanglement to increase the capacity and efficiency of the quantum channel, already exploited in recent experimental applications in both quantum information and quantum computation.
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1. Introduction
Photons can encode quantum information in multiple degrees of freedom (DOF): polarization, path modes, transverse modes, and the energy-time [1]. The energy-time DOF has become increasingly relevant [2–7] due to its intrinsic multi-dimensional encoding space, which shares with DOF such as orbit angular momentum and path mode, its natural resistance to decoherence, its ability to easily generate high visibility interference and compatibility with single mode fibers and waveguides [8,9].
Within the energy-time domain one can encode quantum information in time bins, frequency bins and pulse modes (which are also known as ‘temporal’ or ‘time-frequency’ modes). Both time [10] and frequency bins [2] are particularly relevant in quantum communication and quantum information processing. They have been shown to be a successful tool to experimentally demonstrate the enhancement between quantum and classical communication protocols [11], long-distance quantum communication tasks [6,12], and to decrease the susceptibility to noise and losses and increase key generation rate in quantum key distribution (QKD) protocols [13–15]. Also, recent works have demonstrated new techniques to generate [16] and characterize [3,5] discrete bins in the spectral domain, as well as methods to fully manipulate them with a great level of control [2,9].
The pulse mode DOF refers to the the shape of the wave packet amplitude, which lives in both time and frequency domain. The orthogonality between the modes is preserved by their spectral and temporal phases [8,16]. This natural structure makes them suitable for high-rate transmission, orthogonal-waveform multiplexing, in which signals are distinguished according to the shape of the field [17], or to enhance the performance of quantum networks by enabling transmission of photons encoded in different pulse modes using the same communication channel [18]. The direct generation of pulse modes has been demonstrated via parametric downconversion (PDC) [19] in crystals with tailored non-linearity profile, while their characterization has been shown using sum frequency generation in waveguides [20]. The ability to generate pulse mode entanglement in designed crystals has recently been exploited to also demonstrate entanglement between pulse modes and other high-dimensional DOFs [21]. This is called hyper-entanglement; it allows measurements to be carried out on one DOF without breaking the entanglement structure of the other [22,23]. The generation of hyper-entangled photons has been demonstrated using multiple sources, such as quantum dots [24], ion traps [25] and through PDC crystals and waveguides [21,23,26]. This scheme is particularly used in quantum communication and quantum information processing to enhance both the channel capacity and efficiency [27–29]. The applications include Bell state analysis [30,31], enhanced QKD protocols [32], cluster state generation [4] and the exploration of advanced quantum information concepts in higher-dimensional systems [33].
Here, we report on the generation of a hyper-entangled bi-photon state in pulse mode and frequency bin. We use a domain-engineered crystal to generate pulse-mode entangled photons through parametric downconversion. Subsequently, a polarization-frequency bin mapping technique is used to generate the desired hyper-entanglement structure of the bi-photon state. The characterization of it is performed by reconstructing the joint spectral intensity (JSI) [34], together with the two-photon Hong-Ou-Mandel (HOM) interference measurement [35]. Furthermore, we perform the Schmidt decomposition on our state to retrieve its energy-time structure and we compare it to the simulated one.
2. Methods
In a parametric downconversion process, a pump photon of frequency $\omega _3$ probabilistically generates two photons, traditionally called the signal and idler but here labeled “1” ($\omega _1$) and “2” ($\omega _2$) respectively, in a non-linear medium. To a first-order approximation, the bi-photon state can be expressed as,
The energy-time structure of the bi-photon state can be manipulated by exploiting the relationship between the PEF [2,18,20] and the PMF [16,19,38,39] in a PDC process. In our case, we engineer the domain structure of the crystal in order to generate a first-order Hermite-Gauss polynomial PMF [34]. The resulting bi-photon state is a maximally anti-symmetric Bell state $\vert \psi ^{-} \rangle$ encoded in the pulse mode basis 
Fig. 1. Process outline in the JSA. The plots show the simulated JSAs at intermediate steps of our scheme. (a) Starting from the pulse mode entangled state generated by the crystal, the first step is to displace the lobes along the diagonal indicated by the red arrow. (b) This results in the two downconverted photons being emitted at different wavelengths. The next step is to generate another pair of lobes along the diagonal, noting the lobes are mirrored as indicated by the red arrow. (c) The final JSA represents the four energy-time modes as described in the main text.
To give the state the sought structure, we first displace it in the frequency space, so that the two daughter photons will be emitted at the different wavelengths. The effect on the JSA is a shift along the anti-diagonal, representing a decrease in frequency of the one photon and a commensurate increase in frequency of the other (Fig. 1(b)). The resulting intermediate state is,
where $|{\omega _1}\rangle$ and $|{\omega _2}\rangle$ are two central frequency modes which the two photons are produced. At this stage, the bi-photon state is entangled only in the pulse mode DOF and fully separable in the central frequencies of the produced photons.
To obtain entanglement in frequency bins, we first entangle the state in polarization,
where $\phi$ is the phase associated to the polarization DOF. This intermediate step allows us to map the polarization DOF to the spectral domain [40], and once we then erase the polarization information in state Eq. (6) we finally recover the desired energy-time structure, as in Fig. 1(c). The state produced has a phase $\phi$ associated with the polarization domain which can be manipulated using phase retarders. The final state can be expressed as the tensor product of two pulse modes and two frequency bins: The experimental setup for generating the frequency bin and pulse mode entangled photon pairs is shown in Fig. 2(a). We use a mode-locked Ti:Sapphire laser with a $80$ MHz repetition rate, 1.27 ps pulse duration and central wavelength of 775 nm to pump a collinear type-II PDC process. The pump laser is focused into a 30 mm long aperiodically poled potassium titanyl phosphate (apKTP) crystal, whose PMF has been engineered using our coherence length domain algorithm [19]. We use a custom oven to heat the crystal to $\sim 135^\circ$C. This results in shifting the downconverted photons in frequency and avoiding overlap between the photon spectrum. The shift in the spectral domain is primarily due to the expansion of the crystal, thus the poling domain widths, and the alteration of the refractive indices according to the Sellmeier equation. The final bin separation is expected to be $\sim 11$ nm. This is done to ensure minimal overlap between the frequency bins, which have a nominal bin width of $\sim 3$ nm as measured in Ref. [19]. At this step, we obtain the bi-photon state given in Eq. (5).
Fig. 2. Experimental layout. (a) Entangled photon source (EPS). A Ti-Sapphire laser pumps the aperiodically-poled KTP (apKTP) crystal mounted in a temperature controlled oven. The crystal in embedded in a Sagnac loop using a dual-coated polarizing beamsplitter (PBS) and dual-coated half-wave plate (HWP). Both downconverted photons are sent to single-mode fiber couplers (FC), in one arm a dichroic mirror (DM) is used. The polarization-to-frequency mapping setup consists of polarization fiber controllers (PFC), PBS, quarter-wave plate (QWP), HWP and linear plate polarizers (LPP). (b) JSI measurement. The time-of-flight spectrometer is realized by adding 20 km spooled SMF28 fiber to each arm of the EPS. Photons are detected by superconducting nanowire single photon detectors (SNSPD) and arrival times are timetagged by a fast logic unit (PicoQuant). (c) Two-photon interference measurement. The two-photon interferometer consists of PFCs, PBS, QWP, HWP and LPPs. Additionally, one photon from the EPS is temporally varied, $\tau$, using a motorized stage. Detected photons are recorded using a multi-channel logic unit (UQDevices).
The crystal is embedded in a Sagnac interferometer to entangle the photon pairs in polarization [41], creating hyper-entanglement in the polarization and pulse mode DOF as in Eq. (6). To obtain the desired polarization Bell state, we apply a local unitary transformation on one of the photons using polarization optics. Subsequently, the two photons are sent through two single-mode fibers (SMF) and impinge on a polarizing beam splitter (PBS). In one of the output mode of the PBS, we set up a quarter-wave plate (QWP), half-wave plate (HWP) and linear plate polarizer (LPP) to control the polarization phase $\phi$, while in the other output mode we use a single LPP. Both LPPs are set to project the photons onto the diagonal/anti-diagonal ($|{D}\rangle,|{A}\rangle$) basis, which together with the PBS makes them indistinguishable in polarization, thus translating the polarization entanglement to spectral entanglement. The state produced by the source (labeled EPS in Fig. 2) has the sought hyper-entangled structure in frequency bin and pulse mode, as in Eq. (7).
This polarization mapping technique has a success probability of $25{\% }$, due to the use of the PBS and the LPP for the polarization erasure operation, specifically due to the LPP only transmitting one of the projected outcomes. The overall measured brightness of this source of hyper-entanglement is $\sim 0.6$ kHz/mW. This matches with expectations given the brightness measured for the pulse mode crystal in [19] subject to the success probability of the mapping technique and optical losses in this setup. The photons are detected using commercial superconducting nanowire single photon detectors (SNSPDs).
3. Results
There are several approaches that can ideally allow complete characterization of the energy-time DOF of the target bi-photon state. The use of quantum pulse gates [20] and quantum frequency processors [2] would enable complete tomography of the state in the temporal and frequency domains, respectively. In addition, compressive tomography techniques, which exploits precise knowledge of the system state space, can be applied to both DOF [5]. Other less direct methods use cross-measurement in time-frequency space so that spectral and temporal phases can be reconstructed by measuring JSI and JTI [42–44].
In our experiment, we reconstruct the joint spectral intensity of the state via time-of-flight spectroscopy using fast counting logic for processing detection events, following the procedure in [34]. Additionally, we perform a HOM interference measurement, reconstructing the bi-photon interferogram in time, using a multichannel counting logic. We exploit a priori knowledge of the desired energy-time entanglement structure to combine the results of the JSI and the HOM interference patterns to reconstruct the JSA. Finally, we show that spectral phase in the biphoton state results in a unique set of real and imaginary JSA, allowing us to identify the reconstructed JSA.
3.1 JSI measurement
Our time-of-flight spectrometer (TOFS) is realized by adding dispersive elements to each photon after the source as shown in Fig. 2(b) and measuring the arrival times of each photon. We use two $\sim$20 km single-mode (SMF28) fibers which have a nominal chromatic dispersion of $\sim$20 ps/nm/km at 1550 nm, however the fibers also introduce $\gtrsim 8$ dB loss to each photon. To collect enough statistics for reconstructing the JSI we use measurement integration times of approximately twelve hours to obtain $>1.3 \times 10^{7}$ total coincidences. The coincidence events are established from time-tagged single photon detection events recorded using a Hydraharp 400 which has a $1$ ps resolution. We can then reconstruct the JSI over a $36$ nm spectral range, corresponding to $12.5$ ns which is the pulse separation of the Ti-Sapph laser, with a spectral bin width of 0.0625 nm, corresponding to $25$ ps. In Fig. 3(a), the JSI is shown for a $20$ nm range which contains the main features of the bi-photon spectra. The SNSPD system has a nominal quantum efficiency of $\sim$80${\% }$ and temporal jitter of <50 ps, which ultimately sets the minimal spectral resolution of our TOFS.
Fig. 3. JSA reconstruction. (a) Measured JSI in a 20 nm range, consisting of $320\times 320$ bins with 0.0625 nm width. (b) We apply a $\pi$ phase profile to the measured JSI to infer the JSA of the measured state. (c) The simulated JSA for the ideal frequency bin and pulse mode hyper-entangled state for comparison.
As shown in Fig. 3, the measured JSI exhibits the same behavior as the simulations, respectively panel (a) and (c), displaying two distinct pairs of lobes. The presence of two lobes in each pair is a consequence of the pulse mode entanglement, while the separation of the pairs is indicative of the frequency entanglement.
To characterize the state in terms of energy-time modes, we first need to infer its JSA. The process followed to reconstruct the JSA is explained later in the section. Once we have the JSA of the state, we perform the Schmidt decomposition on it. We will compare the Schmidt number $K_{\text {JSA}}$ so calculated, with the one obtained from the theoretical model $K_{\text {theor}} = 4.02$. To do this, we first need to obtain all the information necessary to infer the JSA. The JSI in fact gives us only information about the amplitude of the state, lacking any phase information.
Based on the results in Ref. [19], we can state the existence of a $\pi$ phase between the two lobes of each pulse-mode pair. This relative phase is associated with the first order Hermite-Gauss function, which is the shape designed in the crystal non-linearity using our domain engineering technique. In addition, we will assume that the frequency bin entanglement has been prepared with a globally symmetric structure, which applies an additional $\pi$ phase difference between the two frequency bins, given the anti-symmetry of the state in the pulse mode DOF. We will verify that this choice is the only one consistent with the observed two-photon interference signature in the next section. We can now apply this relative phase profile and derive the JSA of the state (Fig. 3(b)). We perform a Schmidt decomposition on the reconstructed JSA and obtain a Schmidt number of $K_{\sqrt {\text {JSI}}+\text {phase}}=4.0391 \pm 0.0003$. This uncertainty corresponds to $3\sigma$ standard deviations, obtained from $10^4$ rounds of Monte Carlo (MC) simulation. More precisely, our starting point for the MC is the measured JSI data. Each bin is described by a Poissonian distribution whose mean corresponds to the number of coincidences measured in that bin. In the Monte-Carlo simulation, we generate $10^4$ new JSI matrices by picking random values from the distribution of each bin. We then perform Schmidt decompositions of the matrices so created and we calculate the associated Schmidt numbers. The last step is to calculate the mean and the standard deviation of all the Schmidt numbers calculated in the $10^4$ rounds. Our final Schmidt number and associated standard deviation are the mean of all the Schmidt numbers and standard deviations so calculated, which implies excellent agreement with the expectation based on the energy-time structure of the state in (7).
3.2 Hong-Ou-Mandel interference measurement
In this section we analyze the measurement of state symmetry information, from which we reconstruct the phase information. This is done by performing an HOM style interference measurement and reconstructing the bi-photon interferograms for a set of phase values. The two-photon interference in the case of a two-dimensional system in the frequency DOF provides information on both the dimensionality of the system and its spectral phase [45,46]. Specifically, from the HOM scans we retrieve information on relative phase between the two frequency bins. This is due to the assumption on the phase associated to the pulse mode DOF, which is consider fixed during the state preparation, since we do not use any active component. As mentioned in the previous section, the phase of the pulse mode DOF is assumed to be equal to $\pi$ [19]. From the different HOM patterns we can then conclude coherent interference between the two photons and, furthermore, infer the global symmetry of the bi-photon state.
To measure the HOM patterns we remove the 20 km fibers used for the JSI measurement and send the two photons from the hyper-entanglement source to an interferometer. More specifically, the two photons interfere in a PBS, thus requiring polarizers to project in non-polarization discriminating fashion to correctly obtain the HOM effect, and we measure the two-fold coincidences for a range of time delays, $\tau$ (Fig. 2(d)). To get the analytical expression for the coincidence probability we model the interaction of the two photons on the PBS, after having introduced a time delay $\tau$ to one of them. We then perform the analytical expression of the JSA $f(\omega _1,\omega _2)$ of the state after the interaction, and we calculate the coincidence probability as [47]
Figure 4 shows the plots for the experimentally measured HOM interferograms for four different phase values, alongside the simulated theory curves. For clarity, the fitted interferogram is not shown. For the four values of the phase $\phi$ in (7), $\{ 0,\pi /2, \pi,3\pi /2 \}$, we obtain fits for each interferogram and observe the measured phases to be $\{ 0.031\pm 0.086, 1.61\pm 0.07, 3.19\pm 0.1, 4.61 \pm 0.16 \}$ rad. Notably, for $0$ and $\pi$ phases, the measured interferograms clearly reproduce the typical anti-bunching behaviour for the antisymmetrical state and the bunching for the symmetrical case. We include the results for the two intermediate phase cases, $\phi =\{\pi /2,3\pi /2\}$, which highlights our ability to precisely control the relative phase $\phi$ between the frequency bins of the bi-photon state. In all cases we observe good agreement between the measured phase and the nominal phase setting.
Fig. 4. Two-photon interference patterns. The measured interferograms corresponding to different $\phi$ are shown in plots (a) - (d), with the respective phases denoted above. Raw coincidence data (orange dots) are plotted as a function of the relative temporal delay, $\tau$, evaluated from the delay stage position. Theoretical curves (blue line) based on ideal simulations are scaled to the normalized coincidence probability (c.p.) displayed on right axis. A three-sigma confidence region (green) is shown, assuming Poissonian statistics.
In the case of perfect two-photon interference, the expected visibility is unity. For the four cases in our experiment the fitted visibility values were $\{0.863\pm 0.012, 0.844\pm 0.011, 0.841\pm 0.014, 0.819\pm 0.022\}$. The reduction in visibility is primarily attributed to imperfect mode-matching, particularly the spatial overlap of the two beams on the PBS, which leads to partial distinguishability between the two photons. In our two-photon interferometer the erasure of the polarization information after the PBS also plays a key role, however the wavelength dependence in the polarization optics leads to slightly different waveplate transformations corresponding to each frequency bins.
We use the fitted interferograms to also estimate the spectral separation as well as the spectral bandwidths of the two frequency bins of the bi-photon state. These are captured by the parameters, $\delta$ and $2\sigma$ respectively in (9). Notably, these values correspond to the angular frequencies, thus requires division by $2\pi$. From the fit we obtain a bin separation of $1.37 \pm 0.4$ THz, while the bin width is $0.38 \pm 0.06$ THz, which corresponds to $\sim 11$ nm separation and $\sim 3$ nm bin width, assuming the central wavelength at 1550 nm.
For all the measured HOM scans, the overlap of theoretical and measured HOM interference patterns is well within the $3\sigma$ confidence region, indicating our ability to accurately control the bi-photon state. Errors on the parameters are quoted as standard errors from the fit.
3.3 Inferring phase information from bi-photon interferogram
The JSA provides a complete description of the bi-photon state in the spectral domain. Measuring both the JSI and phase information would allow reconstructing the JSA and fully characterizing the state. From a mathematical perspective, having an amplitude and a phase is equivalent to representing the JSA as a complex function, so that it can be decomposed into its real and imaginary parts. Although conventionally its representation is confined to the real part, this may not be sufficient to characterize the state entirely, as we will show here.
The aggregated information for determining the JSAs in our measurements is shown in Fig. 5. In this experiment, we introduce a phase $\phi$ to the frequency bins by exploiting polarisation optics in the frequency mapping step of the setup. This effectively applies a phase map that is assumed to be uniform across the JSA, with a relative phase difference between the two frequency bin. We note that as this step uses passive linear optical devices, this does not modify the phase structure of the pulse modes. We simulated the ideal JSA for the four different phase values, assuming a perfect crystal with zero non-linearity added phase, as shown in Fig. 5(a). Notably, we can see that for the phase values {$\pi /2$, $3 \pi /2$} when only the real part of the JSA is shown then one of the frequency bins would be missing. This is due to the missing component being completely imaginary. Hence, for completeness we include the plots for the imaginary parts of the JSA, to highlight how the states are uniquely defined. For each of these JSAs the measured JSI is identical, as shown in Fig. 5(b), as this is an intensity measurement that does not depend on the phase information of the biphoton state. In order to retrieve the information of the different phases we need to consider the interferogram from a suitable two-photon interference measurement. We simulate the interferograms corresponding to the four different phases between the frequency-bins in Fig. 5(c), which are uniquely defined.
Fig. 5. State identification through JSA decomposition. The figure is divided in three columns, each of which corresponds to a piece of information needed to uniquely identify the state, and four rows, one for each phase value. (a) Simulated JSA represented in its real and imaginary components. All four set of plots are scaled to the same normalised value ranges. (b) The JSI is identical for all phase values as this is an intensity measurement, i.e., $JSI \doteq \vert JSA \vert ^2$. (c) Simulated bi-photon interferogram for all the four phase values.
We can, therefore, state that the inferred JSA shown in the Fig. 3(b) can be unambiguously deduced even in the absence of direct measurements on the phase of the state.
4. Discussion
We have experimentally implemented and characterized a photon pair source that is hyper-entangled, in frequency bins and pulse mode, using a polarization-mapping technique [40] to create the frequency bin entanglement. In a recent experiment in Ref. [48], an alternative method was introduced for mirroring a joint bi-photon spectrum around a central frequency, using a PDC crystal in a linear double-pass configuration. One advantage of that method is that it does not suffer from the 50${\% }$ loss which is incurred in the entanglement transfer from polarization to frequency, due to the use of LPP. However, the use of the polarization-mapping allowed for precise phase control, by employing waveplates to encode the phase in polarization which is then transferred to the spectral domain.
The intrinsic structure of our hyper-entangled state could be exploited in fundamental tests of quantum mechanics or multiphoton quantum metrology applications [49]. A possible application is the all-optical analogous Stern-Gerlach effect. This has been recently demonstrated through sum frequency generation processes to show the analogy between the spin of the atom and the frequency of light [50]. Using our state, this effect could be shown through PDC process and the hyper-entangled structure could give rise to effects not studied so far.
In order to fully make use of the pulse mode and frequency bin hyper-entanglement generated by our source, it is necessary to employ techniques that act independently on each DOF. For example, one can use quantum pulse gates (QPG), in the degenerate condition, to project each photon in a specific pulse mode [37,51] to preserve the frequency bin structure which can then be completely characterized using quantum frequency processor (QFP) [2]. Similarly using QFPs, or simply diffraction gratings, first to sort each photon into their frequency bin followed by characterization of the pulse-mode entanglement structure using QPGs [51]. By combining these two methods, one can demonstrate the independence of the frequency bin and pulse mode entanglement, i.e., verifying the hyper-entanglement structure directly. In comparison, we indirectly infer the hyper-entanglement structure of the biphoton state using time-of-flight spectroscopy and two-photon interference measurements to reconstruct the JSA. Notably, the JSA embeds the energy-time properties of the state, e.g., frequency bins, time bins and pulse modes. As such, it is crucial to correctly represent the JSA to retrieve the distinct features of each of the encoded DOFs, as we highlight in this work.
The structure of hyper-entanglement in frequency bin and pulse mode may be exploited in quantum communication networks that use wavelength de-multiplexing (WDM) devices to distribute pulse mode entangled qubits to pairs of users [52]. By encoding in pulse modes, one can take advantage of an increased quantum channel capacity by invoking a larger alphabet, as well as improved robustness to noise in transmission through fibers. The feasibility of these applications will require photons with narrower bandwidth for compatibility with devices used in the QFP and QPG, as well as commercial WDMs in existing networks. This may be overcome by applying spectral compression techniques on each photon using chirped upconversion [53] or time lenses [54]. Alternatively, it remains an open challenge to design a tailored non-linear crystal that can simultaneously achieve narrow bandwidth photons and produce hyper-entanglement across multiple energy-time DOFs.
Finally, we remark that there may be disagreement in our use of term hyper-entanglement which historically referred to entanglement across two or more independent DOFs. Increasingly, hyper-entanglement has been used when referring to entanglement in multiple DOFs that do not easily admit strict independence, yet they can be physically acted upon independently without modifying the structure of each other [6]. It is then perhaps worthwhile to extend the definition of hyper-entanglement to cases where the DOFs share commonality in their physical description, i.e., in our case the frequency domain, to better reflect ongoing developments in quantum information processing techniques.
Funding
Engineering and Physical Sciences Research Council (EP/L015110/1, EP/T001011/1).
Acknowledgements
We thank P. Barrow for his technical assistance in the TOFS measurements.
Disclosures
The authors declare no competing interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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