1. Introduction

Characterizing quantum sources and channels is essential for enabling applications relying on quantum correlations such as quantum communication [1], quantum computation [2] and quantum metrology [3]. This task is usually performed by quantum state tomography (QST) [4]. For photonic quantum states that emerge from a photon-pair source, QST involves statistical characterization of coincidence detection between pairs of photons, in various projective measurement configurations. Due to the finite efficiency of detection and necessarily low rates of photon pair generation, characterizing sources in more than one degree of freedom requires prohibitively time-consuming data acquisition, with low signal-to-noise ratio and susceptibility to drift. Thus, with few exceptions [5, 6], QST is performed on one degree of freedom, for example polarization, with the use of broadband single photon detectors [7] that integrate over energy and single-mode fibers that collect in a single spatial mode [8].

While QST is a direct quantum measurement of correlations, the above difficulties limit its ability to investigate the causes of entanglement degradation due to unwanted correlations. Such correlations are generally assumed to exist; for example, broadband polarization-entangled photon-pair sources typically exhibit entanglement degradation unless they are spectrally filtered, which improves the entanglement but reduces count rates. While this is widely accepted and done in practice [9], the underlying correlations have not been measured, and thus our ability to understand and address the causes of unwanted correlations remains a challenge. On the other hand, correlations can be useful for generating different states within a single phase-matching bandwidth[10]. Studying such correlations would benefit from efficient multidimensional tomography techniques.

Recently, a technique called stimulated emission tomography (SET) [11] was proposed to address the practical limitations of traditional QST. By exploiting the link between spontaneous and stimulated processes [12], SET enables efficient source characterization with several orders of magnitude larger signal-to-noise ratio than that which can be achieved in QST. The stimulated signal makes it possible to use classical detectors, enabling the rapid measurement of the joint spectral density [13, 14] and the polarization density matrix [15] of photon pairs. While SET studies to date have demonstrated that SET provides more efficient detection and enhanced resolution compared to approaches based on coincidence measurement, these studies have focused on only one degree of freedom of the photon pairs, either polarization or energy, neglecting correlations between the two.

Here we show that SET is capable of opening up new regimes for the exploration of quantum correlations, far beyond simply enabling improvements in efficiency and resolution. We demonstrate SET can be used to characterize a quantum source in more than one degree of freedom, with much greater efficiency than QST. We show that by reconstructing the polarization density matrix for a pair of photons having energies in a range so small that energy-polarization correlations are negligible, and making such narrowband measurements across the entire photon-pair spectrum, SET enables the reconstruction of the full energy-resolved polarization density matrix, involving an amount of data corresponding to thousands of individual polarization density matrices. This is far beyond what has been achieved with QST. Correlations between different degrees of freedom, such as those between energy and polarization that are revealed here, can be a challenge to the use of photon pairs in any quantum protocol, as they can lead to a lack of fidelity between the produced and target states. But these correlations also can be an opportunity, because they allow for the encoding of more information in the photon pair. The ability to reveal such correlations with SET thus leads to a deeper tomography of sources of quantum correlated states, and hence a better understanding and control over their properties.

2. Methods

We demonstrate the use of SET to determine the energy-resolved polarization-state density matrix of the entangled photons produced using spontaneous four-wave mixing (SFWM) in an optical fiber. In SFWM, a pump pulse interacts with a χ⁽³⁾ medium to generate a signal and idler photon. As proposed [11], SET is implemented by providing a tunable continuous-wave (CW) stimulating idler seed in addition to the pump pulse, and the information necessary to determine the energy-resolved polarization density matrix is extracted by analyzing the stimulated signal.

A schematic of our experimental setup [16] is shown in Fig. 1. Our source is a 20cm polarization-maintaining fiber (PMF) in a Sagnac loop, pumped by ∼ 150fs pulses from a Ti:sapphire laser centered at 715nm. With the pump alone, two sideband photons are created through SFWM: the signal at 629nm and the idler at 829nm. The joint spectral intensity of the photons is characterized by stimulating the four-wave mixing process with the injection of an idler beam [14], and is shown in Fig. 2(a).

 

Fig. 1 Schematic of the experimental setup for stimulated emission tomography of the polarization-entangled photon pairs generated in optical fiber. H, half-wave plate; Q, quarter-wave plate; P, polarizer; BP, bandpass filter; L, lens; PBS, polarizing beam splitter; PMF, polarization maintaining fibre; SMF, single mode fibre; BS, beamsplitter; ND, neutral density filter.

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Fig. 2 (a) Joint spectral intensity of the generated photon pairs, measured by SET. Dashed line indicates the area we average over to obtain the polarization density matrix shown in (b). Real (left) and imaginary (right) parts of (b) the spectrally averaged polarization density matrix reconstructed by SET, and (c) the polarization density matrix reconstructed by quantum state tomography.

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To generate polarization-entangled photon pairs, a half-wave-plate (HWP) is used to set the polarization of the pump at 45° from the horizontal; this balances the power of horizontal and vertical components to ∼ 5mW each. These two components are coupled into opposite ends of the PMF. One end of the fiber is twisted by 90°; ideally, this would generate a polarization-entangled state $|\mathrm{\Psi }〉=\left(|H_s{H}_i〉+\text{exp}(i\phi )|V_s{V}_i〉\right)/\sqrt{2}$ at the other port of the polarizing beam-splitter (PBS), where φ is the relative phase between the two pathways, which is set to zero here.

To perform SET in the polarization degree of freedom, as in [15], an additional tunable continuous-wave Ti:sapphire laser is used to inject a 10mW seed idler beam backward into the Sagnac loop. The seed polarization is controlled by a quarter-wave-plate (QWP) and HWP and monitored at the output by a polarization analyzer consisting of a QWP, HWP, and polarizer. The signal is collected by a single mode fiber, thus reducing the detected spatial degree of freedom to a single mode, and sent to a spectrometer (Andor SR-303i-A). Using the same polarization analyzer as used for QST, we perform SET in the energy degree of freedom by recording the signal spectrum for each signal polarization, once for each output polarization of the idler seed, ranging over the polarization states of the idler that are measured in QST. The seed power is measured by a power meter and used to normalize the detected signal output. We perform SET at different seed frequencies across the idler bandwidth observed in the spontaneous experiment, with a step size of 0.1nm. This sets the idler resolution, while the signal resolution (∼ 0.06nm) is that of the spectrometer. The spectral region used for reconstruction of the density matrix is indicated in Fig. 2(a).

Blocking the pump beam in the setup shown in Fig. 1, we inject the seed backward into the Sagnac loop. For each input seed polarization used in the SET, we scan the frequency and measure the output seed polarization in the idler arm. We use the recorded polarization states of the seed at the output [11] as the corresponding idler polarization state when reconstructing the energy-resolved density matrix.

3. Results

To compare the density matrix obtained by SET with that measured by QST, we perform a trace over the spectral degree of freedom of the spectrally resolved density matrix from SET, which corresponds to averaging over the pairs of energies, each weighted by the probability that each pair of photon energies would be detected. The spectrally averaged SET density matrix is shown in Fig. 2(b). This matrix corresponds to the polarization state of the photon pairs produced by our fiber system. To compare our SET measurements with QST, we replace the detection with single-photon counting modules (PerkinElmer SPCM-AQ4C) and perform QST without the seed. The spectrally averaged density matrix obtained by SET shows a 97.8% fidelity to the QST result (shown in Fig. 2(c)).

Of course, our SET approach can give a much more detailed picture of the state of photon pairs generated in the spontaneous experiment. The energy-resolved polarization density matrix obtained using SET is shown in Fig. 3(a). Note that this plot in fact exhibits 115×40 individual polarization density matrices, one for each pair of photon energies. In the simple picture of a polarization-entangled photon-pair state, the energy and polarization are uncorrelated, meaning that each of these 4600 polarization density matrices should be identical. They are clearly not. As seen in the magnified elements of Fig. 3(b), significant structural features indicate the presence of frequency-polarization correlations.

 

Fig. 3 (a), Real (left) and imaginary (right) parts of the energy-resolved density matrix obtained using SET. (b) One of the 16 elements of the density matrix, magnified to display more detailed structure.

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We can examine these results by applying the data analysis strategies usually employed to characterize entangled states, but now with energy resolution. A standard characterization of the quality of a source is the fidelity F of the generated state to the target state [17]. For each pair of photon energies a different polarization density matrix results, and each of these will in general have a different fidelity with the target state. In Fig. 4 we plot F for a target state $|\mathrm{\Psi }〉=\left(|H_s{H}_i〉+|V_s{V}_i〉\right)/\sqrt{2}$. Near the center of the plot (Fig. 4(b)), where the joint spectral intensity is the largest, the fidelity is the closest to unity (compare Fig. 4(c)). The lobes to either side of the diagonal center lobe in F seem to be associated with the sidelobes in the joint spectral intensity, which arise from the phase-matching constraints [18, 19]. More generally, regions of lower fidelity may arise due to asymmetries between the two pathways through the Sagnac loop, for example due to nonuniform frequency-dependent birefringence in the fiber and other optical elements [20, 21]. From our SET results, we find that entanglement degradation manifests as a combination of several factors: spectral variations of the joint phase (as evident in Fig. 3(b)), spectral variations in the balance of the amplitudes (as seen in the diagonal elements of Fig. 3(a)), and frequency-dependent polarization cross-talk (as seen in Fig. 3(a)). In particular, we calculate the purity and find there is spectrally varying mixedness, with an average weighted purity of 97%. From our SET measurements we deduct the tangle, which has a maximum near the central wavelengths of signal and idler of 0.99; however, when measuring over the full bandwidth of the photon pairs, the overall tangle is reduced. Figure 4 suggests nearly perfect fidelity with the target state can be achieved using tight spectral filtering, a procedure commonly employed in practice. The abundant new information provided by SET thus enables detailed study of correlations between the energy and polarization degrees of freedom in the entangled state that would be difficult to probe using QST. The technique can be applied to other systems and degrees of freedom that require optimization and engineering for quantum applications.

 

Fig. 4 (a) Fidelity F for each pair of photon energies for a target state $|\mathrm{\Psi }〉=\left(|H_s{H}_i〉+|V_s{V}_i〉\right)/\sqrt{2}$ over a wide spectral range. (b,c) Close-ups.

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4. Conclusion

We experimentally realize the energy-resolved reconstruction of the density matrix of a polarization-entangled photon-pair state generated in optical fiber by using SET. This new technique yields results with high resolution that provide abundant useful information, enabling the study of detailed physical phenomena such as energy-polarization correlations in the entangled state, which is usually impossible to investigate using traditional quantum measurement due to the limited amount of photon counts. The approach can be applied to other systems that require optimization and engineering of quantum states. We believe this method should find broader interest and become a useful tool in future quantum applications.