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We propose a multi-photon interferometer with a generalized C-SWAP operation that can estimate Tr[ρ1ρ2 ⋯ρn], a nonlinear functional of n photonic density matrices. The scheme is demonstrated for three single-photon states whose overlap is experimentally measured as the interference visibility of a control qubit encoded into photonic paths. The validity of this method is verified by comparing the visibility with the results of Hong-Ou-Mandel experiments.
© 2013 Optical Society of America
1. Introduction
Single photon quantum states are a key resource for optical quantum information processing. A photon possesses numerous degrees of freedom (DOFs) such as polarization [1], path [2], time-bin [3], and spatial mode [4], each of which can be used as a quantum information carrier. Qubits or qudits [5] are usually encoded into a few DOFs, and other DOFs are disregarded. Indistinguishability with respect to the unused DOFs is essential for quantum interference between multiple photons. Since quantum state tomography (QST) for all the DOFs is impractical, Hong-Ou-Mandel (HOM) interferometry [6] is commonly used for testing the indistinguishability between single photon states [7, 8]. The purity of a photonic state can also be estimated by HOM interferometry under certain conditions [9]. However, HOM interference measurements do not provide further information about the internal structure of the quantum state.
A. K. Ekert et al.[10] and R. Filip [11] have shown that nonlinear functionals of quantum states can be directly estimated by controlled-unitary operations without resorting to QST. For example, a controlled-swap (C-SWAP) operation applied to one control qubit initialized as and two target states, ρ1 and ρ2, can be used to obtain Tr[ρ1ρ2], the overlap of the two states, as the interference visibility of the final control qubit. This overlap measurement can be generalized to an n-state overlap Tr[ρ1ρ2 ⋯ρn] obtained through a controlled-shift (C-SHIFT) operation. Schemes for the C-SWAP operation have been proposed for photonic states [12, 13] and cavity QED systems [14, 15]; to our knowledge, the only experimental realization reported in the literature is a two-qubit experiment in a nuclear-magnetic-resonance system [16].
This paper describes an experimental demonstration of a novel C-SHIFT scheme for photonic quantum states with a photonic path qubit and postselection by coincidence counting at multiple photon counters. Section 2 introduces the operation principle of the interferometer for two target photons and generalizes it to the n-photon case. Section 3 describes the experimental setup for a three-photon C-SHIFT operation and shows the results of overlap estimation for various input states. Section 4 discusses the sources of imperfections by considering the results of HOM experiments using the same photon source. The conclusion of our work is given in section 5.
2. Method
2.1. Scheme for two photons
The simplest example of our method is the two-photon interferometer shown in Fig. 1(a). This structure has been previously used as a path qubit fusion gate [17]. Each input photon passes through a non-polarizing beam splitter (NPBS). The transmission output path from NPBS3 (NPBS1) and the reflection output path from NPBS1 (NPBS3) are superposed at NPBS2 (NPBS4). We postselect only the states that lead to two-fold coincidence counts at photon counters D1 and D2 shown in Fig. 1(a). There are two possible paths for the postselected states: either both photons are respectively transmitted (Case T) or reflected (Case R) at NPBS1 and NPBS3. We can conceptually assume that the path information of the photon going to NPBS2 is first erased by detection at D1. The path of the other photon propagating to NPBS4 represents the path qubit used as the control qubit. When we denote the path state propagating from NPBS3 (NPBS1) to NPBS4 as |0〉 (|1〉), the two-photon state before NPBS4 can be written as , where ψ1 and ψ2 are the initial states of photons 1 and 2, respectively. The first (second) term corresponds to Case R (T). This state is the result of a C-SWAP operation where the path qubit is the control qubit initialized as and |ψ1〉, |ψ2〉 are the target states.
Fig. 1 Interferometers for overlap estimation of (a) two photons and (b) n photons. NPBS: non-polarizing beam splitter, PS: phase shifter, D: single photon counter, CC: coincidence counter.
A phase shifter (PS) is inserted into the |0〉 path enables the measurement of interference fringes between the basis states |0〉 and |1〉 of the control qubit. The unnormalized final state after NPBS4 and the success probability for postselection are:
where ϕ is the phase introduced by the PS, and ρi is the density matrix of photon i. As shown in Eq. (2), the visibility of the interference fringes as ϕ is varied equals the overlap of the two input states. We note that the overlap incorporates all the DOFs of the input photons, and each photon can have multiple spatial and temporal modes.2.2. Scheme for n > 2 photons
The scheme in Fig. 1(a) can be extended to more than two photons. For n input photons, the experimental setup requires 2n NPBSs, one PS, and n detectors aligned in a geometry as shown in Fig. 1(b). The initial paths of the input photons are split by odd-numbered NPBSs; a reflected output and a transmitted output from adjacent odd-numbered NPBSs are superposed at in-between even-numbered NPBSs. Coincidence counting at n detectors postselects the final state. Similarly to the two-photon case, there are two paths for successful postselection: all input photons are transmitted (Case T) or reflected (Case R) at odd-numbered NPBSs. This postselection includes a C-SHIFT operation where the two paths propagating to NPBS2n are the computational basis of the control qubit. The postselected state and its success probability are:
As shown in Eq. (4), the visibility corresponds to the n-fold overlap Tr[ρ1ρ2 ⋯ρn] of the input states. Moreover, Tr[ρn] of a single input state ρ can be measured if n copies of the state are prepared. The eigenvalue spectrum {λi|i = 1, 2,...} of ρ for N dimensional states can be obtained by measuring Tr[ρn] for 2 ≤ n ≤ N and using the relation [10] without full reconstruction of ρ. We note that the structure of photonic temporal/spatial modes does not need to be known for this spectrum analysis. The overall efficiency of the analysis is state-dependent because the number of required measurements depends on the rank of the total density matrix and the success probability of postselection decreases exponentially with n as shown in Eq. (4).The above analysis in this section is valid only when the input state is separable and can be written as a tensor product of matrices for single-photon states. In the general case, the visibility corresponds to the magnitude of the difference between the probabilities that the state is in the symmetric subspace and in the antisymmetric subspace, respectively [7, 11]. If the input photons are in an entangled state, our structure can also be used for entanglement-witness measurements by coincidence counting at specific fixed values of the phase ϕ[11].
3. Experimental setup and results
We use the experimental setup shown in Fig. 2 to directly estimate the overlap of three input photon states. Four horizontally polarized photons are generated by type-I noncollinear spontaneous parametric down-conversion (SPDC) in a β-BaB2O4 crystal pumped by laser pulses (center wavelength 390 nm, pulse duration 200 fs, repetition 76 MHz, average power 250 mW) propagating in a double-pass geometry. Three photons (photons 1 to 3) proceed to the input ports (I1, I2, and I3), and photon 4 is detected by a photon counter (D4) for heralding. Each photon passes through an interference filter (IF) with a half-maximum bandwidth of 5 nm. Rotatable half-wave plates (HWPs) control the input polarizations. Trombone-type (OD1 and OD2) and Babinet-Soleil-type (OD3) variable optical delays (ODs) are inserted to match the optical lengths of interfering paths. The optical delays are adjusted with HOM-type experiments: we maximize photon bunching at the output ports by finding the maximum for coincidence counts at two photon counters concatenated with a fiber beam splitter (FBS) as shown in the dashed box in Fig. 2. A 1-mm-thick glass plate acts as a PS that sweeps the phase by varying its tilt angle. All the photons are coupled into single mode fibers before being counted by avalanche photodiode single photon counters (SPCs).
Fig. 2 Experimental setup. The components in the dashed box are used only during the alignment procedures. BBO: β-BaB2O4, IF: interference filter, ODi: variable optical delay, HWP: half-wave plate, Ii: input port, NPBS: non-polarizing beam splitter, PS: phase shifter, Oi: output port, SMF: single-mode fiber, FBS: fiber beam splitter, Di: single photon counter, CC: coincidence counter.
To verify our method, we first fix the polarizations of photons 1 and 3 to be horizontal and vary the polarization of photon 2 to be 0°, 30°, 60° and 90° with respect to the horizontal axis. Figure 3 shows four-fold coincidence counts as the phase is varied by the PS. The results clearly show the decrease in fringe visibility as the polarization is changed: 0.74 ± 0.06 in (a), 0.49 ± 0.06 in (b), 0.27 ± 0.06 in (c), and 0.09 ± 0.07 in (d). The uncertainties denote the standard deviation estimated from , the statistical uncertainty of coincidence counts. Note the nonclassical nature of the change in visibility: the interference disappears even though the only change in polarization occurs in the photon that does not travel along any interfering path. The nonideal visibility of 0.74 ± 0.06 in Fig. 3(a) can be attributed to the impurity of the photonic temporal/spectral state, which will be discussed in the next section. The relative visibilities of (b), (c), and (d) compared to (a) approximately follow the theory (∝ cos2(angle)) within the errors due to statistical uncertainty and imperfections of optical elements.
Fig. 3 Measured interference fringes. The polarizations of photons 1 and 3 are fixed to be horizontal. The polarization of photon 2 is varied: (a) 0°, (b) 30°, (c) 60°, and (d) 90°.
Birefringence in the optical components can introduce errors when the current setup is applied to arbitrary polarization states. Since the measurement setup in Fig. 3 cannot detect the birefringence between horizontal and vertical polarizations, we test the birefringence effect by setting the polarizations of all input photons along 45°. The measured four-fold coincidence counts show a visibility of 0.75 ± 0.07 as shown in Fig. 4, which agrees with Fig. 3(a) and therefore reveals no significant birefringence effect.
Fig. 4 Interference fringes with the polarizations of all input photons linear along 45° with respect to the horizontal axis.
We also vary the overlap of temporal and spectral profiles of the photons. For the measurements of Figs. 5(a) and 5(b), photon 2 is delayed by +w and −w, respectively, where w is the half-maximum temporal width (26 μm) obtained from the HOM experiments. The measured visibilities are respectively 0.37 ± 0.06 and 0.34 ± 0.07. The measurement of Fig. 5(c) corresponds to the case where interference filters with a greater bandwidth (10 nm) are used for photons 3 and 4. The visibility becomes 0.52 ± 0.05 due to the reduction in the spectral purity of photon 3.
Fig. 5 Interference fringes with varying the temporal/spectral overlap between photons. Photon 3 is temporally delayed by (a) + 26 μm and (b) − 26 μm. (c) The half-maximum filter bandwidth of the interference filters for photons 3 and 4 is changed from 5 nm to 10 nm.
4. Discussion
The nonideal visibility in Fig. 3(a), 0.74 ± 0.06, can be attributed to two kinds of experimental imperfections. The first is minor imperfections in the optical elements such as imbalance between transmission and reflection at NPBSs, finite wavelength bandwidth of HWPs, slight differences between interference filters and phase instability of the interferometer. The reduction in visibility due to these factors is estimated to be less than 10% in the current experimental condition. The other, more major imperfection stems from the SPDC photon source. Photons from SPDC generally are not in a pure spectral/temporal state because of pairwise spectral entanglement [18]. The purity of the biphoton state can be further degraded when the SPDC pump pulses are not Fourier transform-limited. The reduction in visibility due to the spectral impurity of the single photons can be estimated from HOM interference measurements on two photons either from the same SPDC process or from different SPDC processes.
Figure 6 shows the two-fold coincidence counts of the two output ports of the FBS connected to O1 or O2 shown in Fig. 2. The background counts in the graph denote the coincidences due to two photons from the same input port. The visibilities in Figs. 6(a) and 6(b), 0.94 ± 0.03 and 0.78 ± 0.11, correspond to the overlap between photons 1 and 2 and between photons 2 and 3, respectively. Photons 1 and 2 originate from a single SPDC photon pair, therefore the HOM interference visibility is closer to unity regardless of the separability of the biphoton spectrum [18]. In contrast, photons 2 and 3 generated from independent SPDC processes have a lower level of indistinguishability because of the spectral entanglement with other unmeasured photons. The result is interference fringes with a lower visibility and a broader width as shown in Fig. 6(b).
Fig. 6 Hong-Ou-Mandel interferences by two photons from (a) a single pair (photons 1 and 2) and (b) different pairs (photons 2 and 3): Coincidence counts in (a) and (b) are measured by the two photon counters connected to O1 and O2, respectively, as shown in Fig. 2.
The overlap of n-photon states can be decomposed in terms of overlaps between two states if all states are pure:
When the states are not highly mixed states, the n-photon overlap can still be approximated by the product of two-photon overlaps to the first order: Application of the results of the HOM experiments to the above relation shows that the three input photons have an approximate overlap of 0.76 ± 0.08, which agrees with the result of Fig. 3(a). Here we have assumed that Tr[ρ3ρ1] equals Tr[ρ2ρ3]. A simple way to improve the visibility in Fig. 3(a) is to use wavelength filters with a narrower bandwidth while sacrificing the photon generation efficiency. We have chosen the current configuration to maintain the total measurement time to be less than the stability limit of our current setup (a few hours).5. Conclusion
We have experimentally demonstrated an optical interferometer that includes a C-SHIFT operation for three photons. A nonlinear functional Tr[ρ1ρ2ρ3] has been directly estimated for various input states via interference visibility. Our method is applicable to DOFs within single photon states, and may be extended to a multiphoton state that can be merged into a multiple-DOF single-photon state by means of a recently proposed qubit joining technique [19]. Reconfigurable photonic circuits [20] may be a useful approach to realize our C-SHIFT scheme for a greater number of input photons.
Acknowledgments
S. M. Lee thanks Jae Yong Lee for fruitful discussions about variable optical delays. This work has been supported by the KRISS project ‘Convergent Science and Technology for Measurement at the Nanoscale.’
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