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Abstract
The high-dimensional encoding of single photons can offer various possibilities for enhancing quantum information processing. This work experimentally demonstrates the quantum interference of an engineered multidimensional quantum state through the space-division multiplexing of a heralded single-photon state with a spatial light modulator (SLM) and spatial-mode mixing of a single photon through a long multimode fiber (MMF). In our experiment, the heralded single photon generated from a warm 87Rb atomic ensemble was bright, robust, and long-coherent. The multidimensional spatial quantum state of the long-coherent single photon was transported through a 4-m-long MMF and arbitrarily controlled using the SLM. We observed the quantum interference of a single-photon multidimensional spatial quantum state with a visibility of >95%. These results may have potential applications in quantum information processing, for example, in photonic variational quantum eigensolve with high-dimensional single photons and realizing high information capacity per photon for quantum communication.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The various properties of photons, such as their polarization, time, frequency, spatial mode, optical path, and orbital angular momentum, allow for the advantages of photonic quantum states [1–11]. Photons not only afford frying qubits for quantum communication but also robust quantum states for quantum information processing [11]. Furthermore, multidimensional quantum states, namely, qudits of multiple states |1〉, |2〉, …, |d〉 (d ≥ 3), have been achieved by using the temporal, spatial, and polarization modes of photons [9–11]. In particular, the spatial modes of photons are useful for engineering arbitrary multidimensional quantum states using an integrated quantum photonic chip and a spatial light modulator (SLM) [11–13].
Meanwhile, complex media such as various types of scattering media and multimode fibers (MMFs) with spatial inhomogeneities have been applied for the implementation of high-dimensional quantum states in combination with wavefront shaping. Recently, it was reported that high-dimensional quantum states can be implemented via multimode scattering from complex media and wavefront shaping using adaptive quantum optics [13–23]. This approach offers the potential to enhance quantum information processing owing to the large number of controllable channels available in multiple-scattering media.
To control many channels in multiple-scattering media, the input channels are coupled to the output channels, and the mode-mixing process is necessary to afford interference among multiple optical paths. For the efficient multimode interference of many optical channels in scattering media and MMFs, the coherence length of a single photon should be longer than the path-length difference of the optical channels. Since the wavefront-shaping technique was first applied to control the propagation of a heralded single photon through an opaque scattering medium [24], many experiments have been carried out with photon pairs generated via the spontaneous parametric down-conversion (SPDC) process [18–27]. SPDC sources afford highly bright photon pairs; however, the typical coherence length of the SPDC without an optical cavity is from µm to mm [28,29]. To minimize the path-length difference of complex media, the scattering-medium thickness and the MMF length should be short [13–27].
Recently, the generation of correlated photon pairs from atomic ensembles using the spontaneous four-wave mixing (SFWM) process has been intensively studied [30,31]. The coherence length of the superradiant photons from SFWM sources in a warm atomic ensemble is ∼50 cm, significantly longer than that of photons from a typical SPDC source [30]. In particular, a bright and robust photon pair source from Doppler-broadened atomic ensembles has been experimentally realized [30–36]. In the Ref. [35], time–frequency-entangled subnatural linewidth photon pairs have been realized in paraffin-coated atomic vapor cells. The coherence length of subnatural linewidth photons is ∼50 times longer than that of Ref. [30]. However, the experimental demonstration of a multidimensional quantum state using an SFWM source generated from an atomic ensemble has thus far not been reported.
In this work, we experimentally demonstrate the quantum interference of photonic ququarts through the wavefront shaping of a heralded single photon generated from cascade-type warm 87Rb atoms using an SLM. The collective two-photon coherence effect for the 5S1/2–5P3/2–5D5/2 transition of the 87Rb atom ensures that the coherence length of the heralded single photon from the warm atoms is ∼0.5 m of relevance to the Doppler-broadening of the atomic ensemble [36]. The multidimensional quantum states of the single-photon state are achieved by the space-division multiplexing of the SLM. Our work is the first demonstration of the integration of the single-photon source generated from the warm atomic ensemble with the technology of wavefront control using the SLM. We demonstrate the quantum interference of a multidimensional spatial quantum state by controlling the superposition of the multiple states of a single photon.
2. Experimental setup
We briefly describe the heralded single photons generated from a Doppler-broadened cascade-type atomic ensemble based on the 5S1/2–5P3/2–5D5/2 transition of 87Rb. Figure 1(a) shows the energy-level diagram of the 5S1/2–5P3/2–5D5/2 transition of 87Rb and the SFWM process in a cascade-type atomic medium coherently interacting with pump and coupling lasers [30,31]. In this work, we used the same photon pair source as Ref. [31], which experimentally demonstrated the direct generation of photon pairs from a warm atomic ensemble in the cascade decays from atomic ensembles. Under phase-matching conditions, strongly correlated signal and idler photons are emitted via the SFWM process from the two-photon coherent atomic ensemble [31].
Fig. 1. (a) Energy-level diagram of 5S1/2–5P3/2–5D5/2 transition of 87Rb atoms. (b) Experimental schematic for realizing a multidimensional quantum state of the heralded single photon generated from cascade-type warm 87Rb atoms: SMF, single-mode fiber; MMF, multimode fiber; SLM, spatial light modulator; P, polarizer; BS, beam splitter; SPD, single-photon detector; EMCCD, electron-multiplying charged coupled device.
The signal and idler photons via SFWM process are generated under the phase-matching condition corresponding to the energy and momentum conservations. The phase-matching function $\varphi (\theta)$ as a function of the tilting angle of the signal and idler photon relative to the propagating direction of pump and coupling lasers can be expressed as
where Δk the wave-vector mismatch of the four fields, i.e., $\Delta k = {k_p} + {k_c} - {k_i} - {k_s}$, with ${k_{p,c,i,s}}$ denoting the wave vectors, respectively, of the pump and coupling fields, and the idler and signal photons; and L is the length of the 87Rb vapor cell. The frequencies of the emitted signal and idler photons satisfies the energy conservation relation $\omega_p{ + \omega_c}{ = \omega_s}{ + \omega_i}$, where $\omega_s$ and $\omega_i$ represent the angular frequencies of the signal and idler photons, respectively. When the signal and idler photons are collected in two single-mode fibers, the $\varphi (\theta)$ of the idler photon aligned at 1.5° of signal photon is shown in Fig. 2. We can see that the counter-propagating photon pairs travel in nearly single spatial mode. Therefore, the spatial modes of signal and idler photon collected in two single-mode fibers is well overlapped and the spatial coherence of the counter-propagating photon pairs is excellent because of the spatial mode-filtering effect.Fig. 2. Numerically calculated $\varphi (\theta)$ of the idler photon aligned at 1.5° of signal photon and the 12.5-mm-long atomic ensemble.
The experimental schematic for realizing the four-dimensional quantum states of the heralded single photon generated from cascade-type warm 87Rb atoms is shown in Fig. 1(b), where the idler photon is a trigger photon at a single-photon detector (SPD2). In our experiment, when the powers of the pump and coupling lasers were 20 µW pump and 0.3 mW, the single-counting rates of the signal and idler photons were measured to be 250(2) kHz and 196(2) kHz, respectively. The photon-pair coincident count rate was measured to be 3.8(1) kHz with a 3.5 ns coincidence window. To demonstrate a photonic ququart, the signal photons were guided to the wavefront-shaping setup with an SLM and an MMF, as shown in Fig. 1(b). Here, the signal photon from the single-mode fiber (SMF) is spatially filtered and propagated to the center of the SLM. The spatial mode of the signal photon on the SLM is TEM00 (red circle indicated on the SLM) and can be divided and programmed into a four-dimensional quantum state (white dashed line). We can arbitrarily control the phases of each section by the space-division multiplexing of the SLM and easily control the superposition of multiple states.
The MMF functions as a mode-mixing circuit via combining the phase patterns of mode-by-mode wavefront-shaping optimizations [13]. The signal photon reflected by the SLM is launched into a 4-m-long graded-index MMF with a 62.5 µm core diameter and a 0.27 numerical aperture. The transverse spatial and polarization modes at 780 nm were estimated to be ∼1200. There is the maximum path-length difference between the short and long paths in the MMF. While the signal photon propagates through the MMF with random phases and amplitudes, the spatial modes mix, and a speckle pattern is observed at the output. The photons emerging from the MMF are separated using a beam splitter (BS). The output plane of the MMF is imaged on an electron-multiplying charged coupled device (EMCCD; Andor Technology, iXon Life 888) to monitor the wavefront-shaping process. In our setup, the input mode was controlled by ∼900 segments on the SLM, corresponding to 30 × 30 SLM macro-pixels. The optical pattern on the SLM was displayed corresponding to the range from 0 (white) to 2π (black). Here, we optimize the output mode by iterating the optimal phase for each SLM segment, which results in the maximum constructive interference at a point. In the figure, we can observe the output plane images with the EMCCD before and after the iterative optimization of the SLM. In another path of the BS, we can measure the quantum interference of a four-dimensional quantum state using a single-photon detector (SPD1; Excelitas Technologies, SPCM-AQ4C). We directly coupled to the MMF with a 62.5 µm core diameter connected to SPD1. We don’t need a spatial mode filtering for the observation of interference fringe, because we optimize the output mode at a point by iterating the optimal phase for each SLM segments of 900. The optimum wavefront for focusing is included the mode-mixing process of many optical channels in scattering media and MMFs. Therefore, each state of space-division multiplexed single-photon is indistinguishable in the focal image after iterative optimization. However, we should adjust the position of the MMF considering the image plane of the EMCCD.
3. Experimental results and discussion
For realizing the quantum interference of space-division multiplexed single-photon ququarts, we combined a spatial single mode via wavefront optimization with the SLM and the MMF. We used ‘stepwise sequential’ algorithm for the optimum wavefront for focusing [37]. In our experiment, this algorithm in the MMF was implemented by using the phase pattern displayed on the SLM and the image of the EMCCD. The number of typical iterations is ∼500 and the time of iterations is ∼40 minutes. We note here that the brightness and long-coherence of our photon source allow for the implementation of this direct wavefront optimization technique. The phase pattern displayed on the SLM is optimized to focus at a single point. Figure 3(a) shows the output plane images obtained with the EMCCD camera before and after using the iterative optimization phase of the SLM. The single-photon rate after (blue points) iterative optimization shows more than 30-fold enhancement compared with the average single-photon rate before optimization (red points) with a constant phase pattern being displayed on the SLM. We applied to the concept of feedback-based wavefront shaping for focusing, but did not need additional control of the output spatial profile to observe the interference fringes.
Fig. 3. (a) Output plane images recorded with the EMCCD camera before and after wavefront optimization (upper panel). The red and blue dashed lines on the EMCCD output plane images before and after optimization represent the position of the scans of 1D intensity profiles (red points for before and blue points for after optimization. (b) Normalized cross-correlation function, $g_{si}^{(2)}$, between focused signal and trigger idler photons.
The focusing signal photon is measured in coincidence with the trigger idler photon using a time-correlated single-photon counter (TCSPC) in the start-stop mode with a 4 ps time resolution and a 3.5 ns coincidence window. Figure 3(b) shows the normalized cross-correlation function $g_{si}^{(2)}$ between the signal photon detected by SPD1 and idler photons detected by SPD2 (trigger detector), where the x-axis represents the time delay from the signal to the idler photons. We find that $g_{si}^{(2)}$ exhibits a full-width at half maximum (FWHM) of ∼1.7 ns, corresponding to the Doppler-broadening of the warm 87Rb atoms at a temperature of 56 °C. Here, the maximum value of $g_{si}^{(2)}$ was measured to be 27 for an acquisition time of 3 min, which clearly indicates the temporal correlation between signal and idler photons emitted via the SFWM process.
We can deterministically manipulate the phases of each segment by the space-division multiplexing of the SLM, as shown in Fig. 4(a). This space-division multiplexing provides the advantage of the relative ease of controlling the superposition of multiple states. Here, we note that a d-dimensional pure state $|\Psi \rangle$ is a d × 1 column vector in d-dimensional Hilbert space. The four segments (d = 4) of the SLM in Fig. 4(a) are expressed as single-photon four-dimensional quantum states as follows:
Fig. 4. (a) Schematic for the space-division multiplexing of a single photon (red circle) by delineating four segments ($|1 \rangle$, $|2 \rangle$, $|3 \rangle$, and $|4 \rangle$) on the SLM surface. (b) Normalized counts of the heralded single photon in the ${e^{i{\phi _L}}}|L \rangle + {e^{i{\phi _R}}}|R \rangle$ state, when the relative phases (${\phi _L}$ = 0, π/2, π, and 3π/2) of ${\phi _R}$ change from 0 to 2π in 234 steps.
First, when the relative phases of two segments on the SLM surface undergo the same variation from 0 to 2π in 234 steps, we can obtain the quantum interference fringes of the spatial qubit in the case of horizontal space division, as shown in Fig. 4(b), where all the amplitudes are identical. The fringes are programmed to superimpose the input states, ${|\Psi \rangle _{LR}} = {e^{i{\phi _L}}}|L \rangle + {e^{i{\phi _R}}}|R \rangle$, where $|L \rangle = |1 \rangle + |3 \rangle$ and $|R \rangle = |2 \rangle + |4 \rangle$. The interference fringes of the single-photon qubit as a function of ${\phi _R}$ were observed to have >95% visibility in the four cases of ${\phi _L}$. Here, visibility is defined as , where ${N_{\max }}$ and ${N_{\min }}$ denote the maximum and minimum count rates of a heralded single photon, respectively.
Next, as shown in Fig. 5, we consider two different cases (vertical and diagonal) for implementing the spatial qubit as follows: the red curve represents the case of the superposition between the upper and lower segments, and the blue curve represents the case of the superposition between diagonal and anti-diagonal segments $|D \rangle + {e^{i{\phi _A}}}|A \rangle$, where $|u \rangle = |1 \rangle + |2 \rangle$, $|l \rangle = |3 \rangle + |4 \rangle$, $|u \rangle + {e^{i{\phi _l}}}|l \rangle$, $V = ({{N_{\max }} - {N_{\min }}} )/({{N_{\max }} + {N_{\min }}} )$$|D \rangle = |1 \rangle + |4 \rangle$, and $|A \rangle = |2 \rangle + |3 \rangle$. Interference fringes of the single-photon qubit with visibility of 95.5(5)% and 95.3(5)% can be clearly observed for the vertical and diagonal cases, respectively. The error value of each visibility is estimated from the standard deviation of the fitting curve.
Fig. 5. Quantum interferences of single-photon qubit in two different cases: (a) $|u \rangle + {e^{i{\phi _l}}}|l \rangle$ (red curve) and (b) $|D \rangle + {e^{i{\phi _A}}}|A \rangle$ (blue curve).
Next, we measured the interferences of the four-dimensional spatial quantum states in the cases of ${\phi _1} = 0$, ${\phi _2} = \phi$, ${\phi _3} = 2\phi$, ${\phi _4} = 3\phi$, and ${\phi _1} = 0$, ${\phi _2} = 3\phi$, ${\phi _3} = 6\phi$, ${\phi _4} = 9\phi$, as shown in Figs. 6(a) and 6(b), respectively. These measured interferences correspond to the interference fringe from the four slits. The experimental results are in good agreement with the theoretical results. To obtain the interference with high-visibility, all amplitudes of each segment by the space-division multiplexing of the SLM were identical. Also, we used a polarizer as the polarizing filter during the optimization to minimize the polarization mixing effect. The interference visibility is estimated to be ∼95.4(5)%. Considering the cross-correlation function of the photon pair in Fig. 3(b), the measured interference visibility is limited to the accidental count of the photon pair generated from the warm Rb atomic ensemble. Furthermore, we can arbitrarily control the phases of each section by the space-division multiplexing of the SLM and thereby easily control the superpositions of photonic qudits (d = 3, 5, 6, …, 100).
Fig. 6. Quantum interference of four-dimensional spatial quantum states in the two cases of: (a) ${\phi _1} = 0$,, ${\phi _3} = 2\phi$, ${\phi _4} = 3\phi$ and (b) ${\phi _1} = 0$, ${\phi _2} = 3\phi$, ${\phi _3} = 6\phi$, ${\phi _4} = 9\phi$; the red squares denote the experimental results and the red curves are calculated from the ideal ququart states. ${\phi _2} = \phi$
The number of propagation modes of the MMF is limited by the maximum number of spatial modes, the maximum d of qudit in high dimensional quantum states. However, practically, all propagation modes of the MMF can be not completely controlled for implementation of high dimensional quantum states. In our experiment, the number of spatial modes in practice may be limited to approximately 200, corresponding to the steps of the optical pattern on the SLM from 0 to 2π. However, we cannot absolutely confirm the cause of the limit on the number of spatial modes.
4. Conclusion
In conclusion, we experimentally demonstrated the quantum interference of a single-photon multidimensional quantum state generated from a warm 87Rb atomic ensemble through space-division multiplexing with an SLM and the spatial mode mixing of a single photon through a long MMF. We deterministically manipulated the phases of each segment by the space-division multiplexing of the SLM, and we observed the quantum interference of a single-photon multidimensional spatial quantum state with a visibility of >95%. In this study, we could easily control the superpositions of photonic ququarts by adjusting the phases of each section using the SLM. We believe that the realization of a multidimensional photonic quantum state promises a high information capacity per photon for quantum computers and quantum communication. Single quantum particles in higher dimensions can be used for various applications in photonic variational quantum eigensolve [38], in questions about possible representations using hidden variables models [39], and realizing high information capacity per photon for quantum communication.
Funding
Ministry of Science and ICT, South Korea (IITP-2022-2020-0-01606); National Research Foundation of Korea (2020M3E4A1080030, 2021R1A2B5B03002377).
Acknowledgment
We are grateful to Wonshik Choi for supporting the technology of wavefront control using the spatial light modulator (SLM).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper may be available from the corresponding author upon reasonable request.
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