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Abstract
High-dimensional entangled states and quantum repeaters are important elements in efficient long-range quantum communications. The high-dimensional property associated with the orbital angular momentum (OAM) of each photon improves the bandwidth of the quantum communication network. However, the generation of high-dimensional entangled states by the concentration method reduces the brightness of the entangled light source, making extensions to these higher dimensions difficult. To overcome this difficulty, we propose to generate entangled qutrits in the OAM space by loading the pump light with OAM. Compared with the concentration method, our experimental results show that the rate of generation of photon pairs improves significantly with an observed 5.5-fold increase. The increased generation rate provides the system with the ability to resist the noise and improve the fidelity of the state. The S value of the Clauser–Horne–Shimony–Holt inequality increases from 2.48 ± 0.07 to 2.69 ± 0.04 under the same background noise, and the fidelity of the reconstructed density matrix improves from 57.8 ± 0.14% to 70 ± 0.17%. These achievements exhibit the enormous advantages of high-dimensional entanglement generation.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With the rapid development of quantum information science, many traditional methods of secure communication will become less reliable, but quantum communication channels still hold promise for an absolutely secure transmission of encrypted messages [1]. High-dimensional entanglement photon pairs constitute an important resource to achieve this goal. Such pairs are highly compatible with existing quantum repeater schemes [2] and effectively help to suppress the exponential decay of fidelity with distance [3]. At the same time, compared with two-dimensional qubit systems, high-dimensional entangled systems offer a number of advantages such as higher information capacity [4–6] and greater robustness against coherent attacks [7–9]; they can also be used for device-independent quantum key distribution [10]. There are many ways to achieve high-dimensional entangled states through different degrees of freedom (DOFs), such as time-energy [11], paths [12], spatial modes [13–15], and combinations thereof [16]. Recently, photon OAM DOFs with infinite dimensions are attracting increasing attention [17,18]. Moreover, long-distance quantum communications based on OAM in free space and multimode fiber have been experimentally tested [19,20]. The approach for generating high-dimensional entangled photon pairs with OAM DOFs mainly includes spontaneous parametric down-conversion (SPDC) in crystals [21–24] and four-wave mixing (FWM) in atomic ensemble [2,25]. Although there have been many studies on high-dimensional entangled states [24,26] with high fidelity, methods employed to generate high-dimensional entangled states aside from the concentration method are still lacking, which is the main motivation for this work. The preparation of high-dimensional entangled states requires equalizing the different components. The most direct method is to attenuate components with a higher weight [22] through a concentration method, which leads to the loss of photon pairs. To avoid this loss, many studies have been proposed recently to directly generate high-dimensional entangled photon pairs by modulating the pump light beam during the SPDC process [27–30]. However, there is only one pump beam in the SPDC process. If this beam contains different OAM components, crosstalk between different OAM modes in the entangled state increases, thereby reducing the fidelity of the entangled state. Fortunately, there are two pump beams in the FWM process that can be loaded with different OAMs. Moreover, the weights of different components can be modulated while the total OAM remains fixed. In this way, no crosstalk is introduced. Loading OAM on the pump beams in the FWM process has proved that the spiral bandwidth of the generated light can be increased [31]. In addition, our previous work has demonstrated the feasibility of high-dimensional quantum storage based on atomic ensembles [2,32]. Here we propose a method for generating entangled qutrits by loading OAM on the pump fields. Our experiments based on atomic ensembles show that this method provides a much higher rate of generation of paired photons than the concentration method, and these entangled photon pairs are suitable for quantum information processing.
2. Theoretical analysis
We consider an atomic ensemble illuminated by two continuous-wave pump beams propagating in the z-direction and assume that the FWM conditions is met. This produces two highly correlated photons. Since energy is conserved, ${{\omega }_{P1}} + {{\omega }_{P2}} = {{\omega }_{S1}} + {{\omega }_{S2}}$, where the subscripts P1, P2, S1 and S2 refer to the Pump 1, Pump 2, Signal 1 and Signal 2, respectively. Since the photon pairs generated by FWM are entangled as a coherent superpositions of an infinite number of modes with OAM, they are best described by mode functions that are Laguerre-Gauss (LG) modes. The normalized LG modes are given, in cylindrical polar coordinates (ρ, φ, z), by:
Equation (3) allows us to calculate the exact two-photon states generated by two pump with any OAM. The calculation of the coincidence amplitudes can also be used to calculate the number of OAM modes participating in the two-photon state, also known as the spiral bandwidth [31,33]. The theory predicts that the spiral bandwidth of the photon pairs will increase from 1.02 to 2.14 after the pump fields are loaded with opposite OAM (l=±2). It means that more orthogonal modes could be entangled in such a state.
3. Experimental setup
In the simplified experimental setup [Fig. 1(b)], a collinear geometry is adopted for this work. We generate non-classical correlated two-color photon pair through FWM process in hot 85Rb atomic ensemble. The generated photons have wavelengths 780 nm and 1530 nm and are in the transmission window of the atmosphere and optical fiber, respectively. That is, these photons are suitable for hybrid optical communications between space and optical fiber networks [34–36]. There are four energy levels involved in our experiment, 5S1/2 (F=3), 5P1/2 (F’=3), 5P3/2 (F’=3), and 4D5/2 (F”=4), which form a diamond pattern [Fig. 1(c)]. A 4-cm-long 85Rb vapor cell is held at 65 °C, the corresponding atomic density is 4.97×1011 cm−3, and two pump fields are applied to it continuously.
Fig. 1. (a) Two-photon excitation by pump fields with opposite topological number eliminates the phase gradient in the azimuthal direction, thereby maintaining a total OAM of zero. (b) Experimental setup: two-color photon pairs (780 nm and 1530 nm) are generated via four-wave mixing (FWM) in the atomic vapor cell of 85Rb. Pump fields 1 (795 nm) and 2 (1475 nm) are focused and combined in the vapor cell. The FWM is based on a ladder-type atomic system with energy levels of the ground state |1>, intermediate states, |2> and |3>, and excited state |4〉, associated with 5S1/2, 5P1/2, 5P3/2, and 4D5/2 atomic terms, respectively. (c) energy diagram of the atomic system. The lenses (f=200 mm) are used to focus the pump fields and to collimate the diverging signals. Signals 1 and 2 are projected onto a specific OAM basis using SLM 1 and 2, respectively, and then detected by single-photon detectors.
Pump 1 (795 nm) from an external-cavity diode laser (DL100, Toptica) is 1.5 GHz blue-detuned from the transition 5S1/2(F=3)->5P1/2(F’=3), thereby improving the signal-to-noise ratio (SNR). Pump 2 (1475 nm) from another external-cavity diode laser (DL Prodesign, Toptica) is adjusted to excite the two-photon transition 5S1/2(F=3)->4D5/2(F”=4). The horizontal polarized Pump 1 and vertically polarized Pump 2 have power of 25 mW and 18.5 mW, respectively; Each pump beam is controlled by a half-wave plate and polarization beam splitter (PBS). To enhance the nonlinear interaction, the two pump fields are focused on the center of the 85Rb vapor cell with the focal length of 200 mm and a waist width of 50.6 µm and 93.9 µm. They are then combined in a long-pass dichroic mirror (Thorlabs DMLP1000), the transmission band of which is 1020 nm–1550 nm and the reflection band is 520 nm–985 nm. The two pump fields interact with the 85Rb atoms and generate pairs of photons at 780 nm (Signal 1) and 1530 nm (Signal 2).
The generated photon pairs fulfill phase matching and energy conservation as they come from a FWM process, and are collinear with the pump fields. Another long-pass dichroic mirror is used to separate the two-color photon pairs from each other. Two spatial light modulators (SLM Holoeye PLUTO) are used as post-selection tools for projecting Signals 1 and 2 onto OAM modes with different topological numbers, which are then collected by single mode fibers (SMFs) with efficiencies of 85% and 76%, respectively. Signal-1 photon is detected by an avalanche diode (APD) detector (PerkinElmer SPCM-AQR-15-FC, 50% efficiency); Signal-2 photon is delayed by a 100-m long multimode fiber for about 500 ns, and then detected by a gate mode detector (Qasky WT-SPD100, InGaAs Photon Detector with 10% detection efficiency). The infrared single photon detector is triggered by electronic pulses that are delayed by the DG535 after being generated by the APD detector. Because of the collinear geometry, the pump fields are the main sources of noise. In addition to isolating the pump fields under the polarization DOFs using the PBS, we also isolate it under the frequency DOF using three interference filters (Semrock 780 nm maxline) for Signal 1, and three bandpass filters (Semrock 1535nm single band) for Signal 2. According to the luminescence characteristics of the diode laser, the pump fields contain a small amount of wide spectral background fluorescence, which cannot be isolated by the above methods. We need to filter out the background fluorescence in the pump fields before applying them on the atomic ensemble. An interference filter (Semrock 808 nm maxline) for 808 nm is placed in the optical path of Pump 1 with an appropriate angle to isolate the background fluorescence. However, as Pump 2, has no suitable interference filter for 1475 nm, we use a grating to purify it. Through the above filtering methods, noise is suppressed to the dark count level when the pump fields are applied individually. To measure the cross-correlation function, we scan the delay generator DG535 with a step size of 0.1 ns (this step size is much smaller than the coherence time of two-photons). The measured cross-correlation function (Fig. 2) is obtained from a theoretical fit, which was adopted from the method proposed in Refs. [37] and [38]. The full width at half maximum is about 1 ns. The heralded single photon rate is 170 ± 13/s. Benefitting from the effective filtering methods, a SNR of 0.98 ± 0.005 was achieved. The SNR is defined as (S-N)/(S + N), where S denotes the signal coincidence at zero delay, and N the noise coincidence at non-zero delay. Although the cross-correlation counts decrease to 110 ± 10/s when the pump fields convey OAM, the SNR remains high at 0.976 ± 0.006.
Fig. 2. Measurement of the cross-correlation function for the two signals. Black dots mark experimental data; the red line is a theoretical fit to the data.
To obtain the distribution of spatial modes for the generated photon pairs, SLMs we added to act as spatial post-selection tools for projecting Signals 1 and 2 onto different OAM modes. The mode analysis was performed using a set of 49 coincident measurements, the operators of which can be expressed as |l1><l1|⊗|l2><l2|. The ket |l1> (|l2>) for Signal 1 (Signal 2) is chosen from the OAM eigenmodes with topological number l=−3∼3 modes. The SLMs convert the chosen OAM mode into a Gaussian mode, which is subsequently collected by the SMFs.
From the measured intensity distribution matrix of OAM modes [Fig. 3(a)], the probability is mainly distributed in the states where the total OAM of the signal lT= l1+ l2 is equal to that of the pump fields lT= lP1+ lP2 . Clearly, the total OAM is conserved in FWM process, which result in the OAM entanglement between Signals 1 and 2. From Fig. 3(a), the probability that the photon pair is in the Gaussian mode is much higher than that in other OAM modes. Therefore, the main problem of a higher-dimensional quantum system is in equalizing the weights of different components. There are two main ways to solve this problem. One is directly manipulating the quantum state of the generated photon pairs by applying Procrustean filtering [22]; the other is controlling the spatial shape of the pump laser beams [27,28]. These two ways have been investigated and applied for photon pairs generated by SPDC process. Nevertheless, for narrowband photon pairs generated from FWM process in an atomic ensemble, this problem is required to be solved. Similarly, Procrustean filtering is an alternative method used to reduce the amplitude of all OAM components to the lowest value. As the amplitude of component with the higher OAM is much smaller than that of the Gaussian mode, Procrustean filtering reduces the rate of generation of high-dimensional photon pairs considerably. For example, we may filter out the state $1/\sqrt 3 (|- 1\rangle|{1\rangle + |0\rangle} |0\rangle + |1\rangle|- 1\rangle)$ from the generated photon pairs by reducing the amplitude of the Gaussian component. The heralded single photon rate then decreases to 20/s. To generate high-dimensional entangled states at a high rate, we introduce the opposite OAM (l=±2) to the two pump fields, and keep other experimental parameters unchanged. In theory and experiment, arbitrary engineered entangled states can be prepared through SPDC to translate the topological information contained in a pump beam into the amplitudes of the generated quantum states [28]. Because the only pump field in the SPDC process is imprinted with a topological charge, the total OAM of the generated photon pairs is non-zero. Fortunately, we can engineer the entangled states without changing the total OAM, because there are two pump fields in the FWM process. In addition, this operation does not introduce any additional crosstalk, as the total OAM is unique. From the measured distribution of the OAM modes for pump fields imprinted with opposite topological number [Fig. 3(b)], the magnitudes of the three components (|−1>|1>, |0>|0>, |1>|−1>) are nearly equal, and the total OAM of the photon pairs is still zero. Moreover, the rate of generation is as high as 110/s, which is much higher than 20/s. Therefore, by loading OAM on the pump fields, the rate of generation of the high-dimensional entangled photon pairs is improved significantly. The theoretically calculated OAM mode distributions of the photon pairs for Gaussian pump fields [Fig. 3(c)] and for the pump fields with opposite OAM (l=±2) [Fig. 3(d)] show, in a comparison of experimental and theoretical results that the high-order OAM mode amplitude is smaller than the theoretical value because the collection efficiency of the higher order mode is lower. Experimental results that better match the theory may be obtained by adopting more optimized measurement methods [39,40].
Fig. 3. Measured distributions of correlated OAM modes (l=−3∼3) from photon pairs; (a) and (b) present experimental results whereas (c) and (d) are theoretical results: (a, c) When the pump fields are in Gaussian modes, the generated photon pairs are mainly in Gaussian modes. (b, d) When the pump fields convey OAM (l=±2), the generated photon pairs have a greater chance of being in other states.
4. Entanglement measurement
To confirm entanglement, we have to demonstrate that the photon pair state is not just a mixture but a coherent superposition of product states of the various spatial modes which obey angular momentum conservation.
4.1 Two-dimensional entanglement
At first, we simply characterize the entanglement in a two-dimensional subspace (|1> and |-1> basis). The Clauser–Horne–Shimony–Holt (CHSH) inequality is checked before and after the pump fields imprinted with OAM. The CHSH parameter S is defined in Refs. [41,42] as
To obtain maximal S, we select θA=0, θB=π/8, θA’=π/4, θB’=3π/8. The measured S is S=2.48 ± 0.07 when the pump fields are Gaussian and S=2.69 ± 0.04 when the pump fields convey OAM. The CHSH inequality is violated when S>2, which means that there exists entanglement between photon pairs in the two-dimensional subspace.
Moreover, we obtain the two-photon interference curves by setting the phase angle of Signal 1 to θA = 0° or 45° and measuring the coincidence rate at different angles θB of Signal 2. The experimental results [Figs. 4(a) and 4(b)] give the interference curves for the Gaussian pump fields and pump fields with OAM, respectively. If the visibility of the two-photon interference is >70.7%, the CHSH inequality is violated, proving that there is entanglement between the two photons. The visibility we obtain is 84%±4% at θA=0° and 92%±2.5% at θA=45° for Gaussian pump fields, both being larger than 70.7%, which clearly proves that there is entanglement in the two-dimensional subspace. The visibility for the pump fields with OAM is 95%±1% at θA=0° and 94%±1% at θA=45°, which proves the preservation of entanglement. The improvement in the visibilities is because of the increase in the rate of generation of photon pairs.
Fig. 4. Measured correlated coincidence rate for various orientation θB at θA=0° (red curve) and 45° (black curve), which are the interference curves for the two-dimensional subspace (±1) of the OAM entanglement. (a) and (b) correspond to Gaussian pump fields and pump fields with OAM, respectively. Error bars are the ±1 standard deviations. The measurement time for each data point is 10 s.
4.2 Three-dimensional entanglement
To determine the changes of the full states of the photon pairs in a three-dimensional subspace, two qutrit state tomography is performed before and after the pump fields are loaded with OAM. Here, we denote the single photon states with OAM of –ħ, 0, and +ħ by |L>, |G>, and |R>, respectively. The density matrices are reconstructed from a set of 81 coincident measurements on a different basis, which is realized by projecting Signals 1 and 2 onto the basis vectors chosen from |G>, |L>, |R>, $({|{G\rangle + } |L\rangle} )/\sqrt 2 $, $(|G\rangle + |R\rangle)/\sqrt 2 $, $({|{G\rangle + i} |L\rangle} )/\sqrt 2 $, $(|G\rangle + i|R\rangle)/\sqrt 2 $, $(|L\rangle + |R\rangle)/\sqrt 2 $, and $(|L\rangle - i|R\rangle)/\sqrt 2 $ These measurements were performed using SLMs and SMFs [43,44]. The real and imaginary parts of the reconstructed density matrix were plotted for the Gaussian pump fields [Fig. 5(a)] and for the pump fields with OAM (±2ħ) [Fig. 5(b)]. We obtain the fidelity from the reconstructed density matrices ρexp to the maximally entangled states using the expression F=<MES|ρ|MES>, where |MES > is the maximally entangled state $(|L\rangle|{R\rangle + |G\rangle} |G\rangle + |R\rangle|L\rangle)/\sqrt 3 $. The fidelity for the Gaussian pump fields is Fl=0=57.7%±1.4%, and the fidelity for the pump fields with OAM (l=±2) is Fl=±2=70%+1.8%. The Schmidt number of the quantum state is greater than or equal to 3 if the fidelity F>2/3 [43]. When the pump fields are imprinted with OAM, the fidelity improves to more than 2/3, which clearly indicates that the generated high-dimensional entangled state has been optimized. The main reason for the lower fidelity of the three-dimensional entangled state is that, photon pairs are randomly generated everywhere along the 4-cm-long vapor cell. As the photon pairs propagates, different OAM modes will accumulate different Gouy phases, which results in a phase difference between different OAM components. The photon pairs generated at different positions will have different phase differences, which results in the generated quantum states not being very pure, so the measured fidelity is not high. For a two-dimensional entangled state, the Gouy phases of the two components are the same, so there is no phase difference. Therefore, shorter vapor cell can be used to generate high-dimensional entangled states with higher fidelity.
Fig. 5. Real and imaginary parts of the reconstructed density matrices corresponding to the OAM entangled states $\psi = (|L\rangle|{R\rangle + |G\rangle} |G\rangle + |R\rangle|L\rangle)/\sqrt 3 $, when the pump fields correspond to Gaussian modes (a) and OAM modes (b). The photon pairs generated by the pump fields with OAM have a higher fidelity
5. Summary
We increased the preparation efficiency of high-dimensional entangled photon pairs by applying opposite OAM to two pump fields in the cascaded emission process. Because the total OAM of the two pump fields remains zero while the distribution of light power changed, the weights of the different components can be adjusted without changing the total OAM of the entangled photon pairs. In addition, we were able to prepare high-dimensional entangled states experimentally, and the rate of generation of photon pairs was greatly improved compared with the no-concentration method. Moreover, we measured the entanglement characteristics of the entangled photon pairs before and after modulation in two-dimensional and three-dimensional space. The generated high-dimensional entangled state gives a significant improvement in the fidelity of the density matrix relative to the maximally entangled state. Our work provides a method for an efficient preparation of high-dimensional entangled states. We intend to study further how to modulate the pump light field to prepare higher-dimensional entangled photon pairs efficiently in order to improve the information capacity of a single photon.
Funding
National Key Research and Development Program of China (2017YFA0304800); National Natural Science Foundation of China (11604322, 11934013, 61435011, 61525504, 61722510); Anhui Initiative in Quantum Information Technologies (AHY020200); Innovation Fund from CAS; Fundamental Research Funds for the Central Universities; Youth Innovation Promotion Association of the Chinese Academy of Sciences (2018490).
Acknowledgments
We thank Zhi-Yuan Zhou and Yan Li (USTC), for technical assistance.
Disclosures
The authors declare no conflicts of interest.
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