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A high efficiency method for the generation of correlated photon pairs accompanied by reliable means to characterize the efficiency of that process is needed in the study of entangled states, which have important potential applications in quantum information and quantum communication. In this study, we report the first characterization of the efficiency of generation of correlated photon pairs emitted from a CuCl single crystal using the biexciton-resonance hyper-parametric scattering (RHPS) method which is the highly efficient method of generation of correlated photon pairs. In order to characterize the generation efficiency and signal-to-noise ratio of correlated photon pairs using this method, we investigated the pump power dependence on the photon counting rate and coincidence counting rate under resonant excitation. The pump power dependence shows that the power characteristic of the photon counting rates changes from linear to quadratic dependence of the pump power. This behavior represents a superposition of contributions from correlated photon pairs and non-correlated photons. The analysis of the pump power dependence shows that one photon-pair is produced by a pump pulse with 2 x 106 photons. Moreover, the generation efficiency of this method obtained by calculating the number of generated photon pairs per pump power is comparable to that of several methods based on the χ(3) parametric process.
© 2016 Optical Society of America
1. Introduction
Quantum entangled photon pairs are essential to realize various quantum-information processing protocols such as quantum teleportation [1] and quantum cryptography [2]. Entanglement shared by more than three particles, which is referred to as “multipartite entanglement,” is a significant resource for quantum simulation [3–5], quantum lithography [6], and quantum computation [7–9]. In order to study entangled states and apply them to quantum information and communication, highly efficient detection systems and highly efficient sources of entangled photon pairs are required [10, 11].
The generation of entangled photon pairs from a CuCl single crystal via biexciton-resonance hyper-parametric scattering (RHPS) has previously been reported [12, 13]. The polarization entanglement of photons emitted from a CuCl crystal via the RHPS process has been confirmed by investigating the polarization correlation between two photons. The RHPS method applied to a CuCl crystal has been reported to be a highly efficient method to generate entangled photon pairs. However, there are no reports that quantitatively characterize the efficiency of this method. In order to apply the RHPS method using CuCl to the generation source for entangled photon pairs and multipartite entanglement, it is necessary to first clarify the efficiency of the method in producing entangled photon pairs.
The RHPS method is related to a third-order nonlinear parametric process. In the RHPS method using CuCl, the third-order nonlinear susceptibility χ(3) is significantly enhanced by utilizing the resonance of two-photon absorption to the biexciton state of CuCl. It is effective to utilize laser pulses for the measurements of the coincident counts of photons emitted by the RHPS method. However, when the coincidence count rates within a coincidence window are measured by using laser pulses, signals from non-correlated photons, which originate from the Rayleigh scattered light and the luminescence from the bound excitons, are non-negligible. These non-correlated photons are recorded as background counts, which are mostly coincident in the duration time of laser pulses under the measurements of time-correlated histograms. Therefore, the background counts prevent the estimation of the efficiency of generating correlated photon pairs. On the other hand, the number of correlated photon pairs obtained by the RHPS method is proportional to the square of the pump power. This results in a difference in the pump power dependence between correlated photon pairs and non-correlated photons. Thus, the pump power dependence of photon counting rate is expressed as a superposition of contributions from correlated photon pairs and non-correlated photons. Therefore, the investigation of the pump power dependence enables us to estimate the generation efficiency of correlated photon pairs.
In this paper, we report the first quantitative characterization of the generation efficiency of correlated photon pairs using the RHPS method with CuCl. The photon counting signals are obtained by a method that enables us to acquire time-tag data (TTD), which represents a simultaneous measurement of the photon counting rate and the coincidence counting rate. The efficiency of generating correlated photon pairs is characterized by investigating the pump power dependence of the photon counting rate and the coincidence counting rate from TTD. The pump power dependence reveals that the obtained efficiency of correlated photon pairs is comparable to that of several other methods relying on the χ(3) parametric process.
2. Experiment
A CuCl single crystal sample with a size of approximately 5 x 5 x 0.1 mm3 grown from vapor phase was used for the experiment. The temperature of the sample was maintained at 10 K in a cryostat. A schematic of the experimental setup for the RHPS method is shown in Fig. 1(a). The pump pulses were the second harmonic light of a mode-locked Ti:Sapphire pulse laser at a repetition rate of 80 MHz. For excitation at the two-photon resonance of the biexciton state in CuCl, the center wavelength and spectral width of the pump pulses were selected to be λ = 389.0 nm and Δλ = 0.2 nm, respectively, by using a 4f optical system composed of two lenses, two gratings, and a slit. The duration of the pump pulses formed through the 4f optical system was approximately 1 ps. The pump pulses passing through a neutral density filter were focused on the sample, where the spot size of the pump pulses on the sample was approximately 100 μm. The photons emitted from the CuCl single crystal were led into optical multi-mode fibers marked as path-A and path-B in conjunction with two monochromators. The angle between the direction of transmitted pump pulses and the direction of detected photons from RHPS was approximately 45°. The spectrum observed using a charge-coupled device (CCD) camera attached to a monochromator is shown in Fig. 1(b). The two side peaks around the pump pulse at 389.0 nm indicate the polariton modes generated through RHPS. These two peaks observed at 389.9 and 388.1 nm correspond to a high energy polariton (HEP) and low energy polariton (LEP), respectively. Moreover, the two peaks indicate the signals due to correlated photon pairs by taking into account the phase matching condition. The photons emitted from the HEP and LEP were led to two monochromators labeled MA and MB that are connected to photo-multiplier tubes (PMTs) marked as det-A and det-B, respectively. The quantum efficiency of PMTs were approximately 30% around 400 nm. The photon counting signals were detected by each PMT in conjunction with a time correlator. We utilized the acquisition method of TTD for time correlation. The TTD method records all the arrival times of detected photons. It has a large advantage over a time interval analyzer (TIA), because the photon counting rates and time-correlated counting rates can be simultaneously obtained using the TTD method. The temporal resolution and dead time of the TTD method used in this study were approximately 165 ps and less than 10 ns, respectively. The upper limit of the time correlation counting rate of the TTD method is higher than that of a usual TIA.
Fig. 1 (a) Schematic of the experimental setup for the resonant hyper-parametric scattering (RHPS) method. ND: neutral density filter, OF: optical multimode fiber, MA and MB: monochromators, det-A and det-B: photon counting detectors (photo-multiplier tube). (b) RHPS spectrum for a CuCl single crystal. The central peak indicates the scattered light of the pump pulse (389.0 nm). The two side peaks around the pump pulse are the RHPS signals of the high-energy polariton (HEP; 388.1 nm) and the low-energy polariton (LEP; 389.9 nm). The peaks MT and HEP' are other RHPS signals that propagate in counter directions. The peak ML is the emission from the biexciton leaving the longitudinal excitons.
3. Results and discussion
The time-correlation histograms obtained at various peak pump powers are shown in Fig. 2. The coincidence signals at τ = 0 are clearly observed at all pump powers. Here, the difference between the coincidence signals at τ = 0 and τ ≠ 0, marked with CS, indicates the coincidence counting rate of correlated photon pairs. The average value of the coincidence signals at τ ≠ 0, marked with CR, indicates the coincidence counting rate of accidental photons due to background photons and photon pairs. Here, the background photons are the non-correlated photons that originate from the luminescence of bound excitons and photons due to Rayleigh scattered light at the samples, and the number of background photons shows a linear dependence on pump power. The time-correlation histograms at various pump powers clearly illustrate that as the pump power increases, the values of CS and CR increase, while the true-coincidence to accidental-coincidence ratio (CS/CR ≡□CAR) decreases. The obtained variation of CAR with an increase in the pump power is consistent with the theoretical variation of CAR which shows that the value of CAR decreases with increasing the number of correlated photon pairs [14]. At the peak pump power of 100 mW, the value of CS obtained using the TTD method is approximately 200 times greater than that obtained using a TIA and a pulse laser with a repetition rate of 80 MHz [12]. Moreover, the maximum value of CAR is approximately 80, which is four times that obtained using a TIA and a pulse laser with a higher repetition rate of 1 GHz [13]. These results indicate that the simultaneous measurement of photon counting rates and coincidence counting rates using the TTD method is well capable of obtaining the coincidence signals in the wide range of 0.0034 – 1.1 W.
Fig. 2 Time-correlation histograms of the observed photon pairs at various peak pump powers. The dashed line indicates the mean count rate of τ ≠ 0 and arrows indicate the value of CS and CR.
We confirmed the polarization entanglement of correlated photons by investigating the density matrix of the two-photon polarization state for correlated photon pairs emitted by the RHPS method [12, 13]. The value of fidelity which correspond to the entanglement of correlated photon pair was estimated to be 0.91. For correlated photon pairs with the polarization entanglement, we estimated the generation efficiency of correlated photon pairs from the photon counting rates and coincidence counting rates obtained by the RHPS method using CuCl. As shown in Fig. 2, the coincident count rates at τ = 0 obtained by using the pump pulses consist of both correlated photon pairs and non-correlated photons. Since the non-correlated photons emitted from the CuCl single crystal is non-negligible, the non-correlated photons prevent from estimating the generation efficiency of correlated photon pairs from only an observation of the photon counting rate and coincidence counting rate. Then, we estimated the generation efficiency of correlated photon pairs from the pump power dependence of a number of generated photons. The number of correlated photon pairs gpair is proportional to the square of the pump power I because the RHPS process is related to the third-order non-linear parametric process. However, the number of background photons gbg is proportional to the pump power, as mentioned above. gpair and gbg are expressed as
where α is the generation efficiency of correlated photon pairs, and βA and βB are that of background photons emitted in the direction of path-A and path-B, respectively. When the total detection efficiency for the directions of path-A and path-B is denoted by ηS;A and ηS;B, the photon counting rate obtained by each detector of det-A and det-B is represented byThis equation indicates that the photon counting rates obtained by the respective detectors are expressed by the superposition of the pump power dependence of correlated photon pairs and that of background photons, which enables us to quantitatively estimate the generation efficiency of correlated photon pairs. The photon counting rates detected in the directions of path-A and path-B are plotted as a function of pump power in Fig. 3(a). As the pump power is increased, the power characteristic of the photon counting rates gradually changes from linear to quadratic dependence of the pump power. The generation efficiency is estimated by fitting Eq. (2) to the pump power dependence. For this fitting, the total detection efficiency for each detection of path-A and path-B was set to ηS = ηS;A = ηS;B = 2.5%, which was estimated from losses of optical components and quantum efficiency of PMTs. From the fitting procedure, the generation efficiency of correlated photon pairs was estimated as α = 5.1 x 10−7 pairs/pulse/mW2, which demonstrates that one correlated photon pair is produced by a pump pulse that contains 2 x 106 photons. The high generation efficiency is caused by the excitation at the resonance energy of the two-photon absorption to the biexciton state of CuCl. We will later discuss a comparison of the generation efficiency by the RHPS method and several generation methods that use the χ(3) parametric process. Moreover, the generation efficiency of background photons was βA = 2.0 x10−5 photons/pulse/mW, and βB = 4.7 x 10−5 photons/pulse/mW, respectively. The generation efficiency of background photons in the direction of path-B is slightly larger than that for path-A. This difference is caused by the luminescence from the bound excitons at 389.8 nm.Fig. 3 (a) Pump power dependence of photon counting rates CA and CB. The solid lines show the fitted results obtained using Eq. (2). The dotted and dashed lines show the linear and quadratic functions, respectively. (b) Pump power dependence of coincidence counting rates CS, CR, and CAR. The blue solid line is the fitted result. The red and green solid lines are the calculated results using Eq. (5) and (6).
Next, we discuss the pump power dependence of the coincidence counting rate. The time-correlation function of the photon counting rates for det-A and det-B is represented as a function of the delay time τ between photons as
Here, CA(t) and CB(t) indicate the photon counting rates as functions of time t by the detectors of det-A and det-B, respectively. Using the time-correlation function and Eq. (2), the coincidence counting rates of correlated photon pairs CS and accidental photons CR, and the value of CAR are given as a function of pump power I by [15–17] where ηX is the ratio of the experimentally detected number of correlated photon pairs to the ideal detected number of those, corresponding to the detection loss of correlated photon pairs. The experimentally detected number of correlated photon pairs indicates the number of photon pairs simultaneously observed by both the detectors of det-A and det-B. Figure 3(b) shows the pump power dependence of the coincidence counting rates for CS, CR, and CAR. The value of CS is proportional to the square of the pump power, while the value of CR changes from the quadratic dependence to the fourth power dependence with increasing pump power. The pump power dependence of CS and CR is consistent with that represented by Eqs. (4) and (5). Here, it is noted that the generation of multiple pairs will deviate the values of CA, CB, and CS from Eqs. (2) and (4). Then, the generation rate of multiple pairs is estimated to be 0.01 per pulse at the pump power of 1.1 W. This estimated value indicates that the number of multiple pairs is negligibly small in the present study. Therefore, the value of ηX was estimated to be 20.3% by fitting Eq. (4) to the pump power dependence of CS, where the parameters ηS and α were set to the values estimated above. Moreover, the pump power dependence of CR and CAR was calculated by using the values of ηS, α, βA and βB obtained above. The calculated pump power dependence of CR and CAR well reproduces the experimental results, which confirms that the obtained efficiency is valid. Here, as the pump power decreases, the value of CAR increases and converges to a maximum value at the low pump power limit. The reason of this convergence is that the both of the pump power dependence of CS and that of CR show a quadratic dependence in the low pump power limit. Using the low pump power limit in Eq. (6), the maximum value of CAR is expressed asThe value of CARmax was estimated as approximately 100 from the parameters obtained above. Equation (7) shows that the value of CARmax is given by the ratio of the generation efficiency of correlated photon pairs to that of background photons, which is applicable only to the generation method of correlated photon pairs using the χ(3) parametric process. Then, to directly compare the generation efficiency of the RHPS method with that of several generation methods using the χ(3) parametric process, we define CAR'max by taking account of the total number of photon pairs (ηX = 1), as a quantitative indicator presented byIn the present RHPS method, the value of CAR'max was obtained as 550, and was compared with that obtained by an inelastic four-photon scattering (FPS) method and a spontaneous four wave mixing (SFWM) method which are well-known generation methods of correlated photon pairs using the χ(3) parametric process. The values of CAR'max obtained by the FPS method using a dispersion shifted (DS) fiber cooled to 4 K with a 1.6 nm linewidth filter and the SFWM method using a silicon-waveguide (Si-WG) structure operating at room temperature with 0.14 nm linewidth filter were estimated to be 1300 and 93, respectively [18, 19]. The value of CAR'max obtained by the RHPS method using CuCl is comparable to the above-reported values, which clarifies that the background photons in RHPS are as few as those of the two kinds of generation method.Finally, we compare the generation efficiency of correlated photon pairs by the RHPS method with that obtained by several other generation methods that use the χ(3) parametric process, calculating the number of generated photon pairs per peak power of pump pulses. The generation efficiency of correlated photon pairs by the FPS method using a DS fiber and the SFWM method using a Si-WG structure is estimated to 1.7 x 10−6 and 2.2 x 10−7 pairs/pulse/mW2, respectively [18, 19]. The generation efficiency of the RHPS method using CuCl which is 5.1 x 10−7 pairs/pulse/mW2 as mentioned above is comparable to that of the FPS and SFWM methods. Moreover, we try to compare the generation efficiency of the RHPS method with that of the generation method utilizing spontaneous-parametric down-conversion (SPDC) in periodic-poled potassium titanyl phosphate crystals (PPKTP) waveguide, which is frequently utilized to generate correlated photon pairs. Because SPDC is related to the χ(2) parametric process and the generated number of correlated photon pairs via SPDC is proportional to the pump power, it is difficult to compare the generation efficiency of the SPDC method with that of the RHPS method by estimating the number of generated photon pairs per peak power of pump pulses. Then, we estimate the number of photons necessary to generate a pair of correlated photons NP, as simple comparison. The value of NP obtained by using the RHPS method is 2 x 106 as mentioned above. On the other hand, the value of NP for the SPDC method using PPKTP waveguide is estimated to be 2 x 108 [20]. These comparisons demonstrate that the RHPS method using CuCl is a highly efficient method to generate correlated photon pairs.
4. Conclusion
We have investigated the generation efficiency of correlated photon pairs emitted from a CuCl single crystal using the RHPS method by acquisition of TTD which simultaneously provides photon counting rates and coincidence counting rates. The TTD method improves the detection efficiencies of the coincidence counting rate and the true-coincidence to accidental-coincidence ratio, which were 200 and 4 times greater than those obtained using a TIA, respectively. The generation efficiency of correlated photon pairs was evaluated by investigating the pump power dependence of the photon counting rate and coincidence counting rate due to correlated photon pairs and background photons. The evaluated generation efficiency of correlated photon pairs obtained by the RHPS method using CuCl is comparable to that obtained by several generation methods that utilize the χ(3) parametric process. This result demonstrates that the RHPS method using CuCl is an efficient generation method of correlated photon pairs.
Acknowledgments
This work was supported by Research Foundation for Opto-Science and Technology, Support Center for Advanced Telecommunications Technology Research, and JSPS KAKENHI Grants No. 24654092 and No. 26610088. We thank Dr. S. S. Garmon for useful comments.
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