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Abstract
We demonstrate a high brightness (∼2.36 × 105 pairs/s/mW) polarization-entangled photon-pair source at 800-nm via spontaneous parametric down-conversion (SPDC) in a 3-cm long type-II ppKTP crystal pumped unidirectionally in a single-pass geometry. A high coincidences-to-accidentals ratio (CAR ∼ 1200) depicted by our source indicates a strong temporal correlation between the generated photon pairs. This correlated photon source is tunable from collinear to non-collinear emission of the photons and over a range of signal/idler wavelengths ∼8 − 9 nm corresponding to a temperature range of 20–60°C. We measure a quantum state fidelity $F>95{\% }$ with the singlet entangled state $|\psi ^{-}\rangle = \frac {1}{\sqrt {2}}\left (|HV\rangle -|VH\rangle \right )$ along with a violation of the CHSH-Bell’s inequality by ∼485 standard deviations (S = 2.68 ± 0.0014).
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1. Introduction
High flux sources of polarization-entangled photon pairs are lucrative for the implementation of various entanglement-based applications. Spontaneous parametric down-conversion (SPDC) is one of the promising and efficient techniques for generating heralded single and entangled-photons in various degrees of freedom (DOFs) [1], particularly for different applications in quantum information and communication such as quantum key distribution [2], quantum computing [3], quantum teleportation [4,5], etc. It utilizes nonlinear crystals with $\chi ^{(2)} \neq 0$ such as $\beta$–barium borate (BBO), periodically-poled lithium niobate (ppLN) and periodically-poled KTiOPO4 (ppKTP) for generating paired-photons in collinear and non-collinear directions. The photon pair sources utilizing BBO crystals have low brightness ($\sim 10^3-10^4$ pairs/s/mW) due to a relatively small nonlinear susceptibility coefficient [6,7] compared to other sources, based on, e.g., KTP, LN. In addition to the negligible spatial walk-off (between the generated modes) over conventional bulk crystals, the design flexibility of periodically-poled crystals (e.g., ppKTP) offers the controlled generation of desired output state in terms of spectrum, polarization, spatial mode, etc., with the highest brightness achievable at degenerate ($\lambda _s=\lambda _i=2\lambda _p$) photon emission especially in a collinear direction due to a maximum overlap of the interacting fields. Numerous schemes involving polarization Sagnac interferometer (PSI) [8–10], two crystal Sagnac interferometer (TCSI) [11], crossed crystal (CC) [12–14], post-selection [15] and a folded Mach-Zehnder interferometer (MZI) [16,17] have been exploited to generate polarization-entangled photon pair source with ppKTP crystals. Additionally, correlated photon-pair sources based on four-wave mixing in single-mode fibers [18,19] and nonlinear waveguides [20–25] have also advanced significantly. In such sources, a higher effective material nonlinearity is achieved due to the strong spatial confinement of the pump beam, which produces spectrally bright photon pairs compared to the bulk periodically-poled crystals. The tailoring of on-chip quantum state generation has also enabled the interest in quantum optical metasurfaces and nonlinear thin films/waveguides due to its subwavelength thickness leading to the enhancement of spontaneous emission of photons [26–31]. Moreover, the development of quantum networks have also advanced the research in atom-photon coupling experiments that require narrowband (few MHz) single/entangled photon sources [32–34]. Such narrowband photon generation is achieved in cavity-resonant optical parametric oscillator (OPO) allowing the enhancement and narrowing of the spectral distribution of SPDC photon generation, simultaneously [35].
Though waveguides and fiber-coupled sources offer better confinement of modes and higher efficiency, bulk periodically-poled crystals are required for free-space optical (FSO) links, necessary for distant quantum communication among areas that are not suitable for fiber installation or between moving terminals, including satellite-based links. A first attempt to build a polarization-entangled source utilizing a type-II ppKTP crystal in single pass geometry used a simple beamsplitter (BS) to separate the output modes at the cost of an unwanted 50% loss in the coincidences [15]. Although more complicated Sagnac interferometers have been explored to overcome this, however, there is still scope for studying the generation of the polarization-entangled singlet state in a simple single-pass geometry that does not require any path stabilization, etc., especially in thick/long crystals. Moreover, this simple geometry can easily be implemented for developing a bright source of entangled states that could be a potential quantum resource for free-space optical links in quantum communication networks including the generation of rotationally invariant hybrid (polarization and orbital angular momentum) states for its utilization in alignment-free satellite-based quantum key distribution [36].
In this work, we present a high brightness polarization-entangled photon-pair source utilizing a unidirectionally pumped 3-cm long type-II phase-matched ppKTP crystal. It depicts the spatial and spectral tunability of the photons when scanned with respect to temperature and pump-wavelength. The paper is structured as follows: Section 2 briefly discusses the experimental setup used to generate the polarization-entangled photons. In Section 3.1, we discuss the source tunability, and spatially characterize the source in terms of the photon pairs generated in collinear and non-collinear directions with respect to temperature and pump-wavelength, comparing the experimental results with our theoretical calculations. In Section 3.2, we describe the quality of the generated photon pairs by measuring the coincidences-to-accidental ratio and brightness of the source. In Section 3.3, we analyze the spectral characteristics of the source by measuring the joint spectral intensity distribution of the bi-photons generated in the SPDC process around degenerate operation. In Section 3.4, we measure the quality of entanglement in terms of state fidelity, concurrence, and CHSH inequality (S-parameter). Finally, in Section 4, we summarize our results and conclude.
2. Experimental setup
Figure 1 shows the schematic of the experimental setup for generating the entangled photon pairs. We use a tunable (399-401.5 nm, monitored using a wavelength meter), continuous wave (CW) laser to pump the temperature-controlled type-II ppKTP crystal ($1\times 2\times 30$ mm$^3$), phase-matched for $H \rightarrow H+V$. The pump beam is focused onto the ppKTP crystal using a convex lens $L_1$ ($f_1 = 100$ mm) after passing through a tunable power controller comprising a polarizing beam splitter (PBS) and a half waveplate (HWP; $\lambda /2$). The collimated photons using lens $L_2$ ($f_2 = 200$ mm) are imaged by an electron-multiplying charge-coupled device (EMCCD) after separating and reflecting them using a dichroic mirror and a flip mirror (FM), respectively. After passing through a 15-mm-long KTP timing compensating crystal (to eliminate the time difference of the generated orthogonally polarized photons), iris, and a 50-50 beam splitter (BS), the coincidences are recorded using a time-correlated single photon counting (TCSPC) system (C) through a polarization analyzer comprising of a quarter-wave plate (QWP), a HWP, and a PBS in each arm of the BS and a bandpass filter (F, FWHM = 10 nm).
Fig. 1. Experimental setup for polarization-entangled photon-pair generation through SPDC in a type-II ppKTP crystal; $\lambda /2$: half waveplate; PBS: polarizing beam splitter; $M_1, M_2$: mirrors; $L_1, L_2$: convex lenses; DM: dichroic mirror; F: band-pass filter $@800$ nm, full-width half maxima (FWHM) = 10 nm, transmission efficiency $(\eta _T) \sim$ 60%; EMCCD: electron-multiplying charge-coupled device, FM: flip mirror, BS: beam splitter, Analyzer: a combination of QWP, HWP, and PBS.
3. Results and discussion
3.1 Spatial characterization and tunability of SPDC photons: collinear/non-collinear generation
The photon-pair source is tunable over degenerate to non-degenerate operation and over collinear to non-collinear emission of the photons when scanned with respect to temperature and pump wavelength. We first characterize the source for collinear/non-collinear emission of photon pairs by recording the spatial distribution of SPDC photons using an EMCCD as a function of crystal temperature (T) and pump-wavelength $(\lambda _p)$. The yellow region in Fig. 2(a) represents the experimentally measured maximum number of normalized photon counts/brightness (rate of the generated photons per unit pump power) in a collinear direction, signifying perfect quasi-phase matching that corresponds to degenerate photon-pair emission from the ppKTP crystal. The solid black arrows indicate the tuning direction to change the emission of photons from a collinear to a non-collinear geometry and/or vice-versa. It is observed that at high pump-wavelength ($>400$ nm) when the photons are emitted in non-collinear direction, the tuning of the crystal temperature and pump-wavelength have opposite effects on the direction of photon emission, i.e., the photon generation can be tuned from non-collinear to collinear direction by fixing the pump-wavelength (crystal temperature) and increasing (decreasing) the crystal temperature (pump-wavelength) (see insets of Fig. 2(a)). The experimentally measured normalized photon counts are in good agreement with the calculated normalized photon counts (Fig. 2(b)), convoluted with a Gaussian filter function using the expression [37]
Fig. 2. Normalized photon counts generated through SPDC in type-II ppKTP crystal as a function of crystal temperature and pump-wavelength (a) Measured and (b) calculated [37]. The inset shows the EMCCD images captured at different temperatures and pump-wavelength (marked by the black dashed arrows) (c) phase-matching wavelengths of signal/idler (at $\lambda _p$ = 400.2 nm) at different temperatures calculated for our ppKTP crystal with a given poling period of 9.268 $\mu$m.
3.2 Source brightness and coincidences-to-accidentals ratio
Figure 3(a) shows the measured brightness of the source as a function of the crystal temperature. The generated photon pairs are initially optimized after splitting through a PBS (in place of the BS and the analyzers as shown in Fig. 1). The reflected and transmitted photons from the PBS are detected using two single-photon avalanche detectors (SPADs, efficiency $\eta _d \sim 60{\% }$ at $800$ nm), and the corresponding photon coincidences are measured using the TCSPC system for a time window of 512 ps. Considering a photon coupling efficiency of $\eta _c \sim 40{\% }$, bandpass filter efficiency of $\eta _T \sim 60{\% }$ and a transmission loss of $\sim 10-12{\% }$ through the long crystal, lens, dichroic mirror and PBS, in addition to the detector efficiencies, we estimate an overall attenuation of $\sim 13{\% }$, leading to an estimated brightness (spectral brightness) of $\sim 2.36\times 10^5$ pairs/s/mW ($\sim 1.18\times 10^6$ pairs/s/mW/nm) at 28$^{\circ }$C (phase-matching temperature) corresponding to a detected coincidence rate of $\sim 4000$ count/s/mW for degenerate photon pairs. This is the highest reported brightness for a type-II SPDC in ppKTP crystal to the best of our knowledge. Table 1 shows a comparison of our source characteristics with those from previously reported type-II SPDC-based ppKTP sources generating VIS/IR photon pairs. We also measured the coincidences-to-accidentals ratio (CAR) as a function of pump power (P), as shown in Fig. 3(b). The CAR is calculated as $\mathrm {CAR} = \frac {\mathrm {C-A}}{\mathrm {A}}$, where C and A are the measured coincidence and accidental counts, respectively. As shown, the CAR attains a peak value of $\sim 1200$ at P$\sim 0.75$ mW. The accidental counts result from a contribution of detector noise (dark counts) at low pump powers and multi-photon generation in addition to dark counts at high pump powers [38]. A high CAR suggests a strong temporal correlation between the signal and idler photons which is evident in our source.
Fig. 3. Measured (a) brightness as a function of the crystal temperature for a pump power of $\sim 1$ mW; (b) coincidence-to-accidentals ratio (CAR) as a function of the pump power (black circle chain) at a crystal temperature of 26$^{\circ }$C.
Table 1. Comparison of the performance of previously reported FSO type-II SPDC sources exploiting ppKTP crystal for generation of VIS/IR photon pairs.
3.3 Spectral characterization of SPDC photons: joint spectral intensity
Next, we measure the joint spectral intensity (JSI) at the phase-matching temperature that unveils the frequency correlation of the bi-photons emitted in the SPDC process as shown in Fig. 4. It is theoretically calculated as the square of the joint spectral amplitude (JSA), which is the product of the pump envelope function and phase-matching function, described as
where $\mathrm {JSA} = \mathrm {A}(\omega _s, \omega _i) f(\omega _s, \omega _i)$. Here, $\mathrm {A}(\omega _s, \omega _i) = \mathrm {exp}\left [-\left (\frac {\omega _s+\omega _i-\omega _p}{\sigma _p}\right )^2 \right ]$ is the pump envelope function, $f(\omega _s, \omega _i) = \mathrm {sinc}\left (\frac {\Delta k L}{2}\right ) \mathrm {exp} \left ( -\frac {i \Delta kL}{2}\right )$ is the phase matching function. We have recorded the coincidences (in each pixel) with the aid of tunable filters (replacing the bandpass filters in Fig. 1) by fixing the signal wavelength while scanning the idler wavelength in a discrete step of $0.1$ nm. Figure 4 shows the measured JSI which is a negatively inclined elliptical distribution representing spectrally correlated photons. Since the emission bandwidth ($\sim 0.2$ nm, calculated from the phase-matching condition and verified by our measured second harmonic generation data [41]) of the generated photons is significantly narrower than that of the tunable filters ($\sim 0.74$ nm), the resulting JSI is the convolution of multiple spectrally-correlated photons lying within the filter bandwidth. The bandwidth of the measured JSI is $\sim 0.62$ nm, which is also nearly equal to the tunable filter bandwidth used in the experiment.Fig. 4. Experimentally measured normalized JSI of the bi-photons as a function of signal $(\Delta _s)$ and idler $(\Delta _i)$ detunings at $\lambda _p \sim 400.225$ nm, where $\Delta _{s/i} =\lambda _{s/i}-2\lambda _p$, $\lambda _{p/s/i}$ is the central wavelength of the pump/signal/idler.
3.4 Polarization-entanglement of photons: single-pass geometry with unidirectional pumping
The successful generation of correlated photon pairs unveils the path of realizing polarization entanglement. To obtain polarization-entanglement of the orthogonally polarized photon pairs generated in a single-pass geometry, we pass them through a timing-compensating crystal (KTP) followed by a 50-50 (T-R) beam splitter (BS, as shown in Fig. 1). The output joint state of the bi-photons can post-selectively be measured after the BS in a polarization-entangled singlet state by capturing the two-fold photon coincidences. This entanglement can be characterized by first measuring a sinusoidal variation of these coincidences for two non-orthogonal and mutually unbiased projection bases, $H/V$ (horizontal/vertical) and $D/A$ (diagonal/anti-diagonal) as a function of the analyzer angle in one arm of the BS through which one photon is transmitted and another one is reflected, as shown in Fig. 1. The joint bi-photon state after post-selection can be expressed as $|\psi \rangle = \frac {1}{\sqrt {2}}\left (|H\rangle _T|V\rangle _R+e^{i\phi }|V\rangle _T|H\rangle _R\right )$, where $\phi$ is the relative phase. The probability of projecting the entangled state in a projection basis $(\theta _1,\theta _2)$ is given as $P(\theta _1,\theta _2) = \frac {1}{2} \mathrm {sin}^2(\theta _1\pm \theta _2)$ for $|\psi ^{\pm }\rangle$, where $\theta _{1/2}$ is the polarizer angle (twice the HWP angle) in the signal/idler arm [1].
An optimization of the crystal temperature is crucial in obtaining the polarization-entanglement in a single-pass geometry. Figure 5(a) shows the measured sinusoidal variations of coincidences at different crystal temperatures as a function of the idler HWP angle in the transmitted arm of the 50-50 BS when the signal photon is projected in $D$ polarization. The maximum $(C_{\mathrm {max}})$ and minimum $(C_{\mathrm {min}})$ coincidence counts determines the quantum-interference visibility $(V)$, calculated as
Fig. 5(b) shows the measured visibility (black diamond) as a function of crystal temperature. It is evident from the figures that a slight change in the crystal temperature (away from the phase-matching temperature) degrades the visibility of the two-photon interference. After optimizing the phase-matching temperature, we have also verified the two-photon interference in other polarization bases (H/V/A).Fig. 5. (a) Variation of the measured coincidences as a function of the idler HWP angle in the transmitted arm of the 50-50 BS when the signal photon is projected in D polarization basis at different crystal temperatures (T), (b) experimentally measured visibility in the $D$ basis as a function of crystal temperature.
Figure 6(a) shows the two-photon quantum interference of the entangled-photons as a function of the HWP angle of the signal arm (transmitted) when the idler photon is projected in $H/V$ (solid black/black line) and $D/A$ (solid red/green line) polarization bases at phase-matching temperature. As evident from Figs. 5(a) and 6(a), maximum coincidences are observed when $\theta _1-\theta _2 = 90^{\circ }$ (difference of HWPs angle $= 45^{\circ }$ or odd multiples of it.) signifying the orthogonal polarization correlation of the signal/idler photons followed by the $|\psi ^-\rangle$ state generation.
Fig. 6. (a) The recorded coincidence counts (in 20s) as a function of the HWP angle (keeping the angle of QWPs at $0^{\circ }$) of the analyzer in the transmitted arm while fixing the HWP angle (0$^{\circ }$(H), 23$^{\circ }$(D), 45$^{\circ }$(V), 67$^{\circ }$(A)) in the reflected arm of the 50-50 beam splitter in Fig. 1 for a pump power of P = 1 mW. The coincidence time window is set to 512 ps for this measurement. Graphical representation of the (b) real and (c) imaginary components of the experimentally reconstructed density matrix.
We have estimated the interference visibility to be $V > 99{\% }$ in $H/V$ basis and $> 92{\% }$ in $D/A$ basis. The lower bound of the fidelity, which is a measure of the closeness to an ideal maximally entangled bi-photon quantum state, can be calculated as [1]
where $V_{D/A}$ and $V_{H/V}$ are the average visibilities. The fidelity of the generated state is estimated to be $F\ge 95{\% }$ for an average visibility of $99{\% }$ and $93{\% }$ in $H/V$ and $D/A$ bases, respectively. We also confirm the joint quantum state by experimentally reconstructing the density matrix using quantum state tomography measurement, as shown in Figs. 6(b) and (c). The calculated fidelity and concurrence (the probability of yielding an entangled state) from the reconstructed density matrix are $F\sim 95{\% }$ and $C\sim 91{\% }$, respectively for $|\psi ^-\rangle$ state generation [42].Bell’s inequality violation is another measure to quantify quantum entanglement. The CHSH-Bell parameter can be calculated as [43]
4. Conclusion
In conclusion, we first demonstrate that a selective tuning of pump-wavelength and crystal temperature results in a transition of the phase-matching condition from a degenerate and collinear photon-pair generation to a non-degenerate and a non-collinear generation, validated by the theoretical calculations. Next, we report a high source brightness of $\sim 2.36\times 10^5$ photon-pairs/s/mW (within a subnanometer emission bandwidth) for a unidirectionally pumped 3-cm long type-II ppKTP crystal in a single-pass geometry. We also measure a high CAR $(\sim 1200)$ indicating a strong temporal correlation between the generated photon pairs. We further spectrally characterize the source by measuring the JSI that indicates the spectral correlation between the generated photons. Finally, the bi-photon polarization correlation and Bell’s inequality violation $(S = 2.68 \pm 0.0014)$ validate polarization-entanglement with a quantum state fidelity $F>95{\% }$ for the $|\psi ^-\rangle$ singlet state. Such narrowband bright entangled bi-photon states can be a step towards the implementation of rotationally-invariant hybrid-entangled states for free-space quantum communication [36].
Funding
Council of Scientific and Industrial Research, India (09/086(1331)/2018-EMR-I); Defence Research and Development Organisation (DFTM/03/3203/P/07/JATC-P2QP-07/463/D(R&D)).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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