Production of degenerate polarization entangled photon pairs in the telecom-band from a pump enhanced parametric downconversion process

P. J. Thomas; C. J. Chunnilall; D. J. M. Stothard; D. A. Walsh; M. H. Dunn

doi:10.1364/OE.18.026600

1. Introduction

Entangled photon pairs are an important resource for many applications in quantum optical information processes such as quantum teleportation [1], quantum key distribution [2] and implementations of quantum logic gates [3,4]. There has been much interest in the development of polarization entangled photon pair sources based on spontaneous parametric downconversion (SPDC) due to their simplicity and high brightness. Frequently, the states of interest are the four maximally entangled Bell states:

| ψ^{\pm} 〉 = [| V_{1} H_{2} 〉 \pm | H_{1} V_{2} 〉] / \sqrt{2},

| ϕ^{\pm} 〉 = [| V_{1} V_{2} 〉 \pm | H_{1} H_{2} 〉] / \sqrt{2} .

Each ket term in Eqs. (1) and (2) corresponds to the outcome of a simultaneous measurement of the polarization state (H or V) of two photons at positions 1 and 2. High quality polarization entangled states are characterized by ket terms that are indistinguishable with respect to all variables except polarization. Since SPDC entangled photon pair sources capable of producing all four Bell states invariably require type-II phasematching, dispersion between the downconverted photons within the non-linear medium often leads to temporal differences between the ket terms and loss of entanglement. In earlier schemes, such temporal distinguishability was removed by positioning one or two crystals after the downconverter such that dispersive effects from the additional crystals cancelled out those associated with the downconverter [5,6]. In order to preserve the spatial and frequency indistinguishability of the ket terms many early schemes only utilized photons from a small part of the emission cone, substantially reducing count rates. Subsequent schemes for polarization entangled photon pair production incorporated an effective way of suppressing temporal, spatial and frequency distinguishability information without compensating crystals or spatial filters [7]. In such schemes the individual ket terms of the entangled state were formed by the downconversion pairs associated with one of two identical counter-propagating pump beams through a single downconversion medium. In later investigations the counterpropagating pump beam was formed by coupling the light into a Sagnac interferometer containing the downconversion medium, removing the requirement of active phase-stabilization for the output entangled state [8–10].

In separate investigations, a method for increasing the brightness of an SPDC pair source was demonstrated through enhancement of the pump field within the downconversion medium by placing it inside an optically resonant cavity [11–14]. Since a resonant beam will naturally counter-propagate within a linear enhancement cavity, a pump enhanced source of downconverted photons is suitable as the source within a scheme for polarization entanglement, of the type described in Refs [7–10]. The linear pump enhanced photon pair sources presented in Refs [11,12]. would not be suited to this application due to the asymmetric resonant mode of the pump beam which would make the forward and backwardly propagating downconversion emission cones distinguishable. A feature of the source design presented here is a symmetric mode pump enhancement cavity.

Most previous work on polarization entangled photon pair sources was carried out at visible wavelengths, partly because of mature photon detection technology. In addition, a low interaction of visible photons with the atmosphere makes them suitable for free-space technologies such as some QKD implementations, where successful communication over 144 km has been demonstrated [15]. In recent years there has been increasing interest in polarisation entangled photons in the telecom band because such technology would lend itself to quantum systems based on optical fiber and integrated waveguide circuits, as well as distributed quantum communication protocols built upon existing long-distance fiber networks [16]. In addition to techniques based upon SPDC generation [17–21], four wave mixing (FWM) sources have also gained attention for telecom band pair photon generation [22–25]. This is due to the in-fibre characteristic of the generation, meaning fibre coupling losses can be circumvented; however FWM ^χ(3) processes are intrinsically less bright than SPDC ^χ(2) processes.

This paper describes the operational theory and experimental demonstration of a novel source for polarization entangled photon pairs in the telecom band, based upon pump enhanced downconversion processes within a linear enhancement cavity. The downconversion emission out through one output mirror from the enhancement cavity has been characterized elsewhere [26]. The source featured here can be associated with the class of polarization entanglement photon pair sources featuring a counter-propagating pump beam through the downconversion medium [7–10], and therefore concerns the downconversion emission exiting through both enhancement cavity mirrors. The setup was shown to generate the two maximally entangled Bell states $| ψ^{\pm} 〉$ ; the production of the $| ϕ^{\pm} 〉$ states would be possible by adding another wave plate to the setup. Quantum interference visibilities in excess of 0.95 were measured indicating high quality polarization entanglement. Further analysis of an entangled state was performed by testing for violation of the Clauser-Horner-Shimony-Holt (CHSH) form of Bell’s inequality [27]; it was found that the CHSH ‘S-parameter’ was $2.73 \pm 0.12$ , which corresponds to a violation of the classical maximum by about six standard deviations.

2. Theory

The essential components of the scheme are shown in Fig. 1 . The pump beam is coupled into an optical cavity defined by mirrors M₁ and M₂ through a dichroic mirror DM with transmittance of 0.8 at the pump wavelength and high reflectance (>0.99) at the downconversion wavelength. The downconversion medium of length L_C is positioned within the optical cavity that is resonant for the pump, and transmitting for the downconversion such that downconversion pairs emerge from both M₁ and M₂. The mirrors M₃ and DM steer the downconversion towards a polarizing beam splitter PBS, which transmits (reflects) the horizontally (vertically) polarized photons from each pair to separate detectors D₁ and D₂. L_A, L_B, L_D and L_E define the distances denoted by their respective arrows.

Fig. 1 Scheme for producing the two superposition components of the entangled state in Eq. (3). The ket terms are either associated with downconversion photons that are produced while the pump beam is travelling from (a) right to left (forwards) or (b) left to right (backwards).

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The general expression for the entangled state described by Eq. (1) is:

| ψ 〉 = | V_{1} H_{2} 〉 + e^{i ϕ} | H_{1} V_{2} 〉 .

The two ket terms in Eq. (3) can be associated with the measurement of coincident photon arrivals at D₁ and D₂ arising from two distinct events; the first ket is due to the coincident detection of ‘forwardly’ propagating downconversion photons, generated by a pump photon travelling from

M_{1}

to M₂ [Fig. 1(a)], while the second ket is due to ‘backwardly’ propagating downconversion photons generated by a pump photon travelling from M₂ to

M_{1}

[Fig. 1(b)].

In order to form an entangled state that is also suitable for applications requiring quantum interference in the polarisation variable, the component ket terms are required to be indistinguishable with respect to wavelength. Let us define the signal photons to be horizontally polarized, and the idler photons to be vertically polarized at the time of downconversion. Without the half-wave plate (HWP) shown in Fig. 1, the PBS directs the signal photon to either D₁ or D₂ depending on the generation direction through the crystal, similarly for the idler photon. Therefore unless the signal and idler photons are perfectly degenerate, the ket terms of Eq. (3) are distinguishable with respect to wavelength and, therefore, which-path information. Despite the choice of phasematching for degeneracy there will always be a degree of wavelength distinguishability between the signal and idler due to the inherently wavelength anti-correlated nature of the downconversion process. However with the HWP positioned in the ‘forwards’ path as shown in Fig. 1 and oriented at 45° to the polarization axes, the signal and idler polarizations in that path are exchanged. This ensures that both ket terms of the state in Eq. (3) correspond to a signal photon received at D₁ and an idler photon at D₂, and the kets are therefore indistinguishable with respect to wavelength and which-path information.

The $| ψ^{+} 〉$ state is produced from Eq. (3) when $ϕ = n π$ , where n is zero or an even integer, while $| ψ^{-} 〉$ is produced when n is an odd integer. Non-maximally entangled states will be created if photon pairs associated with one ket arrive more frequently at PBS than those associated with the other ket term, i.e. the ket terms have different amplitudes. This may result from different generation rates and/or losses associated with each term. The following discussion seeks to analyse the phases and amplitudes of the $| H_{1} V_{2} 〉$ and $| V_{1} H_{2} 〉$ terms associated with the source design shown in Fig. 1. The analysis shows that the design results in a coherent entangled state and it also gives an insight into how the entangled state may be changed and optimised. First, rewrite Eq. (3) as:

| ψ 〉 = \sqrt{η_{F W}} e^{ϕ_{F W}} | V_{1} H_{2} 〉 + \sqrt{η_{B W}} e^{ϕ_{B W}} | H_{1} V_{2} 〉,

where

η_{F W}

and

η_{B W}

are the classical arrival rates at D₁ and D₂ of paired photons generated in the forward and backward directions through the crystal respectively.

ϕ_{F W}

and

ϕ_{B W}

are the summed phases of the parent pump photon and the daughter pair photons up to the PBS for the forward and backward generation cases respectively. The phases associated with the optical paths from PBS to

D_{1}

and

D_{2}

are common between the ket terms and so are ignored. Since the pump photon can circulate around the cavity any number of times before the downconversion event occurs, the components of the two phase terms

ϕ_{F W}

and

ϕ_{B W}

associated with the enhancement cavity are given respectively by:

ϕ_{F W_{E C}} = k_{p} [L_{D} + n_{y} L_{C}] + [k_{s} + k_{i}] L_{B} + \sum_{N_{e v e n}} N [{L_{B} + n_{y} L_{C} + L_{D}} k_{p} + π],

ϕ_{B W_{E C}} = k_{p} [L_{B} + n_{y} L_{C}] + [k_{s} + k_{i}] L_{D} + \sum_{N_{o d d}} N [{L_{B} + n_{y} L_{C} + L_{D}} k_{p} + π],

where the ‘start’ point for the phase accumulation was taken to be after the pump photon first passes through

M_{1}

. N is the number of complete pump excursions across the cavity before creating a photon pair.

n_{y}

is the refractive index of the pump beam in the PPKTP.

k_{p}

,

k_{s}

and

k_{i}

are the pump, signal and idler wavevectors in vacuum respectively. The π term is due to reflection of the pump beam at a cavity mirror. Equations (5) and (6) assume that the pair photons are generated as the pump beam emerges from

L_{C}

; this is an arbitrary choice because the conversion could happen at any point along

L_{C}

, and since phase matching dictates that

k'_{p} = k'_{s} + k' i

where the

k'

are the crystal wavevectors, the phase accumulated within the crystal is independent of the downconversion event position. Since the enhancement cavity is held on resonance:

[L_{B} + n_{y} L_{C} + L_{D}] k_{p} = q k_{p} \frac{λ_{p}}{2} = q π .

Therefore the pump accumulates a net phase of π or

2 π

for each excursion across the cavity depending on the mode order q. Inserting Eq. (7) into Eqs. (5) and (6) and removing common terms gives:

ϕ_{F W_{E C}} = k_{p} L_{D} + [k_{s} + k_{i}] L_{B} + \sum_{N_{e v e n}} N [q π + π],

ϕ_{B W_{E C}} = k_{p} L_{B} + [k_{s} + k_{i}] L_{D} + \sum_{N_{o d d}} N [q π + π] .

Equation (8) indicates that the summation terms of

ϕ_{F W_{E C}}

differ in phase only by integer numbers of

2 π

, similarly for

ϕ_{B W_{E C}}

in Eq. (9). This allows for each of the summations to be considered as a series of terms

2 π

out of phase with one another, therefore adding constructively.

There are also phase contributions to $ϕ_{F W}$ and $ϕ_{B W}$ associated with the optical paths which are external to the enhancement cavity, i.e. the optical paths from $M_{2}$ / $M_{1}$ to PBS:

ϕ_{F W_{E X T}} = [k_{s} + k_{i}] L_{A} + ϕ_{M_{2} (s, i)} + ϕ_{H W P (s, i)} + 3 π,

ϕ_{B W_{E X T}} = [k_{s} + k_{i}] L_{E} + ϕ_{M_{1} (s, i)} + 3 π .

In Eq. (10),

ϕ_{M_{2} (s, i)}

and

ϕ_{H W P (s, i)}

are the collective signal and idler phases acquired through

M_{2}

and the half wave plate respectively, while in Eq. (11)

ϕ_{M_{1} (s, i)}

is the collective phase acquired by the signal and idler photons when the beam passes through

M_{1}

. The 3π terms are due to the reflection of both downconverted photons at the beam steering mirrors (

M_{3}

and

{DM}_{1}

) and the reflection of one photon at the polarizing beam splitter. The total phase

ϕ_{F W}

in Eq. (4) is given by the sum of

ϕ_{F W_{E C}}

and

ϕ_{F W_{E X T}}

;

ϕ_{B W}

is given by summing

ϕ_{B W_{E C}}

and

ϕ_{B W_{E X T}}

.

The factors $η_{F W}$ and $η_{B W}$ in Eq. (4) are proportional to pump field intensity through the downconversion medium in the forward and backward travelling directions respectively. The pump intensities are given by a summation over N. For a lossless system this leads to:

η_{F W} = {\sum_{r} I_{p} (R_{2} R_{1})}^{r} = \frac{I_{p}}{1 - R_{2} R_{1}},

η_{B W} = {\sum_{r} I_{p} R_{2} (R_{2} R_{1})}^{r} = \frac{I_{p} R_{2}}{1 - R_{2} R_{1}} .

In Eqs. (12) and (13) r is an integer,

I_{p}

is the input pump power to the cavity, and

R_{1}

and

R_{2}

are the pump reflectivities of mirrors

M_{1}

and

M_{2}

respectively. It follows from Eqs. (8) – (13) that the full expression for the produced state outlined in Eq. (4) is given by:

| ψ 〉 = \sqrt{\frac{I_{p}}{1 - R_{2} R_{1}}} | V_{1} H_{2} 〉 + \sqrt{\frac{I_{p} R_{2}}{1 - R_{2} R_{1}}} e^{[k_{s} + k_{i}] \cdot [L_{D} + L_{E} - L_{A} - L_{B}] + k_{p} [L_{B} - L_{D}] - ϕ_{M_{2} (s, i)} + ϕ_{M_{1} (s, i)} - ϕ_{H W P (s, i)}} | H_{1} V_{2} 〉 .

The phase contributions due to the enhancement cavity mirrors and the half wave plate are fixed, also the optical paths

L_{B}

,

L_{C}

and

L_{D}

are fixed by the cavity resonance condition. Therefore in order to form a stable output state, only

L_{E} - L_{A}

needs to be actively fixed.

L_{E} - L_{A}

also needs to be variable in order to switch between the

L_{E} - L_{A}

| ψ^{+} 〉

and

| ψ^{-} 〉

states. When the system is optimised for

| ψ^{+} 〉

, the

| ϕ^{+} 〉

state could be produced by positioning a half wave plate for the downconversion at either output port of PBS, and oriented at 45° to the H/V basis such that H and V are exchanged. Similarly, the half waveplate could be used to switch

| ψ^{-} 〉 \to | ϕ^{-} 〉

, thereby enabling all four Bell states to be produced.

The polarization entanglement source belongs to the same class as those in Ref [7–10]. Unlike the designs in [8–10], the one discussed here does not create a counter-propagating pump beam through the downconverter by coupling the pump into a Sagnac interferometer at the interferometer recombiner. As a result there is the requirement to actively fix $L_{E} - L_{A}$ . However, unlike any of the earlier designs, the downconversion process is pump enhanced.

3. Experimental

The experimental setup is shown in Fig. 2 . CW 532 nm laser radiation from a diode pumped solid state laser DPSSL (Coherent Verdi V10) was used to pump a Ti-sapphire laser (Microlase MBR-100) to obtain single-frequency laser radiation at a measured wavelength of 792 nm. The 792 nm radiation was passed through an optical isolator to prevent perturbation of the laser from back reflections, and a pair of wavelength dispersive prisms (WDP) were used to remove unwanted laser fluorescence. The optical cavity for enhancement at 792 nm was defined by meniscus mirrors M₁ and M₂, with curvature radii of 75 mm. M₁ was coated for reflectance R = 88% at 792 nm and AR (anti-reflection) coated for the 1584 nm downconverted wavelength. M₂ was AR coated for the downconversion and R = 98% at 792 nm. The cavity mirrors were placed 60 mm apart from one another and the periodically-poled potassium titanyl phosphate (PPKTP) nonlinear crystal was centrally located between M₁ and M₂. The crystal was manufactured by Raicol, and was x-cut, 10 mm long and had a grating period of 46.1 μm in order to phase-match the downconversion of the 792 nm pump into two degenerate, orthogonally polarized and collinearly propagating photons at 1584 nm. The crystal temperature was set to 21.5 ± 0.5 °C, which was the temperature at which degenerate downconversion pairs were found to be emitted collinearly to the pump [26].

Fig. 2 Schematic for source of polarization entangled photon pairs in the telecom band. The blue and red lines denote 792 nm pump light and downconversion respectively, while the black lines are electrical connections.

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Using lenses L₁ (30 cm focal length) and L₂ (20 cm focal length), the TEM₀₀ mode of the pump was matched to the lowest order enhancement cavity mode with beam waist diameter ~200 μm located at the mid-point between M₁ and M₂. The enhancement cavity was held on resonance using a Pound-Drever-Hall servo loop [28]. Frequency side bands at ± 50 MHz to the pump frequency ω₀ were applied using a phase modulator (Leysop PM50). The frequency side bands are rejected by the cavity, and form a beat frequency signal with light at ω₀ returning through M₁. The amplitude of the RF beat frequency is dependent on the separation between the centre frequency of the laser beam from the cavity resonance peak. The recombined beam carrying the beat frequency was detected by SiPD₁ (Osram BPX-65 silicon PIN photodiode), from which an error signal was generated. The error signal was input to proportional-integral-derivative (PID) feedback electronics, creating a voltage that was amplified (Piezomechanic SVR 500) before application to the piezo active mirror M₂ such that the cavity was held on resonance.

M₃, M₄ and M₅ are gold beam steering mirrors, while DM₁ and DM₂ are highly reflecting (>0.99) for the downconversion but partially transmitting (0.8) at the pump wavelength. L₃ and L₄ are collimating lenses with focal length 160 mm which are positioned at their focal length along the beam path from the enhancement cavity centre.

In order to form a coherent superposition for the entangled state described in Eqs. (4) and (14), it was necessary to ensure that the optical paths of the downconversion from M₂ to the polarization beam splitter PBS (L_A in Fig. 1) and from M₁ to PBS (L_E in Fig. 1) remain fixed relative to one another. The relative optical paths were monitored against the two beam interference fringe formed at PBS by the transmitted pump beam through the enhancement cavity aligned with the back reflected pump beam from M₁ (that is partially reflected at DM₁). The need for a component of the enhancement cavity input pump beam to be back reflected from M₁ explains why M₁ and M₂ were chosen to have different reflectances: equal reflectances for M₁ and M₂ would result in near impedance-matched coupling of the pump beam to the enhancement cavity whereby all of the input pump (except that component downconverted or otherwise lost) emerges through M₂, and there is no back reflection from M₁. The pump beam back reflected from the enhancement cavity is nominally H polarized at PBS whereas the pump transmitted through the enhancement cavity has H and V polarized components at PBS due to the waveplate. The beam components exiting PBS through the same output port were detected at SiPD₂. PBS was not totally polarizing at the pump wavelength and the small amount of H polarized pump light from DM₁ which is reflected at PBS gave a sufficient signal at SiPD₂. Similarly some V polarised component of the pump beam from M₃ was transmitted through PBS in addition to the H component. It was found that a polarization analyzer positioned in front of SiPD₂ could be oriented to optimize the measured interference fringe visibility. The output from SiPD₂ was input to an electronic side-of-fringe feedback mechanism that generated an error signal dependent on the difference between the detected signal at SiPD₂ and a reference voltage which was set to correspond to the output from SiPD₂ when $| L_{E} - L_{A} |$ is such that the phase delay between the recombined pump beams is N_odd π/2, where N_odd is an odd integer, i.e. where the detected signal at SiPD₂ is changing most rapidly when $| L_{E} - L_{A} |$ is varied. The error signal was converted to a voltage by PID feedback electronics and amplified (Piezomechanic SVR 500) before application to the piezo active mirror M₃ such that the phase delay between the recombined pump beams causes the output voltage from SiPD₂ to equal the reference level. Through design of the feedback electronics, the reference level was made to adjust with fluctuations in the pump power detected at SiPD₃ such that the reference level tracked the associated drift in the overall intensity of the measured fringe pattern. This ensured that the reference level did not correspond to different phase delays between the recombined pump beams with time.

In order to tune the phase term of the entangled state to maximize the measured polarisation entanglement whilst maintaining a phase difference of N_odd π/2 for the recombined pump beams (where the side-of-fringe lock system is most effective), the chromatic dispersion in silica glass between the pump and downconversion wavelengths was utilized. Two 1 mm thick silica microscope slides were placed along L_A, the plates were rotatable at equal and opposite angles to the incident beam in order to cancel out any spatial walk-off effects between the pump and downconversion. The phase delay between the ket terms $| V_{1} H_{2} 〉$ and $| H_{1} V_{2} 〉$ could then be varied by a counter rotation of the two glass slides. The minimum angle between the slide normal and beam for an effective delay mechanism was ~25°. In addition, the glass slides were uncoated, meaning that reflection losses were polarization dependent. Therefore the slides were placed on either side of the HWP plate used for exchanging the signal and idler polarizations in order that the signal and idler photons were transmitted through the two slides with equal probability, which is a useful feature for validating the system alignment.

For M2, R = 0.98, therefore the downconverted field generated from the backward passes of the pump is ~2% less bright than the field generated from the forward passes of the pump. However, due to the Fresnel losses at the glass slide pair, and since $R_{D M_{1}} > R_{M_{3}}$ at the downconversion wavelength, the downconversion generated from the backwardly passing pump is brighter at PBS than the downconversion generated from the forwardly passing pump, and non-maximally entangled states are produced. A variable neutral density filter was introduced into path LE which allowed control over the degree of entanglement, and was set so that $η_{F W} = η_{B W}$ to produce maximally entangled states. Following polarization analyzers for the downconversion and 30 nm bandwidth interference filters centered at 1584 nm, were 11 mm focal length lenses for coupling the downconversion into single mode fiber-coupled InGaAs/InP avalanche photodiode photon counters (Princeton Lightwave PGA 602) D1 and D2. The detectors were synchronously biased with gates of 1 ns at a frequency of ~6.18 MHz. Coincident photon detections between the APDs were recorded by a coincidence counter (PicoQuant HydraHarp 400) cc.

4. Results and discussion

In Ref [26]. we studied the properties of the emission through one output coupler of the enhancement cavity. There we estimated a pair generation rate of 6.24 × 10⁴ s⁻¹ mW⁻¹. Bandpass filters centred at 1584 nm were used to demonstrate that all of the downconversion was within their FWHM bandwidth of 30.8 nm. Although no direct measurements of the downconversion emission spectral profile were made, a phase matching bandwidth of 2.68 nm was inferred from the bandwidth of Hong-Ou-Mandel interference of the photon pairs [29–32] The high visibility (0.94) of the HOM dip at the crystal temperature used here is consistent with the production of highly degenerate pairs. We also verified that the downconversion emission without the enhancement cavity was similar to that obtained previously [31], and was 9.7 times brighter in the presence of the cavity.

The characterization of the device in Fig. 2 involved measurements of the coincident photon detection rate between detectors D₁ and D₂ as a function of their associated polarization analyzer angles θ ₁ and θ ₂ respectively. Figure 3(a) shows data for the coincidence rate measurements when θ ₁ is oriented along H or V, and θ ₂ is varied. The visibility of the modulation for the θ ₁ = 0° (H) curve was determined to be 0.971 ± 0.043 through a least squares fit of a sinusoid to the data, similarly the visibility of the θ ₁ = 90° (V) curve was 0.972 ± 0.036. $R_{g e n}$ , the rate of photon pairs passing out through M₂ per unit pump power input to the cavity $P p$ is given by $R_{g e n} = R_{c} / η_{s} η_{i} P p$ [20]. $R_{c}$ , the detected coincidence rate, is given by the maximum value of the θ ₁ = 90° curve in Fig. 3(a). $η_{s}$ and $η_{i}$ are the overall detection efficiencies of the signal and idler photons that account for: the detector detection efficiencies (0.12); the detector duty cycle (6.2 × 10⁻³); the mirror reflectances (0.92 per surface); the transmittances of the glass plate pair (0.87), spectral filters (0.62), L₄ (0.73), PBS (reflectance ~1, transmittance 0.98) and analyzers (0.99); and the fiber coupling efficiencies (0.21 for the signal, 0.24 for the idler). With the above figures we estimate $R_{g e n}$ = 4.4 × 10⁴ s⁻¹mW⁻¹ for the pair rate through M₂. The rate of pair photons corresponding to those exiting the enhancement cavity through M₁ is similar [the peak values of the θ ₁ = 90° curve in Fig. 3(a) are similar to those for the θ ₁ = 0° curve; see discussion in final paragraph of Section 3], giving an estimate for the total pair photon production rate of 8.9 × 10⁴ s⁻¹mW⁻¹ pump. The rate is much lower than has been measured in systems at visible wavelength, possibly due to a lower effective non-linearity of the PPKTP crystal in the telecom band [33]. However the pair production rate of the system here still corresponds to an order of magnitude increase in brightness relative to comparable systems at telecom wavelengths [26,31].

Fig. 3 (a) Measurements of the number of coincident photon detections at D₁ and D₂ (see Fig. 2) as a function of θ ₂ for θ ₁ = 0° (triangles) and θ ₁ = 90° (circles). (b) Measurements of the number of coincidences as a function of θ₂ with fixed θ ₁ = 45°. Data sets correspond to θ_silica = 30.0° (ϕ = 0 state, hollow circles), θ_silica = 32.1 (ϕ = π/2 state, filled circles) and θ_silica = 34.4° (ϕ = π state, triangles). Measurements at each setting of θ₂ were made over a 300 s time period which corresponds to a detector ‘on time’ of ~1.85 s. For both (a) and (b), the lines are least squares fit to the experimental data, the square data points are the measured singles counts at D₂ over 300 s as a function of θ ₂ . The pump power into the enhancement cavity was 50 mW. The error bars shown were used as weightings for the curve fitting and are based upon Poissonian N uncertainties.

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Optimal alignment is indicated by the constant singles count rate at D₂ with θ ₂, (208 ± 4 s⁻¹, without correction for 51 ± 2 s⁻¹ dark counts). Similarly the singles count rate at D₁ was constant with θ ₁ (179 ± 4 s⁻¹, without correction for 19 ± 1 s⁻¹ dark counts).

The measurement of the $| V_{1} H_{2} 〉$ photons arriving in coincidence at D₁ and D₂ is described by the probability amplitude $\sin θ_{1} \cos θ_{2}$ and the $e^{i ϕ} | H_{1} V_{2} 〉$ coincidence measurement by the amplitude $e^{i ϕ} \cos θ_{1} \sin θ_{2}$ , where ϕ is the collection of phase terms in Eq. (14). The classical expectation value for the coincidence rate is given by adding the sum of the squares of the probability amplitudes. If the two amplitudes are indistinguishable as in the quantum entangled state given by Eq. (3), according to Feynman [34] the expectation value for the coincidence counting rate resulting from the state is given by squaring the sum of the two probability amplitudes: Therefore when θ₁ = 45°, combining the two amplitudes classically gives the coincidence rate:

N_{c l a s s c i a l, θ_{1} = 45 °} (θ_{2}) = \frac{1}{2} [\sin^{2} θ_{2} + \cos^{2} θ_{2}] .

Equation (15) shows that there is no modulation with θ ₂. However, for the entangled state in Eq. (3), the coincidence rate is given by:

N_{e n t a n g l e d, θ_{1} = 45^{\circ}} (θ_{2}) = N_{c l a s s c i a l, θ_{1} = 45 °} (θ_{2}) + \cos θ_{2} \sin θ_{2} \cos ϕ .

Equation (16) shows that when ϕ = 0,π there is a modulation between twice the classical expectation value and zero. The quantum predictions for the measured coincidence counting rate as a function of θ ₂ when θ ₁ = 45° for the ϕ = 0,π states are in good agreement with the experimental data in Fig. 3(b). The data were collected with the side-of-fringe lock feedback loop active and for different angles of the silica plate pair normals to the incident beam, θ_silica. The quantum interference visibility of the curves corresponding to the ϕ = 0 and π states are 0.971 ± 0.041 and 0.932 ± 0.036 respectively, and indicate good quality polarization entanglement. The slightly lower visibility for the ϕ = π curve was probably due to slightly non-optimal θ_silica. In order to tune between ϕ = 0 and π, the required rotation of θ_silica was found to be 4.4°, which is close to the 4.6° predicted using ray-optics geometry and the Sellmeier equation for silica. Taking θ_silica approximately centered between those values required for ϕ = 0 and π, results in a coincidence measurement profile with θ ₂ that corresponds well to that predicted for the ϕ = π/2 entangled state, which mimics the classical result and shows the good system stability.

A further method of entanglement characterisation was carried out by testing for violation of a Bell inequality in the form of the Clauser-Horne-Shimony-Holt (CHSH) S-parameter [27] which is given by a linear combination of four expectation values:

S = | E (θ_{1}, θ_{2}) + E (θ_{1}, θ'_{2}) - E (θ'_{1}, θ_{2}) + E (θ'_{1}, θ'_{2}) |,

where:

E (θ_{1}, θ_{2}) = \frac{N_{+ +} - N_{+ -} - N_{- +} + N_{- -}}{N_{+ +} + N_{+ -} + N_{- +} + N_{- -}} .

The N_s are measured coincidence counting rates, the first N subscript character relates to the analyzer setting of θ ₁; if the character is ‘ + ’ then θ ₁ = θ ₁, if the character is ‘-’, then θ ₁ = θ ₁ + π/2. Similarly, the second character of the N subscript relates to the second analyzer setting. The set

{θ_{1}, θ_{2}, θ'_{1}, θ'_{2}}

are chosen to maximize S. The are many equally optimal sets possible, and the chosen set was

θ_{1} = - π / 4

,

θ_{2} = 0

,

θ'_{1} = 5 π / 8

and

θ'_{2} = 7 π / 8

. Given an optimal set of analyzer settings, for a classical state

S \leq 2

, while for an ideal entangled state

S = 2 \sqrt{2}

. Therefore

S > 2

denotes non-classical behaviour and higher values denote greater entanglement.

From Eqs. (17) and (18), the characterisation of the S parameter required 16 coincidence rate measurements in total, and each were averaged over a 100 second measurement time. When $θ_{s i l i c a}$ was adjusted to give the $ϕ = 0$ entangled state, the measured expectation values and their √N uncertainties were:

E (- π / 4, 0) = - 0. 869 \pm 0.0 68, E (- π / 4, 7 π / 8) = - 0. 546 \pm 0.0 56,

E (5 π / 8, 0) = 0. 503 \pm 0.0 55, E (5 π / 8, 7 π / 8) = - 0. 813 \pm 0.0 64 .

Therefore S = 2.73 ± 0.12, which represents a violation of the Bell inequality by around 6 standard deviations. The measurement time of 100 s corresponds to a detector ‘on time’ of approximately 0.6 s, hence the S-parameter uncertainty is quite large because of the low number of coincidence counts. Further characterisation of the state could be performed by carrying out 2-qubit-state tomography analysis [35], but this has not been done on our system to date.

5. Conclusions

The design and implementation of a novel source of degenerate polarization entangled photon pairs in the telecom band based on a cavity enhanced parametric downconversion process has been presented. The scheme belongs to the class of systems characterized by counter propagating pump beams through the downconversion medium [7–10]. The system allowed for the production of two of the four maximally entangled Bell states, and the remaining two are obtainable through the addition of a further half wave plate into the setup. The relative phase between the two kets of each entangled state was locked by using a side-of-fringe feedback system to fix the optical path difference of a two beam interferometer formed by the 792 nm pump light reflected from/transmitted through the enhancement cavity, aligned at a polarizing beam cube. A rotatable two-glass-slide system in combination with the side-of-fringe locked interferometer was used to vary the relative phase between the ket states to allow for tuning between maximally entangled states. Analysis of the coincidence rate in the diagonal A/D basis between two detectors placed across the output of the device revealed the production of highly entangled states, resulting in quantum interference visibilities ~0.95. The entangled states produced were found to break the CHSH Bell inequality by around 6 standard deviations. The production rate of entangled photon pairs at the output of PBS within a single fiber optic mode was estimated to be 8.9 × 10⁴ s⁻¹ mW⁻¹ pump, which is consistent with those measured in Ref [26], where for the same cavity enhanced downconversion process, the pair generation rate within a single fiber optic mode was estimated to be 6.24 × 10⁴ s⁻¹ mW⁻¹, an order of magnitude greater than for the identical downconversion process without pump enhancement.

Acknowledgements

The financial support of the Department of Business, Innovation and Skills; the Engineering and the Physical Sciences Research Council; and the European Community’s Seventh Framework Programme (ERA-NET Plus, Grant Agreement No. 217257) is gratefully acknowledged.

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Production of degenerate polarization entangled photon pairs in the telecom-band from a pump enhanced parametric downconversion process

Abstract

1. Introduction

2. Theory

3. Experimental

4. Results and discussion

5. Conclusions

Acknowledgements

References and links

Cited By

Figures (3)

Equations (20)

Optics Express