1. Introduction

Polarization-entangled two-photon states, extremely important for quantum optics and quantum information (QI), are most efficiently generated by means of nonlinear optical processes such as type-II SPDC [1], interferometric schemes involving type-I SPDC [2, 3], and various versions of four-wave mixing [4, 5, 6, 7]. In such schemes, the Bell state generated is chosen by setting linear and nonlinear optical elements. To the best of our knowledge, the possibility of generating various Bell states within the phase-matching linewidth was never considered. However, it turns out that at any phase-matched process leading to polarization-entangled two-photon states generation, different Bell states are always produced within the natural bandwidth. The purpose of this Letter is to present this formerly undiscussed phenomenon, which, relating to all current methods for generating two-photon polarization-entangled states, should be considered for a careful characterization of these states, in connection with both QI applications (and related fields as quantum imaging or quantum metrology) and studies on the foundations of quantum mechanics [8].

The two-photon part of the quantum state generated via such a process has the form (see, e.g., [9]), with polarizations P=H,P′=V for type-II and P=P′ for type-I process, respectively,

where Δ(k,k’) is the phase mismatch depending on spatial and frequency modes k,k’, in which a pair is created, ^{a †} _P,_{k,k ’} are photon creation operators and F_NL(r) is the factor containing nonlinearity and the pump (pumps) field distribution over the nonlinear medium. Integration over the volume turns F_NL(r) into its Fourier transform, F_NL(k, k’), but there remains a phase factor, antisymmetric w.r.t. the exchange k↔k’. As a result, in a simplified version, the polarization-entangled state is

where P ₁=P ₄=H,P ₂=P ₃=V or P ₁=P ₂=H,P ₃=P ₄=V, and the phase δ depends on the parameter, additional to polarization, in which the state is entangled (frequency or wavevector, in most cases). Thus, switching between different Bell states occurs within the spectrum allowed by phase-matching.

Here, we discuss this effect for the most striking case, the one of collinear frequency-degenerate type-II SPDC, which is well-known to produce the |Ψ⁺〉 polarization-entangled state but, as we show below, also produces the |Ψ^-〉 state with small non-degeneracy inside the same frequency bandwidth. However, this is rather a general situation: similar effects for the angular lineshape and for the configuration with two type-I crystals [10] have been experimentally observed as well and will be published elsewhere [11].

The singlet state |Ψ^-〉 is special among the four Bell states. It is robust against decoherence and hence can be used in fibre-based quantum communications wherever polarization is involved. It can be filtered out of the Bell set using a 50% non-polarizing beamsplitter (BS) [14, 15]; for this reason it was the basic tool in the first quantum teleportation experiments [16, 17]. This state is unpolarized not only with respect to intensity but also with respect to all its higher-order moments [18]. Its ‘bright’ (high-photon-number) counterpart is predicted to have the properties of ‘scalar light’, for which the fluctuations of all Stokes parameters are suppressed below the shot-noise limit [19].

There are several SPDC-based methods to prepare the polarization-entangled singlet two-photon state. It is important that to achieve polarization entanglement, the schemes based on type-II SPDC always imply a compensation of the e-o delay τ₀ between the orthogonally polarized photons of a single pair [1]. Here we show that the o-e delay τ ₀, which requires certain efforts to be eliminated, produces the phase δ variation within the SPDC natural bandwidth [see Eq.(2)] and is therefore responsible for creating |Ψ^-〉.

2. Theory

The two-photon state generated via collinear frequency-degenerate type-II SPDC from a cw pump can be written, with an account of the frequency spectrum, as

where ω ₀=ω_p/2 and ω_p is the pump frequency. Since orthogonally polarized photons have different group velocities in the crystal, the phase factor $e^{\pm i\Omega {\tau }_0}$ appears, where τ ₀=DL/2 is the mean temporal delay between orthogonally polarized photons, D≡1/uV-1/uH is the difference of the inverse group velocities and L, the length of the crystal. The spectral amplitude is $F\left(\Omega \right)=\frac{\sin \left(\Omega {\tau }_0\right)}{\Omega {\tau }_0}$.

In existing experiments with frequency-degenerate SPDC and frequency selection, it is always the exact-degeneracy frequency, Ω=0, that is selected. Then the resulting twophoton state is a factorized one, ^a† H(ω ₀)^a† V (ω ₀); it can be turned into |Ψ⁺〉 by splitting the beam on a BS. However, consider a small frequency shift from the exact degeneracy condition, Ω=π/2τ ₀. Then, with ω ₁=ω ₀-π/2τ ₀ and ω ₂=ω ₀+π/2τ ₀, (3) becomes |Ψ^-〉≡ F(π/2τ ₀)[^a† H(ω ₁)^a† V (ω ₂)-^a† V (ω ₁)^a† H(ω ₂)]|vac〉. The square modulo of the two-photon amplitude, which gives the total number of photon pairs, is in this case 0.41; it means that the singlet Bell state is produced with the efficiency only 20% smaller than the |Ψ⁺〉 state is produced using a filter and a BS [20]. Therefore, the singlet Bell state is present within the natural bandwidth of the type-II SPDC spectrum and can be filtered out using a proper spectral selection.

3. Experimental setup

 

Fig. 1. Experimental setup. A type-II BBO crystal is cut for collinear frequency-degenerate phasematching; P1 and P2, Glan prisms; D1, D2, single-photon counting modules. Retardation plates (QWP and HWP) are used to study the invariance of the Bell state |Ψ-〉 under polarization transformations. In some measurements, the monochromator is replaced by a 1 km single-mode fibre inserted after the crystal.

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We have demonstrated this phenomenon by means of a set-up (Fig. 1) where two-photon light was generated via collinear frequency-degenerate SPDC by pumping a type-II 0.5 mm BBO crystal with 0.45 W Ar⁺ cw laser beam at the wavelength 351 nm [21]. After the crystal, the pump was eliminated by a UV mirror and the SPDC radiation was addressed to a 50/50 nonpolarizing BS. To perform polarization selection, two Glan prisms were placed at the output ports of the BS. Spectral distribution of the coincidences was analyzed with a diffractive grating monochromator with the resolution 0.8 nm placed in one of the output ports. To reduce the contribution of accidental coincidences, a broadband interference filter with FWHM=40 nm centered around 702 nm was placed after BS. Biphoton pairs were registered by two photodetection apparatuses, consisting of red-glass filters, pinholes, focusing lenses and avalanche photodiodes. The photocount pulses of the two detectors, after passing through delay lines, were sent to the START and STOP inputs of a time-to-amplitude converter (TAC). The output of the TAC was finally addressed to a multichannel analyzer (MCA), and the number of coincidences of photocounts of the two detectors was observed at the MCA output.

4. Results and discussion

First, we studied the dependence of the coincidences counting rate on the wavelength selected by the monochromator. In the absence of the Glan prisms, we obtained the usual type-II SPDC spectrum with a FWHM of 12 nm. If two Glan prisms oriented at angles θ ₁,θ ₂ are inserted in the BS output ports, the coincidence counting rate R_c depends on the selected frequency offset Ω from exact degeneracy as [22, 23]

Typical dependencies of R_c on Ω at θ ₁=π/4 and different values of θ ₂ are shown in Fig. 2(a). This behavior demonstrates two-photon interference, which in this case manifests itself within the lineshape of SPDC frequency spectrum.

The interference is observed most clearly for two orientations of the Glan prism in channel 2: θ ₂=45° and θ ₂=-45°. The experimental dependencies obtained for these cases are shown in Fig. 2(b); in perfect agreement with formula (4), a maximum is observed at the center of the spectrum for the (45°,45°) orientations of the Glan prisms and a minimum, for (45°,-45°) orientations.

 

Fig. 2. (a) Theoretical dependence of the coincidences counting rate on the frequency Ω for various orientations θ ₂ of the Glan prism in channel 2. The Glan prism in channel 1 is fixed at θ ₁=45°. (b) Experimental dependence of the coincidence counting rate on the wavelength selected by the monochromator for two cases: θ ₁=θ ₂=45° (squares, solid line, in black) and θ ₁=45°,θ ₂=-45° (triangles, dashed line, in red). Lines represent a fit with Eq.(4).

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From both theoretical calculation, Fig. 2(a), and the experimental spectra, Fig. 2(b), one can see that, with certain wavelengths selected, a high-visibility polarization interference is observed under the rotation of the Glan prism in channel 2. This is well-known for the case where the selected wavelength is the central one. However, from Fig. 2(a) we see that high-visibilty polarization interference also takes place when the selected wavelength is 695.5 nm or 708.5 nm. Both cases correspond to the selection of the |Ψ^-〉 state.

To confirm this fact experimentally, we have measured the polarization interference for the |Ψ^-〉 state: the wavelength transmitted by the monochromator was fixed at 708.5 nm, and the coincidence counting rate was measured depending on the orientation of polarizer 2, the other polarizer being oriented at θ ₁=45°. The dependence, shown in Fig. 3, demonstrates a visibility of 98%.

To prove the invariance of the produced |Ψ^-〉 state under polarization transformations, we have placed quarter- and half- wave plates (QWP and HWP) at various orientations in front of the BS. In particular, if a HWP orientation is changed, the dependencies similar to those shown in Fig. 2(b) transform as shown in Fig. 4(a,b). As expected from the theoretical prediction, the coincidence counting rate corresponding to the wavelengths 695.5 nm and 708.5 nm (the |Ψ^-〉 state) both at the positions of the Glan prisms (45°,45°) and (45°,-45°) does not change depending on the HWP orientation, i.e., depending on the rotation of the biphoton polarization state before the BS. The resulting visibility of polarization interference exceeds 84% for all settings of the HWP. At all other wavelengths (including the central one, 702 nm), the coincidence counting rate clearly depends on the HWP orientation, and for some orientations, the visibility of polarization visibility becomes zero. Similar behavior is observed for a QWP inserted before the BS.

 

Fig. 3. Polarization interference fringes for the singlet Bell state |Ψ;-〉 (the selected wavelength is λ=708.5nm).

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5. Conclusion

The conclusion is that the frequency spectrum of collinear frequency-degenerate type-II SPDC contains, at about half-width from the center, the decoherence-free singlet Bell state |Ψ^-〉. In order to use this state in quantum communications, one should select a relatively narrow frequency bandwidth. In the above-described experiment, 0.8 nm bandwidth was sufficient for the case of a 0.5 mm crystal, but using a thicker crystal would require a more narrow frequency selection. This difficulty, however, can be overcome easily if the produced state is to be transmitted through optical fibres.

Indeed, in an optical fibre with group-velocity dispersion, the shape of the two-photon spectral amplitude is transformed into the shape of the two-photon time amplitude and hence, to the distribution of the time interval between the arrivals of two correlated photons [22, 23]. The frequency argument of the spectral two-photon amplitude is then transformed into the time argument of the time two-photon amplitude: Ω→τ=2k″ zΩ, k″ being the second derivative of the dispersion law and z the fibre length. Frequency selection of |Ψ^-〉 can be then performed through the time selection of the delay between registering two photons.

This fact can be used in quantum communication whenever propagation of polarization-entangled photons through optical fibres is involved. On the one hand, the receiver can always post-select the singlet state by picking only those coincidence events for which the signal and idler photons come with a fixed nonzero delay τ=πk″z/τ ₀. On the other hand, polarization drift introduced by the fibre will never influence the state |Ψ^-〉. To demonstrate this, we have removed the monochromator and selected the singlet state by means of a 1 km fibre inserted before the BS. Coincidence distributions obtained for the (45°,45°) and (45°,-45°) settings of the Glan prisms are presented in Fig. 5. One can see that at time delays not corresponding to the generation of the singlet state, the dependencies have complicated shapes and do not correspond to the behavior shown in Fig. 2(a), a result of polarization drift in the fibre. At the same time, at points pertaining to the |Ψ^-〉 generation there is a coincidence minimum for the (45°,45°) settings and a maximum for the (45°,-45°) settings of the prisms.

In order to implement any protocol of information transmission (e.g. the one in [13]), one has to generate another entangled state at the same frequencies as |Ψ^-〉. In particular, |Ψ⁺〉 can be easily created at the same frequencies by introducing after the crystal a birefringent material twice increasing the τ ₀ delay. This can be done by inserting a quartz plate with the same thickness as it is necessary to compensate for the τ ₀ delay, but with the optic axis oriented orthogonally [22, 23]. An advantage of this scheme is that the possibility of having the two components at very close frequencies allows to easily cope with depolarization effects of fibers [11].

 

Fig. 4. Experimental dependence of the coincidence counting rate on the wavelength selected by the monochromator for the following orientations of the HWP placed after the crystal: 7°(circles, dotted line, in red); 17° (squares, dashed line, in green); 22,5° (triangles, solid line, in black). Orientations of the Glan prisms are 45°,45° (a) and 45°,-45° (b). Dashed vertical bars show the wavelength where |Ψ^-〉 is generated. Lines represent the theoretical fit.

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Fig. 5. Experimental dependence of the coincidence counting rate on the delay between the arrivals of two photons, with the monochromator replaced by a 1 km optical fibre, for 45°,45° (triangles in black) and 45°,-45° (circles in red) settings of the Glan prisms.

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In summary, we have shown for the first time how different Bell states are contained inside the linewidth of phase matching: a phenomenon that should be always carefully considered in applications of these states. In particular we have demonstrated, both theoretically and experimentally, that for type II SPDC, whilst |Ψ⁺〉 appears in the center of band, |Ψ^-〉 appears on the slopes: a result of utmost relevance for fiber quantum communication.