6.3.1
Two-photon state
In the experiments with twin photons created by the process of SPDC in a non-linear crystal, the photons that are to be detected propagate along the lines connecting the crystal and the detectors. Since then the propagation directions are well-defined, it is advantageous to use a description of the two-photon state in terms of paraxial creation operators. A paraxial creation operator creates a photon in a mode that is localised in a single point in the output plane of the crystal, which simplifies the description greatly, because the photons are created at the same location in the crystal. We derive an expression for the two-photon state created by SPDC in terms of paraxial creation operators and obtain an expression for the coincidence detection amplitude.
We consider the scheme in Fig. 6.1. A pump beam creates collinear twin photons in a non- linear crystal, which is cut such that there is type-I phase matching for collinearly propagating twin photons. After creation the twin photons fall on a 50% : 50% beam splitter. In the output channels of the beam splitter are optical systems, at the end of which the photons are detected. The optical lines between the output plane of the crystal and the detection planes of detector 1 and 2 are referred to as the detection channels 1 and 2, respectively. Only coincidence detection counts are considered for which one photon of a twin is detected in each channel.
In order to arrive at a simple expression for the two-photon state, we apply the paraxial approximation to the electric-field operator that appears in the Hamiltonian for SPDC. In the
interaction picture the Hamiltonian is given by [47] ˆ HI(t) = Z C d~r Ep+(~r,t)Eˆ−(~r,t)Eˆ−(~r,t), (6.17) whereE+p is a component of the positive-frequency part of the classical electric field of the pump beam, and where ˆE−is a component of the negative-frequency part of the electric-field operator. The integration domain ranges over the spaceCthat is occupied by the crystal. In Chapter 8 polarisation aspects of SPDC are discussed. With the vacuum state|0ias the initial state and the evolution governed by the Hamiltonian for SPDC in Eq. (6.17), the state at time
tis, up to first order, the two-photon state
|Ψ(t)i∝
Z t
∞dt
0H
I(t0)|0i. (6.18) The Hamiltonian (6.17) contains the positive-frequency part of the electric field of the pump beam, and the negative-frequency part of the electric-field operator. We apply the paraxial approximation to these fields, and take~εzas the direction of propagation of the twin photons and the pump beam. The output plane of the crystal is located at the planez=0. We consider cases for which the pump beam is monochromatic with frequencyωp, while the photons of a twin are detected in a narrow frequency band centred aroundωp/2. We assume that the crystal is thin so that the transverse profile of the pump field is constant inside the crystal. Then the Rayleigh range of the pump beam must be larger than the thickness of the crystal. For the pump field inside the crystal at the transverse coordinate~ρ in the planezwe write
Ep+(~r,t) =G(~ρ)exp[−iωp(t−npz/c)], (6.19) wherenpis the refractive index, which is determined by the frequencyωpand the polarisation of the pump beam. The negative-frequency part of the electric-field operator in the paraxial approximation is given by the Hermitian conjugate of Eq. (6.11). Since the photons of a twin are detected around the frequencyωp/2, we replaceωin the square-root term inside the integral overωbyωp/2, and take it out of the integral. The transverse profile is assumed to be constant inside the crystal, so we insertz=0 intohf(~ρ,~ρ0;z,ω)and use Eq. (6.14). We drop the polarisation labeling, which is justified since the photons of the twin have identical polarisations and the optical systems are insensitive to polarisation. Instead, we introduce a label that refers to the detection channel. The negative-frequency part of the electric-field operator inside the crystal is then written as
ˆ E−(~r,t)∝ Z ∞ 0 dωexp[iω(t−ntz/c)] h ˆ a†1(~ρ,ω) +aˆ†2(~ρ,ω)i , (6.20) wherent is the refractive index for the twin photons, which is determined by the frequency ωp/2 and the polarisation. In this expression the paraxial creation operator ˆa†i(~ρ,ω)creates a photon in detection channeli=1,2.
We insert Eqs. (6.19) and (6.20) into Eq. (6.18). The integration over the space occupied by the crystal gives rise to the phase-matching condition. The integration over the longitu- dinal coordinatezyields the longitudinal phase-matching condition, which is satisfied in our case for collinear twin photons with the frequencyωp/2. The integration overt0 gives rise
6. Using paraxial quantum operators
to the frequency-matching condition, allowing only states where the frequencies of the pho- tons of the twin add up to the pump frequency. Since the pump beam is monochromatic, the two-photon state is in principle independent of time, and we can replace in Eq. (6.18) the upper limit of the integration overt0 by∞. The two-photon state is then a superposition of all the states where the frequencies of the photons of the twin add up toωp. Since the twin photons are detected with frequencyωp/2 we only retain the part of the superposition where both photons of the twin have this frequency. Since we consider only coincidence detections, the part of the state where both photons exit the beam splitter in the same output channel is dropped. Under these assumptions the two-photon state can be written as
|Ψi∝
Z
d~ρG(~ρ)aˆ†1(~ρ,in)aˆ†2(~ρ,in)|0i, (6.21) where ˆa†i(~ρ,in)creates a photon with frequencyωp/2 in detection channeli=1,2 in the mode that is aδ-function at the location~ρin the output planez=0 of the crystal. By writing the two-photon state (6.21) in this form, it becomes clear that the photons of a twin are created at the same transverse location, but that this location itself is uncertain within the pump spot, which means that the twin photons are spatially entangled (see Chapter 7).
6.3.2
Coincidence detection rate
After creation the photons of a twin propagate through the detection channels. The amplitude for the detection of a photon at position~ρ1in the plane of detector 1, in coincidence with the
detection of a photon at position~ρ2in the plane of detector 2, is given by
A(~ρ1,~ρ2) =h0|aˆ1(~ρ1,out)aˆ2(~ρ2,out)|Ψi. (6.22)
where ˆai(~ρ,out)annihilates a photon with frequencyωp/2 in a mode that is a δ-function at the location~ρ in the plane of detectori=1,2. As in Eq. (6.16) the ’out’ operators are expressed in terms of the ’in’ operators by
ˆ
ai(~ρ,out) = Z
d~ρ0hi(~ρ,~ρ0)aˆi(~ρ0,in) +fˆi(~ρ), (6.23) where hi(~ρ,~ρ0)is the transfer function of detection channel i=1,2 for a light field with frequencyωp/2. Inserting Eq. (6.23) into Eq. (6.22) gives rise to four terms, three of which contain a noise source. The noise operator annihilates a photon in a mode of the outside world. As a consequence, only the term that does not contain a noise source contributes to the coincidence detection amplitude. Accordingly, for coincidence detection the noise sources are irrelevant. It follows that
A(~ρ1,~ρ2)∝
Z
d~ρG(~ρ)h1(~ρ1,~ρ)h2(~ρ2,~ρ). (6.24)
This result was also obtained and employed elsewhere [50–52]. The coincidence detection rate is given by