A microscopic multimode theory of collinear type-I spontaneous parametric downconversion in a cavity is presented. Single-mode and multimode correlation functions have been derived using fully quantized atom and electromagnetic field variables. From a first principles calculation the FWHM of the single-mode correlation function and the cavity enhancement factor have been obtained in terms of mirror reflectivities and the first-order crystal dispersion coefficient. The values obtained are in good agreement with recent experimental results [Phys. Rev. A 62 , 033804 (2000)].
© 2002 Optical Society of America
The optical process of spontaneous parametric down-conversion (SPDC) involves the virtual absorption and spontaneous splitting of an incident (pump) photon in a transparent nonlinear crystal producing two lower-frequency (signal and idler) photons [1–3]. The pairs of photons can be entangled in frequency, momentum and polarization. In type-I SPDC the photons are frequency-entangled and the signal and idler photons have parallel polarizations orthogonal to the pump polarization.
Entangled photons have been used to demonstrate quantum nonlocality [4,5], quantum teleportation [6–8] and more recently quantum information processing [9–11]. Photon pairs have also been used to demonstrate quantum interference phenomena [12–14]. The photon pairs produced in such experiments are separated by less than a picosecond and their correlation properties could only be investigated indirectly, for example by fourth-order interference [2,12]. Ou and Lu [15,16] have recently measured the time separation of photons produced from a nonlinear crystal placed inside a high-Q cavity. Because of the reduced bandwidth of the photons and consequent broadening of the photon correlation functions they were able to measure pair-photon correlations directly. Measurements of single-mode and multimode correlation functions were made for photons with frequencies close to the degenerate frequency for type-I collinear SPDC. Experimental results were modelled using the theory of Collett and Gardiner .
In this paper we propose an alternative microscopic multimode theory to describe the detection of photon pairs produced from a nonlinear crystal placed inside a high-Q cavity. Given the limited number of studies in the sub-threshold regime of operation of the optical parametric oscillator (OPO), we believe our approach will broaden the understanding of the sub-threshold operation of the OPO. We describe the situation in which collinear photon pairs, which experience multiple reflections in the cavity, are produced by pump photons which pass through the crystal once; this is the single-pass case and corresponds to an OPO operating far below threshold . We calculate analytic expressions for both the single-mode and multimode correlation functions in terms of the crystal and cavity parameters. Our approach yields similar results to the theory used by Ou and Lu, but in addition, we have been able to obtain exact expressions for the hitherto phenomological coupling constants introduced in the Ou and Lu analysis. Such constants are important in determining the FWHM of the correlation functions. We consider a high-Q cavity which corresponds to the experimental situation with detectors positioned outside the cavity. Our theoretical simulations compare well with the experimental results of Ou et. al.
2. Spontaneous down-conversion amplitude for a crystal in a cavity
If we first consider a crystal atom at position r⃗3 in free space and for one-atom detectors located at r⃗1 and r⃗2, the generalized amplitude for detecting pairs of down-converted photons from a single crystal atom is given by .
E a,…,e are components of the electric field vector and it should be noted that in (1) repeated superscripts are being summed over. The initial state of the electromagnetic field, |0, α λ0 k0〉, consists of a coherent state with wave-vector k 0 and polarization index λ 0 (the monochromatic pump beam) with other modes in the vacuum state |0〉 . |g 1, g 2 ,g 3> is the wave-function describing the ground state of the detector atoms and the crystal atom and |a 1,a 2,g 1〉 describes the detector atoms in excited states |a 1,a 2〉 and a source atom finally in the ground state |g 3〉 after the two-photon emission process. μ a,…,e(t) denotes components of the interaction-picture electric dipole moment operator for the multi-level atom at position r j.
To obtain the amplitude, , for pair-photon detection in the case of a crystal of length d placed inside a cavity of length d, all embedded in a linear medium of the same refractive index  (see Figure 1), we employ the plane-wave mode functions for the quantized electric field  for the cavity and external reservoir system. Since we are using the complete electromagnetic field which describes modes both inside and outside the cavity, damping effects are already included in the theory and therefore need not be included phenomenologically. For a one-sided cavity in which the amplitude transmission coefficient for photons exiting through mirror M1 from inside the cavity is zero, the quantized electric field is given as
The destruction operators for the modes with spatial function U k⃗j(r⃗) are denoted a k⃗j where εj (k⃗) (j=1,2) denotes the mode polarization vector. The spatial function U k⃗j(r⃗) for inside the cavity is denoted by U in,k⃗j(r⃗) and is defined by the following
t 2o and r 2o being the amplitude transmission and reflection coefficients of M2 respectively for the signal and idler photons; we have taken the reflection coefficient of M1 as r 1o = - 1 consistent with a perfectly reflecting and infinitely thin mirror. The wave vectors k⃗(-) and k⃗(+) describe backward and forward propagating photons and are defined in terms of polar (θ) and azimuthal (ϕ) angles by
The angle θ is measured from the direction of the incident pump beam and therefore θ = 0 corresponds to collinear propagation. In (2) the k integral is defined as
Similarly, the parts of the modes outside the cavity in the region in front of the crystal, i.e., are described by the mode function
After substituting the appropriate expressions for the electric fields in (1) using (3) when r⃗ = r⃗3 and (7) when r⃗ =r⃗1,2 for the spatial mode function, we obtain, on performing a simple integration over the irradiated volume of the crystal
where t 1p is the amplitude transmission coefficient of M1 at the pump frequency. Crystal dispersion has been taken into account with the following wave-vector expansion:
where , are the perfectly phase-matched frequencies and wave-vectors respectively which satisfy the following energy-conservation and phase-matching conditions:
The correlation time τ′ = (t 1 -t 2) + v[z 2 - z 1) and the integration variable x=ω k⃗i *; is the first-order dispersion coefficient and is the second-order dispersion coefficient. In obtaining (9) we have assumed that the pump has a spatial profile which is Gaussian in the x-y plane  with a beam-waist radius of z 1,z 2 are the positions of the detectors along the axis of the cavity. For simplicity, we can assume that the detectors are situated at equal distances from the cavity. For a high-Q cavity we use the following approximation  to the Airy function denominator of the integrand in (9)
The summation in Eq. (12) describes the detection of the degenerate mode plus N non-degenerate modes on either side of the central degenerate mode. The term in Eq. (12) corresponding to l = 0 describes the detection of the degenerate mode. After substituting Eq. (12) in Eq. (9) we obtain to a good approximation the following amplitude
where The single-mode amplitude which describes the detection of the central degenerate mode corresponds to the situation in the above equation when N = 0. We therefore obtain the single-mode amplitude ASM as
3. Multimode pair-photon count rate
The multimode detection probability is equal to the single-mode probability multiplied by a prefactor which is oscillatory. It is instructive to re-express the oscillatory prefactor a
The term with the smallest frequency, has a period ~ 10-12 s. Since detectors cannot measure such rapid oscillations in time, only an average is recorded, and therefore the cosine terms make a vanishing contribution to the count rate. Hence the multimode amplitude is given by
The count rate in the multimode case is therefore larger by a factor of (2N+1) compared to the single-mode count rate, but shows the same time dependence as the single-mode case.
5. Enhancement-factor per mode
The enhancement factor per mode, γ, is defined by the following:
i.e., it is the ratio of the count rate per unit frequency with the crystal in the cavity to the count rate per unit frequency with the crystal in free space. We first of all need to obtain the spectrum of the down-converted light with and without the cavity. In calculating the bandwidth of the light with the cavity we use Eq. (9) for and integrate (using the approximation in Eq. (12) with l = 0) with respect to the correlation time τ′. The FWHM, (Δω)cav, of this function is a reasonable estimate of the bandwidth and is approximately . The numerator in Eq. (18) then works out as where k is a constant. In the absence of the cavity we use the right-hand-side of Eq. (32) in  for the spectrum of the degenerate photons. This gives us an approximate bandwidth The denominator in Eq. (18) to a good approximation is then (count rate/bandwidth)nocav = k . The enhancement factor, γ, is then given by,
If we consider the case of single-mode detection, our analysis predicts a theoretical FWHM of For a 4.1 mm cavity, M2 transmission coefficient of 1.5% and v = 8×C10-9 m-1s  the width is approximately 6 ns. This is in good qualitative agreement with the results of Ou and Lu [15,16] where a 4.00 mm crystal was used and a 3.35 mm external filter cavity was used to filter out the nondegenerate modes so that the measurements correspond to single-mode correlations. Also the enhancement factor is calculated to be 9.3×104 which compares well with the results of Ou et. al. who obtained 5.5 × 104. Any differences between our predictions and that of Ou et. al. is most probably due to the fact that we assumed that M1 was a perfectly reflecting mirror.
A multimode microscopic model of the optical parametric oscillator operating well below threshold has been presented. Effects such as cavity damping and crystal dispersion are both taken into account. We have derived expressions for both the single-mode and multimode amplitudes. For a cavity of a given finesse the calculated value of the FWHM of the single-mode correlation function and the cavity enhancement factor are calculable and compare well with the experimental results of Ou et. el. In a future publication we intend to generalize the theory to describe an oscillator operating close to threshold in which correlated squeezed states can be generated. These sources are have been attracting much attention recently because of their use in quantum information processing systems.
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