1. Introduction

Recently there has been renewed interest in photon-antibunched sources of light due to their possible applications in quantum cryptographic systems [1]. Such sources contain a single photon in a given time interval and can therefore be used for secure communication purposes. Photon bunching and antibunching has been observed recently in fluorescence from a variety of sources like quantum dots [2,3], color centers in diamond [4] and single molecules [5]. In this paper we describe theoretically from first principles the generation, via interference, of two-photon antibunched light from a sub-threshold parametric oscillator. We consider the recent experiments of Ou and Lu [6] in which the output of a sub-threshold parametric oscillator (OPO) is mixed with a continuous-wave local oscillator at a 50/50 beam splitter. Due to quantum interference in the detection of photon pairs from the local oscillator (LO) and the OPO, it has been demonstrated experimentally that the degree of bunching or antibunching can be controlled by the relative phase between the OPO and LO. We apply the recent multimode formalism of the authors [7] which describes the quantum correlations of the output of the OPO and examine the nonclassical effects of homodyning the OPO field with a CW laser field or local oscillator (LO). We first examine the variation of the pair count rate with phase-difference between the LO and the OPO. Our theoretical results are then to used to model the experimental results of Lu and Ou [6]. We then examine the variation of the spectrum of squeezing [8] with phase-difference.

2. Amplitude for pair detection in homodyne

In the experiment of Ou and Lu [6] a coherent state acting as the LO is injected into the OPO and is mixed with the two-photon state. The resultant field then exits the cavity and interacts with a beam splitter outside the OPO (see Fig. 1 in reference [6]). As pointed out in [6], the effect of injecting the coherent state into the cavity is to synchronize the two fields and for spatial filtering of the coherent state. If we assume that a perfect coherent state and a two-photon state in superposition exits the cavity, then such a procedure is equivalent to the following homodyne detection scheme in Fig. 1 since the beam splitter transformation of the electric fields of the OPO and LO is the same in both cases. Therefore, for convenience in describing the experiment of Ou and Lu, consider the following setup (see Fig. 1 below) in which the field from the OPO is mixed with the field of the LO:

 

Fig. 1. Schematic showing the interference of the field from a LO and OPO. BS is a 50/50 beam splitter and D1 and D2 are photon counters.

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If we assume for simplicity that the photon detectors D1 and D2 are single atoms located at r⃗₁ and r⃗₂ respectively, the amplitude for pair detection in the Heisenberg representation is described by

where E⃗_H (r⃗_i, t _i) is the Heisenberg electric field operator at the detectors. We assume that the input pump laser to the OPO and the local oscillator are in single-mode coherent states, |α_LO,α_pump〉, with the LO emitting at the degenerate frequency of the OPO. We also assume single-pass operation of a one-sided OPO as in previous work [7]. In the interaction picture and for an OPO operating well below threshold, we obtain the following amplitude for pair detection to first order in the nonlinear susceptibility χ as

E⃗_I(r⃗_i,t) are the interaction-picture electric field operators and ε_p,φ_p are the amplitude and phase of the pump respectively; $U_{k_i{\lambda }_i}\left({\overrightarrow{r}}_3\right)$ are the spatial mode functions of the electromagnetic field inside the cavity [7,9] with k ₁, k ₂ as the wave vectors of the signal and idler photons and k ₀ the wave vector of the input pump photon. In obtaining (2) we employed beam splitter transformations for the electric field at the detectors:

where E⃗_out and E⃗_LO are the electric fields of the OPO and LO respectively. Equation (2) describes the phenomenon of quantum interference in the detection of pairs from the LO and OPO. In order to simplify (2) we follow similar calculations in previous work [7] and obtain (apart from constant factors), the following:

ε_l,φ_l are the amplitude and phase of the LO, τ is the correlation time, d is the length of the OPO cavity and v is the first-order dispersion coefficient of the nonlinear crystal; S and n ₁ are defined as

where t _2o,r _2o are the amplitude transmission, reflection coefficients of the output mirror of the OPO at the signal and idler frequencies and V the volume of the irradiated crystal. We obtain the count rate which is proportional to ${\mid A^{\left(2\right)}\mid }^2$ as

where $q=\frac{{\epsilon }_p\chi S}{{\epsilon }_l^2}$

3. Spectrum of squeezing

Let us now determine, in general, the nature and extent of squeezing when interference effects are present. In balanced homodyne detection, one measures the difference in the powers of the two beams at r⃗₁ and r⃗₂. The difference in powers of the two beams is obtained from P(t) defined as

Quantum fluctuations around a mean value 〈P(t)〉 is described by

and the spectrum of squeezing, Q(ω), is defined as

The expected value in (9) is calculated to be

where

Δ is the bandwidth of the detected light, $n_{\frac{k_0}{2}}$ is the refractive index of the nonlinear crystal at the degenerate frequency. The other constant γ in (10) is given as

The variable ω in (11) is defined as $\omega =\frac{{\omega }_{k_0}}{2}-{\omega }_k$ , where ω_k is the frequency of the signal photon detected at D1. Taking the Fourier transform of the R.H.S of (10) we obtain the normalized spectrum of squeezing at the degenerate frequency ${\omega }_k=\frac{{\omega }_{k_0}}{2}$ as

where

4. Results

Figures 2a, 2b and 2c are plots of ${\mid A^{\left(2\right)}\mid }^2$ against $\frac{\tau }{n_1\mathit{vd}}$ for q = 2, Δ = 2φ_l - φ_p = 0; q = 2 and Δφ = π and q = 1, Δφ = π respectively. These results agree qualitatively with the experimental results of Lu and Ou in Fig. 3 of their paper [6]. We confirm with our model photon antibunching with q = 1, Δφ = π at zero time delay (Fig. 2c) and at non-zero time delays with q = 2, Δφ = π (Fig. 2b).

 

Fig. 2. Normalized two-photon count rates $\left({\mid A^{\left(2\right)}\mid }^2\right)$ against $\frac{\tau }{n_1\mathit{vd}}$ based on Eq. (7) showing antibunching effects. (a) q = 2 and Δφ = 0, (b) q = 2 and Δφ = π and (c) q = 1 and Δφ = π.

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In Section 3 of our paper we examined the spectrum of squeezing. We found that for the degenerate frequency, the spectrum was given by (12). In the RHS of (13) the first term represents the shot noise, the second is the noise contribution due to the local oscillator and the third is a noise term due to quantum interference. It is clear that noise reduction or squeezing will occur when Δφ = 0 and δ > μ. Based on the results given in Fig. 2, squeezing is then only obtained for highly bunched photon pairs. Using the expressions given in (14) we find that for a 4 mm OPO cavity, degenerate wavelength 700 nm, output mirror transmittivity of 1%, crystal refractive index 1.5, irradiated crystal volume 10^-9 mm³, squeezing is obtained when $\frac{{\epsilon }_p}{{\epsilon }_l^2}>\frac{3\times {10}^{-6}}{\chi }$ , where χ is the nonlinear susceptibility of the crystal. This can certainly be achieved with current technology and therefore such squeezing should be measurable.

5. Conclusion

We presented, from a first principles calculation, a fully quantized multimode theory to describe interference effects of a sub-threshold optical parametric oscillator with a local oscillator. We provided theoretical calculations which describe adequately recent experimental results [6]. In addition we investigated the spectrum of squeezing in the presence of interference and found that a well-defined squeezing threshold exists for zero relative phase difference between the pump and local oscillator and for specific pump and local oscillator intensities given in terms of their respective amplitudes by $\frac{{\epsilon }_p}{{\epsilon }_l^2}>\frac{3\times {10}^{-6}}{\chi }$ .