Introduction to the absolute brightness and number statistics in spontaneous parametric down-conversion

The canonical quantization of the electromagnetic field into plane-wave modes with creation operators ${\hat{a}}_{\vec{k},s}^{\dagger }$ is the first step in the standard quantum treatment of SPDC light. However, it will make subsequent calculations much simpler if we express the transverse momentum components of the field in terms of Hermite–Gaussian modes, since Gaussian pump beams and similar collection modes of down-converted light are valid descriptions of the light generated in SPDC experiments. Moreover, expressing SPDC in terms of Laguerre-Gaussian and Hermite–Gaussian modes of light is physically motivated because of their connection to orbital angular momentum (Allen et al 1992), and the conservation of the same quantity (Walborn et al 2010).

In order to do this, we first introduce some notation. Let $\vec{q}$ denote the projection of the momentum $\vec{k}$ onto the transverse plane, so that $\vec{k}=\vec{q}+{k}_{z}\hat{z}$, and $\hat{z}$ is a unit vector pointing along the optic axis in the direction of propagation. Then, the creation operator ${\hat{a}}_{\vec{k},s}^{\dagger }$ can be expressed as ${\hat{a}}_{(\vec{q},{k}_{z},s)}^{\dagger }$. Since both plane waves and Hermite–Gaussian wavefunctions form a complete basis in 2D space, we can express the plane-wave creation operator ${\hat{a}}_{(\vec{q},{k}_{z},s)}^{\dagger }$ as a sum over transverse mode creation operators ${\hat{a}}_{(\vec{\mu },{k}_{z},s)}^{\dagger }$, where $\vec{\mu }$ is a vector denoting the horizontal and vertical indices of a given Hermite–Gaussian mode;

$$\begin{eqnarray}&&{\hat{a}}_{(\vec{q},{k}_{z},s)}^{\dagger }=\displaystyle \sum _{\vec{\mu }}{\tilde{C}}_{\vec{q},\vec{\mu }}\,{\hat{a}}_{(\vec{\mu },{k}_{z},s)}^{\dagger }.\end{eqnarray} \tag{ 12 }$$

Since this change of basis preserves probability, and is therefore unitary, it follows that the boson commutation relation $[{\hat{a}}_{(\vec{q},{k}_{z},s)},{\hat{a}}_{(\vec{q^{\prime} },{k}_{z},s)}^{\dagger }]={\delta }_{\vec{q},\vec{q^{\prime} }}$ must be preserved in both representations. With this established, we can express the displacement field operator ${\hat{D}}^{-}(\vec{r},t)$ in this new Hermite–Gauss basis.

The transverse spatial dependence of ${\hat{D}}^{-}(\vec{r},t)$ for a given Hermite–Gauss mode indexed by $\vec{\mu }$ relies on the sum:

$$\begin{eqnarray}&&\displaystyle \sum _{\vec{q}}{\tilde{C}}_{\vec{\mu },\vec{q}}\,{e}^{-i\vec{q}\cdot \vec{r}}=\sqrt{{L}_{x}{L}_{y}}\,{g}_{\vec{\mu }}(x,y),\end{eqnarray} \tag{ 13 }$$

where we have defined ${g}_{\vec{\mu }}(x,y)$ to be the normalized ¹⁴ Hermite–Gaussian wavefunction given by the index $\vec{\mu }$. Here, $\vec{\mu }$ is an ordered pair of non-negative integers corresponding to the horizontal and vertical mode index, respectively. Because the momentum components can only take on values that are integer multiples of 2π divided by the respective length of the cavity in each direction, this relation is straightforward to check through normalization. For finite size nonlinear crystals, this relation is approximate, but accurate when the Hermite–Gaussian modes are encompassed by the crystal. Finally, using an element of the paraxial approximation (so that the frequency ω only depends on k_z ), the displacement field operator becomes:

$$\begin{eqnarray}&&{\hat{D}}^{-}(\vec{r},t)=-i\displaystyle \sum _{\vec{\mu },{k}_{z},s}\sqrt{\displaystyle \frac{{\epsilon }_{0}{n}_{{k}_{z}}^{2}{\rm{\hslash }}{\omega }_{{k}_{z}}}{2{L}_{z}}}{\vec{\epsilon }}_{{k}_{z},s}{g}_{\vec{\mu }}(x,y){e}^{-{{ik}}_{z}z}{e}^{i\omega t}{\hat{a}}_{\vec{\mu },{k}_{z},s}^{\dagger }.\end{eqnarray} \tag{ 14 }$$

Here, we point out that ${\hat{a}}_{\vec{\mu },{k}_{z},s}^{\dagger }(t)={\hat{a}}_{\vec{\mu },{k}_{z},s}^{\dagger }{e}^{i\omega t}$.

With the displacement field operators expressed in the Hermite–Gauss basis, we are ready to obtain the nonlinear Hamiltonian (11). In the standard approach for treating SPDC, the pump field is treated as being bright enough that classical electromagnetism is sufficient for its description, and that its intensity is not noticeably diminished due to down-conversion events (also known as the undepleted pump approximation). We use the classical pump approximation throughout most of this paper, but use a more accurate description when discussing the number statistics of the down-converted light.

2.1.1. The classical pump field

Although arbitrary illumination of the nonlinear medium can be expressed as an integral over all frequencies, SPDC occurs only in narrow bands of pump frequencies where phase matching (i.e. momentum conservation) can be achieved due to dispersion ¹⁵ . In light of this, we limit ourselves to the ubiquitous case of a monochromatic pump beam with peak magnitude $| {D}_{p}^{0}|$, frequency ω_p , polarization ${\vec{\epsilon }}_{p}$, and (non-normalized) spatial dependence ${f}_{p}(\vec{r})$ given by:

$$\begin{eqnarray}&&{\vec{D}}_{p}(\vec{r},t)=| {D}_{p}^{0}| {\vec{\epsilon }}_{p}{f}_{p}(\vec{r})\,\cos ({\omega }_{p}t),\end{eqnarray} \tag{ 15 }$$

which can be separated into positive and negative frequency components, giving us:

$$\begin{eqnarray}&&{\vec{D}}_{p}^{-}(\vec{r},t)=| {D}_{p}^{0}| {\vec{\epsilon }}_{p}{f}_{p}(\vec{r})\displaystyle \frac{{e}^{i{\omega }_{p}t}}{2}.\end{eqnarray} \tag{ 16 }$$

The (time averaged) pump intensity ¹⁶ is then:

$$\begin{eqnarray}&&{I}_{p}=\displaystyle \frac{c}{2{\epsilon }_{0}{n}^{3}}\,| {D}_{p}^{0}{| }^{2}| {f}_{p}(\vec{r}){| }^{2}.\end{eqnarray} \tag{ 17 }$$

For later simplification, we factor out the linear phase due to propagating the beam, and get:

$$\begin{eqnarray}&&{f}_{p}(\vec{r})={\tilde{G}}_{p}(\vec{r}){e}^{-{{ik}}_{z}z},\end{eqnarray} \tag{ 18 }$$

where ${\tilde{G}}_{p}(\vec{r})$ is implicitly defined.

Throughout most of this paper, ${\tilde{G}}_{p}(\vec{r})$ will describe the rest of a Gaussian pump beam, so that

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{G}}_{p}(\vec{r}) & \equiv & \displaystyle \frac{{\sigma }_{p}}{\sigma (z)}\exp \left(-\displaystyle \frac{{x}^{2}+{y}^{2}}{4\sigma {\left(z\right)}^{2}}\right)\exp \left(-{{ik}}_{z}\displaystyle \frac{{x}^{2}+{y}^{2}}{2R(z)}\right)\\ & & \times \exp \left(i\,{\tan }^{-1}\left(\displaystyle \frac{z}{{z}_{R}}\right)\right).\end{array}\end{eqnarray} \tag{ 19 }$$

Except in section 3.4 where we consider SPDC using a focused pump beam, we use the simplifying approximation that the pump beam is collimated, so that we may neglect the Guoy phase and curvature of the phase fronts in our calculations. To condense notation, σ(z) is the evolving beam radius (as measured by standard deviation);

$$\begin{eqnarray}&&\sigma (z)\equiv {\sigma }_{p}\sqrt{1+{\left(\displaystyle \frac{z}{{z}_{R}}\right)}^{2}}.\end{eqnarray} \tag{ 20 }$$

R(z) is the evolving radius of curvature of the wavefronts:

$$\begin{eqnarray}&&R(z)\equiv z\left[1+{\left(\displaystyle \frac{{z}_{R}}{z}\right)}^{2}\right]\end{eqnarray} \tag{ 21 }$$

and z_R is the Rayleigh length, such that $\sigma ({z}_{R})=\sqrt{2}{\sigma }_{p};$

$$\begin{eqnarray}&&{z}_{R}\equiv \displaystyle \frac{4\pi {\sigma }_{p}^{2}}{{\lambda }_{p}}.\end{eqnarray} \tag{ 22 }$$

As we can see, the first exponential governs the evolving spatial amplitude of the beam; the second exponential describes the propagation and curvature of the phase fronts, while the last exponential describes the Guoy phase.

With the parameters of a Gaussian pump beam, the amount of energy per second delivered by such a beam (i.e. its power) is expressed as:

$$\begin{eqnarray}&&P=c\displaystyle \frac{| {D}_{p}^{0}{| }^{2}}{{n}^{3}{\epsilon }_{0}}\pi {\sigma }_{p}^{2},\end{eqnarray} \tag{ 23 }$$

which equals the mean intensity of the beam times its effective area ¹⁷ .

2.1.2. Simplifying the nonlinear Hamiltonian

Incorporating our expressions for the displacement field operators and the classically bright pump field, the nonlinear Hamiltonian becomes:

$$\begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{{NL}} & = & \displaystyle \int {d}^{3}r\left({\zeta }_{\mathrm{eff}}^{(2)}(\mathop{r}\limits^{})| {D}_{p}^{0}| {\tilde{G}}_{p}^{* }(\mathop{r}\limits^{}){e}^{{{ik}}_{{pz}}z}{e}^{-i{\omega }_{p}t}\right.\\ & & \times -i\displaystyle \sum _{{\mathop{\mu }\limits^{}}_{1},{k}_{1z}}\sqrt{\displaystyle \frac{{\epsilon }_{0}{n}_{1}^{2}{\rm{\hslash }}{\omega }_{1}}{2{L}_{z}}}{g}_{{\mathop{\mu }\limits^{}}_{1}}(x,y){e}^{-{{ik}}_{1z}z}{e}^{i{\omega }_{1}t}{\hat{a}}_{{\mathop{\mu }\limits^{}}_{1},{k}_{1z}}^{\dagger }\\ & & \times -i\displaystyle \sum _{{\mathop{\mu }\limits^{}}_{2},{k}_{2z}}\sqrt{\displaystyle \frac{{\epsilon }_{0}{n}_{2}^{2}{\rm{\hslash }}{\omega }_{2}}{2{L}_{z}}}{g}_{{\mathop{\mu }\limits^{}}_{2}}(x,y){e}^{-{{ik}}_{2z}z}{e}^{i{\omega }_{2}t}{\hat{a}}_{{\mathop{\mu }\limits^{}}_{2},{k}_{2z}}^{\dagger }\\ & & \left.+{\rm{H}}{\rm{.}}{\rm{c}}.\right).\end{array}\end{eqnarray} \tag{ 24 }$$

Here, we abbreviated ${n}_{{k}_{1z}}$ as n₁, and ${\omega }_{{k}_{1z}}$ as ω₁. Furthermore, we have already performed the sum over the components of the inverse susceptibility. The additional factor of 6 = 3! comes from the permutation symmetry of the nonlinear susceptibility where the total sum is 6 times the value of each term, where all terms are added together. After simplifying, we find:

$$\begin{eqnarray}\begin{array}{rcl}{\hat{{\rm{H}}}}_{{NL}} & = & \displaystyle \frac{{\rm{\hslash }}| {E}_{p}^{0}| }{2{L}_{z}}\displaystyle \sum _{{\vec{\mu }}_{1},{k}_{1z}}\displaystyle \sum _{{\vec{\mu }}_{2},{k}_{2z}}\sqrt{\displaystyle \frac{{\omega }_{1}{\omega }_{2}}{{n}_{1}^{2}{n}_{2}^{2}}}\\ & & \times \,\displaystyle \int {d}^{3}r\left({\chi }_{\mathrm{eff}}^{(2)}(\vec{r}){G}_{p}^{* }(\vec{r}){g}_{{\vec{\mu }}_{1}}(x,y){g}_{{\vec{\mu }}_{2}}(x,y){e}^{-i{\rm{\Delta }}{k}_{z}z}\right)\\ & & \times \,{e}^{i{\rm{\Delta }}\omega t}{\hat{a}}_{{\vec{\mu }}_{1},{k}_{1z}}^{\dagger }{\hat{a}}_{{\vec{\mu }}_{2},{k}_{2z}}^{\dagger }+{\rm{H}}.{\rm{c}}.\end{array}\end{eqnarray} \tag{ 25 }$$

Here we have switched from ${\zeta }_{\mathrm{eff}}^{(2)}$ to the effective nonlinear susceptibility ${\chi }_{\mathrm{eff}}^{(2)}$ using the approximation:

$$\begin{eqnarray}&&-{\epsilon }_{0}^{2}\,{\zeta }_{\mathrm{eff}}^{(2)}\,{n}_{p}^{2}{n}_{1}^{2}{n}_{2}^{2}\approx {\chi }_{\mathrm{eff}}^{(2)},\end{eqnarray} \tag{ 26 }$$

which is satisfied under the same lossless media assumption that allowed us to invoke full permutation symmetry. Since χ⁽²⁾ is what is measured experimentally, and tabulated in handbooks of optical materials, the rest of this tutorial will be expressed in terms of the susceptibility, rather than its inverse.

Since the χ⁽²⁾ nonlinearity is zero outside the nonlinear medium, the spatial integration is carried over the dimensions of the medium. This Hamiltonian describes SPDC in a nonlinear medium of length L_z , unspecified transverse dimensions (but significantly wider than the pump beam), and unspecified poling when illuminated by a monochromatic pump beam directed along the optic axis. Here, the sum over the polarization indices has already been carried out, giving the effective nonlinearity ${\chi }_{\mathrm{eff}}^{(2)}(\vec{r})$. To simplify notation, we defined Δω ≡ ω(k_1z ) + ω(k_2z ) − ω(k_pz ), and Δk_z ≡ k_1z  + k_2z – k_pz . Note that while the Hermite–Gauss modes of the down-converted light ${g}_{\vec{\mu }}(x,y)$ are normalized to have unit norm, the pump spatial dependence ${\tilde{G}}_{p}(\vec{r})$ has a maximum magnitude of unity at $\vec{r}=0$. The peak pump intensity is fixed by the value of the pump field strength $| {D}_{p}|$.

With a general Hamiltonian describing most SPDC processes, we could calculate a general rate of biphoton generation using Fermi's golden rule as shown in Ling et al (2008). Here, we instead show how the direct calculation takes place with first-order time-dependent perturbation theory (from whence Fermi's golden rule originates). We take the initial state of the down-converted fields to be the vacuum state, and the final state to be a biphoton with momenta and Hermite–Gauss mode numbers (${k}_{1z},{\vec{\mu }}_{1}$) and (${k}_{2z},{\vec{\mu }}_{2}$), for the signal and idler photon respectively. The transition probability ${P}_{{k}_{1z},{\vec{\mu }}_{1},{k}_{2z},{\vec{\mu }}_{2}}$ is given by:

$$\begin{eqnarray}&&\begin{array}{l}{P}_{{k}_{1z},{\vec{\mu }}_{1},{k}_{2z},{\vec{\mu }}_{2}}\equiv | \langle {\vec{\mu }}_{1}{k}_{1z},{\vec{\mu }}_{2}{k}_{2z}| {\rm{\Psi }}(t)\rangle {| }^{2}\\ \ \ \approx \,{\left|\langle {\vec{\mu }}_{1}{k}_{1z},{\vec{\mu }}_{2}{k}_{2z}| \left(1-\displaystyle \frac{i}{{\rm{\hslash }}}{\displaystyle \int }_{0}^{t}{dt}^{\prime} {\hat{H}}_{{NL}}(t^{\prime} )\right)| \mathrm{0,0}\rangle \right|}^{2},\end{array}\end{eqnarray} \tag{ 27 }$$

where the expression in parentheses comes from the first-order approximation (a la perturbation theory) ¹⁸ of the time propagation operator. Substituting our expression for the nonlinear Hamiltonian, we obtain:

$$\begin{eqnarray}&&\begin{array}{l}{P}_{{k}_{1z},{\vec{\mu }}_{1},{k}_{2z},{\vec{\mu }}_{2}}=\displaystyle \frac{| {E}_{p}^{0}{| }^{2}}{4{L}_{z}^{2}}\displaystyle \frac{{\omega }_{1}{\omega }_{2}}{{n}_{1}^{2}{n}_{2}^{2}}\\ \ \ \times \,{\left|\displaystyle \int {d}^{3}r\left({\chi }_{\mathrm{eff}}^{(2)}(\vec{r}){G}_{p}^{* }(\vec{r}){g}_{{\vec{\mu }}_{1}}(x,y){g}_{{\vec{\mu }}_{2}}(x,y){e}^{-i{\rm{\Delta }}{k}_{z}z}\right)\right|}^{2}\\ \ \ \times \,{\left|{\displaystyle \int }_{0}^{t}{dt}^{\prime} {e}^{i{\rm{\Delta }}\omega t^{\prime} }\right|}^{2}={W}_{{k}_{1z},{\vec{\mu }}_{1},{k}_{2z},{\vec{\mu }}_{2}}{\left|{\displaystyle \int }_{0}^{t}{dt}^{\prime} {e}^{i{\rm{\Delta }}\omega t^{\prime} }\right|}^{2},\end{array}\end{eqnarray} \tag{ 28 }$$

where ${W}_{{k}_{1z},{\vec{\mu }}_{1},{k}_{2z},{\vec{\mu }}_{2}}$ is defined implicitly to simplify notation. This expression can be further simplified in the limit that t becomes large, and knowing that the magnitude of a complex number is independent of its phase:

$$\begin{eqnarray}&&\begin{array}{l}{P}_{{k}_{1z},{\vec{\mu }}_{1},{k}_{2z},{\vec{\mu }}_{2}}={W}_{{k}_{1z},{\vec{\mu }}_{1},{k}_{2z},{\vec{\mu }}_{2}}\left|t\,\mathrm{sinc}\left(\displaystyle \frac{{\rm{\Delta }}\omega t}{2}\right)\right|\\ \ \,\,\times \,\left|{\displaystyle \int }_{0}^{t}{dt}^{\prime} \,\,{e}^{i{\rm{\Delta }}\omega t^{\prime} }\right|\\ \ \ \mathop{\longrightarrow }\limits_{\mathrm{large}\,t}\,{W}_{{k}_{1z},{\vec{\mu }}_{1},{k}_{2z},{\vec{\mu }}_{2}}\,2\pi \,\delta ({\rm{\Delta }}\omega )\left|{\displaystyle \int }_{0}^{t}{dt}^{\prime} \,\,{e}^{i{\rm{\Delta }}\omega t^{\prime} }\right|.\end{array}\end{eqnarray} \tag{ 29 }$$

In practice, the interaction time t need not be arbitrarily large for this limit to apply. Instead, one only needs t to be significantly longer than the inverse of Δω, which is achieved for times much longer than the picosecond time scales that light takes to travel the length of the nonlinear crystal, but not so large that multiple biphotons are likely to be generated in time t. The range of frequencies defining the width Δω (before this limit is invoked) is known as the phase-matching bandwidth (and is of the order 10¹³–10¹⁴ for most materials). Although ultimately limited by effective nonlinearity ${\chi }_{\mathrm{eff}}^{(2)}$, the phase-matching bandwidth is primarily determined by the dispersion of the material where the condition $| {\rm{\Delta }}{k}_{z}| \lt 2\pi /{L}_{z}$ is satisfied. Since the large time limit for ${P}_{{k}_{1z},{\vec{\mu }}_{1},{k}_{2z},{\vec{\mu }}_{2}}$ can only be nonzero when Δω is zero, the second integral is of a constant term, making ${P}_{{k}_{1z},{\vec{\mu }}_{1},{k}_{2z},{\vec{\mu }}_{2}}$ linear in time. Where the transition rate ${R}_{{k}_{1z},{\vec{\mu }}_{1},{k}_{2z},{\vec{\mu }}_{2}}$ is defined as the time derivative of the transition probability, it levels off to a constant value for large times (e.g. longer than a picosecond):

$$\begin{eqnarray}&&{\mathrm{lim}}_{t\to \infty }{R}_{{k}_{1z},{\vec{\mu }}_{1},{k}_{2z},{\vec{\mu }}_{2}}={W}_{{k}_{1z},{\vec{\mu }}_{1},{k}_{2z},{\vec{\mu }}_{2}}\,2\pi \,\delta ({\rm{\Delta }}\omega ).\end{eqnarray} \tag{ 30 }$$

Of course, the transition probability cannot increase linearly with time indefinitely; the first-order perturbation approximation breaks down. However, in the undepleted pump approximation, and using times of the order of the time it takes light to pass through the crystal, this approximation is valid. To calculate the total transition rate for down-conversion into a single pair of transverse modes ${R}_{{\vec{\mu }}_{1},{\vec{\mu }}_{2}}$, we must add the transition rates for all values of k_1z and k_2z :

$$\begin{eqnarray}&&{R}_{{\vec{\mu }}_{1},{\vec{\mu }}_{2}}=\displaystyle \sum _{{k}_{1z},{k}_{2z}}{R}_{{k}_{1z},{\vec{\mu }}_{1},{k}_{2z},{\vec{\mu }}_{2}},\end{eqnarray} \tag{ 31 }$$

and we can define ${W}_{{\vec{\mu }}_{1},{\vec{\mu }}_{2}}$ similarly. Because the length of the nonlinear medium L_z is much longer than the wavelength of light passing through it, we may approximate the sums over k_1z and k_2z as integrals over k_1z and k_2z , which, in turn, can be expressed as integrals over frequencies ω₁ and ω₂:

$$\begin{eqnarray}&&\displaystyle \sum _{{k}_{1z},{k}_{2z}}\approx {\left(\displaystyle \frac{{L}_{z}}{2\pi }\right)}^{2}\int {{dk}}_{1z}{{dk}}_{2z}\approx {\left(\displaystyle \frac{{L}_{z}}{2\pi }\right)}^{2}\displaystyle \frac{{n}_{g1}{n}_{g2}}{{c}^{2}}\int d{\omega }_{1}d{\omega }_{2},\end{eqnarray} \tag{ 32 }$$

where n_g1 (n_g2) is the group index at the signal (idler) frequency.

With this, the single-mode transition rate ${R}_{{\vec{\mu }}_{1},{\vec{\mu }}_{2}}$ is given by the integral:

$$\begin{eqnarray}&&{R}_{{\vec{\mu }}_{1},{\vec{\mu }}_{2}}=\int d{\omega }_{1}d{\omega }_{2}\,{W}_{{k}_{1z},{\vec{\mu }}_{1},{k}_{2z},{\vec{\mu }}_{2}}\,\displaystyle \frac{{L}_{z}^{2}{n}_{g1}{n}_{g2}}{2\pi {c}^{2}}\delta ({\rm{\Delta }}\omega ),\end{eqnarray} \tag{ 33 }$$

where ${W}_{{\vec{\mu }}_{1},{k}_{1z},{\vec{\mu }}_{2},{k}_{2z}}$ is readily expressed in terms of ω₁ and ω₂. The total rate R, is then the sum over all transverse modes of the single-mode rates.

2.2.1. Transition rate versus the rate of generated biphotons

Let us consider the case where the crystal is cut and tuned to optimize down-conversion such that the spectra of ω₁ and ω₂ are both centered at half the pump frequency ω_p . Then, the momentum mismatch Δk_z can be Taylor-expanded (Fedorov et al 2009) about this central frequency so that:

$$\begin{eqnarray}&&{\rm{\Delta }}{k}_{z}\approx \left(\displaystyle \frac{{\rm{\Delta }}{n}_{g}}{c}\right)\left({\omega }_{1}-\displaystyle \frac{{\omega }_{p}}{2}\right)+\kappa {\left({\omega }_{1}-\displaystyle \frac{{\omega }_{p}}{2}\right)}^{2},\end{eqnarray} \tag{ 40 }$$

where κ is the group velocity dispersion constant at half the pump frequency: $| \tfrac{{d}^{2}k}{d{\omega }^{2}}{| }_{{\omega }_{p}/2}$, and Δn_g is the group index mismatch for the signal and idler photons $| {n}_{g1}-{n}_{g2}|$ at their central frequencies.

In type-0 and type-I SPDC ¹⁹ , the group indices of the signal and idler light are identical because their polarizations are identical. Only the second-order contribution to Δk_z is significant, and we find:

$$\begin{eqnarray}\begin{array}{rcl}{R}_{{SM}} & \propto & \displaystyle \int d{\omega }_{1}{\omega }_{1}({\omega }_{p}-{\omega }_{1}){\mathrm{sinc}}^{2}\left(\displaystyle \frac{{L}_{z}\kappa }{2}{\left({\omega }_{1}-\displaystyle \frac{{\omega }_{p}}{2}\right)}^{2}\right)\\ & = & \displaystyle \frac{({L}_{z}\kappa {\omega }_{p}^{2}-6)\sqrt{2\pi }}{3{\left({L}_{z}\kappa \right)}^{3/2}}\approx \displaystyle \frac{{\omega }_{p}^{2}}{3}\sqrt{\displaystyle \frac{2\pi }{{L}_{z}\kappa }}.\end{array}\end{eqnarray} \tag{ 41 }$$

The approximation holds well for typical crystal parameters and crystal lengths longer than tenths of a millimeter (as is typical). Here, we have also assumed that the portion of the generation rate formula dependent on the indices of refraction is more or less constant over the bandwidth of the down-converted light, which is reasonable for most nonlinear crystals. Making this final simplification, we arrive at the single-mode rate for degenerate type-0 and type-I SPDC:

$$\begin{eqnarray}\begin{array}{rcl}{R}_{{SM}}^{t1} & = & \sqrt{\displaystyle \frac{2}{{\pi }^{3}}}\displaystyle \frac{2}{3{\epsilon }_{0}{c}^{3}}\displaystyle \frac{{n}_{g1}{n}_{g2}}{{n}_{1}^{2}{n}_{2}^{2}{n}_{p}}\displaystyle \frac{{\left({d}_{\mathrm{eff}}\right)}^{2}{\omega }_{p}^{2}}{\sqrt{\kappa }}\\ & & \times {\left|\displaystyle \frac{{\sigma }_{p}^{2}}{{\sigma }_{1}^{2}+2{\sigma }_{p}^{2}}\right|}^{2}\displaystyle \frac{P}{{\sigma }_{p}^{2}}{L}_{z}^{3/2},\end{array}\end{eqnarray} \tag{ 42 }$$

where ${d}_{\mathrm{eff}}\equiv {\chi }_{\mathrm{eff}}^{(2)}/2$ is the more common convention for defining the effective nonlinear susceptibility, and we substituted the relation for the power of the Gaussian pump beam (23).

3.1.1. Degenerate SPDC with negligible group velocity dispersion

Under ordinary circumstances, describing degenerate type-0/I SPDC only requires knowing the momentum mismatch Δk_z up to second order in the frequency, which is governed by the group velocity dispersion κ. However, it is possible to achieve degenerate SPDC where the down-conversion wavelength happens to be close enough to an inflection point in the dispersion that κ is at or near enough to zero, so that we must go out to fourth order in frequency to describe the phase matching (Nasr et al 2005). In degenerate type-0/I SPDC, the first and third order terms of Δk_z are zero. If we let the fourth-order dispersion constant ${\rm{\Upsilon }}\equiv | \tfrac{{d}^{4}k}{d{\omega }^{4}}{| }_{{\omega }_{p}/2}$, then the single-mode rate R_SM is governed by:

$$\begin{eqnarray}&&{R}_{{SM}}\propto \int d{\omega }_{1}{\omega }_{1}({\omega }_{p}-{\omega }_{1}){\mathrm{sinc}}^{2}\left(\displaystyle \frac{{L}_{z}{\rm{\Upsilon }}}{24}{\left({\omega }_{1}-\displaystyle \frac{{\omega }_{p}}{2}\right)}^{4}\right)\end{eqnarray} \tag{ 43 }$$

$$\begin{eqnarray}&&\approx \,{\omega }_{p}^{2}\displaystyle \frac{2({2}^{3/4})\cos (\pi /8){\rm{\Gamma }}(1/4)}{21}{\left(\displaystyle \frac{24}{{L}_{z}{\rm{\Upsilon }}}\right)}^{1/4}.\end{eqnarray} \tag{ 44 }$$

The resulting bandwidth and brightness of the down-converted light can exceed what occurs in ordinary degenerate SPDC by an order of magnitude, resulting in an ultra-broadband source of entangled photon pairs spanning hundreds of nanometers (Nasr et al 2005) ²⁰ .

3.1.2. Multimode degenerate SPDC

Although many experiments make use of photon pairs coupled into single-mode fiber, this coupling destroys the transverse spatial correlations and the high-dimensional entanglement in that degree of freedom. In experiments that involve coupling down-converted light into multi-mode fiber, or ones using a large-area photon detector, the relevant rate of biphoton generation is the rate of generation into all transverse Hermite–Gaussian modes. Ordinarily, the total rate would be the sum of the single-mode rates over all pairs of signal and idler modes (38). However, directly evaluating this sum yields non-physical results, as the formula for the single-mode rate is contingent on the paraxial approximation. For a given beam waist, Hermite–Gaussian beams with sufficiently large transverse momentum (or high mode index) are non-paraxial. Instead, it is much simpler to calculate the relative probability that the biphoton will be emitted into the zeroth order signal and idler Gaussian modes, and from this, determine the ratio of the total rate to the single-mode rate. Where the idler mode radius σ₁ defining the Hermite–Gauss basis is a free parameter, we set it equal to the pump radius σ_p to simplify calculation. For types 0 and I degenerate collinear down-conversion, the ratio is given by:

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{R}_{{SM}}}{{R}_{T}} & = & | \langle {\vec{\mu }}_{1}=\vec{0},{\vec{\mu }}_{2}=\vec{0}| {\rm{\Psi }}\rangle {| }^{2}=\displaystyle \frac{4a{\sigma }_{p}^{2}}{{\left({\sigma }_{p}^{2}+\sqrt{{a}^{2}+{\sigma }_{p}^{4}}\right)}^{2}}\\ & \approx & \displaystyle \frac{{L}_{z}{\lambda }_{p}}{4\pi {n}_{p}{\sigma }_{p}^{2}},\end{array}\end{eqnarray} \tag{ 45 }$$

such that $a=\tfrac{{L}_{z}{\lambda }_{p}}{4\pi {n}_{p}}$, and,

$$\begin{eqnarray}&&\langle {x}_{1},{y}_{1},{x}_{2},{y}_{2}| {\vec{\mu }}_{1},{\vec{\mu }}_{2}\rangle ={g}_{{\vec{\mu }}_{1}}({x}_{1},{y}_{1}){g}_{{\vec{\mu }}_{2}}({x}_{2},{y}_{2});\end{eqnarray} \tag{ 46 }$$

and the approximation is valid for large pump beam widths and thin crystals.

Deriving the transverse wavefunction of a biphoton generated in collinear SPDC is generally more involved than the case where we also consider only degenerate frequencies (Schneeloch and Howell 2016). Instead, one must integrate the biphoton wavefunction over the frequency spectrum of the down-converted light, and renormalize accordingly, resulting in a substantially broadened wavefunction. However, we may still approximate the accurate biphoton wavefunction as a scaled representation of the biphoton wavefunction in the degenerate frequency case. We scale a by a constant factor ϕ, and find for type-I SPDC:

$$\begin{eqnarray}&&{R}_{T}^{(t1)}=\displaystyle \frac{32\sqrt{2{\pi }^{3}}}{27{\epsilon }_{0}c}\left(\displaystyle \frac{{n}_{g1}{n}_{g2}}{{n}_{1}^{2}{n}_{2}^{2}}\right)\displaystyle \frac{{d}_{\mathrm{eff}}^{2}}{{\lambda }_{p}^{3}\sqrt{\kappa }}\displaystyle \frac{P\sqrt{{L}_{z}}}{\phi }.\end{eqnarray} \tag{ 47 }$$

Here, we approximate ϕ ≈ 0.335 by matching the peaks of the degenerate and more accurate biphoton wavefunction in the same fashion as one can obtain a double-Gaussian approximation to the biphoton wavefunction (Schneeloch and Howell 2016). An interesting qualitative point here discussed in other references (Süzer and Goodson 2008) is that although the single-mode brightness increases with focusing (i.e. decreasing σ_p ), the overall brightness does not increase this way, unless the focusing is strong enough that the curvature of the phase fronts of the pump beam affects phase matching.

3.1.3. Degenerate type-II SPDC

In type-II SPDC, the signal and idler photons are of orthogonal polarizations, and experience different indices of refraction. In this regime, the linear contribution to Δk_z about the signal and idler photons' central frequencies (40) is nonzero, and cannot be ignored. In this case:

$$\begin{eqnarray}\begin{array}{rcl}{R}_{{SM}} & \propto & \displaystyle \int d{\omega }_{1}{\omega }_{1}({\omega }_{p}-{\omega }_{1})\times {\mathrm{sinc}}^{2}\left(\displaystyle \frac{{L}_{z}| {n}_{g1}-{n}_{g2}| }{2c}\left({\omega }_{1}-\displaystyle \frac{{\omega }_{p}}{2}\right)\right.\\ & & +\,\left.\displaystyle \frac{{L}_{z}\kappa }{2}{\left({\omega }_{1}-\displaystyle \frac{{\omega }_{p}}{2}\right)}^{2}\right).\end{array}\end{eqnarray} \tag{ 48 }$$

For most nonlinear optical materials, the quadratic contribution to the argument of the sinc function is negligible relative to the linear contribution because the group index difference Δn_g is large enough (of the order 10⁻² or greater for most materials) in comparison to the group-velocity dispersion κ. This integral cannot be done analytically, but can be bounded from above. Because the square of the sinc function is a non-negative function, and ${\omega }_{1}({\omega }_{p}-{\omega }_{1})\leqslant {\omega }_{p}^{2}/4$, the rate is bounded above by an integral that can be done analytically. Indeed:

$$\begin{eqnarray}&&{R}_{{SM}}\propto \displaystyle \frac{c\pi {\omega }_{p}^{2}}{2{L}_{z}{\rm{\Delta }}{n}_{g}}.\end{eqnarray} \tag{ 49 }$$

The approximate proportionality is valid, when the width of the sinc function in ω₁ is much less than the pump frequency (typically, less than a quarter in most nonlinear media). Consequently, the approximation is an over-estimate (typically by less than 7%). From this, we can get the single-mode rate for type-II degenerate SPDC:

$$\begin{eqnarray}&&{R}_{{SM}}^{t2}=\displaystyle \frac{1}{\pi {\epsilon }_{0}{c}^{2}}\displaystyle \frac{{n}_{g1}{n}_{g2}}{{n}_{1}^{2}{n}_{2}^{2}{n}_{p}}\displaystyle \frac{{\left({d}_{\mathrm{eff}}\right)}^{2}{\omega }_{p}^{2}}{{\rm{\Delta }}{n}_{g}}{\left|\displaystyle \frac{{\sigma }_{p}^{2}}{{\sigma }_{1}^{2}+2{\sigma }_{p}^{2}}\right|}^{2}\displaystyle \frac{P}{{\sigma }_{p}^{2}}{L}_{z}.\end{eqnarray} \tag{ 50 }$$

Interestingly, one may compare this to the corresponding single-mode rate for collinear type-II SPDC derived in Ling et al (2008), and see that our formula differs by a near-unity factor of the ratio of the indices of refraction n_g1 n_g2/n₁ n₂, amounting to only a 3% difference in prediction using their experimental parameters. For a description of the absolute biphoton generation rate into non-collinear Gaussian modes, such as is useful when using type-II SPDC as a source of polarization-entangled photon pairs, their reference provides an invaluable discussion.

In order to get the total rate for type-II SPDC, one can use the inner product between the zeroth order Hermite–Gaussian modes, and biphoton wavefunction for type-II SPDC as was done previously (47) for type-I SPDC. However, the biphoton wavefunction for type-II SPDC is not as straightforward to derive or approximate, due to transverse walk-off between the signal and idler light ²¹ . For a thorough analysis of the biphoton wavefunction in type-II SPDC, see Walborn et al (2010).

3.1.4. Degenerate SPDC with narrow frequency filtering

By angle and temperature tuning the crystal, it is possible that the signal and idler frequency spectra no longer overlap, having different central frequencies that add up to the pump frequency. The Taylor expansion for Δk_z is taken with respect to the signal beam's center frequency ω₁₍₀₎. In this case:

$$\begin{eqnarray}&&{\rm{\Delta }}{k}_{z}\approx \left(\displaystyle \frac{{\rm{\Delta }}{n}_{g}}{c}\right)\left({\omega }_{1}-{\omega }_{1(0)}\right)+\displaystyle \frac{({\kappa }_{1}+{\kappa }_{2})}{2}{\left({\omega }_{1}-{\omega }_{1(0)}\right)}^{2}.\end{eqnarray} \tag{ 51 }$$

Here, ${\kappa }_{1}$ and κ₂ are the group velocity dispersion constants at the signal and idler central frequencies, respectively. When the central frequencies are different enough that the group index mismatch Δn_g is significant (e.g. greater than 10⁻²), the rate of biphoton generation is qualitatively identical for both type-I and type-II SPDC.

Thus far, we have examined the absolute brightness of SPDC in isotropic crystals (i.e. where χ⁽²⁾ is a constant throughout the crystal volume). This is a fine regime when perfect phase matching is achievable (that is, where tuning the crystal allows the indices of refraction to be such that Δk_z  = 0). However, this is not always possible. The general dependence of biphoton brightness on crystal length L_z is given by:

$$\begin{eqnarray}&&{R}_{{SM}}\propto \int \,d{\omega }_{1}{\omega }_{1}({\omega }_{p}-{\omega }_{1}){\left|{\int }_{-\infty }^{\infty }{dz}\bar{\chi }(z){e}^{-i{\rm{\Delta }}{k}_{z}z}\right|}^{2},\end{eqnarray} \tag{ 52 }$$

where for an isotropic crystal $\bar{\chi }(z)$ is unity inside the crystal, and zero outside. When perfect phase matching is not achievable (i.e. when the indices of refraction are not compatible for SPDC at the desired pump and signal/idler frequencies), the magnitude square of the integral over z oscillates with crystal length between zero and $4/{\rm{\Delta }}{k}_{z}^{2}$. The value of Δk_z is set by the frequencies of the pump, signal, and idler light, and the indices of refraction at their respective frequencies. For a given set of pump, signal, and idler frequencies, imperfect phase matching can be ameliorated by periodically poling the nonlinear crystal. If one switches the poling direction (changing $\bar{\chi }$ from 1 to −1) just as the amplitude is maximum (i.e. when L_z  = π/Δk_z ), the amplitude grows further as though it were at a minimum. See figure 1 for comparison with and without periodic poling. By switching the poling periodically at these intervals such that the poling period Λ_pol = 2π/Δk_z , one can achieve significant brightness without perfect phase matching. This technique is known as quasi-phase matching.

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As shown in Boyd (2007), the poling profile $\bar{\chi }(z)$ can be broken up into a Fourier series with fundamental momentum $\tfrac{2\pi }{{{\rm{\Lambda }}}_{\mathrm{pol}}}$:

$$\begin{eqnarray}&&\bar{\chi }(z)={X}_{0}+\displaystyle \sum _{n\ne 0}{X}_{n}{e}^{{{ik}}_{n}z}:\qquad {k}_{n}=\displaystyle \frac{2\pi }{{{\rm{\Lambda }}}_{\mathrm{pol}}}n,\end{eqnarray} \tag{ 53 }$$

where n runs from $-\infty$ to $\infty$ excluding zero, X₀ = 0, and

$$\begin{eqnarray}&&{X}_{n}=\mathrm{sinc}\left(\displaystyle \frac{n\pi }{2}\right).\end{eqnarray} \tag{ 54 }$$

From this, we see the length dependence (52) simplifies to:

$$\begin{eqnarray}&&{R}_{{SM}}\propto \int \,d{\omega }_{1}{\omega }_{1}({\omega }_{p}-{\omega }_{1}){\left|{\int }_{-\displaystyle \frac{{L}_{z}}{2}}^{\displaystyle \frac{{L}_{z}}{2}}{dz}\displaystyle \sum _{n}{X}_{n}{e}^{-i({\rm{\Delta }}{k}_{z}-{k}_{n})z}\right|}^{2}.\end{eqnarray} \tag{ 55 }$$

In performing this quasi-phase matching, typically only one Fourier component X_n will contribute to the brightness because only one value of k_n will be close enough to offset Δk_z to achieve quasi-phase matching. The range of values of Δk_z over which phase-matching is favorable is approximately 4π/L_z , while the shift in Δk_z between different orders of quasi-phase matching is 4π/Λ_pol, which is larger often by multiple orders of magnitude. Since X_n decreases with n, first-order phase matching (i.e. n = 1 or −1), is most desirable for maximum brightness. The calculation for R_SM follows the same steps with periodic poling as with an isotropic crystal. Δk_z is still Taylor-expanded about the signal and idler central frequencies. The only difference is that the zero-order terms for Δk_z added to −k_m gives zero instead. As such, the single-mode rate of biphoton generation when periodic poling with nth order quasi-phase matching, ${R}_{{SM}}^{{PP}(n)}$ is multiplied by the factor ${X}_{n}^{2}$:

$$\begin{eqnarray}&&{R}_{{SM}}^{{PP}(n)}=\displaystyle \frac{4}{{n}^{2}{\pi }^{2}}{R}_{{SM}}.\end{eqnarray} \tag{ 56 }$$

This correction holds for all types of down-conversion, and will work for the multi-mode regime (discussed previously) as well. It is important to note that where published values for d_eff differ between isotropic and periodically poled nonlinear crystals of the same material, these factors of $\tfrac{2}{n\pi }$ are already included.

Although periodic poling is accomplished by switching the crystal orientation (and therefore the sign of ${\chi }^{(2)}$) periodically over the length of the crystal, this is not the only fashion in which quasi-phase matching can be achieved. If one instead periodically dopes the crystal, changing its composition periodically over its length, and therefore periodically changing χ⁽²⁾, quasi-phase matching may be achieved in precisely the same regimes. Alternatively, in a waveguide, one can produce a sinusoidal variation in the pump intensity by sinusoidally varying the width of the waveguide, which can also be used to achieve quasi-phase matching (Rao et al 2017). As one final note, poling periods in some materials can be made so small that the fundamental momentum completely offsets the pump momentum. In this regime, it is possible to produce counter-propagating photon pairs (Pasiskevicius et al 2008, 2012) in SPDC.

In all the situations considered thus far, the pump beam was considered collimated. However, if one wants to maximize the number of biphotons generated per second that couple into a single-mode fiber, a focused beam offers significant improvement (as discussed previously). In order to see how the single-mode rate changes in the regime of tight focusing, we turn to the work of Bennink (Bennink 2010), who treats this situation in detail.

The dependence of the rate of biphoton generation on the spatial aspects of the pump beam is given by the overlap integral

$$\begin{eqnarray}&&{R}_{{SM}}\propto {\left|\int {d}^{3}r\left({\chi }_{\mathrm{eff}}^{(2)}(\vec{r}){E}_{p}^{* }(\vec{r}){E}_{1}(\vec{r}){E}_{2}(\vec{r})\right)\right|}^{2},\end{eqnarray} \tag{ 57 }$$

where in our approximations, $D\approx {\epsilon }_{0}{n}^{2}E$. In order to properly treat collinear SPDC into the zeroth-order signal/idler Gaussian modes when the pump beam is focused strong enough that its width changes significantly over the length of the crystal, Bennink uses a slightly different expression for the signal/idler spatial modes. Instead of being Gaussian in transverse dimensions, and constant along the optic axis (i.e. collimated), Bennink considers the signal/idler fields as focused Gaussian beams with their own beam parameters in addition to the pump beam. While a full discussion of his calculations is beyond the scope of this tutorial (and redundant), he finds the joint pair-collection probability, which is proportional to the biphoton generation rate. For type-II SPDC, and non-degenerate type-I SPDC, for near-perfect phase matching, and assuming identical beam focal parameters ξ for the pump, signal and idler modes, one can show:

$$\begin{eqnarray}&&{R}_{{SM}}^{t2}\propto \displaystyle \frac{{d}_{\mathrm{eff}}^{2}{\omega }_{p}^{3}}{{\rm{\Delta }}{n}_{g}}{\mathrm{Tan}}^{-1}(\xi )P,\end{eqnarray} \tag{ 58 }$$

where the (pump) beam focal parameter ξ is defined as the ratio of the crystal length L_z divided by twice the Rayleigh range, z_R . Thus, a small focal parameter indicates a nearly collimated beam. In the limit of a nearly collimated beam, Bennink's formula coincides with the single-mode formula derived previously (50) up to constant factors.

To date, no calculations have obtained the absolute coincidence rates in the regime of focused pump beams, but Bennink's work captures the salient qualitative behavior of the biphoton generation rate on changing pump focal parameter. In addition, the work of Dixon et al (2014) expands on these results, and shows how one may sacrifice absolute brightness in exchange for a greatly improved heralding efficiency, as is useful in developing SPDC as a source of heralded single photons.

Having found a formula for the time evolution of the annihilation operators, the number of photon pairs can be calculated by finding the expectation value $\langle {\hat{a}}_{{k}_{1}}^{\dagger }{\hat{a}}_{{k}_{1}}\rangle$ and summing over all modes k₁:

$$\begin{eqnarray}&&{N}_{{SM}}(t)=\displaystyle \sum _{{k}_{1}}\langle {\hat{a}}_{{k}_{1}}^{\dagger }{\hat{a}}_{{k}_{1}}\rangle (t)=\displaystyle \sum _{{k}_{1}{k}_{2}}{\left|\sinh {\left(\sqrt{{\hat{N}}_{p}}{Gt}\right)}_{{k}_{1}{k}_{2}}\right|}^{2},\end{eqnarray} \tag{ 68 }$$

so that in the same limits where the first-order approximation is valid:

$$\begin{eqnarray}&&{N}_{{SM}}(t)\approx \displaystyle \sum _{{k}_{1}{k}_{2}}{\left|{G}_{{k}_{1}{k}_{2}}\right|}^{2}\langle {\hat{N}}_{p}\rangle {t}^{2}.\end{eqnarray} \tag{ 69 }$$

Here, the mean pump photon number $\langle {\hat{N}}_{p}\rangle$ will be the average number of pump photons in the nonlinear medium at any given time:

$$\begin{eqnarray}&&\langle {\hat{N}}_{p}\rangle =\displaystyle \frac{P}{{\rm{\hslash }}{\omega }_{p}}\cdot \displaystyle \frac{{L}_{z}{n}_{p}}{c},\end{eqnarray} \tag{ 70 }$$

where P is pump power.

Considering the simple case of a collimated Gaussian pump beam coupled to a pair of Gaussian signal and idler modes, and that the length of the crystal is much larger than the wavelength of the pump light, the only contributions to the sum over k₁ and k₂ are those such that Δk_z  = 0. This is then a sum over one variable, which we may approximate as an integral, and express in terms of frequency. For a given function f(k₁, k₂):

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \sum _{{k}_{1}}f({k}_{1},{k}_{p}-{k}_{1})\left(\displaystyle \frac{2\pi }{{L}_{z}}\right)\approx \displaystyle \int {{dk}}_{1}f({k}_{1},{k}_{p}-{k}_{1})\\ \ \ =\,\displaystyle \int {{dk}}_{1}{{dk}}_{2}f({k}_{1},{k}_{2})\delta ({k}_{1}+{k}_{2}-{k}_{p})\\ \ \ =\,\displaystyle \frac{{n}_{g1}{n}_{g2}}{{c}^{2}}\displaystyle \int d{\omega }_{1}d{\omega }_{2}f({\omega }_{1},{\omega }_{2})\delta \left(\displaystyle \frac{{n}_{1}{\omega }_{1}+{n}_{2}{\omega }_{2}-{n}_{p}{\omega }_{p}}{c}\right)\\ \ \ =\,\displaystyle \frac{{n}_{g1}{n}_{g2}}{{n}_{1}c}\displaystyle \int d{\omega }_{1}d{\omega }_{2}f({\omega }_{1},{\omega }_{2})\delta \left({\omega }_{1}+\displaystyle \frac{{n}_{2}}{{n}_{1}}{\omega }_{2}-\displaystyle \frac{{n}_{p}}{{n}_{1}}{\omega }_{p}\right).\end{array}\end{eqnarray} \tag{ 71 }$$

For type-0 and type-I phase matching, the δ function simplifies to δ(Δω), which gives us:

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{N}_{{SM}}(t)}{{t}^{2}} & \approx & \displaystyle \frac{2}{{\pi }^{2}{\epsilon }_{0}{c}^{2}}\displaystyle \frac{{n}_{g1}{n}_{g2}}{{n}_{p}{n}_{1}^{3}{n}_{2}^{2}}{\left|\displaystyle \frac{{\sigma }_{p}^{2}}{{\sigma }_{1}^{2}+2{\sigma }_{p}^{2}}\right|}^{2}\displaystyle \frac{P\,{d}_{\mathrm{eff}}^{2}{L}_{z}}{{\sigma }_{p}^{2}}\\ & & \times \,\displaystyle \int d{\omega }_{1}{\omega }_{1}\left({\omega }_{p}-{\omega }_{1}\right){\mathrm{sinc}}^{2}\left(\displaystyle \frac{{\rm{\Delta }}{k}_{z}{L}_{z}}{2}\right).\end{array}\end{eqnarray} \tag{ 72 }$$

These integrals over frequency can be evaluated or approximated with the same methods discussed in the previous section. For type-I degenerate SPDC,

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{N}_{{SM}}(t)}{{t}^{2}} & \approx & \sqrt{\displaystyle \frac{2}{{\pi }^{3}}}\displaystyle \frac{2}{3{\epsilon }_{0}{c}^{2}}\displaystyle \frac{{n}_{g1}{n}_{g2}}{{n}_{p}{n}_{1}^{3}{n}_{2}^{2}}\displaystyle \frac{{d}_{\mathrm{eff}}^{2}\,{\omega }_{p}^{2}}{\sqrt{\kappa }}\\ & & \times {\left|\displaystyle \frac{{\sigma }_{p}^{2}}{{\sigma }_{1}^{2}+2{\sigma }_{p}^{2}}\right|}^{2}\displaystyle \frac{P}{{\sigma }_{p}^{2}}{L}_{z}^{1/2}.\end{array}\end{eqnarray} \tag{ 73 }$$

For type-II phase matching, the δ function does not simplify to δ(Δω), but the same upper bound approximation may be taken. The value of N_SM (t) obtained will be the same as if one let the δ function be δ(Δω), but with an additional factor of (2n_p – n₁)/n₂, which is of the order unity.

Finally, to obtain the single-mode rate, we point out that N_SM (t) is the mean number of biphotons generated as a function of time. We will use the time t = T_DC as the time it takes either the pump or down-converted light to travel the length of the crystal ²² . Then, the rate R_SM is the ratio of N_SM (T_DC ) over T_DC , which is L_z n₁/c, giving us:

$$\begin{eqnarray}\begin{array}{rcl}{R}_{{SM}} & \approx & \sqrt{\displaystyle \frac{2}{{\pi }^{3}}}\displaystyle \frac{2}{3{\epsilon }_{0}{c}^{3}}\displaystyle \frac{{n}_{g1}{n}_{g2}}{{n}_{1}^{2}{n}_{2}^{2}{n}_{p}}\displaystyle \frac{{\left({d}_{\mathrm{eff}}\right)}^{2}{\omega }_{p}^{2}}{\sqrt{\kappa }}\\ & & \times \,{\left|\displaystyle \frac{{\sigma }_{p}^{2}}{{\sigma }_{1}^{2}+2{\sigma }_{p}^{2}}\right|}^{2}\displaystyle \frac{P}{{\sigma }_{p}^{2}}{L}_{z}^{3/2}\end{array}\end{eqnarray} \tag{ 74 }$$

which agrees precisely with the formula for the rate of generated biphotons we obtained earlier via first-order perturbation theory.

Previously, we solved for the time evolution of the signal and idler annihilation operators. However, using that relation to obtain the actual quantum state of SPDC light takes one additional step.

If we define U as a unitary transformation diagonalizing the matrix G, the same transformation will define eigenmodes of the two-mode squeezing operator.

Let Λ be the diagonalized matrix of G:

$$\begin{eqnarray}&&{\rm{\Lambda }}={{UGU}}^{\dagger }.\end{eqnarray} \tag{ 75 }$$

Furthermore, let the annihilation operators ${\vec{\hat{b}}}_{1}$ be defined as $U\cdot {\vec{\hat{a}}}_{1}$, (i.e. the annihilation operators of the eigenmodes of the SPDC Hamiltonian). Then, the linear system of equations for the annihilation operators separates into independent linear equations for the annihilation operators of the eigenmodes:

$$\begin{eqnarray}&&{\vec{\hat{b}}}_{1}(t)=\cosh (\sqrt{{\hat{N}}_{p}}{\rm{\Lambda }}t)\cdot {\vec{\hat{b}}}_{1}-i\sinh (\sqrt{{\hat{N}}_{p}}{\rm{\Lambda }}t)\cdot {\vec{\hat{b}}}_{2}^{\dagger }.\end{eqnarray} \tag{ 76 }$$

The quantum state of SPDC light is obtained from these eigenmodes (Lvovsky 2016), and is a product of multiple two-mode squeezed states (one for each correlated pair of eigenmodes) when the pump beam is in a coherent state. For reference, the two-mode squeezed state between modes 1 and 2 with squeezing amount r is given by:

$$\begin{eqnarray}&&| {TMSV}\rangle =\displaystyle \frac{1}{\cosh (r)}\displaystyle \sum _{n=0}^{\infty }{\tanh }^{n}(r)| n{\rangle }_{1}| n{\rangle }_{2}.\end{eqnarray} \tag{ 77 }$$

With the two-mode squeezed vacuum state properly scaled to fit experimental parameters, we can explore what we expect to measure as we increase the intensity of the pump. In a time interval equal to the length of time it takes light to pass through the nonlinear crystal, the state of the field has probabilities to be in a zero biphoton state, a one-biphoton state, a two-biphoton state, and so on. Light whose number statistics obey this exponentially decaying photon number distribution is known as thermal or super-Poissonian light because its variance is larger than its mean. In contrast, coherent light (as from dipole radiation or laser light) has Poissonian number statistics. That said, it may seem surprising that coincidence counting measurements show Poissonian statistics for the down-converted light (Avenhaus et al 2008). However, realistic experiments exhibit photodetection across multiple pairs of modes; the empirical number statistics are those of a mixture of multiple exponentially distributed random variables, which is better described with a Poisson distribution.

In order to serve as a viable source for heralded single photons, the number of higher-order biphoton states generated must be small, relative to the single-biphoton state. Fortunately, the expression for the relative likelihood of higher-order biphoton number states is quite simple:

$$\begin{eqnarray}&&\displaystyle \frac{P(2\,\mathrm{or}\,\mathrm{more})}{P(1)}={\sinh }^{2}(r).\end{eqnarray} \tag{ 78 }$$

When considering the SPDC state as a product of multiple two-mode squeezed vacuum states, the ratio of events of multi-biphoton generation to events of single biphoton generation is straightforward to estimate. First, the total ratio of multi-biphoton generation events to single biphoton generation events is approximately the mean of the ratios of multi-biphoton to single biphoton events in each pair of modes. We can estimate this as the sum of the ratios over all modes (which happens to equal N_SM (T_DC )) times the mean probability over all mode pairs. For type-I SPDC, we find:

$$\begin{eqnarray}&&\displaystyle \frac{P(2\,\mathrm{or}\,\mathrm{more})}{P(1)}\approx {N}_{{SM}}({T}_{{DC}})* \displaystyle \frac{12}{35}(4-\sqrt{2})\displaystyle \frac{c}{{n}_{1}}\sqrt{\displaystyle \frac{\kappa \pi }{{L}_{z}}},\end{eqnarray} \tag{ 79 }$$

where, again, κ is the group velocity dispersion constant for the down-converted light. In order to obtain this formula, we used the large signal-idler correlations to estimate the marginal frequency probability density ²³ , and converted it to momentum to calculate the mean probability as the integral of the square of the probability density times the mode spacing $2\pi /{L}_{z}$. For typical experimental parameters in bulk, this ratio of multi-biphoton events to single biphoton events is of the order 10⁻⁸ per Watt of pump power. For CW beams of typical intensities, multi-biphoton events would be exceedingly rare. However, using pulsed lasers with a moderate mean power, but small pulse length, it is possible to achieve the high (peak) power levels necessary at the picosecond time scales near T_DC (i.e. how long light takes to travel through the crystal). Indeed, when using pulsed SPDC in improved heralded single photon sources, multi-photon events are significant enough to limit the overall system efficiency, so that new strategies (such as in Broome et al 2011) are being developed to reduce both the number and impact of these events.

In a single-mode waveguide, the rate of biphoton generation in SPDC would be straightforward to calculate if the pump were in the same single spatial mode as the signal and idler light. This would represent the ideal case of SPDC from a single-mode pump beam down-converting into a single spatial mode calculated in section 3. Moreover, because high-intensity pump light is maintained over the entire length of the waveguide due to spatial mode confinement, SPDC can be made much more efficient than in bulk crystals, reaching record values in excess of 10⁹ pairs per second per mW of pump power (Jechow et al 2008, Bock et al 2016). As in a single-mode fiber, one pump transverse spatial mode can propagate through the waveguide, in addition to one transverse signal and idler mode. In the bulk crystal regime, we decomposed the down-converted light into Hermite–Gaussian spatial modes, but we could just as easily decompose them into any basis of modes fitting a particular waveguide. Indeed, we may approximate the spatial modes of the waveguide with Hermite–Gaussian modes by setting the standard deviations σ_p and σ₁ as equal to a fourth of the mode field diameters appropriate to those waveguides at the appropriate wavelengths.

However, because the pump light and down-converted light are a full octave of frequency apart, a waveguide that is single-mode for the down-converted light will be multi-mode at the much shorter pump wavelength. Ordinarily, the multi-mode pump light adds a degree of complication due to modal dispersion ²⁴ , which makes phase matching more challenging with each spatial mode experiencing a different effective index of refraction. However, with a graded index profile (as is the case with waveguides produced by diffusing a dopant into a nonlinear medium) this effect can be mitigated, since the range of indices over the spatial modes can be made small. In section 7, we test our theoretical prediction for type-II SPDC into a single-spatial mode using a periodically poled potassium titanyl phosphate (PPKTP) waveguide.

When considering SPDC in optical cavities and resonators, it becomes necessary to accommodate loss (over possibly many round trips) to have even a qualitatively accurate description. This is the case, even when the material is sufficiently lossless to exploit the symmetries of the nonlinear susceptibility for later calculation. While the unitary evolution of a closed quantum system does not permit any loss of energy (by say, absorption), it is straightforward to describe loss as a coupling between modes of an extended quantum system-plus-environment, with a correspondingly extended unitary evolution. In doing this, we remain able to treat SPDC in a lossy medium with our standard nonlinear Hamiltonian, but where the signal, idler, and pump creation and annihilation operators experience a continuous series of couplings (theoretically, with generalized beamsplitters (BSs)) to scattering modes over the length of the medium, as discussed further in this section.

In our treatment of SPDC in cavities and resonators, we begin with a brief discussion for how the photon creation/annihilation operators evolve when passing through a lossy medium. Following this, we give an abbreviated introduction describing how the modes in a single-bus MRR are coupled to one another as a prototypical example of an optical cavity (see figure 4 for diagram). With this understanding, we then proceed to describe SPDC in a MRR, where the nonlinear medium is the resonator itself. We find the Heisenberg equation of motion for the photon creation/annihilation operators in the lossy MRR, and use the relationship between the fields inside and outside the MRR to obtain the state of the down-converted light in the output bus, where such light can be directed and collected in a variety of experiments. With the state of the exiting SPDC light, we calculate the generation rate of exiting photon pairs, as well as isolated singles due to loss, among other factors, and compare the two to see what factors impact the relative quality (i.e. heralding efficiency) of cavity-based SPDC photon sources. We conclude with a brief discussion on how the time correlations between photon pairs are affected by the MRR. For a thorough discussion of nonlinear optics in MRRs, we recommend the PhD theses (Vernon 2017) and (Gentry 2018).

To keep notation simple, we assume a 'particle-in-a-box' mode expansion versus the more realistic Hermite–Gaussian decomposition, as discussed above. For simplicity, we will also assume near-perfect phase matching and negligible dispersion. This is a valid approximation when the phase matching bandwidth is much wider than the linewidth of the cavity, and where the optical properties of the material are also essentially constant over this linewidth. With this, we can concentrate on the effects that the passive feedback of the MRR cavity has on photon-pair generation.

5.2.1. BS, propagation loss and cavities

5.2.1.1. Beam splitters

Before discussing cavities, let us first discuss the simplest of all passive optical elements, the BS through which fields will enter and exit a cavity. In figure 2, we illustrate the standard BS with input modes ${\hat{a}}_{{in}},{\hat{b}}_{{in}}$ and output modes ${\hat{a}}_{{out}},{\hat{b}}_{{out}}$, related by the unitary matrix U_bs with transmission and refection coefficients τ, ρ such that $| \tau {| }^{2}+| \rho {| }^{2}=1;$

$$\begin{eqnarray}&&\left(\begin{array}{c}{\hat{a}}_{{out}}\\ {\hat{b}}_{{out}}\end{array}\right)=\left(\begin{array}{cc}\tau & \rho \\ -{\rho }^{* } & {\tau }^{* }\end{array}\right)\,\left(\begin{array}{c}{\hat{a}}_{{in}}\\ {\hat{b}}_{{in}}\end{array}\right),{\hat{\vec{a}}}_{{out}}={U}_{{bs}}\,{\hat{\vec{a}}}_{{in}}.\end{eqnarray} \tag{ 80 }$$

Typically, one often encounters τ real with $\rho =i\,\sqrt{1-{\tau }^{2}}$. The significance of the unitarity of U_bs is that it preserves the commutation relations between fields from input to output, so that $[{\hat{a}}_{{in}},{\hat{a}}_{{in}}^{\dagger }]=1\Rightarrow [{\hat{a}}_{{out}},{\hat{a}}_{{out}}^{\dagger }]=1$, and similarly for the $\hat{b}$ mode. This is just the statement of conservation of probability, i.e. that all signals have been accounted for, and no parts of the signals have been lost.

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5.2.1.2. Loss

To incorporate propagation (or scattering) loss in the system, one can use a model developed by Loudon (Loudon 2000, Alsing et al 2017) where in the frequency domain one has

$$\begin{eqnarray}&&{\hat{a}}_{r+1}(\omega )=T(\omega )\,{\hat{a}}_{r}(\omega )+R(\omega )\,{\hat{s}}_{r}^{({in})}(\omega ),\end{eqnarray} \tag{ 81a }$$

$$\begin{eqnarray}&&{\hat{s}}_{r}^{({out})}(\omega )=R(\omega )\,{\hat{a}}_{r}(\omega )+T(\omega )\,{\hat{s}}_{r}^{({in})}(\omega ),\end{eqnarray} \tag{ 81b }$$

as illustrated in figure 3. The attenuated signal (of interest) ${\hat{a}}_{r}$ and the scattering sites (unobserved, 'lost' modes) ${\hat{s}}_{r}$ satisfy the usual boson commutation relations $[{\hat{a}}_{r}(\omega ),{\hat{a}}_{r}^{\dagger }(\omega ^{\prime} )]\,=$ $[{\hat{s}}_{r}^{({in},{out})}(\omega ),{\hat{s}}_{r}^{\dagger ({in},{out})}(\omega ^{\prime} )]$ $=\delta (\omega -\omega ^{\prime} )$.

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Successive iteration of (81a ) yields,

$$\begin{eqnarray}&&{\hat{a}}_{N+1}(\omega )={T}^{N}(\omega )\,{\hat{a}}_{1}(\omega )+R(\omega )\,\displaystyle \sum _{r=1}^{N}{T}^{N-r}(\omega )\,{\hat{s}}_{r}^{({in})}(\omega ).\end{eqnarray} \tag{ 82 }$$

In the limit of having an infinite series of BSs with infinitesimal coupling, we obtain the relationship for how loss is treated in a continuous medium. We now take the continuum limit: $N\to \infty ;$ Δz = L/N→0; and ${\sum }_{r=1}^{N}\to {\left({\rm{\Delta }}z\right)}^{-1}{\int }_{0}^{L}{dz}$. Because an individual BS in this infinite series has infinitesimal coupling (i.e. $| R(\omega ){| }^{2}\to 0$), we define the independent attenuation constant ${\rm{\Gamma }}(\omega )=| R(\omega ){| }^{2}/{\rm{\Delta }}z$. Then, using $| T(\omega ){| }^{2}\,+| R(\omega ){| }^{2}=1$ we have,

$$\begin{eqnarray}&&{\left|T(\omega )\right|}^{2N}={\left(1-{\left|R(\omega )\right|}^{2}\right)}^{N}={\left(1-{\rm{\Gamma }}(\omega )L/N\right)}^{N}\to {e}^{-{\rm{\Gamma }}(\omega )L},\end{eqnarray} \tag{ 83 }$$

for which we define,

$$\begin{eqnarray}&&T(\omega )\equiv {e}^{i\xi (\omega ){\rm{\Delta }}z}={e}^{in(\omega )(\omega /c)-\displaystyle \frac{1}{2}{\rm{\Gamma }}(\omega ){\rm{\Delta }}z},\end{eqnarray} \tag{ 84 }$$

$$\begin{eqnarray}&&\xi (\omega )\equiv \beta (\omega )+i{\rm{\Gamma }}(\omega )/2,\end{eqnarray} \tag{ 85 }$$

$$\begin{eqnarray}&&\beta (\omega )\equiv n(\omega )(\omega /c).\end{eqnarray} \tag{ 86 }$$

In (84), we have chosen the phase of T(ω) to incorporate the free propagation constant (i.e. wavenumber) β(ω) ≡ n(ω) (ω/c) through a medium of index of refraction n(ω). In addition, we have defined the complex propagation constant as ξ(ω) ≡ β(ω) + iΓ(ω)/2.

To complete our treatment of loss in a continuous medium, we use (N − r)Δz = L − z, and convert from discrete to continuous modes to obtain Loudon's expression for an attenuated traveling beam (Loudon 2000):

$$\begin{eqnarray}&&{\hat{a}}_{L}(\omega )={e}^{i\xi (\omega )L}\,{\hat{a}}_{0}(\omega )+i\sqrt{{\rm{\Gamma }}(\omega )}\,{\int }_{0}^{L}{dz}\,{e}^{i\xi (\omega )(L-z)}\,\hat{s}(z,\omega ).\end{eqnarray} \tag{ 87 }$$

For convenience, we have introduced the shorthand notation for the input field at $z=0\,$ (${\hat{a}}_{1}$ in figure 3), as ${\hat{a}}_{0}(\omega )\,=\hat{a}(z=0,\omega )$ and for the output field at z = L as ${\hat{a}}_{L}(\omega )$. An explicit computation (Alsing et al 2017) shows that $[{\hat{a}}_{L}(\omega ),{\hat{a}}_{L}^{\dagger }(\omega ^{\prime} )]=\delta (\omega -\omega ^{\prime} );$ the expression for the attenuated traveling wave ${\hat{a}}_{L}(\omega )$ explicitly preserves the output field commutation relations.

To connect our expressions to alternative treatments of lossy media, we can rewrite (87) in a Langevin form (Walls and Milburn 1994, Scully and Zubairy 1997, Orszag 2000) as,

$$\begin{eqnarray}&&{\hat{a}}_{L}(\omega )={e}^{i\xi (\omega )L}\,{\hat{a}}_{0}(\omega )+i\sqrt{1-{e}^{-{\rm{\Gamma }}(\omega )L}}\,\hat{f}(\omega ),\end{eqnarray} \tag{ 88a }$$

$$\begin{eqnarray}&&\hat{f}(\omega )\equiv \sqrt{\displaystyle \frac{{\rm{\Gamma }}(\omega )}{1-{e}^{-{\rm{\Gamma }}(\omega )L}}}\,{\int }_{0}^{L}{dz}\,{e}^{i\xi (\omega )(L-z)}\,\hat{s}(z,\omega ),\end{eqnarray} \tag{ 88b }$$

where the Langevin noise operators $\hat{f}(\omega )$ satisfy the commutation relations,

$$\begin{eqnarray}&&[\hat{f}(\omega ),{\hat{f}}^{\dagger }(\omega ^{\prime} )]=\delta (\omega -\omega ^{\prime} ).\end{eqnarray} \tag{ 89 }$$

One could deduce (88a ) by phenomenologically introducing loss as ${\hat{a}}_{L}(\omega )\sim {e}^{[i\beta (\omega )-{\rm{\Gamma }}(\omega )/2]L}\,{\hat{a}}_{0}(\omega )$, assuming that ${\hat{a}}_{L}(\omega )$ takes the form of ${\hat{a}}_{L}(\omega )={ \mathcal A }\,{\hat{a}}_{0}(\omega )+{ \mathcal B }\,\hat{f}(\omega )$, and requiring by quantum mechanics that $[{\hat{a}}_{L}(\omega ),{\hat{a}}_{L}^{\dagger }(\omega ^{\prime} )]\,=\delta (\omega -\omega ^{\prime} )$. This deduction is the essence of the Langevin approach, where the inclusion of loss requires the introduction of additional noise operators $\hat{f}(\omega )$ to ensure that the quantum-mechanical commutation relations are preserved. This is also an embodiment of the fluctuation-dissipation theorem (Mandel and Wolf 1995b). What is not obtained from this procedure is the actual form of $\hat{f}(\omega )$ as given by (88b ).

Alternatively, one can treat loss in an optical medium as the Hamiltonian evolution of an extended quantum system. If we consider the total Hamiltonian as the sum of the system Hamiltonian $({\hat{H}}_{{sys}}={\hat{H}}_{L})$ (see (9)), an environment Hamiltonian of free photons ${\hat{H}}_{{env}}={\int }_{-\infty }^{\infty }d\omega \,{\rm{\hslash }}\,\omega \,{\hat{e}}^{\dagger }(\omega )\,\hat{e}(\omega )$, and a coupling interaction between the two; ${\hat{H}}_{{int}}=i\,{\rm{\hslash }}{\int }_{-\infty }^{\infty }d\omega \,\kappa (\omega )\,\left({\hat{e}}^{\dagger }(\omega )\,\hat{a}(\omega )-\hat{e}(\omega )\,{\hat{a}}^{\dagger }(\omega )\right)$ the Heisenberg equation of motion for this system in a reference frame rotating with respect to the central frequency of the light approximates to the Heisenberg–Langevin equation (Walls and Milburn 1994, Orszag 2000):

$$\begin{eqnarray}&&\dot{\hat{a}}(t)=-\displaystyle \frac{i}{{\rm{\hslash }}}\,[\hat{a},{\hat{H}}_{{sys}}]-\displaystyle \frac{{\gamma }_{a}}{2}\,\hat{a}(t)+\sqrt{{\gamma }_{a}}\,{\hat{f}}_{a}(t),\end{eqnarray} \tag{ 90 }$$

where γ_a  = Γ_a c/n_ga , is the attenuation constant in time. The solution to this equation also yields (88a ). Here, it is also understood that $\hat{a}(t)$ is the time evolution of a single mode of the electromagnetic field $\hat{a}(\omega )$ in a lossy medium. In the lossless case (i.e. γ_a  = 0), the cavity mode evolves unitarily under the system Hamiltonian ${\hat{H}}_{{sys}}$. When loss is present, the mode is damped by the operator loss term $-({\gamma }_{a}/2)\,\hat{a}$, but the total evolution remains unitary; it is preserved due to the additional noise term $\sqrt{{\gamma }_{a}}\hat{f}$.

In the section where we specifically tackle the problem of SPDC in a lossy cavity, we use a Heisenberg–Langevin equation similar to (90), but where ${\hat{H}}_{{sys}}$ includes both ${\hat{H}}_{L}$ and ${\hat{H}}_{{NL}}$. Moreover, we use a rotating frame of reference so that the total time derivative of the propagating mode, $\dot{\hat{a}}$, is given as $({\partial }_{t}+(c/{n}_{g}){\partial }_{z})\hat{a}$. Once the equation of motion is solved to find $\hat{a}$ as a function of position in the MRR, this expression is incorporated into the interaction-picture Hamiltonian to find the state of the down-converted light.

5.2.1.3. Cavities and MRR

We now use the above results to examine the output mode ${\hat{a}}_{{out}}$ of a cavity subject to an input mode driving field ${\hat{a}}_{{in}}$, with the internal cavity mode $\hat{a}$. Without loss of generality, we take the cavity to be a MRR as illustrated in figure 4, which also corresponds to a Fabry–Perot cavity with one input/output semitransparent mirror, and one fully reflecting mirror.

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In analogy with a classical field derivation (Alsing et al 2017), the output mode ${\hat{a}}_{{out}}$ is a function of the sum over all possible trajectories from the input mode ${\hat{a}}_{{in}}$, as it makes an arbitrary number (including zero) of circulations around the cavity:

$$\begin{eqnarray}&&{\hat{a}}_{{out}}=\rho \,{\hat{a}}_{{in}}\end{eqnarray} \tag{ 91a }$$

$$\begin{eqnarray}&&+\,{\left(-{\tau }^{* }\right)}_{{a}_{{in}}\to {a}_{0}}{\left({\hat{a}}_{L}\right)}_{{a}_{0}\to {a}_{L}}{\left(\tau \right)}_{{a}_{L}\to {a}_{{out}}}\end{eqnarray} \tag{ 91b }$$

$$\begin{eqnarray}&&+\,{\left(-{\tau }^{* }\right)}_{{a}_{{in}}\to {a}_{0}}{\left({\rho }^{* }{\hat{a}}_{2L}\right)}_{{a}_{0}\to {a}_{2L}}{\left(\tau \right)}_{{a}_{2L}\to {a}_{{out}}}\end{eqnarray} \tag{ 91c }$$

$$\begin{eqnarray}&&+\,{\left(-{\tau }^{* }\right)}_{{a}_{{in}}\to {a}_{0}}{\left({\left({\rho }^{* }\right)}^{2}{\hat{a}}_{3L}\right)}_{{a}_{0}\to {a}_{3L}}{\left(\tau \right)}_{{a}_{3L}\to {a}_{{out}}}\end{eqnarray} \tag{ 91d }$$

$$\begin{eqnarray}&&+\,\ldots ,\end{eqnarray} \tag{ 91e }$$

$$\begin{eqnarray}&&=\,\rho \,{\hat{a}}_{{in}}-| \tau {| }^{2}\,\displaystyle \sum _{n=0}^{\infty }{\left({\rho }^{* }\right)}^{n}\,{\hat{a}}_{(n+1)L},\end{eqnarray} \tag{ 91f }$$

$$\begin{eqnarray}&&=\,\left(\rho -\alpha \,{e}^{i\theta }\,| \tau {| }^{2}\,\displaystyle \sum _{n=0}^{\infty }{\left({\rho }^{* }\alpha {e}^{i\theta }\right)}^{n}\right)\,{\hat{a}}_{{in}}\end{eqnarray} \tag{ 91g }$$

$$\begin{eqnarray}&&-\,i| \tau {| }^{2}\,\sqrt{{\rm{\Gamma }}}\displaystyle \sum _{n=0}^{\infty }{\left({\rho }^{* }\right)}^{n}\,{\int }_{0}^{(n+1)L}{dz}\,{e}^{i\xi (\omega )[(n+1)L\mbox{--}z]}\hat{s}(z,\omega ),\end{eqnarray} \tag{ 91h }$$

$$\begin{eqnarray}&&=\,\left(\displaystyle \frac{\rho -\alpha \,{e}^{i\theta }}{1-{\rho }^{* }\,\alpha \,{e}^{i\theta }}\right)\,{\hat{a}}_{{in}}\end{eqnarray} \tag{ 91i }$$

$$\begin{eqnarray}&&-\,i| \tau {| }^{2}\,\sqrt{{\rm{\Gamma }}}\displaystyle \sum _{n=0}^{\infty }{\left({\rho }^{* }\right)}^{n}\,{\int }_{0}^{(n+1)L}{dz}\,{e}^{i\xi (\omega )[(n+1)L\mbox{--}z]}\hat{s}(z,\omega ).\end{eqnarray} \tag{ 91j }$$

First, the output photon can arrive directly from the input bus by 'reflection' off of the MRR (as described in (91a )). Next, (as written diagrammatically in (91b )), the photon can couple into the MRR, acquiring factor $-{\tau }^{* }$, evolve through one circulation (circumference L of the resonator) as described in (87), and couple out of the resonator acquiring factor $\tau$. Successive paths involve multiple circulations within the resonator, acquiring additional factors of ${\rho }^{* }$ from self-coupling (i.e. 'reflection') after each circulation. To simplify notation, we have used the definition e^iξL ≡ α e^iθ defining $\alpha ={e}^{-\tfrac{1}{2}{\rm{\Gamma }}L}$ to be the internal loss factor in one circulation of the resonator, and θ ≡ βL to be the phase gained in free propagation over the same distance.

As derived in Alsing et al (2017), an explicit calculation of the output field commutation relation yields,

$$\begin{eqnarray}&&[{\hat{a}}_{{out}}(\omega ),{\hat{a}}_{{out}}^{\dagger }(\omega ^{\prime} )]=\delta (\omega -\omega ^{\prime} ).\end{eqnarray} \tag{ 92 }$$

This preservation of unitarity allows us to write

$$\begin{eqnarray}&&{\hat{a}}_{{out}}(\omega )={G}_{{out},{in}}(\omega )\,{\hat{a}}_{{in}}+{H}_{{out},{in}}(\omega )\,{\hat{f}}_{a}(\omega ),\end{eqnarray} \tag{ 93a }$$

$$\begin{eqnarray}&&| {H}_{{out},{in}}(\omega )| =\sqrt{1-| {G}_{{out},{in}}(\omega ){| }^{2}},\end{eqnarray} \tag{ 93b }$$

where G_out,in (ω) is the coefficient preceding ${\hat{a}}_{{in}}$ in (91j ), whose magnitude is always less than or equal to unity. This defines the Langevin quantum noise operator ${\hat{f}}_{a}(\omega )$ from the unitary requirement of the preservation of the free field output commutator. Interestingly, G_out,in (ω) is identical in form to the classical transmission coefficient (Yariv 2000), as would be expected. It is important to note that in treating loss in a MRR, we implicitly assumed the medium is isotropic. However, as shown in Alsing et al (2017), this assumption can be relaxed and the commutation relations (92) still hold for multiple, piecewise defined propagation wavevectors and losses along the ring resonator of circumference L.

5.2.2. Biphoton generation within the MRR

For biphoton generation arising from either the ${\chi }^{(2)}$ process of SPDC, or the χ⁽³⁾ process of spontaneous four-wave mixing (SFWM), Alsing and Hach (Alsing and Hach 2017a) consider a signal mode $\hat{a}$, and an idler mode $\hat{b}$ circulating within the MRR, and here, we do the same.

In the non-depleted pump approximation, one can arrive at the Hamiltonian:

$$\begin{eqnarray}\begin{array}{rcl}{\hat{{\rm{H}}}}_{{NL}} & = & \displaystyle \int {dzd}{\omega }_{1}d{\omega }_{2}\,g({\omega }_{p}{\omega }_{1}{\omega }_{2}){e}^{-i{\rm{\Delta }}{k}_{z}z}{e}^{i{\rm{\Delta }}\omega t}\\ & & \times \left(\alpha (z,{\omega }_{p}){\hat{a}}^{\dagger }(z,{\omega }_{1}){\hat{b}}^{\dagger }(z,{\omega }_{2})\right)+{\rm{H}}.{\rm{c}}.,\end{array}\end{eqnarray} \tag{ 94 }$$

where for SPDC:

$$\begin{eqnarray}&&{g}_{\mathrm{spdc}}=-i\left(\displaystyle \frac{{\chi }_{\mathrm{eff}}^{(2)}}{4\pi c\sqrt{L}}\right)\sqrt{\displaystyle \frac{{n}_{g1}{n}_{g2}}{{n}_{1}^{2}{n}_{2}^{2}{n}_{p}^{2}}}\sqrt{\displaystyle \frac{{\left({\rm{\hslash }}{\omega }_{p}\right)}^{3}}{2{\epsilon }_{0}}}{{\rm{\Phi }}}_{{xy}}^{\mathrm{SPDC}}\end{eqnarray} \tag{ 95 }$$

and for SFWM ²⁵ :

$$\begin{eqnarray}&&{g}_{\mathrm{sfwm}}=-\left(\displaystyle \frac{3{\chi }_{\mathrm{eff}}^{(3)}}{4\pi {cL}}\right)\sqrt{\displaystyle \frac{{n}_{g1}{n}_{g2}}{{n}_{1}^{2}{n}_{2}^{2}{n}_{p}^{4}}}\left(\displaystyle \frac{{\left({\rm{\hslash }}{\omega }_{p}\right)}^{2}}{{\epsilon }_{0}}\right){{\rm{\Phi }}}_{{xy}}^{\mathrm{SFWM}}.\end{eqnarray} \tag{ 96 }$$

In more accurate treatments of SPDC and SFWM in a MRR, the mode functions ${g}_{\vec{\mu }}(\vec{r})$ would be calculated given the geometry of the material, and how the index of refraction varies spatially (e.g. step-index versus graded index). Here, we define the spatial overlap integrals ${{\rm{\Phi }}}_{{xy}}^{\mathrm{SPDC}}$ and ${{\rm{\Phi }}}_{{xy}}^{\mathrm{SFWM}}$ as:

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{xy}}^{\mathrm{SPDC}}=\int {dxdy}\,{g}_{\vec{{\mu }_{p}}}^{* }(x,y){g}_{\vec{{\mu }_{1}}}(x,y){g}_{\vec{{\mu }_{2}}}(x,y)\end{eqnarray} \tag{ 97 }$$

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{xy}}^{\mathrm{SFWM}}=\int {dxdy}\,{g}_{\vec{{\mu }_{p1}}}^{* }(x,y){g}_{\vec{{\mu }_{p2}}}^{* }(x,y){g}_{\vec{{\mu }_{1}}}(x,y){g}_{\vec{{\mu }_{2}}}(x,y).\end{eqnarray} \tag{ 98 }$$

In the 'particle-in-a-box' mode basis, the coupling constants are given by:

$$\begin{eqnarray}&&{g}_{\mathrm{spdc}}=-i\left(\displaystyle \frac{32{\chi }_{\mathrm{eff}}^{(2)}}{9{\pi }^{3}c}\right)\sqrt{\displaystyle \frac{{n}_{g1}{n}_{g2}}{{n}_{1}^{2}{n}_{2}^{2}{n}_{p}^{2}}}\sqrt{\displaystyle \frac{{\left({\rm{\hslash }}{\omega }_{p}\right)}^{3}}{2{\epsilon }_{0}{V}_{{ring}}}}\end{eqnarray} \tag{ 99 }$$

and

$$\begin{eqnarray}&&{g}_{\mathrm{sfwm}}=-\left(\displaystyle \frac{27{\chi }_{\mathrm{eff}}^{(3)}}{16\pi c}\right)\sqrt{\displaystyle \frac{{n}_{g1}{n}_{g2}}{{n}_{1}^{2}{n}_{2}^{2}{n}_{p}^{4}}}\left(\displaystyle \frac{{\left({\rm{\hslash }}{\omega }_{p}\right)}^{2}}{{\epsilon }_{0}{V}_{{ring}}}\right).\end{eqnarray} \tag{ 100 }$$

In order to obtain this approximate Hamiltonian, we have used the lowest-order plane-wave cavity modes (i.e. 'particle-in-a-box' modes) instead of the Hermite–Gaussian modes to describe ${g}_{\vec{\mu }}(x,y)$, and integrated over both transverse dimensions. We let L = 2πR, the circumference of the ring, essentially treating the ring as a conformal mapping of a rectangular nonlinear waveguide ²⁶ . With this, we also let V_ring ≡ L_x L_y L using the dimensions of the deformed rectangular medium. Furthermore, where the pump is undepleted and in a coherent state, we have replaced the pump annihilation operator ${\hat{a}}_{p}$ with its corresponding coherent state amplitude α_p . We have taken the same steps used before to express the Hamiltonian as an integral over frequency, and we make the approximation that $\sqrt{{\omega }_{1}{\omega }_{2}}$ is approximately equal to the corresponding square root product of their central values. In SFWM, we let α(ω_p , z) represent the square of the pump coherent state amplitude. For the rest of this section, we will focus on SPDC, but it is instructive to be aware that besides issues related to different phase matching, dependence on pump intensity, and the much smaller value of ${\chi }_{\mathrm{eff}}^{(3)}$ relative to ${\chi }_{\mathrm{eff}}^{(2)}$, the physics of photon pair generation in a cavity is very similar for both SPDC and SFWM.

For further simplification, and to arrive at the essential aspects of SPDC in a MRR, we first use the simplifying approximation of near-perfect phase matching, so that ${e}^{-i{\rm{\Delta }}{k}_{z}z}\approx 1$. Next, we use the approximation of interaction times long enough to enforce energy conservation so that ${e}^{i{\rm{\Delta }}\omega t}\to (\sqrt{2\pi }/{T}_{{DC}})\delta ({\rm{\Delta }}\omega )$. This judicious substitution allows us to abbreviate the calculations done to calculate the biphoton rate in first-order perturbation theory as in previous sections. The interaction time, T_DC , is the round-trip time of light at the signal/idler frequencies. With these substitutions, the Hamiltonian simplifies to:

$$\begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{{NL}} & = & \displaystyle \int {dz}\,d{\omega }_{1}\,\displaystyle \frac{\sqrt{2\pi }}{{T}_{{DC}}}g({\omega }_{p},{\omega }_{1},{{\rm{\Omega }}}_{p}-{\omega }_{1})\\ & & \times \left(\alpha (z,{{\rm{\Omega }}}_{p}){\hat{a}}^{\dagger }(z,{\omega }_{1}){\hat{b}}^{\dagger }(z,{{\rm{\Omega }}}_{p}-{\omega }_{1})\right)+{\rm{H}}{\rm{.}}{\rm{c}}.,\end{array}\end{eqnarray} \tag{ 101 }$$

where ${{\rm{\Omega }}}_{p}={\omega }_{p}\,(\mathrm{alt}.\,2\,{\omega }_{p})$ for SPDC (alt. SFWM) such that the signal frequency is at Ω_p /2 + ν and the idler frequency is at Ω_p /2 − ν. Note that for later convenience, we define $({\rm{\hslash }}\,g)\equiv (\sqrt{2\,\pi }/{T}_{{DC}})\,{g}_{\mathrm{spdc}(\mathrm{sfwm})}$. To simplify the Hamiltonian even further, we shift to a reference frame rotating at the central frequency Ω_p /2. Then, in the following, the frequency ν represents an offset from Ω_p /2, so that $\hat{a}({{\rm{\Omega }}}_{p}/2+\nu )\to \hat{a}(\nu )$ and $\hat{b}({{\rm{\Omega }}}_{p}/2-\nu )\to \hat{b}(-\nu )$. We will further use the common quantum-optical shorthand notation ${\hat{b}}^{\dagger }(\nu )\equiv {[\hat{b}(-\nu )]}^{\dagger }$ (Orszag 2000). Thus, in the non-depleted pump approximation, we obtain the Hamiltonian:

$$\begin{eqnarray}&&{\hat{{\rm{H}}}}_{{NL}}=\int {dz}\,d\nu \,{\rm{\hslash }}g\left({\alpha }_{p}\,{\hat{a}}^{\dagger }(z,\nu )\,{\hat{b}}^{\dagger }(z,\nu )\right)+{\rm{H}}.{\rm{c}}.,\end{eqnarray} \tag{ 102 }$$

where we will take α_p  ≡ α_p (z, Ω_p /2) = constant throughout the MRR.

As was discussed previously, the signal and idler modes satisfy the Heisenberg–Langevin equation of motion in the frequency domain (using ${\partial }_{t}\,\hat{a}(t)=-i\nu \hat{a}(t)$) (Raymer and McKinstrie 2013, Alsing and Hach 2017a, 2017b), where this time, ${\hat{H}}_{{NL}}$ is included in ${\hat{H}}_{{sys}}$. In the rotating reference frame, the equations for the signal and idler modes are given by:

$$\begin{eqnarray}&&\begin{array}{l}\left(-i\,\nu +\displaystyle \frac{c}{{n}_{{ga}}}\,{\partial }_{z}\right)\,\hat{a}(z,\nu )=-i\,g\,L\,{\alpha }_{p}(z,{{\rm{\Omega }}}_{p}/2)\,{\hat{b}}^{\dagger }(z,\nu )\\ \quad -\,\displaystyle \frac{\gamma {{\prime} }_{a}}{2}\,\hat{a}(z,\nu )+{\alpha }_{{polz}}\,{\hat{f}}_{a}(z,\nu ),\,\end{array}\end{eqnarray} \tag{ 103a }$$

$$\begin{eqnarray}&&\begin{array}{l}\left(-i\,\nu +\displaystyle \frac{c}{{n}_{{gb}}}\,{\partial }_{z}\right)\,{\hat{b}}^{\dagger }(z,\nu )=i\,g\,L\,{\alpha }_{p}^{* }(z,{{\rm{\Omega }}}_{p}/2)\,\hat{a}(z,\nu )\\ \quad -\,\displaystyle \frac{\gamma {{\prime} }_{b}}{2}\,{\hat{b}}^{\dagger }(z,\nu )+{\alpha }_{{polz}}\,{\hat{f}}_{b}^{\dagger }(z,\nu ),\end{array}\end{eqnarray} \tag{ 103b }$$

where $\gamma {{\prime} }_{k}$ is the internal propagation loss for mode $k\in \{a,b\}$, and ${\hat{f}}_{k}$ are corresponding Langevin noise operators added to preserve the canonical form of the output commutators. The constant α_polz is a Langevin coupling constant to the scattered modes required to preserve the unitary evolution of the fields in the lossy MRR.

By expressing the relations between the input, cavity, and output fields in terms of matrices, we greatly simplify the subsequent algebra used to find the state of the output fields. In particular, the input–output boundary conditions are given by:

$$\begin{eqnarray}&&\left(\begin{array}{c}{\hat{a}}_{{0}_{+}}\\ {\hat{b}}_{{0}_{+}}^{\dagger }\end{array}\right)=\left(\begin{array}{cc}-{\tau }_{a}^{* } & 0\\ 0 & -{\tau }_{b}\end{array}\right)\left(\begin{array}{c}{\hat{a}}_{{in}}\\ {\hat{b}}_{{in}}^{\dagger }\end{array}\right)+\left(\begin{array}{cc}{\rho }_{a}^{* } & 0\\ 0 & {\rho }_{b}\end{array}\right)\left(\begin{array}{c}{\hat{a}}_{{L}_{-}}\\ {\hat{b}}_{{L}_{-}}^{\dagger }\end{array}\right),\end{eqnarray} \tag{ 104a }$$

$$\begin{eqnarray}&&\left(\begin{array}{c}{\hat{a}}_{{out}}\\ {\hat{b}}_{{out}}^{\dagger }\end{array}\right)=\left(\begin{array}{cc}{\tau }_{a} & 0\\ 0 & {\tau }_{b}^{* }\end{array}\right)\left(\begin{array}{c}{\hat{a}}_{{L}_{-}}\\ {\hat{b}}_{{L}_{-}}^{\dagger }\end{array}\right)+\left(\begin{array}{cc}{\rho }_{a} & 0\\ 0 & {\rho }_{b}^{* }\end{array}\right)\left(\begin{array}{c}{\hat{a}}_{{in}}\\ {\hat{b}}_{{in}}^{\dagger }\end{array}\right).\end{eqnarray} \tag{ 104b }$$

Defining vector and matrix notation implicitly, the boundary conditions (104a ) and (104b ) may be written in simplified form:

$$\begin{eqnarray}&&{\vec{\hat{a}}}_{{0}_{+}}=-{\bf{X}}\cdot {\vec{\hat{a}}}_{{in}}+{\bf{T}}\cdot {\vec{\hat{a}}}_{{L}_{-}}\end{eqnarray} \tag{ 105a }$$

$$\begin{eqnarray}&&{\vec{\hat{a}}}_{{out}}={{\bf{T}}}^{* }\cdot {\vec{\hat{a}}}_{{in}}+{{\bf{X}}}^{* }\cdot {\vec{\hat{a}}}_{{L}_{-}}.\end{eqnarray} \tag{ 105b }$$

Equations (103a ) and (103b ) in matrix notation are given by:

$$\begin{eqnarray}&&{\partial }_{z}\vec{\hat{a}}(z,\nu )={\bf{M}}\cdot \vec{\hat{a}}(z,\nu )+\displaystyle \frac{{n}_{g}{\alpha }_{{polz}}}{c}\vec{\hat{f}}(z,\nu ),\end{eqnarray} \tag{ 106 }$$

where

$$\begin{eqnarray}{\bf{M}}\equiv \left(\begin{array}{cc}i\displaystyle \frac{{n}_{g}\nu }{c}-\displaystyle \frac{{{\rm{\Gamma }}}_{a}}{2} & -\displaystyle \frac{{n}_{g}}{c}| g| L| {\alpha }_{p}| {e}^{i{\theta }_{p}}\\ -\displaystyle \frac{{n}_{g}}{c}| g| L| {\alpha }_{p}| {e}^{-i{\theta }_{p}} & i\displaystyle \frac{{n}_{g}\nu }{c}-\displaystyle \frac{{{\rm{\Gamma }}}_{b}}{2}\end{array}\right).\end{eqnarray} \tag{ 107 }$$

The solutions of (103a ) and (103b ) are then:

$$\begin{eqnarray}&&{\vec{\hat{a}}}_{{L}_{-}}={e}^{{\bf{M}}L}\cdot {\vec{\hat{a}}}_{{0}_{+}}+\displaystyle \frac{{n}_{g}{\alpha }_{{polz}}}{c}{\int }_{0}^{L}{dz}\,{e}^{{\bf{M}}(L-z)}\cdot \vec{\hat{f}}(z).\end{eqnarray} \tag{ 108 }$$

Although the solution requires taking the matrix exponential, we use the approximation of equal loss (Γ_a  = Γ_b  = Γ), and equal group index for signal and idler (as in type-I SPDC) to obtain the solution:

$$\begin{eqnarray}&&\begin{array}{rcl}\left(\begin{array}{c}{\hat{a}}_{{L}_{-}}\\ {\hat{b}}_{{L}_{-}}^{\dagger }\end{array}\right) & \approx & \alpha {e}^{i\theta }\left(\begin{array}{cc}\cosh (r) & -{e}^{i{\theta }_{p}}\sinh (r)\\ -{e}^{-i{\theta }_{p}}\sinh (r) & \cosh (r)\end{array}\right)\left(\begin{array}{c}{\hat{a}}_{{0}_{+}}\\ {\hat{b}}_{{0}_{+}}^{\dagger }\end{array}\right)\\ & & +\,\left(\begin{array}{cc}{B}_{11} & 0\\ 0 & {B}_{22}\end{array}\right)\left(\begin{array}{c}{\hat{f}}_{a}\\ {\hat{f}}_{b}^{\dagger }\end{array}\right),\end{array}\end{eqnarray} \tag{ 109 }$$

or in vector notation:

$$\begin{eqnarray}&&{\vec{\hat{a}}}_{{L}_{-}}={\bf{R}}\cdot {\vec{\hat{a}}}_{{0}_{+}}+{\bf{B}}\cdot \vec{\hat{f}},\end{eqnarray} \tag{ 110 }$$

where R and B are defined implicitly. The coefficients ${B}_{11}\,=\sqrt{1-{\alpha }^{2}}$, and ${B}_{22}=\sqrt{1-{\alpha }^{2}}$, where α = e^{−Γ L/2}. These coefficients are determined by requiring preservation of the commutation relations $[{\hat{a}}_{L-},{\hat{a}}_{L-}^{\dagger }]=[{\hat{b}}_{L-},{\hat{b}}_{L-}^{\dagger }]=[{\hat{a}}_{0+},{\hat{a}}_{0+}^{\dagger }]\,=[{\hat{b}}_{0+},{\hat{b}}_{0+}^{\dagger }]$. Here, we have used the notation: $r=| g| L| {\alpha }_{p}| {T}_{{DC}}$, and ${\alpha }_{p}=| {\alpha }_{p}| {e}^{i{\theta }_{p}}$, and θ_p  = (1/2)Ω_p T_DC , and Γ = γn_g /c. It is interesting to point out that, in the limit of zero loss, the squeezing transformation is essentially identical to that derived in the previous section, though now expressed in terms of length instead of time.

In addition, we can consider many circulations within the resonator to examine the net relationship between gain and loss. While ${\rm{\Gamma }}/2$ represents the amplitude loss per unit length in the resonator, the quantity r/L represents the amplitude gain per unit length due to SPDC. Incorporating out-coupling loss $| \rho {| }^{2}$ into the total loss per round trip, we find that in order to have a net exponential gain of SPDC light (i.e. parametric gain), the pump intensity must be high enough that r exceeds the threshold:

$$\begin{eqnarray}&&{r}_{{thresh}}\geqslant \displaystyle \frac{{\rm{\Gamma }}L}{2}+\mathrm{ln}\left(\displaystyle \frac{1}{| \rho | }\right).\end{eqnarray} \tag{ 111 }$$

This is also known as the threshold for optical parametric oscillation, where over many cycles, intensities of down-converted light may be bright enough to be comparable to the pump. This is distinct from the pump power levels where multi-biphoton events become significant. When using SPDC as a source of heralded single photons, we operate well below this threshold, because multi-biphoton events would overwhelm the photon pair statistics at such high intensities. For typical MRR parameters, r_thresh corresponds to input pump powers of the order 1–10 milliwatts, though higher-Q resonators will lower this threshold further. These approximations are liberal and numerous, as an accurate result requires knowing what the actual spatial modes of the waveguide are, what the effective index of refraction of the propagating spatial modes are, and how much of the pump power in the MRR is in the lowest order spatial mode. In particular, this is important because an MRR that is single-mode at the down-converted wavelength will be multi-mode at the pump wavelength. Due to conservation of momentum, only the lowest-order pump mode in such an MRR can drive photon pair generation if it is single-mode at the down-converted wavelength.

5.2.3. The output two-photon signal-idler state

To obtain the state of the down-converted light outside the resonator, we may use the matrix expressions in (104a ) and (104b ), to express the output fields ${\vec{\hat{a}}}_{{0}_{+}}$ in terms of ${\vec{\hat{a}}}_{{out}}$ and $\vec{\hat{f}}$. To do this, we can express the output field operator as a sum over the possible number of circulations in the MRR, as was done previously in relating ${\vec{\hat{a}}}_{{out}}$ to ${\vec{\hat{a}}}_{{in}}$. In this case, there are no photons in the input field at the frequency of the down-converted light, as the down-converted light is being generated within the MRR. When relating ${\vec{\hat{a}}}_{{0}_{+}}$ to ${\vec{\hat{a}}}_{{out}}$, we find:

$$\begin{eqnarray}&&{\vec{\hat{a}}}_{{0}_{+}}(\nu )={\bf{D}}(\nu )\cdot {\vec{\hat{a}}}_{{out}}(\nu )+{\bf{J}}(\nu )\cdot \vec{\hat{f}}(\nu ),\end{eqnarray} \tag{ 112 }$$

where

$$\begin{eqnarray}&&{\bf{D}}={\left({\bf{R}}-{{\bf{T}}}^{* }\right)}^{-1}{\bf{X}}\end{eqnarray} \tag{ 113a }$$

$$\begin{eqnarray}&&{\bf{J}}=-{\left({\bf{R}}-{{\bf{T}}}^{* }\right)}^{-1}{\bf{B}}.\end{eqnarray} \tag{ 113b }$$

Assuming the parameters are the same for a and b, these matrices have relatively simple expressions:

$$\begin{eqnarray}{\bf{D}}=\tau \left(\begin{array}{cc}\displaystyle \frac{(\cosh (r)(\alpha {e}^{i\theta })-\rho )}{{ \mathcal D }} & \displaystyle \frac{{e}^{i{\theta }_{p}}(\alpha {e}^{i\theta })\sinh (r)}{{ \mathcal D }}\\ \displaystyle \frac{{e}^{-i{\theta }_{p}}(\alpha {e}^{i\theta })\sinh (r)}{{ \mathcal D }} & \displaystyle \frac{(\cosh (r)(\alpha {e}^{i\theta })-\rho )}{{ \mathcal D }}\end{array}\right)\end{eqnarray} \tag{ 114 }$$

$$\begin{eqnarray}&&{\bf{J}}=-\,\displaystyle \frac{\sqrt{1-{\alpha }^{2}}}{\tau }\,{\bf{D}},\end{eqnarray} \tag{ 115 }$$

where

$$\begin{eqnarray}&&{ \mathcal D }\equiv {\left(\alpha {e}^{i\theta }\right)}^{2}+{\rho }^{2}-2(\alpha {e}^{i\theta })\rho \,\cosh (r),\end{eqnarray} \tag{ 116 }$$

and the dependence on ν is given by θ = νT_DC .

For later convenience, we also define the notation:

$$\begin{eqnarray}{\bf{D}}(\nu )=\left(\begin{array}{cc}{D}_{{aa}}(\nu ) & {D}_{{ab}}(\nu )\\ {D}_{{ba}}(\nu ) & {D}_{{bb}}(\nu )\end{array}\right),\end{eqnarray} \tag{ 117a }$$

$$\begin{eqnarray}{\bf{J}}(\nu )=\left(\begin{array}{cc}{J}_{{aa}}(\nu ) & {J}_{{ab}}(\nu )\\ {J}_{{ba}}(\nu ) & {J}_{{bb}}(\nu )\end{array}\right).\end{eqnarray} \tag{ 117b }$$

For weak but classically bright pump fields, the state of the down-converted fields is well approximated to first order in ${\hat{H}}_{{NL}}$, and given by:

$$\begin{eqnarray}&&| {\rm{\Psi }}({T}_{{DC}}){\rangle }_{{ab}}={e}^{-(i/{\rm{\hslash }}){\hat{H}}^{{NL}}{T}_{{DC}}}\,| {\rm{\Psi }}{\rangle }_{{in}}\approx \left(1-\displaystyle \frac{i}{{\rm{\hslash }}}\,{\hat{H}}^{{NL}}\,{T}_{{DC}}\right)| {\rm{vac}}\rangle \end{eqnarray} \tag{ 118a }$$

$$\begin{eqnarray}&&\begin{array}{l}=\,\left[1-{\displaystyle \int }_{-\infty }^{\infty }\,d\nu \,| g| {T}_{{DC}}{\displaystyle \int }_{0}^{L}\,{dz}\,| {\alpha }_{p}| \left({e}^{i{\theta }_{p}(\nu )}\,{\hat{a}}^{\dagger }(z,\nu )\,{\hat{b}}^{\dagger }(z,\nu )\right.\right.\\ \quad \left.\left.+\,{e}^{-i{\theta }_{p}(\nu )}\,\hat{a}(z,\nu )\,\hat{b}(z,\nu )\right)\right]| {\rm{vac}}\rangle \end{array}\end{eqnarray} \tag{ 118b }$$

$$\begin{eqnarray}&&\begin{array}{l}\approx \,\left[1-{\displaystyle \int }_{-\infty }^{\infty }\,d\nu \,{r}_{{ab}}(\nu )\,\left({e}^{i{\theta }_{p}(\nu )}\,{\hat{a}}^{\dagger }({0}_{+},\nu )\,{\hat{b}}^{\dagger }({0}_{+},\nu )\right.\right.\\ \quad \left.\left.+\,{e}^{-i{\theta }_{p}(\nu )}\,\hat{a}({0}_{+},\nu )\,\hat{b}({0}_{+},\nu )\right)\right]| {\rm{vac}}\rangle ,\end{array}\end{eqnarray} \tag{ 118c }$$

where ${\hat{H}}^{{NL}}$ is given in (102), and for simplicity: ${r}_{{ab}}(\nu )\,\equiv | g| \,| {\alpha }_{p}| \,L\,{T}_{{DC}}$. Note that in this estimation of the quantum state, we use the interaction picture, where the state evolves according to ${\hat{H}}_{{NL}}$, while the creation and annihilation operators evolve according to ${\hat{H}}_{L}$ and the interaction Hamiltonian accounting for loss. In this case, we treat the evolution of the operators as in the Heisenberg–Langevin equation (108), but without the contribution of ${\hat{H}}_{{NL}}$, effectively setting r = 0. In this picture, we can relax the assumption of near-perfect phase matching, so that r_ab (ν) acquires an additional factor of sinc(Δk_z L/2) after integrating over z. The integration over z is approximated under the assumption that the damping over z is slow enough that the exponential damping can be approximated to first order (i.e. linearly) from 0 to L₋. For r = 0, the output relation matrices D and J are greatly simplified to:

$$\begin{eqnarray}{\bf{D}}(\nu ,r=0)=\displaystyle \frac{\tau }{\alpha {e}^{i\theta }-\rho }\left(\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right),\end{eqnarray} \tag{ 119a }$$

$$\begin{eqnarray}{\bf{J}}(\nu ,r=0)=\displaystyle \frac{-\sqrt{1-{\alpha }^{2}}}{\alpha {e}^{i\theta }-\rho }\left(\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right).\end{eqnarray} \tag{ 119b }$$

Although these matrices blow up in the limit of critical coupling (i.e. $\rho \to \alpha$) where r = 0, first-order perturbation theory is no longer accurate in such regimes. When calculating expectation values using the solutions to the Heisenberg–Langevin equation (where r > 0) to get a more accurate estimate, the number of generated biphotons exiting the resonator is maximum at critical coupling, but finite.

Now that the state of the SPDC light inside the resonator has a straightforward form, the output state $| {\rm{\Psi }}{\rangle }_{{out}}$ is obtained from the internal state $| {\rm{\Psi }}({T}_{{ab}}){\rangle }_{{ab}}$ as the Heisenberg operators ${\vec{\hat{a}}}_{{0}_{+}}$ evolve through the resonator and couple out, becoming ${{\bf{X}}}^{* }{\bf{R}}\,{\vec{\hat{a}}}_{{0}_{+}}$. The scattered light given by creation operator $\vec{\hat{f}}$ has exited the system, and does not enter into the Heisenberg propagation of the down-converted light from inside to outside the resonator. Then, using our expression for ${\vec{\hat{a}}}_{{0}_{+}}$ in terms of ${\vec{\hat{a}}}_{{out}}$ and $\vec{\hat{f}}$. The output state of the fields has a straightforward expression (with ν argument suppressed to save space):

$$\begin{eqnarray}\begin{array}{rcl}| {\rm{\Psi }}{\rangle }_{{out}} & = & | \mathrm{vac}\rangle -{\displaystyle \int }_{-\infty }^{\infty }d\nu \,{\alpha }_{a}^{* }{\alpha }_{b}^{* }{\tau }_{a}^{* }{\tau }_{b}^{* }{r}_{{ab}}\,\mathrm{sinc}\left(\displaystyle \frac{{\rm{\Delta }}{k}_{z}L}{2}\right)\\ & & \times \,\left[\left({e}^{i{\theta }_{p}}{D}_{{aa}}^{* }{D}_{{bb}}+{e}^{-i{\theta }_{p}}{D}_{{ab}}{D}_{{ba}}^{* }\right){\hat{a}}_{{out}}^{\dagger }{\hat{b}}_{{out}}^{\dagger }\right.\\ & & +\left({e}^{i{\theta }_{p}}{D}_{{aa}}^{* }{J}_{{bb}}+{e}^{-i{\theta }_{p}}{D}_{{ba}}^{* }{J}_{{ab}}\right){\hat{a}}_{{out}}^{\dagger }{\hat{f}}_{b}^{\dagger }\\ & & +\left({e}^{i{\theta }_{p}}{D}_{{bb}}{J}_{{aa}}^{* }+{e}^{-i{\theta }_{p}}{D}_{{ab}}{J}_{{ba}}^{* }\right){\hat{f}}_{a}^{\dagger }{\hat{b}}_{{out}}^{\dagger }\\ & & \left.\left.+\left({e}^{i{\theta }_{p}}{J}_{{aa}}^{* }{J}_{{bb}}+{e}^{-i{\theta }_{p}}{J}_{{ab}}{J}_{{ba}}^{* }\right){\hat{f}}_{a}^{\dagger }{\hat{f}}_{b}^{\dagger }\right]\Space{0ex}{1.35em}{0ex}\right|\mathrm{vac}\rangle ,\end{array}\end{eqnarray} \tag{ 120 }$$

where annihilation operators acting on the vacuum state yield a null result. In the previous matrix expressions, we let τ_a  = τ_b  = τ, and let τ be real to simplify notation. Interestingly, the phase of τ can be incorporated as a contribution to the phase ${e}^{i{\theta }_{p}}$ because although the previous expression contains terms associated to ${e}^{i{\theta }_{p}}$ and ${e}^{-i{\theta }_{p}}$, closer examination of the coefficients associated to these terms reveals a global phase dependence of ${e}^{i{\theta }_{p}}$.

With the state of the down-converted light exiting the resonator $| {\rm{\Psi }}{\rangle }_{{out}}$ known, we see that it is readily decomposed into four elements. The amplitude for biphoton production precedes ${\hat{a}}_{{out}}^{\dagger }{\hat{b}}_{{out}}^{\dagger }$, while the amplitude for a signal photon with scattered idler precedes ${\hat{a}}_{{out}}^{\dagger }{\hat{f}}_{b}^{\dagger }$. The corresponding amplitudes for scattered idlers, and both scattered photons are straightforward as well.

5.2.4. Rate and heralding efficiency of biphotons exiting cavity

As was discussed previously for bulk crystals, the rate of biphotons coupling out of the resonator is given by the probability for the existence of the biphoton from $| {\rm{\Psi }}{\rangle }_{{out}}$, divided by the round-trip time T_DC , where $| {\rm{\Psi }}{\rangle }_{{out}}$ is obtained from $| {\rm{\Psi }}({T}_{{DC}}){\rangle }_{{ab}}$. As a function of ν, the biphoton rate per unit frequency ${{ \mathcal R }}_{{ab}}(\nu )$ is given by:

$$\begin{eqnarray}&&{{ \mathcal R }}_{{ab}}(\nu )=\displaystyle \frac{2\pi {L}^{2}}{{{\rm{\hslash }}}^{2}{T}_{{DC}}}| {g}_{{spdc}}{| }^{2}| {\alpha }_{p}{| }^{2}\,| {\psi }_{{ab}}(\nu ){| }^{2}{\mathrm{sinc}}^{2}\left(\displaystyle \frac{{\rm{\Delta }}{k}_{z}L}{2}\right),\end{eqnarray} \tag{ 121 }$$

where

$$\begin{eqnarray}\begin{array}{rcl}{\psi }_{{ab}}(\nu ) & \equiv & {\alpha }_{a}^{* }{\alpha }_{b}^{* }{\tau }_{a}^{* }{\tau }_{b}^{* }\\ & & \times \left({e}^{i{\theta }_{p}}{D}_{{aa}}^{* }(\nu ){D}_{{bb}}(\nu )+{e}^{-i{\theta }_{p}}{D}_{{ab}}(\nu ){D}_{{ba}}^{* }(\nu )\right).\end{array}\end{eqnarray} \tag{ 122 }$$

P is the input pump power (in the bus) and B is the cavity buildup factor at the pump wavelength, approximately ²⁷ equal to the finesse ${ \mathcal F }$ divided by π/2. In figure 5, we have plotted $| {\psi }_{{ab}}(\nu ){| }^{2}$ to examine the shape of the spectrum of down-converted light when the cavity linewidth is much narrower than the phase matching bandwidth. Where we have assumed strict energy conservation, this spectrum represents a subset of the detected biphotons, i.e. the spectrum of the signal light over one linewidth of the cavity. When reflected about ν = 0, this is the idler spectrum. With the rate ${{ \mathcal R }}_{{ab}}(\nu )$ known, we integrate over the area of a single resonance to obtain the coincidence rate due to emission into a single pair of frequency peaks ${R}_{{ab}}^{({peak})}$, and find:

$$\begin{eqnarray}&&{R}_{{ab}}^{({peak})}\approx \displaystyle \frac{8192}{81{\pi }^{4}{\epsilon }_{0}{c}^{2}}\,\displaystyle \frac{{n}_{g1}{n}_{g2}}{{n}_{1}^{2}{n}_{2}^{2}{n}_{p}^{2}}\,\,\displaystyle \frac{{d}_{\mathrm{eff}}^{2}{\omega }_{p}^{2}}{{L}_{x}{L}_{y}}L(1-{\rho }^{4})\,B\,P,\end{eqnarray} \tag{ 123 }$$

where the approximation assumes α ≈ 1 for the integration of $| \psi (\nu ){| }^{2}$, and we are sufficiently far from critical coupling that $| \psi (\nu ){| }^{2}$ is not significantly altered when assuming r ≈ 0. In the limit of zero self-coupling ($\rho \to 0$) the down-converted light can only make one round trip around the MRR, and the formula becomes identical to the single-mode rate in the bulk crystal (i.e. waveguide) regime. The pump buildup factor B approaches unity, $| {\psi }_{{ab}}(\nu ){| }^{2}$ grows wider than the phase-matching bandwidth of the light so that it is near unity over the bandwidth of the sinc function, and we must carry out the same phase-matching integrals as in previous sections. In this same limit, we see that the effect of loss is that R_ab scales as $| \alpha {| }^{4}$, or by two factors of the power loss; one for the signal photon and one for the idler photon.

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It is interesting to point out that, here, the total rate ${R}_{{ab}}^{({peak})}$ scales linearly with L, even though the narrow frequency filtering of the MRR would suggest a quadratic dependence. This is due to the linewidth of the MRR itself depending on L, where longer resonators have a corresponding narrower linewidth.

As an example of the utility of this formula, consider the following. Let us assume type-I SPDC in a MRR of aluminum nitride with radius 30 μm, with transverse horizontal and vertical thicknesses of 1.0 μm and 0.3 μm, respectively. The effective nonlinearity d_eff ≈ 4.7 pm V⁻¹. Let the quality factor at the pump wavelength be 10⁴ which gives a buildup factor B of about 12.3. Let the pump wavelength λ_p  = 775 nm. We will let n_g1 = n_g2 = 2.19 and n₁ = n₂ = 2.16 and n_p  = 2.14. With these parameters, we obtain an astonishingly high rate ${R}_{{ab}}^{({peak})}$ of approximately 3.0 × 10⁷ pairs per second per mW of pump power between correlated resonances in the cavity. In the limit of no self coupling ($\rho \to 0$) and no cavity buildup $B\to 1$, $| \psi (\nu ){| }^{2}\approx 1$ over all frequency so that an accurate treatment must explicitly consider phase-matching (i.e. ${e}^{i{\rm{\Delta }}{k}_{z}}\rlap{/}{\approx }1$), and an accurate treatment is well described in the bulk crystal regime. While the ideality of our approximations (including our choice of basis modes) makes it unrealistic that this formula provides an accurate estimate of the number of exiting photon pairs per second, it does illustrate the potential single-bus MRRs have as a bright source of photon pairs via SPDC.

In more practical implementations of SPDC in MRRs, a dual-bus configuration may be used so that one waveguide may be dedicated to coupling in/out pump light, and the other for outcoupling SPDC photon pairs. Alternatively, in type-II SPDC, the coupling between bus and MRR can be strongly polarization-dependent, and it may be possible to well-separate the signal and idler photons from one another instead of tolerating the reduction in coincidences relative to singles that comes with separation with a non-polarizing BS.

In order to gauge the utility of the photon pairs exiting the resonator, it is not enough to simply know the photon pair rate. Because of loss in the resonator among other places, the number of signal photons without matching idlers exiting the resonator is significant enough, that its dependence on experimental parameters is important to know. Although usually discussed in the context of spatial correlations, here, we shall define the resonator heralding efficiency η_R to be the ratio of the signal photon rate coming from exiting photon pairs (equal to the photon pair rate discussed previously), divided by the sum of this rate and the rate of signal photons exiting the resonator, where the idler has been lost. Where common factors in the ratio cancel out, we find:

$$\begin{eqnarray}&&{\eta }_{R}(\nu )\approx \displaystyle \frac{| {D}_{{bb}}(\nu ){| }^{2}}{| {D}_{{bb}}(\nu ){| }^{2}+| {J}_{{bb}}(\nu ){| }^{2}},\end{eqnarray} \tag{ 124 }$$

where we take the same approximations for calculating the individual rates as before. In this case, we find the heralding efficiency is nearly constant over the FSR of the resonator, and arrive at the approximation:

$$\begin{eqnarray}&&{\eta }_{R}\approx \displaystyle \frac{1-{\rho }^{2}}{2-{\rho }^{2}-{\alpha }^{2}}.\end{eqnarray} \tag{ 125 }$$

For the single-bus MRR studied here, we see a tradeoff between enhancing either brightness or heralding efficiency due to the parameters of the resonator. While lower intrinsic loss (i.e. $\alpha \to \approx 1$) is an absolute improvement, increasing the self-coupling ρ only increases brightness at the expense of lowering heralding efficiency. Indeed, η_R is maximized in the limit of no self coupling (i.e. where $\rho \to 0$), and only approaches 50% at critical coupling. In the limit of strong self-coupling, where $\rho \to 1$ for constant loss α, the heralding efficiency decreases towards zero, since it becomes progressively more and more likely that a photon in the resonator will be scattered out as loss rather than couple into the output bus.

5.2.5. Time correlations of biphotons exiting cavity

In order to accurately treat the time correlations between the signal and idler photons exiting the cavity, it is necessary to include phase matching. Energy conservation allows us to say ${\psi }_{{ab}}(\nu )\,\approx {\psi }_{{ab}}({\nu }_{-}/\sqrt{2})$, where ${\nu }_{-}\equiv ({\nu }_{1}-{\nu }_{2})/\sqrt{2}\,=({\omega }_{1}-{\omega }_{2})/\sqrt{2}$. We are interested in this model only for discussion of the behavior of the time correlations between biphotons exiting a MRR. For type-I SPDC, the overall phase matching function is given by $\mathrm{sinc}(L\kappa {\nu }_{-}/\sqrt{8}){\psi }_{{ab}}({\nu }_{-}/\sqrt{2})$, which can be broken up into three different terms The sinc function is a broad envelope function multiplying ${\psi }_{{ab}}({\nu }_{-}/\sqrt{2})$, and ${\psi }_{{ab}}({\nu }_{-}/\sqrt{2})$ is well approximated as the convolution of a Dirac comb with spacing $2\sqrt{2}\pi /{T}_{{DC}}$ with a Lorentzian 'tine' of FWHM $| \alpha \,-\rho | \sqrt{8}/({T}_{{DC}}\sqrt{\alpha \rho })$ (see figure 6(a) for diagram of $| \psi ({\nu }_{-}){| }^{2}$). Because of the simplicity of our expression, we can readily take the inverse Fourier transform to examine the time correlations. Using the convolution theorem to our advantage, we see that in time (as in figure 6(b)), the amplitude of ${t}_{-}$ has a similar breakdown to the corresponding function of ν₋. The 'envelope' in time is given by the inverse transform of the 'tine' function in frequency, and the tine function in time is given by the inverse transform of the envelope function in frequency. The spacing of the comb in t₋ is given by ${T}_{{DC}}/(2\sqrt{2}\pi )$.

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In the time domain, the inverse transformed Lorenzian is an exponential spike with decay constant in t₋ of $| \alpha -\rho | \sqrt{2}/({T}_{{DC}}\sqrt{\alpha \rho })$, which serves as an envelope for a comb of inverse-transformed sinc resonances, with spacing equal to ${T}_{{DC}}/(2\sqrt{2}\pi )$. The exact shape of the 'inverse-transformed sinc resonances' is determined by the type of phase matching, as discussed in previous sections. Where T_DC is on the order of a few picoseconds, experimental measurements of the time correlations by coincidence counting are not yet capable of resolving individual peaks, but may have sufficient range to capture the breadth of these time correlations. Indeed, the number of tines in t₋ until the exponential envelope decays to 1/e of its peak value is directly proportional to the finesse of the resonator at the down-conversion frequency ²⁸ . As an example, when T_DC is of the order of 2 picoseconds, coincidence counting setups with range of 20 nanoseconds will adequately capture the time correlations in single-bus micro ring resonators with a finesse of the order 10³.

Regardless of the initial quantum state of the pump, we can use the differential equation for ${\hat{N}}_{1}(t)$ (129) to find the rate of photon pair generation. For a given quantum state of the field $\hat{\rho }$, the mean number of photon pairs, also given by $\langle {\hat{N}}_{1}(t)\rangle$ is:

$$\begin{eqnarray}&&{N}_{{SM}}(t)=\mathrm{Tr}[\hat{\rho }\,{\hat{N}}_{1}(t)].\end{eqnarray} \tag{ 136 }$$

Because the signal and idler fields are initially in the vacuum state, for times before significant pump depletion, this simplifies to:

$$\begin{eqnarray}&&{N}_{{SM}}(t)=\mathrm{Tr}[\hat{\rho }\,{\sinh }^{2}(\sqrt{{\hat{N}}_{p}}\,{gt})].\end{eqnarray} \tag{ 137 }$$

At smaller pump powers or smaller times, this simplifies further to:

$$\begin{eqnarray}&&{N}_{{SM}}(t)\approx \mathrm{Tr}[\hat{\rho }\,({\hat{N}}_{p})\,{g}^{2}{t}^{2}]=\langle {\hat{N}}_{p}\rangle \,{g}^{2}{t}^{2}.\end{eqnarray} \tag{ 138 }$$

Therefore, at small times, and pump powers, the average number of generated biphotons depends only on the mean pump power, regardless of whether it is in a coherent state, Fock state, or any other state. For higher pump powers, where this approximation no longer applies, there is some qualitative difference between the efficiency of SPDC with different pump photon statistics and same mean pump power.

If we take the trace in the photon number basis, and let P(n_p ) be the probability of measuring n_p photons at time t, then the number of generated biphotons N_SM (t) is expressible as:

$$\begin{eqnarray}&&{N}_{{SM}}(t)=\displaystyle \sum _{{n}_{p}=0}^{\infty }P({n}_{p})\,{\sinh }^{2}(\sqrt{{n}_{p}}\,{gt}).\end{eqnarray} \tag{ 139 }$$

Since the function $f(x)={\sinh }^{2}(\sqrt{x})$ is a convex, monotonically increasing function ³² of x for all positive values of x, the mean value of the function $\langle f(x)\rangle$ is larger than the function of the corresponding mean value of x, $f(\langle x\rangle )$. Consequently, pump beams with larger fluctuations of photon number will have larger biphoton generation efficiency solely by the virtue of there being probable events of larger photon number. Whether this is due to power instability in the pump, or a fundamental difference in the quantum number statistics of the pump, the overall effect on biphoton generation rate will remain the same. Even so, comparing the mean number of biphotons generated for a Fock state pump, a coherent state pump, and a thermal state pump with same mean photon number yields an inconsequential discrepency. Even at pump intensities approaching the damage threshhold of many nonlinear materials (e.g. 1 MW mm⁻²), the estimated difference in N_SM (T_DC ) between a Fock pump, a coherent pump, and a thermal pump is less than 1%.

When entering the regime of significant pump depletion and long interaction times, the efficiency of SPDC can vary significantly. Although we showed earlier that coherent state pumps incident on simple nonlinear media have a maximum down-conversion efficiency of approximately 50%, it has been shown (Niu et al 2017) that a 1-photon Fock state pump can have 100% down-conversion efficiency, while n-photon Fock states up to n = 50 have maximum efficiencies above 77%.

Efficiency aside, it is a very interesting question how the quantum state of the down-converted fields changes with the quantum state of the pump, and has been discussed since the early days of the field (Giallorenzi and Tang 1968), where in general the coherence of the pump field is mapped to the coherence of the down-converted field with the generated biphoton intensity remaining constant for constant pump power. The two-mode squeezed vacuum state for SPDC light assumes a coherent state pump, but the state of the down-converted fields for a Fock state pump, or a thermal state pump will differ greatly. The nature of the down-converted fields as a function of exotic quantum pump states remains a rich field for further development.

The first experiment (figure 8) tests the single-mode rate for degenerate type-0 SPDC with a periodically poled nonlinear crystal. We used a 40 mm periodically poled lithium niobate (PPLN) crystal manufactured by Covesion with a 1 mm (transverse) width, and $19.5\,\mu {\rm{m}}$ poling period, temperature tuned to 107.2 °C for degenerate SPDC from 782.09 to 1564.18 nm. Our pump laser was an OBIS laser with measured wavelength of 782.09 nm and bandwidth of approximately 0.01 nm. The pump laser light was directed into the crystal through a single-mode fiber, triplet fiber collimator, and focusing lens to obtain a well-approximated Gaussian beam with spot size σ_p  = 52.6 ± 2 μm at the center of the crystal. Using corresponding collection optics for the down-converted light, we obtain a mode-matched down-converted beam radius of σ₁ = 55.1 ± 2 μm also at the center of the crystal. Using the Sellmeier equations for lithium niobate, and published values for d_eff (Gayer et al 2008), we obtained the necessary phase and group indices of refraction, as well as the group velocity dispersion constant κ.

To simplify the initial alignment of our setup, we input 1 564.18 nm light into the back end of the experiment, and coupled the second harmonic generation light into the fiber that would later be connected to the pump laser. Since the exit fiber tip is in an image plane of the center of the crystal, the down-converted light is spatially correlated at the fiber tip, and we let the coupling loss through the exiting fiber collimator to be the coupling efficiency C. Since the down-converted light was too dim to be seen in free space with ordinary power meters, we estimated C using laser light at 1564 nm shining through an experiment with identical focusing optics and found C to be approximately (0.807 ± 0.025) though the coupling to an ideal mode-matched Gaussian beam may be higher. We estimate the heralding effiency η ≈ (0.862 ± 0.022) with our experimental beam parameters, and the formula for the heralding efficiency for SPDC with focused Gaussian beams in Dixon et al (2014). Per milliwatt of pump power per second, we measured singles rates of 16.00 ± 0.21 million and 17.99 ± 0.22 million for the signal and idler detectors, and background noise levels Φ₁ ≈ 0.05 × 10⁶ and Φ₂ ≈ 0.06 × 10⁶. We measured a coincidence count rate of 2.93 ± 0.05 million with accidentals rate A₁₂ ≈ 0.02 × 10⁶, giving coincidence to singles ratios of 16.1% and 18.1%, respectively, which in turn gives us a raw pair generation rate N of (95.63 ± 2.71) million pairs per second per mW of pump power.

With our experimental parameters, our formula (42) predicts a rate of (94.86 ± 10.89) × 10⁶ coincidence counts per second per mW of pump power. The raw pair generation rate obtained from our experiment was approximately (95.63 ± 2.71) × 10⁶ per second per mW of pump power, differing from our prediction by less than 1%, or 0.1 standard deviations. The relatively large uncertainty in the theoretical prediction is due to the propagation of uncertainties of multiple variables. The individually large 5% uncertainty in d_eff is due to imperfections between different manufacturing process of otherwise identical crystals. To have such a small disagreement between theory and experiment is subject to multiple caveats, namely, that the true coupling efficiency is unmeasured. Because we only measure the maximum coupling in a parallel experiment, we can only assume that this represents the coupling efficiency in the experiment if it too is optimally coupled. Though much effort was devoted to maximizing the coupling of the down-converted light into the single-mode fiber, it is likely that the experimental coupling efficiency is less than 0.805 by possibly as much as 10%–20%, which would then increase our estimate of N by 10%–20%, significantly exceeding the theoretical value.

In the next experiment (figure 9), we performed tests our formula for the total biphoton generation rate for collinear type-I SPDC in an isotropic crystal (47). Here, we used a 1 mm crystal of bismuth barium borate (BiBO) manufactured by Newlight Photonics. We used a 405 nm OBIS laser to produce down-converted photon pairs centered at 810 nm. We separated the photons with a 50/50 BS, and focused the light onto large-area single photon detectors. Because we are sampling over all modes, extracting the raw pair generation rate N from the singles and coincidences is simpler; we can set η and C equal to unity. Given our experimental parameters, we predict a pair generation rate of (53.87 ±10.87) × 10⁶ per second per mW of pump power. The experiment measured singles rates per mW of pump power of (6.16 ± 0.05) × 10⁵ and (6.02 ± 0.05) × 10⁵ per second, with respective background rates of (6.04 ± 0.15) × 10⁴ and (6.40 ± 0.10) × 10⁴ per second. We recorded a coincidence rate of (2.71 ± 0.06) × 10³ per second and an accidentals rate of (4.39 ± 2.79) per second. From these statistics, we obtain a raw pair generation rate of (64.68 ± 1.69) million pairs per second per mW of pump power, exceeding our theoretical prediction by 20%, though this is still within the large range of uncertainty due to limited knowledge of the biphoton wavefunction, among other factors.

For our third experiment (figure 10), we used a waveguide of PPKTP manufactured by AdvR, for type-II SPDC from 773 to 1546 nm poled for first-order quasi-phase matching. This experiment was pumped with a Newport NewFocus tunable laser centered at 773 nm. Here, we separated the signal and idler photons completely with a polarizing BS. Moreover, we may set C = 1 since both pair-generation, and collection occur in a single optical mode. The waveguide we used was 21.2 mm long, with values for σ_p and σ₁ being (0.875 ±0.125) μm, and (1.875 ± 0.125) μm, respectively. Per mW of pump power, we measured singles rates of (3.71 ± 0.05) ×10⁶ s⁻¹ and (4.51 ± 0.05) × 10⁶ s⁻¹, with a coincidence count rate of (4.71 ± 0.07) × 10⁵ s⁻¹. From these rates, we obtain a raw pair generation rate of approximately (35.5 ± 0.8) × 10⁶ s⁻¹ per mW of pump power.

Using our single-mode formula for type-II SPDC in a periodically poled medium and the given experimental parameters, we estimate a rate of $(23.58\pm 5.60)\times {10}^{6}$ per second per mW of pump power. This differs from the experimental rate by as much as 33%, but due to asymetries in the eigenmodes of the waveguide (Fiorentino et al 2007, Shukhin et al 2015), a simple Gaussian mode of equal widths in both transverse dimensions cannot be assumed to be what couples into the exit fiber. Indeed, given the rubidium doping needed to create the waveguide, the waveguide itself has different effective widths in each transverse dimension. Assuming a 30% difference between the different transverse widths of the eigenmodes is reasonable (see diagram in Fiorentino et al (2007)), and is sufficient to produce a theoretical preciction that agrees well with experimental data. The theoretical estimate is also subject to the relatively large uncertainties in the pump and signal/idler radii inside the waveguide (of approximately 0.18 μm), whose value is generally more difficult to determine than in step-index optical fibers. Moreover, the waveguide is small enough that modal dispersion may noticeably change the effective index of refraction in comparison to bulk media. In addition, there is a rather large (≈10%) uncertainty in d_eff, which varies significantly between different PPKTP crystals, likely due to thermal stress patterns in the manufacturing process. For our theoretical prediction, we used the d₂₄ coefficient responsible for type-II SPDC given in Fiorentino et al (2007) (so that d_eff = d₂₄ not counting quasi-phase matching factors), which treats SPDC in a PPKTP waveguide. Where they list d₂₄ = 3.92 pm/V for SPDC for a 405 nm pump, we use Miller's rule ³³ to obtain d₂₄ ≈ 3.18 pm/V for SPDC with a 773 nm pump. To describe our waveguide adequately, it is single-mode at the down-conversion wavelength, but it is multi-mode at the pump wavelength. The mode field diameter at the pump wavelength is given as the diameter of the light entering the crystal from a single-mode fiber fused to the waveguide, which is not the diameter of the TEM00 mode accepted by the waveguide.