Signatures of Hong–Ou–Mandel interference at microwave frequencies

Suppose that the two fields incident on the input ports of the beam splitter are described by dimensionless, time-dependent quantum fields denoted by $\hat {a}'(t)$ and $\hat {b}'(t)$. Unitarity of the (balanced) beam splitter implies that the output fields, $\hat {a}(t)$ and $\hat {b}(t)$, are then given by the Heisenberg picture evolution [24]

$$\begin{equation} \left[\begin{array}{@{}c@{}} \hat{a}(t) \\ \hat{b}(t) \end{array} \right] = \frac{1}{\sqrt{2}} \left[ \begin{array}{@{}cc@{}} 1 & -1 \\ 1 & 1 \end{array} \right] \left[ \begin{array}{@{}c@{}} \hat{a}'(t) \\ \hat{b}'(t) \end{array} \right]. \end{equation} \tag{ 1 }$$

These fields are depicted in the Hong–Ou–Mandel interference experiment schematic of figure 1. Note that, in general, a variable phase may be included in the matrix of equation (1), though the results described here shall be independent of this phase. We are interested, primarily, in the statistics of the photons output from the beam splitter. These are described by correlation functions of the number operators of the output modes, the number operators being denoted by $\hat {n}_c (t) \equiv \hat {c}^\dagger (t) \hat {c} (t)$. Then, the quantities we seek are the normally ordered correlation functions of the output mode number operators:

$$\begin{equation} \langle : \hat{n}_a (t+\tau ) \hat{n}_{r} (t) : \rangle = \langle \hat{r}^\dagger (t) \hat{a}^\dagger (t+\tau ) \hat{a} (t+\tau ) \hat{r} (t) \rangle \end{equation} \tag{ 2 }$$

with $\hat {r}$ representing either $\hat {a}$ (for the output intensity auto-correlation function) or $\hat {b}$ (for the output intensity cross-correlation function). These correlation functions allow one to discriminate between quantum and classical fields [24]. Since the intensity of a field is proportional to the number of photons it contains, henceforth we shall refer to correlation functions of the form of equation (2) as intensity correlation functions. The term second-order correlation function shall be reserved here for functions of the difference in detection times (τ) alone, to be introduced below. The correlation functions $\langle : \hat {n}_b(t+\tau ) \hat {n}_r (t) : \rangle$ give the same results due to the symmetry of equation (1).

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Clearly, we can obtain the intensity correlation functions of equation (2) in terms of the input fields by substituting the expressions in equation (1). The result includes 16 fourth-order moments of the time-dependent input fields. Now, if it is assumed that the sources produce separable (i.e. not entangled) fields (clearly a reasonable assumption for independent sources), the four-term cross-correlation functions of input fields may each be decomposed into products of two auto-correlation functions of the input fields. Therefore, the output intensity auto-correlation and cross-correlation functions are

$$\begin{eqnarray} &&\fl \langle : \hat{n}_a (t+\tau ) \hat{n}_{a(b)} (t) : \rangle = {\textstyle\frac{1}{4}} \sum_{ (\hat{c}, \skew3\hat{d} ) \in \Pi (\hat{a}', \hat{b}' ) } [ \langle \hat{c}^\dagger (t) \hat{c}^\dagger (t+\tau ) \hat{c} (t+\tau ) \hat{c} (t) \rangle + \langle \hat{c}^\dagger (t) \hat{c} (t) \rangle \langle \skew3\hat{d}^\dagger (t+\tau ) \skew3\hat{d} (t+\tau ) \rangle \nonumber \\ &&\pm \langle \hat{c}^\dagger (t+\tau) \hat{c} (t) \rangle \langle \skew3\hat{d}^\dagger (t ) \skew3\hat{d} (t+\tau ) \rangle \pm \langle \hat{c}^\dagger (t+\tau) \hat{c}^\dagger (t) \rangle \langle \skew3\hat{d} (t+\tau ) \skew3\hat{d} (t ) \rangle \nonumber \\ &&+ \langle \skew3\hat{d}(t) \rangle \langle \hat{c}^\dagger (t) \hat{c}^\dagger (t+\tau ) \hat{c}(t+\tau ) \rangle + \langle \skew3\hat{d}^\dagger (t) \rangle \langle \hat{c}^\dagger (t+\tau ) \hat{c}(t+\tau ) \hat{c} (t) \rangle \nonumber \\ &&\pm \langle \skew3\hat{d}(t+\tau) \rangle \langle \hat{c}^\dagger (t+\tau ) \hat{c}^\dagger (t ) \hat{c} (t) \rangle \pm \langle \skew3\hat{d}^\dagger (t+\tau) \rangle \langle \hat{c}^\dagger (t ) \hat{c} (t+\tau ) \hat{c} (t) \rangle ], \end{eqnarray} \tag{ 3 }$$

where the + and − signs correspond to the auto-correlation and cross-correlation functions, respectively. Note that the summation is over the set of permutations of the input mode operators $\hat {a}'$ and $\hat {b}'$; that is, $\Pi ( \hat {a}', \hat {b}' ) \equiv \{ (\hat {a}', \hat {b}'), ( \hat {b}', \hat {a}') \}$. The output intensity correlation functions depend on the input intensity correlation functions (first term in equation (3)), the products of the intensities of the two inputs (second term), the products of the first-order correlation functions of the two inputs (third term) and on a number of phase-dependent correlation functions of the inputs (terms four to eight).

For the purpose of demonstrating Hong–Ou–Mandel interference, the contributions of the fourth through to the eighth terms to the intensity correlation functions of equation (3) are neither necessary nor desirable. We will consider their contribution using the example of the fourth term, though the same arguments may be applied to the fifth through to the eighth terms.

Suppose that the fields corresponding to the input modes, $\hat {c}$ and $\skew3\hat{d}$ in equation (3), can be written as slowly varying fields modulating oscillations at carrier frequencies centred on ω₀ and with a difference Δ, and with some relative phase offset θ: $\hat {c}(t) \equiv \bar {c}(t) \mathrm {e}^{+\mathrm {i} (\omega _{0}+\Delta /2 ) t + \mathrm {i}\theta }$ and $\skew3\hat{d}(t) \equiv \skew3\bar{d}(t) \mathrm {e}^{+\mathrm {i} (\omega _{0}-\Delta /2 ) t}$. Accordingly, we can rewrite the fourth term in equation (3) as

$$\begin{equation} \fl \langle \hat{c}^\dagger (t+\tau ) \hat{c}^\dagger (t) \rangle \langle \skew3\hat{d}(t+\tau ) \skew3\hat{d} (t) \rangle = \langle \bar{c}^\dagger (t+\tau ) \bar{c}^\dagger (t) \rangle \langle \skew3\bar{d}(t+\tau ) \skew3\bar{d} (t) \rangle \mathrm{e}^{- \mathrm{i} \Delta (2t+\tau)}\mathrm{e}^{-2\mathrm{i}\theta}. \end{equation} \tag{ 4 }$$

There are a number of situations under which the contributions from such terms are negligible. If the inputs are single-photon pulses then the phase-dependent second moments in equation (3) will always evaluate to zero. For example, see equation (13), we can write

$$\begin{equation} \langle 1 | \hat{c}^\dagger (t+\tau ) \hat{c}^\dagger (t) | 1 \rangle = \zeta^*(t+\tau ) \zeta^* (t) \langle 1 | \hat{c}^{\dagger 2}_0 | 1 \rangle = 0. \end{equation} \tag{ 5 }$$

Even if the inputs are not single photons, the contribution of these terms may still be made to vanish by uniformly varying the relative phase θ of the two input fields over repeated runs of the experiment, and then averaging over these repeated runs. That is, for example,

$$\begin{equation} \overline{ f(t,t + \tau ) \mathrm{e}^{-2\mathrm{i}\theta} } = f(t,t+\tau ) \sum^{M}_{j=1} \mathrm{e}^{-2\mathrm{i}\theta_j} = 0 , \end{equation} \tag{ 6 }$$

where θ_j = 2πj/M with M being the number of measurements averaged over. Therefore, in all cases of interest to us, the phase-dependent moments will vanish and the expression in equation (3) reduces to

$$\begin{eqnarray} &&\fl \overline{ \langle : \hat{n}_a (t+\tau ) \hat{n}_{a(b)} (t) : \rangle } = \frac{1}{4} \sum_{ ( \hat{c}, \skew3\hat{d} ) \in \Pi ( \hat{a}', \hat{b}' ) } [\langle \hat{c}^\dagger (t) \hat{c}^\dagger (t+\tau ) \hat{c} (t+\tau) \hat{c} (t) \rangle \nonumber \\ &&+ \langle \hat{c}^\dagger (t) \hat{c} (t) \rangle \langle \skew3\hat{d}^\dagger (t+\tau ) \skew3\hat{d} (t+\tau ) \rangle \pm \langle \hat{c}^\dagger (t+\tau) \hat{c} (t) \rangle \langle \skew3\hat{d}^\dagger (t ) \skew3\hat{d} (t+\tau ) \rangle ]. \end{eqnarray} \tag{ 7 }$$

Here the bar over the correlation function indicates that it is to be evaluated over many runs of the experiment. Henceforth, this shall be assumed for the correlation functions quoted and we drop this notation. The output intensity correlation functions are completely determined by the input intensity correlation functions and products of the input first-order correlation functions. For the output intensity cross-correlation function, the two-photon interference is seen in the cancellation of the second and third terms at τ = 0 in equation (7). On the other hand, the second and third terms will add constructively for the output auto-correlation function at τ = 0, corresponding to photon coalescence. Whether or not this constitutes Hong–Ou–Mandel interference in the usual sense is determined by the input intensity correlation functions in the first term of equation (7). Although the preceding statements are true in general, the signature of the interference for non-zero τ will depend on the form of the input fields.

The output intensity correlation functions, as expressed in equations (3) and (7), are explicitly dependent on two detection times, t and t + τ. The second-order correlation functions, giving the likelihood of detecting a photon at some time τ assuming detection at time zero, may be obtained either by integrating or by taking a limit

$$\begin{eqnarray} &&G^{(2)}_{cd} (\tau ) = \int \mathrm{d}t \langle : \hat{n}_c (t+\tau ) \hat{n}_d (t ) : \rangle \ \ \ {\rm or } \ \ \ G^{(2)}_{cd} (\tau ) = \lim_{t \rightarrow \infty} \langle : \hat{n}_c (t+\tau ) \hat{n}_d (t ) : \rangle, \end{eqnarray} \tag{ 8 }$$

respectively. Whether one integrates or takes a limit depends on whether the system's evolution is described in a (non-rotating) laboratory or a rotating frame. The quantities in equation (8) are conventionally referred to as second-order correlation functions [24]. Similarly, we introduce the first-order correlation functions

$$\begin{eqnarray} &&G^{(1)}_c (\tau ) = \int \mathrm{d}t \langle \hat{c}^\dagger (t) \hat{c} (t+\tau ) \rangle \ \ \ {\rm or} \ \ \ G^{(1)}_c (\tau ) = \lim_{t \rightarrow \infty} \langle \hat{c}^\dagger (t) \hat{c} (t+\tau ) \rangle. \end{eqnarray} \tag{ 9 }$$

In the case where the second-order correlation functions are obtained by taking limits, the output second-order correlation functions follow immediately from equation (7) as

$$\begin{equation} G^{(2)}_{aa(ab)}( \tau ) = \frac{1}{4} \sum_{(c,d) \in \Pi (a',b')} \{ G^{(2)}_{c} (\tau ) + G^{(1)}_{c}(0) G^{(1)}_{d} (0) \pm [ G^{(1)}_{c} (\tau ) ]^* G^{(1)}_{d} (\tau ) \}, \end{equation} \tag{ 10 }$$

where we have introduced the short-hand notation G⁽²⁾_c(τ) ≡ G⁽²⁾_cc(τ). If both sources exhibit identical statistics, meaning G⁽¹⁾_a'(τ) = G⁽¹⁾_b'(τ) ≡ G⁽¹⁾_i(τ) and G⁽²⁾_a'(τ) = G⁽²⁾_b'(τ) ≡ G⁽²⁾_i(τ), the output second-order correlation functions are simply

$$\begin{equation} G^{(2)}_{aa(ab)} (\tau ) = \frac{1}{2} \{ G^{(2)}_i (\tau ) \pm | G^{(1)}_i (\tau ) |^2 + [ G^{(1)}_i(0) ]^2 \}. \end{equation} \tag{ 11 }$$

As for equation (7), the second and third terms of the second-order cross-correlation functions cancel at τ = 0, corresponding to photon interference at the beam splitter. We now consider the evaluation of these correlation functions for particular forms of the input fields.

The case relevant to the experiment described in [1] is that in which each source produces a highly pure, microwave-frequency single-photon state. We shall evaluate the correlation functions of equation (7) by introducing temporal mode functions to describe these single photons. First, the time-dependent quantum fields introduced above, $\hat {c}(t)$ in general, may be represented in the frequency domain as $\hat {c}(\omega ) = \frac {1}{\sqrt {2\pi }} \int \mathrm {d}t \ \mathrm {e}^{+\mathrm {i}\omega t} \hat {c} (t)$. Then a single-photon pulse is conventionally represented using the spectral density Φ(ω), that is,

$$\begin{equation} \left| 1 \right\rangle = \int \mathrm{d}\omega \ \Phi (\omega ) \hat{c}^\dagger (\omega ) \left| 0 \right\rangle. \end{equation} \tag{ 12 }$$

One can show, by considering the action of $\hat {c}(t)$ on the state $\left | 1 \right \rangle$, that the time-dependent quantum field $\hat {c}(t)$ may be represented as

$$\begin{equation} \hat{c}(t) = \zeta (t) \hat{c}_0, \quad {\rm where} \quad \zeta (t) = \frac{1}{\sqrt{2\pi}} \int \mathrm{d}\omega \ \Phi (\omega ) \mathrm{e}^{-\mathrm{i}\omega t} \end{equation} \tag{ 13 }$$

is the temporal mode function of the single-photon pulse, and $\hat {c}_0$ is the annihilation operator corresponding to this temporal mode. It is then clear that $\langle \hat {c}^\dagger (t) \hat {c} (t) \rangle = | \zeta (t) |^2$, giving the probability of the photon being detected at time t.

In the experiment of [1], there are two independent microwave-frequency sources producing highly pure single-photon states. The sources are characterized via measurements of their second-order correlation functions. We can write the beam splitter input fields (from each source) in terms of temporal mode functions as

$$\begin{equation} \hat{a}' (t) = \zeta_a (t) \hat{a}'_0 , \quad \hat{b}' (t) = \zeta_b (t) \hat{b}'_0 . \end{equation} \tag{ 14 }$$

Accordingly, the two-mode state input to the beam splitter may be written as $\hat {a}'^\dagger (t) \hat {b}'^\dagger (t) \left | 0 \right \rangle$. From equation (7), the output intensity correlation functions are

$$\begin{equation} \langle : \hat{n}_a (t+\tau ) \hat{n}_{a(b)} (t) : \rangle = \frac{1}{4} | \zeta_a (t+\tau ) \zeta_b (t) \pm \zeta_b (t+\tau ) \zeta_a (t) |^2. \end{equation} \tag{ 15 }$$

Here, as in equation (7), the potential for cancellation of the terms on the right-hand-side in the case of the intensity cross-correlation function is clear. Indeed, at τ = 0, we see that the intensity cross-correlation function vanishes irrespective of the precise form of the temporal mode functions. Since there are now no additional contributions to the output intensity cross-correlation functions, the intensity correlation function vanishes at τ = 0. This suppression is the signature of Hong–Ou–Mandel interference. On the other hand, the terms in equation (15) add constructively for the intensity auto-correlation function, indicative of photon coalescence. Of course, the form of the intensity correlation functions for non-zero τ depends on the form of the mode functions, and we evaluate this now for Gaussian and Lorentzian photons.

One form of mode function commonly produced by single-photon sources are Gaussian modes. For example, this is the case for trapped atom sources in which linewidth broadening mechanisms are dominant [8]. In the subsequent section we consider the case of Lorentzian mode functions, which arise naturally for circuit QED systems. Given the recently demonstrated ability to shape the waveforms of single photons [25–27], we could consider more general mode functions as well.

Two Gaussian modes, with a carrier frequency difference Δ (centred on ω₀), a relative temporal offset δτ (centred on t = 0), and a common pulse width specified by σ, are described by the temporal mode functions:

$$\begin{equation} \zeta_{a(b)} (t) = \displaystyle\sqrt[4]{\frac{1}{\pi \sigma^2}} \exp \left[ - \frac{\left( t \pm \delta \tau /2 \right)^2}{2\sigma^2} - \mathrm{i} \left( \omega_0 \pm \Delta /2 \right) t \right]. \end{equation} \tag{ 16 }$$

Substituting equation (16) into equation (15) gives the explicit form of the output intensity correlation functions. Subsequently, integrating over the first detection time t leads to the second-order correlation functions

$$\begin{equation} G^{(2)}_{aa(ab)}(\tau ) = \frac{ \cosh \left( \tau \ \delta \tau /\sigma^2 \right) \pm \cos \left( \tau \Delta \right) }{ 2 \sqrt{2\pi \sigma^2} } \exp \left[ - \left( \delta \tau^2 + \tau^2 \right) / 2\sigma^2 \right] . \end{equation} \tag{ 17 }$$

The total correlation probabilities, P_aa and P_ab, are then obtained by integrating over the detection time τ. The correlation probabilities P_cd simply give the probability of detecting one photon in mode c and one photon in mode d, for one pair of input photons, irrespective of the detection times. They are

$$\begin{equation} P_{aa(ab)} = \frac{1}{4} \left[ 1 \pm \exp \left( - \frac{\Delta^2 \sigma^2}{2} - \frac{\delta \tau^2}{2\sigma^2} \right) \right] . \end{equation} \tag{ 18 }$$

Note that P_ba = P_ab and P_bb = P_aa, such that $\sum _{c,d \in \{a,b\}} P_{cd} = 1$, as required for a beam splitter that is assumed to be lossless. The total probability for photon coalescence at one output port is P_aa + P_bb = 2P_aa, while the total probability for one photon in either output port, the coincidence probability, is P_c ≡ P_ab + P_ba = 2P_ab. The correlation probabilities may be regarded as probability distributions over the controlled parameters δτ and Δ. For indistinguishable photons (δτ = 0 and Δ = 0) the coincidence probability is P_c = 0, and for fully distinguishable photons (i.e. $\left | \delta \tau \right | \gg \sigma$ and/or $\left | \Delta \right | \gg 1/\sigma$) we have P_c → 1/2. The coincidence probability, plotted as a function of either δτ or Δ, exhibits the classic Hong–Ou–Mandel dip [2].

The same calculation may be performed for Lorentzian photons. In the experiment of [1] the temporal mode function of the released photon is determined by the decay rate of the coupled circuit QED system. Accordingly, these photons have the shape of a truncated exponential decay. Two such mode functions, with a relative temporal offset δt and a carrier frequency difference Δ, are given by

$$\begin{equation} \zeta_{a(b)}(t) = \sqrt{2\gamma}\,\mathrm{e}^{-\gamma \left( t \pm \delta t /2 \right)} u(t \pm \delta t/2) \mathrm{e}^{-\mathrm{i} (\omega_0 \pm \Delta/2)t} , \end{equation} \tag{ 19 }$$

where γ is the field decay rate of the coupled system, ω₀ is the frequency about which the two carrier frequencies are centred and u(...) denotes the unit step function. Note that the field decay rate is the characteristic decay rate of the electric field, in contrast to the commonly quoted (cavity) decay rate of the energy stored in the cavity. According to equation (19), the first photon (in mode a') is 'released' at t = −δt/2, while the second photon (in mode b') is released at t = + δt/2. In the frequency domain, the mode functions correspond to the spectral densities

$$\begin{equation} \Phi_{a(b)} ( \omega ) = \sqrt{ \frac{\gamma}{\pi} } \frac{1}{ \gamma - \mathrm{i} [ \omega - ( \omega_0 \pm \Delta/2 ) ] } \exp \{ - \mathrm{i} [ \omega - ( \omega_0 \pm \Delta/2 ) ] \delta t/2 \}. \end{equation} \tag{ 20 }$$

This representation makes it clear why the exponentially decaying single-photon pulses described by equation (19) are referred to as Lorentzian photons. Of course, the mode function thus defined is unphysically sharp. The filtering effects of a finite measurement bandwidth shall be addressed below.

Substituting equation (19) into equation (15), assuming that δt > 0 and τ > 0, we find the output intensity correlation functions to be

$$\begin{eqnarray} &&\langle : \hat{n}_{a}(t+\tau ) \hat{n}_{a(b)}(t) : \rangle = \gamma^2 \mathrm{e}^{-2\gamma (2t+\tau )} u(t+\tau-\delta t/2) u(t+\delta t/2) \nonumber \\ &&- \gamma^2 \mathrm{e}^{-2\gamma (2t+\tau )} u(t-\delta t/2 ) \left\{ \begin{array}{@{}l@{}} 1 - 4 \cos^2 (\tau \Delta /2), \\ 1 - 4 \sin^2 (\tau \Delta /2), \end{array} \right. \end{eqnarray} \tag{ 21 }$$

where the upper (lower) result corresponds to the auto-correlation (cross-correlation) function. Equation (21) consists of two contributions, corresponding to different combinations of the unit step functions switching on. The first product of step functions in equation (21) switches on when both the first detection (t) is after the release of the first photon (− δt/2) and the second detection time (t + τ) is after the release of the second photon (+ δt/2). The second unit step function is switched on when the first, and therefore the second detection time (τ > 0) is after the release of the second photon. Integrating over the first detection time t leads to the output second-order correlation functions

$$\begin{eqnarray} &&\fl G^{(2)}_{aa(ab)}(\tau ) = \frac{\gamma}{2} \mathrm{e}^{-2\gamma \delta t} \left[ \sinh 2\gamma \tau \ u (\delta t - \tau ) \vphantom{\begin{array}{@{}c@{}} \sin^2 \\ \sin^2 \end{array}} + 2\mathrm{e}^{-2\gamma \tau} u (\delta t - \tau ) \left\{ \begin{array}{@{}l@{}} \cos^2 \left( \tau \Delta /2 \right) \\ \sin^2 \left(\tau \Delta /2 \right) \end{array} \right. \right] \nonumber \\ &&+ \frac{\gamma}{2} \mathrm{e}^{-2\gamma \tau} \left[ \sinh 2\gamma \delta t \ u(\tau - \delta t) \vphantom{\begin{array}{@{}c@{}} \sin^2 \\ \sin^2 \end{array}} + 2\mathrm{e}^{-2\gamma \delta t} u(\tau - \delta t) \left\{ \begin{array}{@{}l@{}} \cos^2 \left(\tau \Delta /2 \right) \\ \sin^2 \left(\tau \Delta /2 \right) \end{array} \right. \right]. \end{eqnarray} \tag{ 22 }$$

Again, equation (22) is split into two terms, depending on whether the difference in measurement times (τ) is greater or less than the temporal delay (δt) between the release of photons from either source. Integrating over the difference in detection times τ (positive and negative) we obtain the correlation probabilities

$$\begin{equation} P_{aa(ab)} = \frac{1}{4} \left( 1 \pm \frac{4 \gamma^2}{ 4\gamma^2 + \Delta^2 } \mathrm{e}^{-2\gamma \delta t} \right) . \end{equation} \tag{ 23 }$$

As before, we have P_bb = P_aa, P_ba = P_ab, $\sum P_{cd}=1$, and the coincidence probability is P_c = P_aa + P_bb = 2P_aa. Again, these probabilities may be regarded as probability distributions over δt and Δ. If the photons are indistinguishable (δt = 0 and Δ = 0), P_c = 0, while for fully distinguishable photons (δt ≫ 1/γ and/or $\left | \Delta \right | \gg \gamma$), P_c → 1/2.

The experiment in [1] is performed with trains of pulsed single photons. In general, this can lead to features in the correlation functions at (and around) integer multiples of the pulse period. These are evaluated here. In order to determine the output intensity correlation functions in this case, we write the time-dependent quantum fields at the inputs as the sum of the fields corresponding to each temporal mode. For 2N + 1 photons in the pulse train we can write

$$\begin{equation} \hat{a}' (t) = \sum^{N}_{k=-N} \zeta_{ak}(t) \hat{a}'_{0k}, \ \ \ \hat{b}' (t) = \sum^{N}_{k=-N} \zeta_{bk}(t) \hat{b}'_{0k} \end{equation} \tag{ 24 }$$

with the temporal mode operator commutation relations being $[ \hat {c}'_{0k}, \hat {c}'^\dagger _{0k'} ] = \delta _{kk'}$, where $\hat {c}'$ denotes an input mode lowering operator, $\hat {a}'$ or $\hat {b}'$. The appropriate temporal mode functions for the kth photon of each input (assumed Lorentzian here) are

$$\begin{equation}\fl \zeta_{ak(bk)}(t) = \sqrt{2\gamma } \mathrm{e}^{-\mathrm{i} \left( \omega_0 \pm \Delta/2 \right) t} \mathrm{e}^{ -\gamma \left( t - kt_{\mathrm{p}} \pm \delta t/2 \right) } u\!\left( t - kt_{\mathrm{p}} \pm \delta t/2 \right) \end{equation} \tag{ 25 }$$

with t_p denoting the pulse period. The Kronecker delta in the commutation relations is justified by the fact that the temporal mode functions are normalized and are approximately orthogonal in the limit that the pulse period is much greater than the pulse width. Substituting the field decompositions of equation (24) into the intensity correlation functions of equation (7) leads to

$$\begin{eqnarray} &&\fl \langle : \hat{n}_a (t+\tau ) \hat{n}_{a(b)} (t) : \rangle = \frac{1}{4} \sum_{(c,d) \in \Pi (a,b)} \left\{\! \left\langle \left( \sum_k \zeta^*_{ck}(t) \hat{c}^\dagger_{0k} \right) \left( \sum_l \zeta^*_{cl}(t+\tau) \hat{c}^\dagger_{0l} \right) \left( \sum_m \zeta_{cm}(t+\tau) \hat{c}_{0m} \!\right) \right. \right. \nonumber \\ &&\left. \left. \times \left( \sum_n \zeta_{cn}(t) \hat{c}_{0n} \right) \right\rangle + \left\langle \left( \sum_k \zeta^*_{ck}(t) \hat{c}^\dagger_{0k} \right) \left( \sum_l \zeta_{cl}(t) \hat{c}_{0l} \right) \right\rangle \left\langle \left( \sum_m \zeta^*_{dm}(t+\tau) \skew3\hat{d}^\dagger_{0m}\! \right) \right. \right. \nonumber\\ &&\left. \left. \times \left( \sum_n \zeta_{dn}(t+\tau) \skew3\hat{d}_{0n} \right) \right\rangle \pm \left\langle \left( \sum_k \zeta^*_{ck}(t+\tau) \hat{c}^\dagger_{0k} \right) \left( \sum_l \zeta_{cl}(t) \hat{c}_{0l} \right) \right\rangle \right.\nonumber\\ &&\times \left.\left\langle \left( \sum_m \zeta^*_{dm}(t) \skew3\hat{d}^\dagger_{0m} \right) \left( \sum_n \zeta_{dn}(t+\tau) \skew3\hat{d}_{0n} \right) \right\rangle \right\}. \end{eqnarray} \tag{ 26 }$$

Using the assumption that the input state is separable with respect to the basis of temporal modes introduced, equation (26) reduces to

$$\begin{eqnarray} &&\fl \left\langle : \hat{n}_a(t+\tau ) \hat{n}_{a(b)}(t) : \right\rangle = \frac{1}{4} \sum_{(c,d) \in \Pi (a,b)} \left\{ \sum_m [ \zeta^*_{cm}(t) \zeta^*_{cm}(t+\tau ) \zeta_{cm}(t+\tau ) \zeta_{cm}(t)\langle \hat{c}^\dagger_{0m} \hat{c}^\dagger_{0m} \hat{c}_{0m} \hat{c}_{0m} \rangle ] \right. \nonumber \\ &&+ \sum_{m,n (m \neq n)} [ \zeta^*_{cm}(t) \zeta_{cm}(t+\tau ) \zeta^*_{cn}(t+\tau ) \zeta_{cn}(t) \langle \hat{c}^\dagger_{0m} \hat{c}_{0m} \rangle \langle \hat{c}^\dagger_{0n} \hat{c}_{0n} \rangle ] \nonumber \\ &&\left. + \sum_{m,n (m \neq n)} [ \zeta^*_{cn}(t) \zeta_{cn}(t) \zeta^*_{cm}(t+\tau ) \zeta_{cm}(t+\tau ) \langle \hat{c}^\dagger_{0m} \hat{c}_{0m} \rangle \langle \hat{c}^\dagger_{0n} \hat{c}_{0n} \rangle ] \right. \nonumber \\ &&+ \left( \sum_k [ \zeta^*_{ck}(t) \zeta_{ck}(t)\langle \hat{c}^\dagger_{0k} \hat{c}_{0k} \rangle ] \right) \left( \sum_l [ \zeta^*_{dl}(t+\tau ) \zeta_{dl}(t+\tau ) \langle \skew3\hat{d}^\dagger_{0l} \skew3\hat{d}_{0l} \rangle ] \right) \nonumber \\ &&\pm \left. \left( \sum_k [ \zeta^*_{ck}(t+\tau) \zeta_{ck}(t) \langle \hat{c}^\dagger_{0k} \hat{c}_{0k} \rangle ] \right) \left( \sum_l [\zeta^*_{dl}(t ) \zeta_{dl}(t+\tau ) \langle \skew3\hat{d}^\dagger_{0l} \skew3\hat{d}_{0l} \rangle ] \right) \right\} . \end{eqnarray} \tag{ 27 }$$

The output second-order correlation functions, G⁽²⁾_aa(ab)(τ), are then obtained by integrating over the first detection time t, as in equation (8).

These correlation functions are plotted in figure 2. Note that we actually calculate circular correlation functions, meaning that we evaluate these sums for a finite pulse train, and then wrap the time index around the pulse train. This means that the peak heights at later pulse periods will be at the same height as those at earlier pulse periods, rather than showing an artificial decay in peak height that would otherwise be produced due to the finite nature of the pulse train. Furthermore, the correlation functions are normalized such that the peaks at ±kt_p in G⁽²⁾_ab(τ) are unity for indistinguishable photons input.

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As expected, the signature of interference in figure 2 is the suppression of the output second-order cross-correlation function for small delays τ for indistinguishable photons (δt = 0). As the distinguishability of the photons is increased through increasing δt, one clearly sees corresponding peaks in the first repeat period. The peaks in the correlation functions at ±kt_p (k an integer) exhibit side peaks at ±δt. This is easily understood: a given photon of each pair may be correlated with the first or second photon of the subsequent pair, as well as with the first or second photon of the preceding pair.

Integrating the intensity correlation function of equation (27) results in second-order correlation functions with the very sharp features seen in figure 2, due to the abrupt switching of the single-photon pulses described by the unit step and exponential function in equation (19). However, such features are not observed in experimental data due to filtering in the detection scheme. To facilitate comparison with experimental data, we incorporate this filtering into our calculations. The filtering is performed on the time-dependent signals obtained by the data acquisition scheme in [1], denoted by S_a(b)(t). The second-order correlation functions of the fields are obtained from the filtered version of these signals, S^f_a(b)(t), given by the convolution integral

$$\begin{equation} S^{\mathrm{f}}_{a(b)} (t ) = \int^{+\infty}_{-\infty} S_{a(b)}(t') f (t - t' ) \mathrm{d}t' , \end{equation} \tag{ 28 }$$

where f is a filter function in the time domain. Now the correlation functions of the output fields of interest may be obtained from the correlation functions of these measured signals, and so for the sake of calculations, we may implement the filtering directly on the temporal mode functions, that is,

$$\begin{equation} \zeta^{\mathrm{f}}_{a(b)k} (t ) = \int^{+\infty}_{-\infty} \zeta_{a(b)}(t') f (t - t' )\, \mathrm{d}t'. \end{equation} \tag{ 29 }$$

The filtering in [1] was implemented as a finite-impulse-response filter [28]. For the purpose of calculations, equation (29) was implemented with a filter function representative of that done experimentally. The intensity correlation functions are then given by equation (27), evaluated using the filtered temporal mode functions of the form of equation (29). The second-order correlation functions are obtained by integrating over the first detection time t, as in equation (8). These are shown in figure 3, along with data from [1]. Very good agreement with the experimental data is obtained.

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Rather than assuming the circuit QED systems are configured as pulsed single-photon sources, we can assume that they are continuously driven and operated in the regime of resonant photon blockade [23]. The output microwave field from each source will exhibit sub-Poissonian and anti-bunched photon statistics [29, 30], leading to the possibility of observing Hong–Ou–Mandel interference at the beam splitter [12]. Treating only the lowest transition of the coupled circuit QED system [13], we consider the source simply as a driven quantum two-level system (TLS). The physics of each source is then essentially the physics of resonance fluorescence [31, 32], with additional dephasing included. Note that this TLS is in fact formed from the coupled system, and is therefore not the 'qubit' typically employed for superconducting quantum information processing.

We denote the TLS transition frequency by ω_c (where c denotes the corresponding input mode, either a' or b'), and assume that it is subject to driving by a resonant monochromatic, large-amplitude microwave field. Each system is described, in a frame rotating with respect to H₀ = (ℏω_c/2)σ_z, by the Hamiltonian $H_{\mathrm {I}} = \hbar \left ( \Omega \sigma _+ + \Omega ^* \sigma _- \right )/2$ where Ω is the driving strength. The coupling of the TLS to its environment may be fully described, in the Markovian white noise approximation, by damping at a rate γ₁ and dephasing at a rate γ_p. Accordingly, the evolution of the TLS density operator ρ may be given in the form of a Lindblad master equation [24] as

$$\begin{equation} \dot{\rho} = - \frac{\mathrm{i}}{2} \left[ \Omega \sigma_+ + \Omega^* \sigma_-, \rho \right] + \gamma_1 \mathcal{D}\left[ \sigma_- \right] \rho - \frac{\gamma_p}{4} \left[ \sigma_z, \left[ \sigma_z , \rho \right] \right], \end{equation} \tag{ 30 }$$

where $\mathcal {D}[ \skew3\hat{A} ] \equiv \skew3\hat{A}\rho \skew3\hat{A}^\dagger - \frac {1}{2} \skew3\hat{A}^\dagger \skew3\hat{A} \rho - \frac {1}{2} \rho \skew3\hat{A}^\dagger \skew3\hat{A}$ is the so-called dissipative superoperator. Note that in this model each source is fully characterized by the parameters γ₁,γ_p,Ω and ω_c.

As in the case of pulsed sources, the quantities of interest are the second-order correlation functions of the microwave fields at the outputs of the beam splitter. The master equation (30) facilitates the evaluation of correlation functions of TLS operators, and from the theory of atomic spontaneous emission [32], we know that the output field will be proportional to the TLS lowering operator

$$\begin{equation} \hat{c}(t) \sim {\sigma}^c_- (t), \end{equation} \tag{ 31 }$$

where c denotes the input mode a' or b'. The output intensity correlation functions then follow from equation (3), to within a constant factor, by making the substitution in equation (31). Rather than explicitly evaluating the coefficient in equation (31), we shall ultimately calculate normalized (field) correlation functions for which the coefficients will cancel.

First we calculate the output correlation functions in terms of the input correlation functions. Suppose that the TLS transition frequencies are ω₀ ± Δ/2 (for the sources corresponding to the input modes a' and b', respectively), and that we are working in an interaction picture with respect to $H_0 = \sum _{c \in \{a',b' \} }(\hbar \omega _0/2) \sigma ^c_z$. Then, from equations (7) and (31), the output intensity correlation functions (to within a constant factor) are

$$\begin{eqnarray} &&\fl \langle : \hat{n}_a (t+\tau ) \hat{n}_{a(b)} (t) : \rangle \ \sim \frac{1}{4} \sum_{(c,d) \in \Pi (a',b')} [ \langle \sigma^{c}_+ (t) \sigma^{c}_+ (t+\tau ) \sigma^{c}_- (t+\tau ) \sigma^{c}_- (t) \rangle \nonumber \\ &&+ \langle \sigma^{c}_+(t) \sigma^{c}_-(t) \rangle \langle \sigma^{d}_+(t+\tau ) \sigma^{d}_-(t+\tau ) \rangle ] + \frac{1}{4}[ \pm \mathrm{e}^{\mathrm{i}\tau\Delta } \langle \sigma^{b'}_+ (t+\tau ) \sigma^{b'}_- (t) \rangle \nonumber \\ &&\times \langle \sigma^{a'}_+(t) \sigma^{a'}_-(t+\tau ) \rangle \pm \mathrm{e}^{-\mathrm{i}\tau\Delta } \langle \sigma^{a'}_+ (t+\tau ) \sigma^{a'}_-(t) \rangle \langle \sigma^{b'}_+(t) \sigma^{b'}_-(t+\tau) \rangle]. \end{eqnarray} \tag{ 32 }$$

Note that the validity of this expression depends on averaging out explicitly phase-dependent contributions by randomizing the relative drive phases over repeated runs of the experiment, as discussed in section 2.2. Taking the limit t → ∞ of equation (32) we have the output second-order correlation functions

$$\begin{eqnarray} &&G_{aa(ab)}(\tau ) \sim \frac{1}{4} \{ G^{(2)}_{a'} (\tau ) + G^{(2)}_{b'} (\tau ) + 2 G^{(1)}_{a'}(0) G^{(1)}_{b'}(0) \nonumber \\ && \pm \mathrm{e}^{\mathrm{i}\tau\Delta } G^{(1)}_{a'} (\tau ) [ G^{(1)}_{b'} (\tau ) ]^* \pm \mathrm{e}^{-\mathrm{i}\tau\Delta } [ G^{(1)}_{a'} (\tau )]^{*} G^{(1)}_{b'} (\tau ) \}, \end{eqnarray} \tag{ 33 }$$

where the correlation functions on the right-hand-side are now TLS correlation functions:

$$\begin{equation} G^{(1)}_c (\tau ) \equiv \lim_{t \rightarrow \infty } \langle \sigma^c_+ (t) \sigma^c_- (t+\tau ) \rangle , \end{equation} \tag{ 34a }$$

$$\begin{equation} G^{(2)}_c (\tau ) \equiv \lim_{t \rightarrow \infty } \langle \sigma^c_+ (t) \sigma^c_+ (t+\tau ) \sigma^c_- (t+\tau ) \sigma^c_- (t) \rangle. \end{equation} \tag{ 34b }$$

$$\begin{equation} G^{(2)}_{aa(ab)}(\tau ) \sim \frac{1}{2} \{ G^{(2)}_i (\tau ) \pm \cos \tau \Delta | G^{(1)}_i (\tau ) |^2 + [ G^{(1)}_i (0) ]^2 \}. \end{equation} \tag{ 35 }$$

Again, the second and third terms in the output cross-correlation function of equation (35) cancel at τ = 0. For non-zero τ, the interference is oscillatory, an effect sometimes referred to as a quantum beat [8].

Now we turn to the evaluation of the input TLS correlation functions. The system of equations for the expectations of TLS operators are readily calculated from equation (30). They are commonly referred to as the optical Bloch equations, and may be expressed as the inhomogeneous system [24]

$$\begin{equation} \frac{\mathrm{d}}{\mathrm{d}t} \left\langle \vec{\sigma} \right\rangle = A \left\langle \vec{\sigma } \right\rangle + \vec{b} , \end{equation} \tag{ 36 }$$

where we have introduced the vector of expectations of TLS (Pauli) operators, $\left \langle \vec {\sigma } \right \rangle = \left ( \left \langle \sigma _+ \right \rangle , \left \langle \sigma _- \right \rangle , \left \langle \sigma _z \right \rangle \right )^{\mathrm {T}}$. The inhomogeneity is given by $\vec {b} = \left ( 0,0,-\gamma _1 \right )^{\mathrm {T}}$, and the system matrix is given by

$$\begin{equation} A = \left[ \begin{array}{@{}ccc@{}} - \gamma_2/2 + \mathrm{i} \Delta & 0 & -\mathrm{i}\Omega^*/2 \\ 0 & - \gamma_2/2 - \mathrm{i} \Delta & \mathrm{i}\Omega/2 \\ -\mathrm{i}\Omega & \mathrm{i}\Omega^* & - \gamma_1 \end{array} \right] , \end{equation} \tag{ 37 }$$

where we have also introduced the total dephasing rate, γ₂ ≡ γ₁ + 2γ_p. One can find the steady-state solution of equation (36), $\left \langle \vec {\sigma } \right \rangle _{\mathrm { ss}}$, and subsequently recast equation (36) as the homogeneous system, $\mathrm {d} \left \langle \vec {\sigma } \right \rangle '/\mathrm {d}t = A \left \langle \vec {\sigma } \right \rangle '$ where $\left \langle \vec {\sigma } \right \rangle ' \equiv \left \langle \vec {\sigma } \right \rangle - \left \langle \vec {\sigma } \right \rangle _{\mathrm { ss}}$. Now the time-dependent solution of the homogeneous system may be obtained by diagonalization, and subsequently the solution to equation (36) may be written in the form

$$\begin{equation} \left\langle \vec{\sigma} (t+\tau) \right\rangle = C(\tau ) \left\langle \vec{\sigma} (t) \right\rangle + \vec{d}(\tau ) , \end{equation} \tag{ 38 }$$

where we have introduced the notation

$$\begin{equation} C(\tau ) \equiv X\,\mathrm{e}^{ X^{-1} A X \tau } X^{-1}, \end{equation} \tag{ 39a }$$

$$\begin{equation} \vec{d} (\tau ) \equiv (1 - X\,\mathrm{e}^{ X^{-1} A X \tau} ) \langle \vec{\sigma} \rangle_{\rm ss} \end{equation} \tag{ 39b }$$

The form of the solution in equation (38) is amenable to application of the quantum regression theorem [32]. In particular, for an element of the solution in equation (38), we can write

$$\begin{equation} \langle \sigma_i (t+\tau ) \rangle = \sum_j C_{ij} (\tau ) \langle \sigma_j (t) \rangle + d_i (\tau ) , \end{equation} \tag{ 40 }$$

where i∈{ + ,−,z}, the summation index j runs over this set, and C_ij(τ) and d_i(τ) denote elements of the matrix C(τ) and vector $\vec {d} (\tau )$, respectively. The quantum regression theorem allows us to evaluate correlation functions as

$$\begin{equation}\fl \langle \sigma_k (t) \sigma_i (t+\tau ) \sigma_l (t) \rangle = \sum_j C_{ij} (\tau ) \langle \sigma_k(t) \sigma_j(t) \sigma_l(t) \rangle + d_i (\tau ) \langle \sigma_k(t) \sigma_l(t) \rangle. \end{equation} \tag{ 41 }$$

The TLS correlation functions in equation (32) are obtained using equation (41) and the algebraic properties of Pauli operators.

According to equation (35), the output second-order correlation functions are fully determined by the input first-order and second-order correlation functions. We can write out G⁽¹⁾_i(τ) and G⁽²⁾_i(τ), which are well-known results from the physics of resonance fluorescence [32]. For the first-order correlation functions we have

$$\begin{equation}\fl G^{(1)}_i (\tau ) = \frac{\left| \Omega \right|^2}{\gamma_1 \gamma_2 + 2\left| \Omega \right|^2} \left[ \frac{\gamma^2_1}{\gamma_1 \gamma_2 + 2\left| \Omega \right|^2} + \frac{1}{2} \mathrm{e}^{-\gamma_2 \tau/2}+ \frac{1}{4} \mathrm{e}^{-\left( 2\gamma_1 + \gamma_2 \right) \tau/4} \left( \lambda_+ \mathrm{e}^{ + \kappa \tau} + \lambda_- \mathrm{e}^{ - \kappa \tau} \right) \right], \end{equation} \tag{ 42 }$$

where we have introduced the notation

$$\begin{equation}\fl 4\kappa \equiv \sqrt{ \left( 2\gamma_1 - \gamma_2 \right)^2 - 16 \left| \Omega \right|^2 } ,\end{equation} \tag{ 43a }$$

$$\begin{equation}\fl \lambda_\pm = \{ 2 | \Omega |^2 - 2\gamma^2_1 + \gamma_1 \gamma_2 \pm [ 2 | \Omega |^2 ( 6\gamma_1 - \gamma_2 ) - \gamma_1 ( 2\gamma_1 - \gamma_2 )^2 ] /4\kappa \} \big/ ( \gamma_1 \gamma_2 + 2 | \Omega |^2 ). \end{equation} \tag{ 43b }$$

$\left | \Omega \right | \gg \gamma _1,\gamma _2$

$$\begin{equation} \fl G^{(2)}_i (\tau ) = \left( \frac{\left| \Omega \right|^2}{\gamma_1 \gamma_2 + 2\left| \Omega \right|^2} \right)^2 \left[ 1 - \mathrm{e}^{ -\left( 2\gamma_1 + \gamma_2 \right) \tau /4 } \left( \cosh \kappa \tau + \frac{2\gamma_1 + \gamma_2}{4\kappa} \sinh \kappa \tau \right) \right] . \end{equation} \tag{ 44 }$$

It is more common to quote the normalized result, which is

$$\begin{eqnarray} &&g^{(2)}_i (\tau ) \equiv \frac{ G^{(2)}_i (\tau ) }{ [ G^{(1)}_i (0) ]^2}= 1 - \mathrm{e}^{-\left( 2\gamma_1 + \gamma_2 \right) \tau /4}\left( \cosh \kappa \tau + \frac{2\gamma_1 + \gamma_2 }{4\kappa} \sinh \kappa \tau \right). \end{eqnarray} \tag{ 45 }$$

Of course, this is a normalized TLS correlation function. Here, we shall normalize the output field correlation functions by imposing the requirement that they asymptotically approach unity.

Now the output second-order correlation functions are given by equation (35), using the results in equations (42) and (44). They are shown in figure 4. As expected, the interference effect is observed in the suppression of the cross-correlation function for small delays τ, while introducing distinguishability through the frequency difference Δ leads to an oscillation in the observed interference for non-zero τ.

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In the continuously driven case just considered, photon distinguishability can only be introduced through a difference in the carrier frequencies. An alternative approach, relevant in the trapped atom case [12] or in a three-dimensional circuit QED architecture [33], is to introduce distinguishability through a polarization degree of freedom. To treat this problem, we decompose the time-dependent fields into their (horizontal and vertical) polarization components

$$\begin{equation} \hat{c} (t) = \hat{c}_{\mathrm{h}} (t) + \hat{c}_{\mathrm{v}} (t), \end{equation} \tag{ 46 }$$

where the subscripts h and v denote the horizontally and vertically polarized components of the field, respectively. Generalizing the expression of equation (2) to include polarization indices, and then summing over these indices, we can write the output intensity correlation functions as

$$\begin{equation} \langle : \hat{n}_a ( t + \tau ) \hat{n}_{c} (t) : \rangle = \sum_{ {k,l} \in \left\{ \mathrm{h,v} \right\} } \langle \hat{c}^\dagger_k (t) \hat{a}^\dagger_l (t+\tau ) \hat{a}_l(t+\tau ) \hat{c}_k (t) \rangle \end{equation} \tag{ 47 }$$

with c representing either output mode (a or b), as before.

Now we take the components of the input fields to be related to the TLS lowering operators by

$$\begin{equation} \hat{a}'_{\mathrm{h}} (t) \sim \sigma^{a'}_-(t) , \quad \hat{a}'_{\mathrm{v}} (t) = 0, \end{equation} \tag{ 48a }$$

$$\begin{equation} \hat{b}'_{\mathrm{h}} (t) \sim \sigma^{b'}_- (t) \cos \phi , \quad \hat{b}'_{\mathrm{v}} (t) \sim \sigma^{b'}_- (t) \sin \phi . \end{equation} \tag{ 48b }$$

$$\begin{eqnarray} &&\fl \langle : \hat{n}_a (t+\tau ) \hat{n}_{a(b)} (t) : \rangle \sim \frac{1}{4} \sum_{(c,d) \in \Pi ( a',b' ) } \left\{ \left\langle \sigma^c_+ (t) \sigma^c_+ (t+\tau ) \sigma^c_- (t+\tau) \sigma^c_- (t) \right\rangle \right. \nonumber \\ &&\left. + \left\langle \sigma^c_+ (t) \sigma^c_- (t) \right\rangle \left\langle \sigma^d_+ (t+\tau ) \sigma^d_- (t+\tau ) \right\rangle \pm \cos^2 \phi \left\langle \sigma^c_+ (t+\tau ) \sigma^c_- (t) \right\rangle \left\langle \sigma^d_+ (t) \sigma^d_- (t+\tau ) \right\rangle \right\}.\nonumber\\ \end{eqnarray} \tag{ 49 }$$

The photon distinguishability, and therefore the presence or absence of two-photon quantum interference, is determined by the relative polarization angle ϕ. For ϕ = 0, the photons at either input are indistinguishable, and the second and third terms in equation (49) cancel for the cross-correlation function, corresponding to interference. For the auto-correlation function, the second and third terms will add constructively, indicative of photon coalescence. For ϕ = π/2, the photons at either input are fully distinguishable, and the last term in equation (49) makes no contribution. Now taking the limit t → ∞, as per equation (8), yields the second-order correlation functions

$$\begin{eqnarray} &&\fl G^{(2)}_{aa(ab)} ( \tau ) \sim \frac{1}{4} \sum_{(c,d) \in \Pi (a',b')} \{ G^{(2)}_{c}(\tau ) + G^{(1)}_{c}(0) G^{(1)}_{d}(0) \pm \cos^2 \phi \ G^{(1)}_{c} (\tau ) [ G^{(1)}_{d}(\tau )]^* \} . \end{eqnarray} \tag{ 50 }$$

Under the additional assumption that the two sources (polarization aside) have identical properties, meaning G⁽²⁾_a'(τ) = G⁽²⁾_b'(τ) ≡ G⁽²⁾_i(τ) and G⁽¹⁾_a'(τ) = G⁽¹⁾_b'(τ) ≡ G⁽¹⁾_i(τ), then we find

$$\begin{eqnarray} &&G^{(2)}_{aa(ab)}(\tau )\sim \frac{1}{2} \{ G^{(2)}_i (\tau ) + [ G^{(1)}_i (0) ]^2 \pm \cos^2 \phi | G^{(1)}_i (\tau ) |^2 \}. \end{eqnarray} \tag{ 51 }$$

As for equation (49), the dependence of photon interference on the relative polarization angle ϕ is clear. These output correlation functions are shown in figure 5. Clearly, access to a polarization degree of freedom allows one to smoothly interpolate between the cases of distinguishable and indistinguishable photons in the continuously driven case.

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