Glauber two-photon correlation function

After we understood the frequency-mode behavior of cavity-enhanced SPDC by evalu-ating the first-order correlation function, now we start to think about how to observe the frequency-mode behavior(e.g., the bandwidth) of biphotons in the experimental method?

Actually, this is not a trivial problem since the power of biphotons is too weak to de-tect that in a simple way. In order to find some clues for getting the bandwidth of the biphotons, we observed Eqs.2.39and found that the bandwidth is only dependent on the decay rate of signal and idler fields from the cavity. That is straightforward to reminiscent that the relation between the decay rates and the time-domain correlation between signal and idler photons. Therefore, here we turn our attention to consider the temporal corre-lation between the signal and idler photons. The temporal correcorre-lation is described by the

Glauber two-photon correlation function,

G⁽²⁾_s,i(τ ) =D ˆE_s^(+)†(x, t) ˆE_i^(+)†(x, t + τ ) ˆE_i⁽⁺⁾(x, t + τ ) ˆE_s⁽⁺⁾(x, t)E

, (2.40)

which can be measured by a simple measurement way of the coincidence counts in the experiment. To calculate the two-photon correlation function, we can imitate the process of G⁽¹⁾s (τ ) calculation. First, we consider the ket part of Eqs.2.40. After sub Eqs.2.6into Eqs.2.40and using Eqs.2.30, then we have

X The commutation relation of field operators contributes the Dirac delta function into the integration, that makes the situation become concise,

X

With the similar process, we also can get the bra side of G⁽²⁾_s,i(τ ). By combining these two parts, we find

Notice that Eqs.2.44showed the two-photon correlation function just is the Fourier trans-form of the state amplitude from Ω-domain to τ -domain. The integration in Eqs.2.44can be solved by considering the residue theorem. Final, we have the two-photon correlation function:

Here we have found an obvious feature of the double exponential decay in τ of the two-photon correlation function in Eqs.2.45. This behavior is because of the fields is decay from an optical cavity. However, the decay rate of signal and idler fields are different

doi:10.6342/NTU201901825

-4 -2 0 2 4 6 8

0.0 0.2 0.4 0.6 0.8 1.0

Spectral power density (arb. unit)

Delay time(ns)

-40 -20 0 20 40

Figure 2.4: Two-photon correlation function of cavity-SPDC. In this figure, we show both single-mode(gray dash line) and multi-mode(yellow and blue line) operation. To empha-size the effect of mode-number-dependent in the multi-mode operation, we compare the cases of 11- and 2-modes that are presented by the yellow and blue line, respectively. The sub-figure show the full view of the two-photon correlation function. That is clear to see that whether the single-mode or multi-mode operation, the profile of G⁽²⁾_s,i(τ ) are an expo-nential decay. Here we set the decay rate of γ_s(i) = 0.2(0.16) GHz. The mode spacing of signal and idler is at 1 GHz.

and that causes the asymmetry profile in the delay time τ . On the other hand, the profile of two-photon correlation function is dependent on the measurement order of signal and idler photons. When τ > 0 (detected idler photon first), the decay rate is dominated by the signal photons and vice versa.

In Fig.2.4, we show the two-photon correlation function in the cases of the different mode number. In those cases, we can observe that the two-photon correlation function showed the interference phenomenon between the different frequency-modes. For exam-ple, G⁽²⁾_s,i(τ ) response the comb-like profile when we considered the numerous number of modes, which similar to the comb laser behavior. Furthermore, when the mode number is decreased to the single-mode operation, the interference pattern will disappear and leave the double exponential decay profile. This big difference can be used to distinguish if the source is in the single-mode operation or not.

In the real experiment, the detection of G⁽²⁾_s,i(τ ) relies on the coincidence count mea-surement. In the measurement process, one of the photons of pair will be assigned to be the trigger. By counting the time delay between the trigger and another photon, and

recording the probability of detection rate in time delay, we can reconstruct the two-photon correlation function. Since the profile of two-two-photon correlation function is only dependent on the decay rate of signal and idler fields, the bandwidth of biphoton(Eqs.2.39) can be estimated by the two-photon correlation function. In addition, we also can intu-itional define the correlation time of 1/γ_s+ 1/γ_ito weigh the temporal length of biphoton wavepacket. The advantage of longer correlation time is that corresponds to the longer correlation length which has better visibility of interference. And there is also provide a longer manipulation time of biphoton waveform.

Now we know that the two-photon correlation function shows the information of pho-ton pairs, such as the bandwidth and the mode-number of phopho-ton pairs. However, the two-photon correlation function didn’t provide any quantum features for the light source after all. Thus, that is necessary to introduce a benchmark for verifying the quantum nature of light. On the other hands, the measurement two-photon correlation function is based on the conditional measurement which means the quantum state already be selected to the conditional state. What if, we are measuring the signal field or idle field without the conditional measurements, what kind of quantum state will we obtain?

To answer those questions, in the next section, we introduce the auto-correlation func-tion and Cauchy–Schwarz inequality for illustrating the quantum nature of generated fields.