1.4 Introductory this thesis
2.1.3 Biphoton generation and it’s wavefunction
The heart of the SPDC process is the nonlinear medium. The medium provides the non-linearity then force the pump photon is split into signal and idler. To picture the process, we introduce the interaction Hamiltonian of SPDC, that is given by[57,58]
Hˆint= χ 2l
Z 0
−l
dx( ˆEp(+)Eˆs(+)†Eˆi(+)†+ H.C.), (2.17) where χ is second-order electric susceptibility tensor that dependent on the frequency of each field. In the interaction Hamiltonian, the operator of ˆEs,idenoted the field operators inside the cavity as given by Eqs.2.15and2.16 since the interaction has happened inside the optical cavity. The form of ˆHintshows a clear picture of the SPDC process. The mean-ing of operator ˆEp(+)Eˆs(+)†Eˆi(+)†denotes the pump photon is annihilated and further create the signal and idler photons. Furthermore, the term of hermitian conjugate represents the anti-process as given above(e.g., the sum-frequency generation).
So far, we have assembled all the tools used to describe the generation of photons and their properties, including the Hamiltonian of the system and the operators of the field.
This is the time to turn our attention to the quantum state of generated fields. Based on the quantum mechanics, that is directly to depict the quantum state evolution by finding a unitary operator, ˆU (t) = exp(−iR dt ˆHint/¯h). Furthermore, the wavefunction at time t can be expressed by the wavefunction at t = 0 of |ψ(t)i = ˆU (t) |ψ(0)i. In the SPDC process, the initial state is a vacuum state for both signal and idler, |0s, 0ii. Therefore, we can write down the evolution of the state vector as:
|ψ(t)i = e−iR dt ˆHint/¯h|0s, 0ii . (2.18) Now let’s stop and take a look at this equation. First, we consider the ˆHint part. As we depicted before, the pump is relatively strong and therefore that can be expressed in the classical region as ˆEp → Ep. In this case, the Hamiltonian can be rewritten as
Hˆint∝ Z 0
−l
dx(ξ ˆEs†Eˆi†+ H.C.), (2.19) where ξ is a complex number. That is easy to discern that the evolution operator of this process is the two-mode squeezing operator[59]. Therefore, for the case of an initial
state of the vacuum, the evaluated quantum state will present the two-mode squeezed state(TMS)[59]. The TMS involves the multi-photon process, the large photon number state makes the correlation between signal and idler fields are expanded to the higher di-mension such as the quantum fluctuations of fields. That property of continuous-variable entanglement lets the TMS become a very powerful tool in quantum information science.
Notwithstanding this, here our goal is to generate the photon pairs or biphotons. To re-alize this, the standard way is reducing the pump fields to the far-below threshold and further avoid the multi-photon process. In this case, the interaction can be seen as a per-turbation. Hence, by following the perturbation theory, the evolution of the state vector is represented by That is clear to see, the first term denotes the vacuum state which corresponds to the non-interaction process. Therefore, the signal and idler modes preserve the vacuum state.
The vacuum state occupies most population in the state vector. That is only a few pump photons will be converted into the signal and idler photons which are presented by the second term. According to Eqs.2.19, the second term shows the photon-pair generation.
This state also is the state what we case about: the biphoton state. To pick up the biphoton state(and also avoid to get the vacuum state), the conditional measurement provides a way to let us can measure the biphoton(or single-photon) more efficient. Based on the process of conditional measurement, the vacuum state can be ignored. Now we can focus on to evaluate the conditional state of biphoton. The first step is to deal with the interaction Hamiltonian. By considering the Eqs.2.15, 2.16 and 2.17, the interaction Hamiltonian becomes[57,58]
The function ∆km,n(Ω, Ω0) in the function Fm,n is the phase-matching condition that can
doi:10.6342/NTU201901825
here we already consider the Taylor expansion at a frequency of ωs,i for the refractive index of signal and idler field to first order, respectively. The parameters vg,s(i)is the group velocities of signal(idler) fields at the frequency of ωs(i)which equal to c/(ns+ ωs∂ωns).
After we rectified the Hamiltonian, now we back to the biphoton state vector. Accord-ing to the form of ˆHintof Eqs.2.17, the biphoton state can be written as
|ψ(δt)i = α
Here, we need to notice that, the state vector is a non-normalized state since we consider the conditional measurement. To solve the normalization issue, we assumed there is a normalized constant called N that absorbed the constants that independent on the integral, tentatively. Therefore, we can consider the normalization after we calculated the detail of the state vector. Furthermore, the state vector can be expressed by
|ψ(δt)i ∝
Now, let us observe this form of the state vector. The factor δt is the interaction time of pump fields applied. If we consider the steady-state case of δt → ∞, the sinc-function will approach to the delta function. By using this characteristic, we can further settle the integration of Ω0. Then the state vector becomes
|ψ(δt)i ∝
As you see, since the integral of the delta function, the phase-matching condition in Fm,n
where τ0 is the time delay between signal and idler photons. We then consider Fm,n, after the integration, we have the Fm,n as
Fm,n(Ω, −m∆ωs− n∆ωi− Ω) ≈ sinc[(m∆ωs+ Ω)τ0/2]e−i/2(m∆ωs+Ω)τ0. Notice that the ratio of susceptibility tensor has been replaced by 1 since we assume the susceptibility is slowly-varying in the SPDC gain profile. By substituting this result into the state vector Eqs.2.25, we obtain the biphoton state vector as
|ψ(δt)i ∝
Furthermore, the sinc-function depict the gain profile of SPDC. If the bandwidth of SPDC is much broader than cavity linewidth(i.e. γs,i ∆ωs,i |τ0|−1), the sinc-function in integration of Ω can be seen like a constant. Therefore, we can further write down the biphoton state vector as[57,58]
|ψ(δt)i = N where N is the normalization constant and
φm,n(Ω) = sinc(m∆ωsτ0/2)
γs
2 − iΩ γi
2 + i(m∆ωs+ n∆ωi+ Ω) e
−im∆ωsτ0/2. (2.28) The function φm,n(Ω) shows state amplitude in the frequency domain. After Fourier trans-form of φm,n(Ω) to the time domain, that corresponds to the temporal mode of biphoton state. The Eqs.2.27 implied the frequency-entanglement between signal and idler pho-tons since the frequency correlation of the field operators. Furthermore, the state vector Eqs.2.27carried more information of the biphoton state. That can help us to understand more behaviors of biphoton by using Eqs.2.27to calculate some physical quantities.