Manipulation of light by EIT-QMs

Figure 4.13: Quantum fidelity of EIT-QM with the single-photon input. The frequency distribution of the single-photon is assumed as Gaussian. The plotting parameters are the same as the Fig.4.7. The yellow lines represent the quantum fidelity and the black dash lines are correspond efficiency. Notice the value of fidelity always lower than efficiency in all parameters.

strength of control of the reading phase then offset the broadening effect. Here we called the process as manipulation process. In the next section, we will start to calculate the case of manipulation process and demonstrate how this process to affect the quantum fidelity of EIT-QMs.

4.4 Manipulation of light by EIT-QMs

In the previous section, we have talked about the quantum fidelity of EIT-QM. The broad-ening effect of EIT-medium severe affects quantum fidelity. A way to retard the reduction of quantum fidelity due to broadening effect is the manipulation process. As we despite as given above, the manipulation process is built on the conversion process. By using different strength to reconstruct the light fields and further manipulate the light fields.

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4.4.1 Light field converted from the atomic coherence in the reading phase

In order to analyze this process, we first focus on the storage phase. Unlike the slow light process, in this phase, the ground state coherence has been determined by the incident sig-nal light field during the writing process. Therefore, the ground state coherence becomes an initial condition in this phase. Here we temporarily use ˆσ_s(z) to represent it. On the other hand, as we said earlier, the signal light field has been converted to atomic coher-ence, so the initial condition of the light field is 0. With those initial conditions, now we back to the Eqs.4.35to evaluate the evolution of light fields from the atomic coherence.

First, we reconsider Eqs.4.37. But this time, we place the complete solution of ˆσ_gs(z, t) of Eqs.4.35into the Eqs.4.37,

 ∂ After Fourier transform, we can calculate the equation as given above in the frequency domain, then we find

The notation r at the superscript in the equations means those equations are for the reading phase. Ω_r is the Rabi frequency of the control field in the reading phase. Eqs.4.53 guide the light fields evolution from the atomic coherence. That can see the light field be gen-erated from the source term of the ground state coherence ˆσ_s(z). On one hand, the factor A^r(ω) lead the oscillation of source, on the other hand, that also affect the group velocity of generated signal fields by Λ^r(ω). The Eqs.4.53is a first-order non-homogeneous ODE and therefore we can refer the method which is used in Eqs.4.10. Different from the case in Eqs.4.10, now the initial condition of light is 0. Therefore, after finding the integral factor, we have the solution as

ˆ

a^r_s(z, ω) = 2g_sN

cΩ^∗_r A^r(ω) [ˆσ_s(z) ∗ exp (−Λ^r(ω)z)] (z). (4.54) In the solution Eqs.4.54, we can see again the oscillation factor A^r(ω) leads the ampli-tude grows of field and also see exp(−Λ^r(ω)z) guide the propagation of the field. For a given initial ground state coherence ˆσ_s(z), we can solve the field behavior by Eqs.4.54.

However, σsis determined by the condition of the writing phase. In order to get a suitable

ˆ

σ_sto describe the manipulation process, now we start to consider the case of the writing phase.

Ground state coherence in the writing phase

In the writing phase, the initial condition is the same as the condition of slow light. Since our goal is to find the ˆσ_sthat comes from the writing phase, we use the Eqs.4.35and find the relation between light field and atomic coherence. After Fourier transform, we find the field-atom relation in the frequency domain as

ˆ

σ^w_gs(z, ω) = A^w(ω)g_sˆa^w_s(ω, z)

Ω_w . (4.55)

About the field in the equation as given above, we already solved that in Eqs.4.11. There-fore, we can directly sub this solution into the atomic coherence. However, at the time T_c, we turn off the writing field Ω_w let the light field information transfer to the ground state coherence and solidify in the medium. Therefore, by substituting Eqs.4.11into Eqs.4.55 then doing the inverse Fourier transform to Tc, we have the ground state coherence that we store in the medium as

ˆ

σ^w_gs(z, T_c) = g_s

Ω_wF⁻¹[A^w(ω)e^−Λw(ω)zˆa^w_s(0, ω)](T_c). (4.56) Eqs.4.56 describes the distribution of ground state coherence in space which is corre-sponding to the initial condition of the reading process and therefore ˆσ_s(z) = ˆσ_gs^w(z, T_c).

Now we have prepared the sufficient conditions for evaluating the light field in the reading phase, except the input field of ˆa^w_s(0, ω). By choosing a suitable ˆa^w_s(0, ω), we can sub ˆσ^w_gs(z, T_c) into Eqs.4.54, final we will get the output field solution. But having said that, this calculates is extremely complicated. Thus, here we give some supposes and condition for simplifying the case.

First, we consider the input signal pulse is a Gaussian function in the time domain ˆ

a^w_s(z = 0, t) = ˆa^w₀(z = 0)exp(−2ln2(t/T_s)²) or corresponding to the frequency domain as ˆa^w₀(z = 0)T_s/√

4ln2exp(−(ωT_s)²/8ln2) with the bandwidth of ∆ω = 2√

2ln2/T_s. On the other hand, we consider Λ^w(ω) to the second order term of ω(refer to Eqs.4.20). Fi-nal, we consider the control is switched-off adiabatically which means the high-frequency response in A^w(ω) can be vanished. Therefore, the factor A^w(ω) is replaced as −1. With those conditions and assumption, then we have

σ_gs^w(z, T_c) = g_sˆa^w₀T_s

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. We can see that even though the signal pulse is a Gaussian, however, since the broadening effect is dependent on the propagation distance in the medium, the broadening extent are different in the different location. Therefore, ground-state coherence is not a Gaussian distribution in the space. This result also already showed in the Fig.4.10(a).

The non-Gaussian behavior obstruct our calculation again, therefore, here we give another approximation. According to Eqs.4.57, we understand the main contribution of ˆ

With those conditions, the ground state coherence approximates a Gaussian distribu-tion and also having the Hau length[62] of L^w_h = v_wT_p. The center of ˆσ_gs^w(z, T_c) is at L_c= v_wT_cin the medium. On other hands, as the discussion as given before, to store all information of the signal field into the medium, we suppose the signal field be mapped perfectly onto atomic ensemble. In other words, the spacial length of σ_smust shorter than medium length L at t = T_c, i.e. η = L/L^w_h > 1. Final, we have the initial condition of ˆ

σ_s(z). Next we back to the reading phase and further use this condition to solve the light field in the reading phase.

Back to the reading phase

Let’s further pursue the signal field in reading phase. Here we sub Eqs.4.58into Eqs.4.54 then consider Λ^r(ω) to the second order term of ω. Final vanishing the high-frequency response in A^r(ω), we have

where process provide two functions for controlling the behavior of output light pulse, including a controllable storage time and the controllable pulse shape. In the whole process, the storage time is defined as the time difference between writing and reading phases. On the other hand, the factor ξ² dominant the pulse shape of the output field. We can see that, in the case of the Gaussian pulse input, the bandwidth of output pulse is ∆ω/ξ. Therefore, we can manipulate the bandwidth of output pulse by this process also. Notice that, for a given condition of the atomic system and writing phase, the bandwidth of output pulse only dependent on the reading field or ξ_r/w.

4.4.2 Quantum fidelity in manipulation process

Now we have proved the manipulation process gives a new degree of freedom to control the output fields. Here we back to the case of the quantum fidelity that we care about.

Since we know the quantum fidelity of signal photons is basically that represents the like-ness between input and output frequency distributions, to preserve the pulse shape of input is a way to enhance the fidelity. To explain that, here we calculate the quantum fidelity again. According to Eqs.4.59and Eqs.4.46, we can simply get the quantum fidelity in the manipulation process as

F_M = 2

ξ_r/w+ ^β_ξ^2(L)

r/w

. (4.60)

As the discussion as given above, we understood the ξr/w-dependent is because of the bandwidth of pulse is controlled. Therefore, we can improve the quantum fidelity by the manipulation process. On the other hands, the factor β(L) represent the F_M not only react the varying of the bandwidth of output field but also show the efficiency will affect the quantum fidelity.

According to Eqs.4.60, we can easy to observe that the F_M has a maximum value.

This value can be obtained by differentiating the Eqs.4.60 and find the point of slope equal to 0. Or, we can observe the Eqs.4.59to get some inspiration. In the Eqs.4.59, the

doi:10.6342/NTU201901825 pulse shape is only dependent on the factor 1 − ξ, therefore, if the factor 1 − ξ = 0, we

will have an output pulse that exactly same with input. Based on our argument, we solve the 1 − ξ = 0 and find the relation of

ξ = 1 → |Ω_r|² = β_w(L)|Ω_w|². (4.61) This relation means that for the given writing conditions, we always can find a specific reading control field to correct the fidelity of the output pulse. By taking this relation into the field solution of Eqs.4.59, we have the light pulse with an optimal condition as

ˆ

The Eqs.4.62very clearly show the reading pulse is only be shifted a time delay but with-out any broadening effect. However, the amplitude of the field is reduced by finite EIT-bandwidth β_w in the writing phase. Next, that is very straightforward to sub the condition of Eqs.4.61 to the F_M, then we find the optimal quantum fidelity in the manipulation process:

F_opt = 1

β_w(L) = T. (4.63)

Just like our discussion in Section.4.3.1, the quantum fidelity of single-photon is limited by the efficiency, in other words, the optimal quantum fidelity is equal to the efficiency of EIT-QM. In Fig.4.14, we show the improvement of fidelity by manipulation process.

Since the optical depth of the system leads the bandwidth of EIT in the case of a given η, we can see the fidelity is low in the low optical depth region. Therefore, we have more rooms for improvement in low optical depth region as shown in Fig.4.14.