Normalized cross-correlation function

The behavior of non-classical correlation g_s,i⁽²⁾(τ_d) in the manipulation process is related to the signal-to-noise ratio of coincidence counts measurement. Therefore, value of g_s,i⁽²⁾(τ_d) react the temporal-mode of G⁽²⁾_s,i(τ ) is centralized or not. On the other hands, the ability to manipulate the environment also affect the quantity of g⁽²⁾_s,i. That is to say, g⁽²⁾_s,i describe the ability of the whole system for detecting the non-classical nature of light. Therefore, if there is a process which can enhance the quantity of g_s,i⁽²⁾, that will be very valuable.

We have shown the function of the manipulation process for controlling the temporal-mode of G⁽²⁾_s,i(τ ). For an illuminating case of ξ_r/w > 1 in Fig.5.17, we have see the temporal-mode of G⁽²⁾_s,i(τ ) are compressed in the time domain. Thus, centralized G⁽²⁾_s,i(τ ) gives an illustrious coincidence counts within a time bins at delay time τd. We also can

understand the behavior by an interpretation of that: since high-ξ_r/w increase the group velocity of biphotons in the reading phase, that makes the photon number flux enhanced in the biphoton wave-package. Therefore, we can observe a brighter nonclassical correlation of biphotons.

In the ideal case, the enhancement of g_s,i⁽²⁾(τ_d) is proportional to pξ_r/w, since the temporal-mode of biphoton wavefunctions is squeezed by the factor ofpξ_r/w. However, in our experimental setup, the control beam leakage and photon-switching effect in cesium D₂-line(Ref.[6] and Appendix A.2) both are unavoidable in our system. Those negative issues will enter the behavior of g⁽²⁾_s,i(τd) and further suppress the g⁽²⁾_s,i(τd) growth. Notice those negative issues are follow the ξ_r/wincrease as increase, moreover, the enhancement of g_s,i⁽²⁾(τ_d) also proportional to pξ_r/w. Therefore, we can expect there is an optimum condition for g⁽²⁾_s,i(τ_d) in our quantum memory system. In order to model this behavior, we give an equation for g⁽²⁾_s,i(τ_d) which simply described those issues in the manipulation process. The equation is given by

g⁽²⁾_s,i(τ_d) = Ns,ipξ_r/we^−γsξr/w aξr/w+ Nb

, (5.6)

where Ns,i denotes the coincidence counting within an integration time, Nb is the count-ing from the background noise photons which include the incoherence photons from the source and the detector dark counts. γ_s is the decay rate from photon-switching effect.

The leakage of the control field is proportional to the ratio between reading to writing powers(ξ_r/w) with a slope a. Now, with this prior knowledge, let us start to analysis the non-classical correlation in the manipulation process of Fig.5.17. The results are shown Fig.5.20. In addition, the noise photons from the leakage of the control beam also are shown in Fig.5.20.

The data in Fig.5.20shows that is an optimal condition for g_s,i⁽²⁾(τ_d). In order to fit the behavior of g_s,i⁽²⁾(τ_d) in Fig.5.20, we first find the noise photons parameters of α = 0.43 and Nb = 2.8. On the other hands, the photon-switching decay of γ_s ∼ 0.055 which is determined in Fig.5.18. After subbing those parameters into Eqs.5.6, we have the fitting curve which shows a good agreement with the experimental data in the Fig.5.20.

First, all results are shows that the EIT-QM maintains the quantum nature of g⁽²⁾_s,i > 2 which violate the classical limit. About the behavior of g_s,i⁽²⁾(τ_d), we can see that even though g⁽²⁾_s,i(τd) are lower than slow light in the region of ξr/w < 1 in Fig.5.20, but by increasing ξ_r/w, an optimum g⁽²⁾_s,i(τ_d) ∼ 7.5 has be found at ξ_r/w ∼ 3.5 which greater than the case of ξ_r/w = 1 of 5.8 and the slow light case of 7.2. However, after passing the region of ξ_r/w > 3.5, that can observe the decreasing of g_s,i⁽²⁾ due to the issues of photon-switching and leakage. In order to show the effect by the photon-photon-switching, here we show the theoretical curve of the case of no-photon switching(e.g. the QM in the transition of

doi:10.6342/NTU201901825 cesium D₁-line) by a gray dash line as shown in Fig.5.20. We can see the big difference

in the high-ξr/w regime. That also demonstrates the importance of choosing a suitable transition for atomic QM.

In this section, we show the result of the non-classical correlation can be maintained in the manipulated procedure. Furthermore, the enhancement of non-classical correlation is achieved by this process. These results demonstrate the stability of the system and the ability to measure non-classical properties. Next, we discuss further the ability of the EIT system of quantum memory to preserve the quantum state of photons. In the next section, we start to use the quantum fidelity to picture the system in order to quantitative this feature.

2 4 6 8 10

0 2 4 6 8 10

R/W power ratio

g(2) s,i

0 80 160 240 320 400



τ d

Slow light

Storage

Figure 5.20: Manipulation of non-classical correlation of biphoton wavefunction after the convert from atomic coherence. The yellow squares represent the g_s,i⁽²⁾(τ_d) of retrieval photons with different reading power. The yellow solid line is theoretical fitting based on Eqs.5.6with the consideration of the decay rate γ_s ∼ 0.055, the noise fitting of α = 0.43 and Nb = 2.8. The background noise rate is presented by the gray line(linear fitting) and circles(experimental data). The red dashed line shows the cross-correlation of slow light. The gray line denoted the classical limit of g_s,i⁽²⁾ = 2. The gray dash line is the the-oretical curve without photon-switching effect. The error bar is considered by assuming Poissonian statistics.

5.8.2 Quantum fidelity in manipulation process

The quantum fidelity, which is a more rigorous quantity for characterizing the quantum storage process. In the section of slow light fidelity, we have discussed the properties of

fidelity and find the relation between the EIT-bandwidth. In that time, we understood the requirement to get the high quantum fidelity is to prepare a high EIT-bandwidth. That is in order to maintain the frequency-mode of the quantum state of light and further preserve the temporal-mode of biphotons. However, for a given optical depth of the system and the fixed η for the storage condition, that is determined the EIT-bandwidth and also makes the quantum fidelity be limited. In order to overcome this limitation, we calculated the manipulation process in Chapter4which provides another degree of freedom to control the temporal-mode of biphotons. By processing the manipulation, quantum fidelity has an opportunity to improve and further promote the performance of EIT-QMs. In this section, we follow the process in the section of slow light fidelity to analyze the quantum fidelity in the manipulation process.

Before estimating the quantum fidelity F for experimental data, we first try to con-jecture the behavior of F in the manipulation process. So far, we have understood our system involve two imperfect cases of the photon-switching effect and the noise photons from the control field leakage. Since the F care about the varying of frequency-mode of biphoton wavefunctions, the uncorrelated noise photon from the control field can be ignored. Follow the ξr/w increase, the manipulation process gradually correct the broad-ening effect and also improve the quantum fidelity. However, the photon-switching effect also increases further inhibit the advancement of F , which is very similar to the behavior of g⁽²⁾_s,i. Therefore, we also expect that has an optimal value and condition for the quantum fidelity in the manipulation process.

To describe the quantum fidelity in the manipulation process, we refer the theoretical model in Chapter 4. In the discussion, we derived the field solution of Eqs.4.59 in the manipulation process and also got the quantum fidelity in the manipulation process, F_M as

F_M = 2

ξ_r/w+ ^β_ξ^2(L)

r/w

. (5.7)

In that case, for the given conditions of writing phase, the bandwidth of output field is proportional to ξ_r/w, and therefore we have the F_M as shown in Eqs.4.60. However, the same reason as given in the section of bandwidth controlling, the bandwidth of output biphotons is no longer proportional to ξr/w. Here we must to correct the behavior to pξ_r/w-proportion. On the other hands, we also need to consider some imperfect effects, such as the photon-switching effect and the leakage in the writing phase. For the case of photon-switching effect, we can refer to the Fig.5.18 and picture that of an experiential decay as exp(−γ_sξ_r/w). For the part of the leakage in the writing phase, that give the upper limitation of FM since the quantum fidelity also relates to the efficiency. By combining

doi:10.6342/NTU201901825 these effects, then we have a more realistic behavior of F_M as

F_M = 2l₀e^−γsξs,i ξ^1/2_r/w+ ^β2(L)

ξ^1/2_r/w

, (5.8)

where l0 represents the leakage in the writing phase. In our case, this factor is around l0 ∼ 0.8. So far, we have roughly predicted the quantum fidelity in the manipulation process. Next we estimate the quantum fidelity in the cases of Fig.5.17 by using the Eqs.4.66. The result is shown Fig.5.21.

2 4 6 8 10

0.15 0.20 0.25 0.30 0.35

R/W power ratio

Quantum fidelity

Figure 5.21: Manipulation of quantum fidelity. The gray squares represents the estimation of quantum fidelity of the cases in Fig.5.17. The solid line is the theoretical curve based on Eqs.5.8. The fitting parameters are given by: ξl = 0.8, α = 55, η = 4 and γs= 0.055.

Notice that all values are lower than storage efficiency as shown Fig.5.18.

We can observe the quantum fidelity are manipulated in the process and notice all values are lower than storage efficiency in Fig.5.18. Pleases also notice that efficiency be-havior in Fig.5.18show the better efficiency are performed in low-ξ_r/wregime. However, that is not reacted onto the behavior of F_M in the same regime. Since the quantum fidelity more strictly to evaluate the quantum state of biphotons, even we have higher efficiency in the low-ξr/w regime, however, the manipulation process severely affects the quantum state of light and further change the composition of the frequency distribution of bipho-tons. Final, that result in the reduction of quantum fidelity in low-ξ_r/wregime. Follow ξ_r/w increasing, we can observe the optimal fidelity of ∼ 0.34 at ξ_r/w ∼ 3.5. This condition is determined by the photon-switching effect and the improvement of the manipulation process. For higher ξr/w, the more serious photon-switching effect and over-manipulating

by the process both progressively change the quantum state of biphotons, and therefore, that can see the decay of quantum fidelity in high-ξr/wcase.

In the part of our theoretical model, the Fig.5.21present a good agreement between the theoretical fitting and the experimental results. That shows the rationality of our procedure for fidelity modification. Moreover, that agreement illustrates our theoretical model can provide a good physical description of this experimental process.

5.9 Summary

In this chapter, we describe in detail the experimental setup of the quantum storage and manipulation system. The system combines two sub-systems of the photon-pair source based on cavity-enhanced SPDC and the atomic quantum memory based on cold cesium in the D2-line transition. By measuring the EIT-spectrum by using the SPDC photons, we verify the interaction between the photons and the EIT-medium. Furthermore, we recon-struct the biphoton wavefunctions that after interacting with the medium by the coinci-dence counts measurement. The result clearly shows the biphoton wavefunction is slowed down by EIT-medium(Fig.5.14). The demonstration proved the photon source possesses a nice ability to provide the non-classical photon pairs for atomic quantum memories. In order to execute the quantum storage and manipulation, we developed a simple switching setup for controlling the writing and reading phases(Fig.5.9). After sending the switching signal to the control field, we successfully achieved the quantum storage and manipulation processes(Fig.5.17). Based on the experimental data, we further analysand the properties of retrieval biphotons in both classical and quantum information. For the classical part, a series of storage efficiency is evaluated. Since the photon-switching effect in the D₂-line, the average efficiency is ∼ 32%. On the other hands, we also demonstrated the ma-nipulation of the bandwidth of biphotons(Fig.5.19) and observed the bandwidth can be controlled from 1.6 to 6.8MHz. In the analysis for the quantum properties, we estimated the non-classical correlation g_s,i⁽²⁾. The result showed the retrieval biphotons violated the classical limit of 2(Fig.5.20), which verified our QM maintain the quantum nature of pho-ton pairs. Moreover, the manipulation process also demonstrated improvement of g⁽²⁾_s,i from 5.8(storage) to 7.5(manipulation). The result showed the process can provide an-other degree of freedom to improve the non-classical correlations. To more precise to picture the performance of QM, we introduce the quantum fidelity to quantify the process of the quantum storage. Since the imperfect issues in the real experiment(e.g., photon-switching and the leakage), the fidelity of the storage process is only ∼ 24%. However, after the manipulation process, as the prediction as given by our theoretical model, we can improve the quantum fidelity to the upper limit of efficiency of ∼ 32% in our

experimen-doi:10.6342/NTU201901825 tal conditions(Fig.5.21). Furthermore, for those experimental results, we all provide the

theoretical models and fittings for those results and show the good agreements between our theory and experiment.

Chapter 6 Conclusion

6.1 Summary

After the long discussion, now is the time to summarize this thesis. In Chapter 2 of this thesis, we introduced the theoretical background of cavity-enhanced SPDC. Follow the theoretical model, we have discussed such as the frequency spectrum and the two-photon correlation. Not only classical quantities, the quantum nature of generated fields but also are introduced: for instance, the auto-correlation gave the tools for measuring the properties of the single field. On the other hands, to measure the quantum nature of multi-fields correlation, the Cauchy–Schwarz inequality showed the classical limit and further give the quantum benchmark for the generated fields.

With the theoretical background that introduced in Chapter 2, we further presented the experimental design of photon source based on cavity-SPDC in Chapter3. In order to solve an important problem of cavity-SPDC photon source: to maintain the resonant condition of the cavity and lock the generated photons at atomic transition, simultane-ously. In this work, we developed the locking system for the photon source and further cleverly solve this issue. The concept of locking system is using the time-multiplexing to separate the OPO-locking mode and photon-pair generation. The generation of OPO pro-vides a nice guideline for optimizing the system and further maintain the stability of the system. By measuring the two-photon correlation function as given in Chapter2, we have estimated the bandwidth of generated photon pairs of 6.6 MHz. Thanks for the locking system for maintaining the double resonant conditions, the generation rate is relatively high of 7.24 × 10⁵s⁻¹mW⁻¹, and the count rate is 7840 s⁻¹mW⁻¹. On the other hands, to picture the energy purity of generated photon pairs, we introduced the spectral bright-ness and that is estimated of 1.06×10⁵ s⁻¹ mW⁻¹ MHz⁻¹ which is the highest record in that time(2018). By checking the frequency modes on a Fabry-Perot etalon, we verified the single-mode operation. Furthermore, by measuring the transmit spectrum of the

ce-doi:10.6342/NTU201901825 sium cell, we checked the locking operation locked the frequency of photon pairs and also

matched the atomic transition(see table.3.1).

After the peroration and verifying the compatibility of the atomic transition of photon-pair source, we next enter to the theoretical model of atomic quantum memories in Chap-ter4. By considering a Λ-type EIT energy diagram, we solved the solution of field op-erator and further got the input-output relation. The relation helps us to understand the behavior of light after interacting with the EIT-QMs. Using the input-output relation, we further presented the classical and quantum properties of light. In order to give the quantum benchmark of EIT-QMs, the quantum fidelity of single-photon state has been discussed. By a simple inference, we comprehend the broadening effect is the main effect make a reduction. Therefore, we further considered the manipulation process to remedy the broadening effect in the EIT-QMs. Our theoretical analysis showed the process can enhance the quantum fidelity to the upper limit of the storage efficiency.

Based on the theoretical model introduced in Chapter4, we furthermore demonstrate the quantum storage of heralded single photons sent by the cavity-enhanced SPDC in the atomic quantum memories in Chapter 5. In order to verify the interaction between photons and atoms, we first test the biphoton slow light experiments and also analyzed the quantum fidelity. The result showed a good agreement between experiment and theory.

Once the biphotons slow light effect is verified, we furthermore presented the quantum storage and manipulation process of biphoton wavefunctions. Our quantum memories achieve the quantum storage with a storage efficiency of about 40% and the non-classical correlation of g_s,i⁽²⁾ = 5.87. Furthermore, we demonstrated the manipulation process and showed the non-classical correlation can be increased to g⁽²⁾_s,i = 7.5. On the other hands, the quantum fidelity can be effectively raised to the maximum limit of 32% in our case.