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We propose a new scheme of the resonant four-wave mixing (FWM) for the frequency up or down conversion, which is more efficient than the commonly-used scheme of the non-resonant FWM. In this new scheme, two control fields are spatially varied such that a probe field at the input can be converted to a signal field at the output. The efficiency of probe-to-signal energy conversion can be 90% at medium’s optical depth of about 100. Our proposed scheme works for both the continuous-wave and pulse cases, and is flexible in choosing the control field intensity. This work provides a very useful tool in the nonlinear frequency conversion.
© 2016 Optical Society of America
1. Introduction
As a versatile tool in nonlinear optics, the process of four-wave mixing (FWM) is of broad interest. It is directly relevant to important research subjects such as frequency conversions or sum frequency generations of classical light [1–19] and single photons [20–22], productions of squeezed light [23–27] and biphotons [28–30], coherent optical storage or quantum memory [31–38], low-light-level nonlinear optics and quantum information manipulation [39–41], etc. As for the frequency conversion of FWM, utilizing the effect of electromagnetically induced transparency (EIT) [42–44] can further enhance conversion efficiency due to the suppression of linear absorption as well as enhancement of nonlinear susceptibility. The transition scheme of the EIT-based FWM is illustrated in Fig. 1. A weak probe field (denoted as Ωp1) can be converted to a signal field (Ωp2) or vice versa under the presence of two strong control fields (Ωc1 and Ωc2). The four fields form the double-Λ configuration. When all the laser fields are resonant with respect to their corresponding transitions, i.e. Δ = 0 in Fig. 1, the conversion efficiency is however limited to 25% in theory [8,11] and less than 10% observed in experiments [9, 12]. To break such limit prompts the motivation of this work.
Fig. 1 Transition scheme of the EIT-based FWM. The probe field Ωp1 and the control field Ωc1 form one Λ configuration; the signal field Ωp2 and the control field Ωc2 form another.
Here, we propose a new scheme of resonant FWM in which the two control fields are spatially varied. The probe-to-signal energy conversion in the new scheme is more efficient than that in the commonly-used scheme of non-resonant FWM, in which a pair of transitions in the Λ-type configuration have an optimum one-photon detuning [19], i.e. Δ ≠ 0 in Fig. 1. The proposed new scheme is inspired by an experimental observation and the physical picture of normal modes. This paper is organized as the followings. We firstly show the experimental observation that the total output energy in the resonant FWM process can be enhanced by temporally switching on the control field Ωc2 late. Then, we describe the physical pitcure of normal modes and explain the observation. In the basis of the physical picture, we next illustrate the idea of the new scheme. Furthermore, two examples of the new scheme are used to demonstrate that a conversion efficiency of 90% can be achieved at medium’s optical depth of about 100. Finally, we discuss the merits of the new scheme, compare the achieved conversion efficiencies of the new scheme with those of the non-resonant FWM scheme, and give the conclusion.
2. Experimental observation
We performed two measurements of the resonant FWM with cold 87Rb atoms. The states of |1〉, |2〉, |3〉 and |4〉 in Fig. 1 correspond to |5S1/2, F = 1, m = 1〉, |5S1/2, F = 2, m = 1〉, |5P3/2, F = 2,m = 2〉 and |5P1/2, F = 2, m = 2〉 in the experiment, respectively. Details of the experiment are similar to those reported in [45,46]. We sent only the probe pulse to the input of the atom cloud, and detected the generated signal pulse and the remaining probe pulse at the output. In the first measurement, the two control fields had already been stably present before the probe pulse was fired. The experimental data of probe and signal pulses are shown by the circles in Figs. 2(a) and 2(b). In the second measurement, only the control field Ωc1 was initially present and, after the entire probe pulse had entered the medium, Ωc1 was turned down and the control field Ωc2 was switched on. We employed such switching process to maintain the value of in the second measurement approximately constant and equal to that in the first one, because the EIT bandwidth is proportional to . In Figs. 2(c) and 2(d), the circles represent the experimental data of probe and signal pulses. The energy losses due to the EIT bandwidth in the two measurements should be about the same. However, it can be clearly seen that the total output energy in the second measurement is about twice of that in the first one.
Fig. 2 Outcomes of the FWM processes as the probe pulse being the only input. Black, blue and red circles are the experimental data of the input and output probe pulses and the output signal pulse, respectively. The input probe is scaled down by a factor of 0.1. Blue and red dashed lines are the experimental data of the two control fields Ωc1 and Ωc2. The data in (a) and (b) were taken synchronously, and so were those in (c) and (d). Solid lines are the predictions. In the theoretical calculation, OD = 47, Ωc1 = 0.35Γ [its initial value being 0.48 in (c)], Ωc2 = 0.35Γ, and γ = 3×10−4Γ. We determined these parameters with the slow-light data of the single-Λ EIT system [47].
3. Theoretical model
To make the theoretical predictions, we numerically solved the following equations that describe the time evolution of density-matrix operator of the atoms and the propagation of the probe and signal fields.
where ρij is the element of the density-matrix operator, γ is the ground-state decoherence rate, and α and L are the optical depth (OD) and length of the medium. The probe and signal fields were treated as the perturbation. The blue and red solid lines in Fig. 2 are the theoretical predictions and agree with the experimental data well.We introduce the normal modes of the double-Λ system that will be used to explain the results of Fig. 2 and to illustrate the idea of the proposed new scheme. Two normal modes ΩpT and ΩpD are the superpositions of the probe and signal fields given by
where . We also define a new basis of states |T〉 and |D〉 as Using Eqs. (1)–(5), we achieved the result that the transition scheme of ΩpT and ΩpD under the basis of |T〉 and |D〉 is shown by Fig. 3. The derivation that achieves Fig. 3 can be found in the Appendix. The two normal modes ΩpT and ΩpD do not couple to each other. ΩpT interacts with a three-level EIT system and can be seen as the transparency mode due to the EIT transparency. ΩpD interacts with a two-level system and can be seen as the dissipation mode due to the absorption induced by the two-level transition.Fig. 3 A transition scheme equivalent to the resonant double-Λ scheme shown in Fig. 1. ΩpT and ΩpD are the superpositions of Ωp1 and Ωp2 given by Eq. (6); |T〉 and |D〉 are those of |3〉 and |4〉 given by Eq. (7). While Ωc1 and Ωc2 are the Rabi frequencies of the + two control fields shown in Fig. 1, Ωc,tot here is equal to .
Utilizing the normal modes, we explain the results shown in Fig. 2. Remember that only the probe pulse is sent to the input. In Figs. 2(a) and 2(b), Ωc1 and Ωc2 were both present and had the same magnitude. At the input, half of the probe energy became ΩpD and the other half became ΩpT at the input according to Eq. (6). ΩpD interacted with the two-level system, and dissipated energy completely due to a large OD. ΩpT interacted with the three-level EIT system, and propagated through the medium. At the output, the total energy all came from ΩpT which initially carried about 50% of the input energy. In Figs. 2(c) and 2(d), only Ωc1 was present and Ωc2 = 0 initially. At the input, the entire probe energy became ΩpT which is essentially the probe field, and ΩpD = 0. As Ωp1 had completely entered the medium, adiabatically switching on Ωc2 can keep the energy always in ΩpT and suppress the excitation of ΩpD. At the output, the total energy all came from ΩpT which initially carried 100% of the input energy. This explains that the total output energy in Figs. 2(c) and 2(d) is about twice of that in Figs. 2(a) and 2(b). Therefore, the suppression of the excitation of dissipation mode ΩpD is the key factor to enhance the output energy in the resonant FWM process.
4. New scheme
Inspired by the experimental observation and the physical picture of normal modes, we now propose a new scheme that can achieve a high probe-to-signal conversion efficiency with a moderate OD and work for both the pulse and continuous-wave cases. The idea behind the scheme is to spatially vary the two control fields in the way that one field is increased adiabatically and another is decreased adiabatically in the medium. Hence, the excitation of dissipation mode ΩpD is suppressed. While only the transparency mode ΩpT propagates in the medium, its Ωp1 component is gradually converted to its Ωp2 component. The probe light goes into the medium and only the signal light comes out of it.
To demonstrate how the proposed scheme works, we consider the continuous-wave case and use the following spatially-varied control fields:
where β = π/(2L). At the input of the medium (i.e. z = 0), Ωc1 is equal to its maximum and Ωc2 = 0; at the output (i.e. z = L), Ωc1 = 0 and Ωc2 reaches its maximum. In the steady state and with the above control fields, we set γ = 0 and reduce Eqs. (1)–(5) to Because only the probe field Ωp1 is present at the input, ΩpT (0) = Ωp1(0) = 1 and ΩpD(0) = Ωp2(0) = 0 according to Eq. (6). The solution of the above equations is given by where . If η ≫ β, Eq. (11) becomes Thus, η ≫ β is the adiabatic condition or criterion for varying the two control fields to suppress the dissipation mode. Under the adiabatic condition, the signal field or the transparency mode at the output is approximately given by With a moderate OD (i.e. α), one can convert nearly all of the input probe field to the output signal field with little energy loss.Using the control fields defined by Eq. (8), we show the numerical results calculated from Eqs. (1)–(5) in Fig. 4. At OD = 100, the powers of the transparency and dissipation modes (|ΩpT|2 and |ΩpD|2) as well as those of the probe and signal fields (|Ωp1|2 and |Ωp2|2) are plotted against the position z in Fig. 4(a). Normalized to the input power, the power of dissipation mode |ΩpD|2 is always below 0.001 and that of the transparency mode |ΩpT|2 gradually decays to about 0.91. As Ωc1 is decreasing and Ωc2 is increasing in the medium, the transparency mode’s probe component is converted to its signal component. The numerical results are consistent with the analytical derivation shown in the previous paragraph. Figure 4(b) shows the transmissions of the probe and signal fields as functions of OD. The conversion efficiencies or ratios of the output signal to input probe powers are 70%, 80% and 90% at OD = 26, 43 and 92, respectively. As indicated by Fig. 1, the FWM process induces the transition of |1〉 → |3〉 → |2〉 → |4〉 → |1〉 to convert a probe photon to a signal photon. There is no gain mechanism for the total energy of the probe and signal fields. The maximum efficiency is limited to 100%, i.e. all the probe photons being completely converted to the signal photons. Hence, the conversion efficiency or signal field transmission is initially enhanced by the increasing OD, and exhibits saturation and asymptotically approaches to 100% at the large OD as shown in Fig. 4(b).
Fig. 4 (a) Powers of the transparency mode |ΩpT |2 (magenta), dissipation mode |ΩpD|2 (green), probe field |Ωp1|2 (blue) and signal field |Ωp2|2 (red) as functions of the position z inside the medium at OD = 100. Values of power are normalized to the input power. (b) Power transmissions of |Ωp1|2 and |Ωp2|2 as functions of OD. In (a) and (b), the Rabi frequencies of the two control fields vary like Ωc cos[πz/(2L)] and Ωc sin[πz/(2L)] and γ = 0, where the magnitude of Ωc does not affect the calculation results. With a non-negligible decoherence rate (e.g. γ = 0.001Γ), the power transmission of the signal field decreases merely by about 1% (OD = 50) or 2% (OD = 100) at Ωc = 3Γ.
We discuss a feasible implementation of the spatially-varied control fields. The Rabi frequencies of the control fields are given by
As shown in Fig. 5(a), the two control fields propagate in the direction normal to the z axis, where the z axis is the propagation direction of the probe and signal fields. The center of one control beam is aligned to the entrance of the medium (z = 0) and that of another is aligned to the exit (z = L). The Gaussian intensity profiles of laser fields produce the spatial patterns described by the above equations. Figures 5(b) and 5(c) show the ratio of the output signal (or the output probe) power to the input probe power at different values of w. A smaller (larger) w or equivalently smaller (larger) Ωc2(0) makes the input probe field converted less (more) to the dissipation mode at the entrance of the medium, but degrades (improves) the adiabatic condition. Therefore, there is an optimum w at a given OD. Note that the experimental implementation of the above FWM scheme is similar to that of the stimulated Raman adiabatic passage (STI-RAP) scheme [48] as suggested by one reviewer. The two control fields in the STIRAP scheme are arranged in the similar way as those in the FWM scheme mentioned above. Propagating along the z axis, the light beam in the FWM scheme is in analogy with the atomic or molecular beam in the STIRAP scheme. The electromagnetic wave of the light beam is converted from the probe to signal fields and, correspondingly, the population of the atomic or molecular beam is transferred from one state to another. Although the physical mechanisms behind the FWM and STIRAP schemes are different, the experimental implementations of the two schemes can be put in nearly one-to-one correspondence.Fig. 5 (a) Diagram of the new FWM scheme with the spatially-varied control fields given by Ωc exp(−z2/w2) and Ωc exp[− (z − L)2/w2]. (b) and (c) are the transmissions of the probe (blue) and signal (red) fields as functions of OD. In the calculation, the two control fields in (a) are used and γ = 0. Magnitude of Ωc does not affect the calculation results in the continuous-wave case. Solid, dashed and dotted lines represent w/L = 0.47, 0.66 and 0.91. In the pulse case, as w/L = 0.66 the probe-to-signal conversion efficiency decreases merely by about 1% (OD = 50) or 2% (OD = 100) at Ωc = 2.4Γ and the e−2 half width of the input Gaussian pulse being 30Γ−1.
5. Discussion and conclusion
There are a few notes about the proposed new scheme. First of all, the results shown in Figs. 4 and 5 do not depend on the amplitude of the control fields, i.e. the value of Ωc in Eq. (8) or (15), in the contiuous-wave case and at a negligible decoherence rate γ. In the pulse case, the normal-mode picture illustrated in Fig. 3 is still valid. However, the transparency mode now suffers an energy loss depending on the pulse bandwidth. Nevertheless, as long as the EIT transparency bandwidth is much larger than the pulse bandwidth, this energy loss can be negligible. The EIT bandwidth is approximately equal to . One can increase Ωc,tot to make the conversion efficiency nearly intact in the pulse case. An example can be found in the caption of Fig. 5. Secondly, if γ is not negligible, using a larger Ωc can also reduce the energy loss induced by γ and make the results presented in Figs. 4 and 5 nearly unchanged. This is because the energy decay due to γ is approximately equal to ( ). An example can be found in the caption of Fig. 4. Thirdly, this new scheme of spatially-varied the control fields can work for both the continuous-wave and pulse cases. On the other hand, the method of temporally-varied Ωc2 as shown in Fig. 2(c) does not work for the continuous-wave or long-pulse case. Although the temporal method can work for the pulse case, Ωc2 must be switched on after the entire probe pulse has entered the medium. This requires a rather large OD to minimize the energy loss due to the EIT bandwidth [45].
We compared the conversion efficiency of this new scheme of resonant FWM with that of the commonly-used scheme of non-resonant FWM, in which a pair of transitions in the Λ type configuration have an optimum one-photon detuning (i.e. Δ ≠ 0 in Fig. 1) [19]. In the new scheme, the control fields given by Eq. (15) are used and w is optimized. The conversion efficiencies of 70%, 80% and 90% can be achieved at OD of 26, 45 and 110 with the new scheme, but at much larger values of OD of 44, 77 and 175 with the non-resonant FWM scheme. Furthermore, the conversion efficiency of new scheme is rather insensitive to the amplitude imbalance of the two control fields, but that of the non-resonant FWM scheme is sensitive to this imbalance. For example, in the new scheme one amplitude being twice of the other changes the efficiency merely from 90% to 88% at OD of 110, and in the non-resonant FWM scheme one amplitude being only 20% larger than the other changes the efficiency from 90% to 79% at OD of 175. The new scheme is more efficient and robust than the non-resonant FWM scheme.
In conclusion, the double-Λ transition scheme shown in Fig. 1 is equivalent to the transition scheme of the normal modes shown in Fig. 3. Inspired by the physical picture of normal modes, we propose a new scheme that the conversion from the probe to signal fields is made by two spatially-varied control fields. As long as the adiabatic condition for the suppression of the excitation of dissipation-mode is well satisfied, the conversion efficiency or ratio of the output signal energy to the input probe energy can be close to 100% as indicated by Eq. (14). Regarding the frequency up or down conversion, this new scheme of resonant FWM is more efficient than the commonly-used scheme of non-resonant FWM. Our work advances the technology of nonlinear optics and may lead to novel applications in optical frequency conversion.
Appendix
In the basis of |T〉 and |D〉, two new variables ρT and ρD of the density-matrix elements are defined as the followings:
With ρT and ρD, we can transform Eqs. (1)–(5) into two independent groups of equations for ΩpT and ΩpD as the followings: and where η = α/(2L). Thus, ΩpT and ΩpD do not couple to each other and represent the two normal modes of the system. The above equations show that ΩpT interacts with the three-level EIT subsystem and ΩpD interacts with the two-level subsystem as depicted in Fig. 3.Acknowledgments
This work was supported by the Ministry of Science and Technology of Taiwan under Grant Nos. 101-2112-M-007-008-MY3 and 104-2119-M-007-004. GW thanks to the financial supports from the Ministry of Science and Technology of Taiwan and from National Natural Science Foundation 11104111 grant of China.
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