1. Introduction

Preparation and manipulation of a well-defined quantum state is of fundamental importance since it is extensively applied to many branches of physics ranging from laser physics [1] to quantum information processing [2].

In quantum information processing, using highly excited Rydberg states allows one to switch on and off strong interaction that are necessary for engineering many-body quantum states, towards implementing quantum simulation with large arrays of Rydberg atoms [3]. However, the transition frequency from Rydberg states to ground state falls generally in the ultraviolet region, which makes the transition difficult with a single laser. To complete the transition, two lasers (two-photon processes) are required [4, 5]. The drawback by this method is that the transfer efficiency may not be high. This gives rise to a question that does the transition from the ground state to Rydberg states occur by lasers with frequency far-off-resonant with respect to the transition frequency?

Long-lived excited nuclear states (also known as isomers) can store large amounts of energy over longer periods of time. Release on demand of the energy stored in the metastable state called isomer depletion together with nuclear batteries [6–9], have received great attention in the last one and a half decades. Depletion occurs when the isomer is excited to a higher level, which is associated with freely radiating states and therefore releases the energy of the metastable state. Coherent population transfer between nuclear states would therefore not only be a powerful tool for preparation and detection in nuclear physics, but also especially useful for control of energy stored in isomers.

In atomic physics, a successful and robust technique for atomic coherent population transfer is the stimulated Raman adiabatic passage (STIRAP) [10]. The transfer of such techniques to nuclear systems, although encouraged by progress of laser technology, has not been accomplished due to the lack of γ-ray lasers.

To bridge the gap between X-ray laser frequency and nuclear transition energies, a key scheme is to combine moderately accelerated target nuclei and novel X-ray lasers [11]. Using this proposal, the interaction of X-rays from the European X-ray Free Electron Laser (XFEL) [12] with nuclear two-level systems was studied theoretically [11, 13]. However, the coherent control of isomers have never been addressed, partially because of the poor coherence properties of the XFEL. This raises again the question that whether the population transfer can be performed with high fidelity by off-resonant drivings. In this work, we present a proposal for population transfer between two states using lasers off-resonant with the transition energies of the two states. Note that there also exist several work [14–16] investigating population transfer by using only non-resonant driving fields recently, which can be explained by the superoscillation phenomenon. However, the major difference is that the driving fields is periodic in our work. In particular, we consider two types of pulse, i.e., square-well pulse and Gaussian pulse, as the driving fields.

2. Physical mechanism for population transfer

Consider a two-level system described by a Hamiltonian with general form (h̄ = 1)

In this work we focus on the system dynamics driven by periodic square-well driving field. That is, the system is governed by the Hamiltonian H₁ = d⃗₁ · σ⃗ in the time interval [0, t₁], while the Hamiltonian is H₂ = d⃗₂ · σ⃗ in the time interval (t₁, T]. T is the period of the square-well driving field, t₂ = T − t₁. The evolution operator within one period of time can be written as,

Note that, at an arbitrary time t = t′ + nT (n is the number of evolution periods), the evolution operator can be written as

3. Examples

As an example, we show how to achieve population inversion |0〉 → |1〉 in a two-level atomic system to which a classical field is applied. The system Hamiltonian reads

 

Fig. 1 The population as a function of time where P₀ (P₁) is the population of state |0〉 (|1〉). (a) The intensity modulation, Δ/Ω₁ = 30, Ω₂/Ω₁ = 2. (b) The frequency modulation, Δ₁/Ω = 20, Δ₂/Ω = 30. (c) The constant Rabi frequency and detuning, Δ₁/Ω = 20. The dash lines represent the function |sinΛt|².

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The total time needed to achieve population inversion is,

 

Fig. 2 The minimal time 𝒯_f as a function of (a) the coupling constant Ω₁ in the intensity modulation, Δ/Ω₂ = 30; (b) the detuning Δ₁ in the frequency modulation, Δ₂/Ω = 30. The solid and dash lines are the exact and approximate results, respectively. The horizontal line (dot-dash line) is for ${𝒯}_f^{\prime }=\frac{{\pi }^2}{4\mathrm{\Omega }}$. (c) The population with different detunings Δ_k in the intensity modulation, where the square-well pulse is described by Ω₁/Ω₂ = 3 and Δ₁/Ω₂ = 100 for the transition |0〉 ↔ |1〉.

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4. Robustness of the scheme and its extension to three-level systems

For a two-level system, the detuning (labelled as Δ₁) might be different from that used in the calculation. One then asks whether this dismatch affects population inversion in this scheme. In Fig. 2(c), we plot the population on the levels other than the target, where the detunings of these levels to the level |0〉 are denoted by Δ_k. We find in Fig. 2(c) that the effect is remarkable. E.g., the population is less than 0.1 when the detunings have 5% deviation from the detuning Δ₁. Fig. 2(c) also shows that deviations in coupling constants Ω₁ [the deviation in the coupling constant is defined as Ω′₁ = (1 + δ)Ω₁] has small influence on the population inversion |0〉 → |1〉.

Note that it may be difficult to obtain a perfect square-well field in practice. Thus we should analyze how deviations of the applied field from the perfect square-well one affect the population transfer. Taking the intensity modulation as an example, we replace the perfect square-well field with an approximate square-well field to study this effect. Assume that the approximate square-well field takes

 

Fig. 3 The population P₁ as a function of time with different γ in the intensity modulation. (a) γ = 10000. (b) γ = 300. (c) γ = 100. (d) γ = 50. The bottom panels (e)–(h) are the square-well field corresponding to the top panels (a)–(d), respectively. The parameters of perfect square-well field are Ω₂/Ω₁ = 3 and Δ/Ω₁ = 50.

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In actual modulation, there might exist errors such as perturbations or noises in the coupling constant or the detuning. Taking the intensity modulation as an example again, we add noise terms ε(t) into the perfect square-well field, which are taken randomly in a certain interval. The results are plotted in Fig. 4, which suggests that the system dynamics is robust against noises.

 

Fig. 4 The population P₁ as a function of time with noisy square-well field. (a) The noise ε(t) is generated randomly in the interval [−0.05, 0.05]. (b) The noise ε(t) is generated randomly in the interval [−0.3, 0.3]. The bottom panels (c) and (d) are the noisy square-well fields corresponding to the top panels (a) and (b), respectively. For perfect square-well fields, we employ Ω₂/Ω₁ = 3, and Δ/Ω₁ = 50 in the intensity modulation.

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So far, we do not consider the effect of environment yet. In the presence of decoherence, we can use the following master equation to describe the system dynamics [18],

 

Fig. 5 The population versus the dissipation rate γ₀₁ and the dephasing rate γ₁₁. The initial state is |0〉, Δ₁/Ω = 30, Δ₂/Ω = 300.

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Next we study the situation that the detuning is not very large. That is, in the intensity manipulation, one of coupling constants (say, Ω₁) is not very small compared to the detuning. If the system parameters satisfy one of the following equations,

 

Fig. 6 (a) The population as a function of time and detuning, Ω₁/Ω₂ = 10. (b) The population as a function of time when Δ/Ω₂ = 30 (the dash line), where the parameters approximately satisfy Eq. (10).

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This technique can be extended to achieve an arbitrary coherent superposition state, for example, |Ψ_T〉 = cosΘ|0〉 + sinΘ|1〉 in the Λ-type (or V -type) three-level system, where two lower levels couple to another through classical fields with coupling constant Ω₁ and Ω₂, respectively. Both detunings are assumed to be the same Δ. At first, we should choose the ratio of Ω₁/Ω₂ to equal tan(Θ) given in |Ψ_T〉. By periodically changing the detunings Δ_a and Δ_b with fixed coupling constant Ω₁ and Ω₂, one can achieve the state |Ψ_T〉. If the three-level atom is placed in a cavity, and one of the transition is driven by the cavity field while the other is driven by a classical field, we can realize single photon storing by manipulating the frequency and the intensity of the classical field (for details, see Appendix).

In experiments, the coupling might not be a constant for laser pulse drivings. Further investigation shows that the scheme works well provided that the pulse takes a Gaussian form $\mathrm{\Omega }(t)=𝒜e^{-\frac{{\left(t-4\xi \right)}^2}{2{\xi }^2}}$. Here 𝒜 and ξ denote the amplitude and the width of the Gaussian pulse, respectively. In this case the period T is the time width of full Gaussian pulse, and we truncate the width of Gaussian pulse with T = 8ξ in the following. The evolution operator can be written as $U(T,0)=𝕋e^{-i{\int }_0^{T}H(t^{\prime })d{t}^{\prime }}$ with 𝕋 denoting the time-ordering operator. It is difficult to find an analytical expression for the evolution operator. Fortunately, we can prove that after n Gaussian pulses the system evolution operator takes following form

 

Fig. 7 The population versus {𝒜, ξ} of the Gaussian pulse, Δ/Ω = 2.

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5. Extension to Rabi model

The Rabi model [19–23] describes a two-level atom (system) coupled to a single-mode electromagnetic field, which provides us with a simplest example for light-matter interactions. The Hamiltonian describing such a system reads

Suppose the two-level system is in the ground state ρ^s(0) = |g〉 〈g| and the field is in a coherent state initially, namely

 

Fig. 8 (a) The population of |e〉 and |g〉 as a function of time with resonant atom-field couplings (i.e., Δ = ω₀ − ω_b = 0). (b) The probability that there are n photons in the field at different time. The field is initially in a coherent state given by Eq. (14) with the average photon number 〈n〉 = 20. The photon number is truncated at 51 in numerical calculations.

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In contrast, by the intensity modulation, we can realize the population transfer as shown in the following. Keeping the detuning (Δ = ω_b − ω₀) fixed, and modulating the coupling constant Ω₁ and Ω₂, we plot the evolution of the population in Fig. 9(a). We find that population transfer from |g〉 to |e〉 occurs at Ω₁t = 2.5. An inspection of Fig. 9(b) shows that the state of the field (the coherent state) remains unchanged in the dynamics [see Fig. 9(d) as well]. Furthermore, we find that population transfer can occur for a wide range of initial states of the field, as shown in Fig. 10, where the initial state of the field is ${\rho }_{nn}^b(0)={𝒫}_n$. Here 𝒫_n are a set of random numbers in interval [0, 1] and satisfy the normalization condition ∑_n 𝒫_n = 1.

 

Fig. 9 (a) The population of |e〉 and |g〉 as a function of time in the intensity modulation, Δ/Ω₁ = 300, Ω₂/Ω₁ = 3. (b) The probability that there are n photons in the field at different time. The field is initially in a coherent state described by Eq. (14) with 〈n〉 = 20. (c) The occupation of No. m basis as a function of time, where the basis are ordered as {|g, 0〉, |e, 0〉, |g, 1〉, |e, 1〉, ..., |g, 50〉, |e, 50〉, |g, 51〉}. (d) The probability of n photons in the field as a function of time.

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Fig. 10 (a) The population at |e〉 and |g〉 as a function of time in the intensity modulation. (b) The probability that there are n photons in the field at different time. The field is in random state initially.

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6. Discussion and conclusion

On the other hand, this method can be applied to study Landau-Zener-Stückelberg interference [27–31], which has been observed in a superconducting qubit under periodic modulation [32–38]. Obviously, the Landau-Zener transition can also be fore-sight by using our method. In addition, the scheme can be used to realize population inversion for Rydberg states, e.g., from the ground state |g〉 ≡ 5S_1/2|F = 2, m = 2〉 to the Rydberg state |r〉 ≡ 60S_1/2 of the cold ⁸⁷Rb atom. Since this atomic transition frequency is in the ultraviolet region, it conventionally employs 780-nm and 480-nm lasers via the intermediate state |e〉 ≡ 5P_3/2|F = 3, m = 3〉 under two-photon resonance condition [39]. In our scheme, only 780-nm lasers can work. The detuning between atomic transition frequency and 780-nm laser frequency is about Δ = 6.25 × 10⁸ MHz, which is in the far-off-resonant regime. The two coupling strengths we employ are Ω₁ = 2π × 30 MHz and Ω₂ = 2π × 300 MHz. We can set t₁ ≃ t₂ ≃ 5.03 ps, and the addressing period T ≃ 10.06 ps. Note that the period of driving field should be modulated with high precision in the far-off-resonant regime. As a result the total time of realizing this transition is about 2.9 μs.

The other possible application is to achieve long-lived excited nuclear states. The transition energy between the nuclear states is much larger than the energy of X-ray lasers. To compensate the energy difference between nuclear transition and X-ray laser, one envisages accelerated nuclei interacting with x-ray laser pulses. In the strong acceleration regime, the resonance condition cannot be satisfied very well for requiring high degree of accuracy of relativistic factor [11,40]. As a result it significantly influences population transfer. Our calculations shows that this problem can be effectively solved by a series of Gaussian pulses. Take the E1 transition in ²²³Ra as an example. It is impossible to transfer the population to excited state with only a single pulse when the detuning Δ is about 0.1 eV [11]. However, if we take a laser intensity I = 9 × 10²² W/cm² and the width of Gaussian pulse is 31 fs, the transition can be obtained by 5 Gaussian pulses.

In conclusion, a scheme to realize population transfer by far-off-resonant drivings is proposed. By two sequentially applied lasers with different frequency or intensity, population transfer can be perfectly achieved in two-level systems. This proposal can be extended to N-level systems and to the case with cavity fields instead of classical fields. Furthermore, it can be exploited to achieve population transfer in the Rabi model regardless the form of bosonic field, which has been experimentally explored in the photonic analog simulator [41].

The scheme can be applied to population transfer from ground state to Rydberg states or Rydberg state preparation. In X-ray quantum optics [42], it may also find applications, since the lack of γ-ray lasers and huge transition energy inside the nuclei. This scheme can work with both square-well pulse and Gaussian pulse, and it is robust against dissipation and dephasing in the sense that the life-time of Rydberg states as well as the nuclear excited state is long. By manipulating the system parameters, we can make the system stay at the excited level longer. This work paves a new avenue in preparation and manipulation of quantum states with off-resonant driving fields, and might find potential applications in the field of quantum optics.

Appendix: Three-level system in periodic square-well fields

We now demonstrate how to apply the periodic modulation to the other structure of quantum systems–three-level system. As shown in Fig. 11(a), we consider a Λ-type three-level system (it also works in V -type system) that two levels coupled to the third with coupling constants Ω₁ and Ω₂, respectively. Δ denotes the detuning in this system. Without loss of generality, we assume Ω₁ and Ω₂ to be real. In the interaction picture, the Hamiltonian in the Hilbert space spanned by {|0〉, |1〉, |2〉} reads

(15)$$H=\left(\begin{array}{ccc}0& 0& {\mathrm{\Omega }}_1\\ 0& 0& {\mathrm{\Omega }}_2\\ {\mathrm{\Omega }}_1& {\mathrm{\Omega }}_2& \mathrm{\Delta }\end{array}\right).$$

The evolution operator becomes

(16)$$U(t,0)=e^{-iHt}=e^{-\frac{i\mathrm{\Delta }t}{2}}\left(\begin{array}{ccc}{B}_1& B_2& B_4\\ B_2& B_3& B_5\\ B_4& B_5& B_6\end{array}\right),$$

where $B_1=\frac{{\mathrm{\Omega }}_2^2}{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2}{e}^{\frac{i}{2}\mathrm{\Delta }t}+\frac{{\mathrm{\Omega }}_1^2}{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2}\left(\text{cos}\frac{yt}{2}+i\frac{\mathrm{\Delta }}{y}\text{sin}\frac{yt}{2}\right)$, $B_2=-\frac{{\mathrm{\Omega }}_1{\mathrm{\Omega }}_2}{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2}{e}^{\frac{i}{2}\mathrm{\Delta }t}+\frac{{\mathrm{\Omega }}_1{\mathrm{\Omega }}_2}{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2}\left(\text{cos}\frac{yt}{2}+i\frac{\mathrm{\Delta }}{y}\text{sin}\frac{yt}{2}\right)$, $B_3=\frac{{\mathrm{\Omega }}_1^2}{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2}{e}^{\frac{i}{2}\mathrm{\Delta }t}+\frac{{\mathrm{\Omega }}_2^2}{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2}\left(\text{cos}\frac{yt}{2}+i\frac{\mathrm{\Delta }}{y}\text{sin}\frac{yt}{2}\right)$, $B_4=-2i\frac{{\mathrm{\Omega }}_1}{y}\text{sin}\frac{yt}{2}$, $B_5=-2i\frac{{\mathrm{\Omega }}_2}{y}\text{sin}\frac{yt}{2}$, $B_6=\text{cos}\frac{yt}{2}-i\frac{\mathrm{\Delta }}{y}\text{sin}\frac{yt}{2}$, and $y=\sqrt{4{\mathrm{\Omega }}_1^2+4{\mathrm{\Omega }}_2^2+{\mathrm{\Delta }}^2}$. When the initial state is |2〉 and in the large detuning limit (i.e., Δ ≫ Ω₁, Ω₂), we find that B₄ ≃ B₅ ≃ 0 in the evolution operator. Thus the population transfer can not be realized in this case.

 

Fig. 11 The level configuration of a three-level system.

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Now we turn to study the system dynamics under periodic frequency modulation. That is, the Hamiltonian is H_a in the time interval [0, t₁] and H_b in the time interval (t₁, T], where $H_j=\left(\begin{array}{ccc}0& 0& {\mathrm{\Omega }}_1\\ 0& 0& {\mathrm{\Omega }}_2\\ {\mathrm{\Omega }}_1& {\mathrm{\Omega }}_2& {\mathrm{\Delta }}_j\end{array}\right)$, j = a, b. Note that both Δ_a and Δ_b are chosen to be very large, so the system dynamics is frozen with only a single driving. The evolution operator of one period can be calculated as

(17)$$U(T,0)=e^{-i{H}_b{t}_2}{e}^{-i{H}_a{t}_1}=\frac{e^{-\frac{i}{2}\left({\mathrm{\Delta }}_a{t}_1+{\mathrm{\Delta }}_b{t}_2\right)}}{y_a{y}_b}\left(\begin{array}{ccc}{B}_1^{\prime }& B_2^{\prime }& B_4^{\prime }\\ B_2^{\prime }& B_3^{\prime }& B_5^{\prime }\\ -B_4^{\prime }& -B_5^{\prime }& B_6^{\prime }\end{array}\right),$$

where $B_1^{\prime }=\frac{1}{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2}\left[y_a{y}_b{\mathrm{\Omega }}_2^2{e}^{\frac{i}{2}\left({\mathrm{\Delta }}_a{t}_1+{\mathrm{\Delta }}_b{t}_2\right)}-{\mathrm{\Omega }}_1^2\left(4{\mathrm{\Omega }}_1^2+4{\mathrm{\Omega }}_2^2+{\mathrm{\Delta }}_a{\mathrm{\Delta }}_b\right)\right]$, $B_2^{\prime }=-\frac{1}{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2}\left[y_a{y}_b{\mathrm{\Omega }}_1{\mathrm{\Omega }}_2{e}^{\frac{i}{2}\left({\mathrm{\Delta }}_a{t}_1+{\mathrm{\Delta }}_b{t}_2\right)}+{\mathrm{\Omega }}_1{\mathrm{\Omega }}_2\left(4{\mathrm{\Omega }}_1^2+4{\mathrm{\Omega }}_2^2+{\mathrm{\Delta }}_a{\mathrm{\Delta }}_b\right)\right]$, $B_3^{\prime }=\frac{1}{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2}\left[y_a{y}_b{\mathrm{\Omega }}_1^2{e}^{\frac{i}{2}\left({\mathrm{\Delta }}_a{t}_1+{\mathrm{\Delta }}_b{t}_2\right)}-{\mathrm{\Omega }}_2^2\left(4{\mathrm{\Omega }}_1^2+4{\mathrm{\Omega }}_2^2+{\mathrm{\Delta }}_a{\mathrm{\Delta }}_b\right)\right]$, B′₄ = 2Ω₁(Δ_b − Δ_a), B′₅ = 2Ω₂(Δ_b − Δ_a), and $B_6^{\prime }=-4\left({\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2\right)-{\mathrm{\Delta }}_a{\mathrm{\Delta }}_b$. We have set y_at₁ = y_bt₂ = π, $y_j=\sqrt{4{\mathrm{\Omega }}_1^2+4{\mathrm{\Omega }}_2^2+{\mathrm{\Delta }}_j^2}$, j = a, b. Since the evolution operator is time-dependent, it is difficult to calculate the system dynamics analytically for n evolution periods. Further observations demonstrate that if the system state |Ψ(t)〉 = c₀(t)|0〉 + c₁(t)|1〉 + c₂(t)|2〉 satisfies the condition $\frac{c_0(t)}{c_1(t)}=\frac{{\mathrm{\Omega }}_1}{{\mathrm{\Omega }}_2}$, the evolution operator would be time-independent. Then the evolution operator after n evolution periods becomes

$$U(t,0)=U(nT,0)=\frac{e^{-\frac{in}{2}\left({\mathrm{\Delta }}_a{t}_1+{\mathrm{\Delta }}_b{t}_2\right)}}{{\left(y_a{y}_b\right)}^n}\left(\begin{array}{ccc}\frac{{\mathrm{\Omega }}_1^2}{2\left({\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2\right)}\left(s_1^n+s_2^n\right)& \frac{{\mathrm{\Omega }}_1{\mathrm{\Omega }}_2}{2\left({\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2\right)}\left(s_1^n+s_2^n\right)& \frac{i{\mathrm{\Omega }}_1}{2\sqrt{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2}}\left(s_1^n-s_2^n\right)\\ \frac{{\mathrm{\Omega }}_1{\mathrm{\Omega }}_2}{2\left({\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2\right)}\left(s_1^n+s_2^n\right)& \frac{{\mathrm{\Omega }}_2^2}{2\left({\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2\right)}\left(s_1^n+s_2^n\right)& \frac{i{\mathrm{\Omega }}_2}{2\sqrt{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2}}\left(s_1^n-s_2^n\right)\\ \frac{-i{\mathrm{\Omega }}_1}{2\sqrt{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2}}\left(s_1^n-s_2^n\right)& \frac{-i{\mathrm{\Omega }}_2}{2\sqrt{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2}}\left(s_1^n-s_2^n\right)& \frac{1}{2}\left(s_1^n+s_2^n\right)\end{array}\right),$$

where s₁ = d₁ + id₂ = |s|e^iφ, s₂ = d₁ − id₂ = |s|e^−iφ, $\text{tan}\phi =\frac{d_2}{d_1}$, $\left|s\right|=\sqrt{d_1^2+d_2^2}$, $d_1=-4{\mathrm{\Omega }}_1^2-4{\mathrm{\Omega }}_2^2-{\mathrm{\Delta }}_a{\mathrm{\Delta }}_b$, and $d_2=2({\mathrm{\Delta }}_a-{\mathrm{\Delta }}_b)\sqrt{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2}$. When the initial state is |Ψ(0)〉 = |2〉, the final state becomes

(18)$$|\mathrm{\Psi }(t)〉=U(nT,0)|\mathrm{\Psi },0〉=𝒩\left(\begin{array}{c}{\mathrm{\Omega }}_1\text{sin}(n\phi )\\ {\mathrm{\Omega }}_2\text{sin}(n\phi )\\ -i\sqrt{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2}\text{cos}(n\phi )\end{array}\right),$$

where 𝒩 is a normalization constant. Note that whether the system evolves or not depends only on the phase φ, regardless of the large detuning condition, and the Rabi-like oscillation frequency is determined by the phase φ.

We next demonstrate how to realize an arbitrary coherent superposition state |Ψ_T〉 = cosΘ|0〉 + sinΘ|1〉 in the Λ-type (or V -type) three-level system. We regulate the detunings Δ_a and Δ_b and fix the coupling constants Ω₁ and Ω₂. Especially, the ratio of Ω₁/Ω₂ is modulated to equal tanΘ in |Ψ_T〉. Again, we study in the large detuning regime. According to Eq. (18), if we set $N\phi =\frac{\pi }{2}$, the system would arrive at the coherent superposition state |Ψ_T〉 and the total time of achieving |Ψ_T〉 is about

(19)$$𝒯=\frac{\pi }{2{\text{tan}}^{-1}\left|\frac{2({\mathrm{\Delta }}_a-{\mathrm{\Delta }}_b)\sqrt{{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2}}{4{\mathrm{\Omega }}_1^2+4{\mathrm{\Omega }}_2^2+{\mathrm{\Delta }}_a{\mathrm{\Delta }}_b}\right|}\left(\frac{\pi }{\sqrt{4{\mathrm{\Omega }}_1^2+4{\mathrm{\Omega }}_2^2+{\mathrm{\Delta }}_a^2}}+\frac{\pi }{\sqrt{4{\mathrm{\Omega }}_1^2+4{\mathrm{\Omega }}_2^2+{\mathrm{\Delta }}_b^2}}\right).$$

For comparison, we plot the system dynamics under constant driving field with the detuning Δ/Ω₁ = 50 in Fig. 12(a) and Δ/Ω₁ = 100 in Fig. 12(b), where the fixed coupling constants satisfy Ω₂/Ω₁ = 2. One can find that the system dynamics is frozen due to the large detuning condition. However, as shown in Fig. 12(c), it is completely different in the frequency modulation, where two tunable detunings are also Δ_a/Ω₁ = 50 and Δ_b/Ω₁ = 100. Namely, the population transfer can be realized in the later case.

 

Fig. 12 The population as a function of time where the initial state is |2〉. (a) Δ/Ω₁ = 50, Ω₂/Ω₁ = 2. (b) Δ/Ω₁ = 100, Ω₂/Ω₁ = 2. (c) Δ_a/Ω₁ = 50, Δ_b/Ω₁ = 100, Ω₂/Ω₁ = 2.

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Note that our theory can also be generalized into N-level systems, where one level couples to the reminder levels with the corresponding coupling constants Ω₁, Ω₂,..., Ω_N, and all the detunings are Δ. By a similar procedure shown above in the frequency modulation, we can obtain the following coherent superposition state,

(20)$$|{\mathrm{\Psi }}_T〉=𝒩\left({\mathrm{\Omega }}_1|0〉+{\mathrm{\Omega }}_2|1〉+\cdots +{\mathrm{\Omega }}_N|N-1〉\right),$$

where the coefficients Ω₁, Ω₂,..., Ω_N are determined by coupling constants and 𝒩 is a normalization constant.

Next, we show how to modulate the detunings Δ or δ to realize population transfer |0〉 → |2〉 in Fig. 11(b). Numerical simulation results are presented in Fig. 13. We regulate two detunings Δ_α and Δ_β in Fig. 13(a), where two Hamiltonians read $H_j=\left(\begin{array}{ccc}0& 0& {\mathrm{\Omega }}_1\\ 0& \delta & {\mathrm{\Omega }}_2\\ {\mathrm{\Omega }}_1& {\mathrm{\Omega }}_2& {\mathrm{\Delta }}_j\end{array}\right)$, j = α, β. Fig. 13(b) demonstrates the dynamics by regulating two detunings δ_α and δ_β, where two Hamiltonians read $H_j=\left(\begin{array}{ccc}0& 0& {\mathrm{\Omega }}_1\\ 0& {\delta }_j& {\mathrm{\Omega }}_2\\ {\mathrm{\Omega }}_1& {\mathrm{\Omega }}_2& \mathrm{\Delta }\end{array}\right)$, j = α, β. It is shown that the system evolves with time in both modulations, and one can obtain population transfer |0〉 → |2〉 by choosing the evolution time 𝒯.

 

Fig. 13 The population as a function of time in frequency modulation. The initial state is |0〉. (a) Δ_α/Ω₁ = 40, Δ_β/Ω₁ = 20, δ/Ω₁ = 10, Ω₂ = Ω₁. (b) δ_α = 0, δ_β/Ω₁ = 30, Δ/Ω₁ = 10, Ω₂/Ω₁ = Ω₁.

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We should emphasize that the population transfer |2〉 → |1〉 can also be realized by intensity modulation. That is, the Hamiltonian is $H_1^{\prime }=\left(\begin{array}{ccc}0& 0& {\mathrm{\Omega }}_1\\ 0& \delta & {\mathrm{\Omega }}_2\\ {\mathrm{\Omega }}_1& {\mathrm{\Omega }}_2& \mathrm{\Delta }\end{array}\right)$ in the time interval [0, t₁], while the Hamiltonian is $H_2^{\prime }=\left(\begin{array}{ccc}0& 0& {\mathrm{\Omega }}_1\\ 0& \delta & {\mathrm{\Omega }}_2^{\prime }\\ {\mathrm{\Omega }}_1& {\mathrm{\Omega }}_2^{\prime }& \mathrm{\Delta }\end{array}\right)$ in the time interval (t₁, T]. The results are plotted in Fig. 14. If we put the three-level atom in a cavity, the transition between |0〉 ↔ |2〉 is coupled by the cavity field with the detuning Δ, and the transition between |1〉 ↔ |2〉 is coupled by classical field with the detuning Δ − δ. The Hilbert space of the atom-cavity system is spanned by {|0, 1〉, |2, 0〉, |1, 0〉}, where |m, n〉 denotes the atom state |m〉 and n photons in the cavity. For the initial state |0, 1〉, by combining the frequency and intensity modulation of classical field to reach the state |1, 0〉, one can use it to realize single photon storing or releasing.

 

Fig. 14 The population as a function of time in intensity modulation. The initial state is |2〉, Δ/Ω₁ = 40, δ = 0, Ω₂ = Ω₁, Ω′₂/Ω₁ = −1.

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Acknowledgments

We thank Y. M. Liu for helpful discussions. This work is supported by the National Natural Science Foundation of China (NSFC) under Grants No.11534002 and No.61475033.

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