1. Introduction

Efficient control of atomic coherence and population transfer is a basic requirement for many experimental studies in the field of quantum optics and therefore has attracted great attention in the past two decades. Accordingly, a lot of significant progress has been made in different parametric regions corresponding, respectively, to coherent population trapping [1], electromagnetically induced transparency [2], laser without population inversion [3], spontaneous emission cancellation [4], and spectral line narrowing [5]. So far stimulated Raman adiabatic passage (STIRAP) [6] has been proven to be an efficient and robust technique for the complete population transfer from one state to another state of atoms or molecules. On the other hand, fractional STIRAP (a slightly modified version of STIRAP) can be explored to transfer a part of population from one atomic state into another atomic state and simultaneously create an arbitrary coherence between them [7]. As is well known, many works on STIRAP or fractional STIRAP are done with the simplest three-level Λ system to attain different goals [8,9]. Apart from the three-level Λ system, the four-level tripod system is also extensively studied in theory [10,11] or in experiment [12,13] as far as STIRAP or fractional STIRAP is concerned.

Recently the STIRAP or fractional STIRAP technique has been extended by several groups to the three-level ladder system, which has potential applications in achieving short wavelength lasing and quantum logic gates based on the dipole blockade effect [14]. In particular, Sangouard et al. proposed an interesting technique to prepare coherent superpositions of two non-degenerate quantum states in a three-level ladder system driven by two coherent light pulses [15]. This technique utilizes an adiabatic passage assisted by dynamic Stark shifts induced by a third laser field and can be applied to enhance certain nonlinear optical processes. Moreover, Camp et al. examined by numerical calculations the population dynamics of a coherently driven three-level ladder system without assuming the adiabaticity of atom-field interaction [16]. Last but not least, Fernandez et al. demonstrated a STIRAP-like process to efficiently control the vibro-rotational states of Na₂ molecules in the case where lifetimes of the intermediate state and the highest state are much shorter than the atom-field interaction duration [17].

In this paper, we present a new scheme for achieving efficient population transfer and coherence generation in a four-level ladder system with the STIRAP technique, the fractional STIRAP technique, and the π pulse technique. This scheme depends on the application of three coherent light pulses with carefully chosen Rabi frequencies, time widths, and delay times. Our numerical results show that, for a realistic four-level ladder system with all spontaneous decay rates suitably considered, about 90% atomic population can be coherently transferred into the highest state from the ground state by combining the STIRAP technique and the π pulse technique; atomic coherence between the ground state and the highest state may exceed 0.43 when the fractional STIRAP technique and the π pulse technique are adopted instead. In particular, the highest state is assumed here to be a Rydberg state so that its atomic population and atomic coherence can be kept for at least several microseconds, which is much longer than the atom-field interaction time (typically several hundreds of nanoseconds). Compared with the three-level ladder system, the four-level ladder system has two special advantages: it can be controlled by modulating the applied light pulses with more freedoms; it has an electric-dipole allowed transition between the ground state and the highest state.

2. Theoretical model and equations

We illustrate in Fig. 1 a four-level ladder-type atomic system, in which a pump pulse of central frequency ω_p and a Stokes pulse of central frequency ω_s may coherently transfer atoms from state |1〉 to state |3〉 without populating state |2〉 via the STIRAP or fractional STIRAP technique. In addition, a π pulse of central frequency ω_π is applied to further transfer atoms from state |3〉 to state |4〉, which will be assumed to be a long-lived Rydberg state in what follows. We also assume that the pump pulse is detuned from transition |1〉 ↔ |2〉 by Δ₂ = ω ₂₁ – ω_p, the Stokes pulse is detuned from transition |2〉 ↔ |3〉 by Δ₃ = ω ₃₂ – ω_s, while the π-pulse is always resonant with transition |3〉 ↔ |4〉 (i.e. ω ₄₃ = ω_π).

 

Fig. 1 Energy-level diagram of a four-level ladder-type atomic system with Ω_p, Ω_s, and Ω_π being, respectively, Rabi frequencies of the applied pump, Stokes, and π pulses.

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For the purpose of efficient population transfer from state |1〉 to state |4〉, we will set the three light pulses mentioned above to have Gaussian profiles in the time domain so that their Rabi frequencies can be described by ${\Omega }_s\hspace{0.17em}\left(t\right)\hspace{0.17em}=\hspace{0.17em}{\Omega }_{s0}{e}^{-t^2/T_s^2}$, ${\Omega }_p\hspace{0.17em}\left(t\right)\hspace{0.17em}=\hspace{0.17em}{\Omega }_{p0}{e}^{-{\left(t\hspace{0.17em}-\hspace{0.17em}{\tau }_p\right)}^2/T_p^2}$ and ${\Omega }_{\pi }\hspace{0.17em}\left(t\right)\hspace{0.17em}=\hspace{0.17em}{\Omega }_{\pi 0}{e}^{-{\left(t\hspace{0.17em}-\hspace{0.17em}{\tau }_{\pi }^2\right)}^2/T_{\pi }^2}$. Here τ_i (i = p, π) is the delay time of the pump pulse or the π pulse relative to the Stokes pulse; T_i (i = s, p, π) is the time width of the Stokes pulse, the pump pulse, or the π pulse; Ω_{i 0} (i = s, p, π) is the peak Rabi frequency of the Stokes pulse, the pump pulse, or the π pulse. Note, in particular, that the STIRAP or fractional STIRAP technique requires a counter-intuitive order of the pump and Stokes pulses with τ_p > 0 and the π pulse should satisfy the following equation

In the framework of the semiclassical theory, under the electric-dipole approximation and the rotating-wave approximation, we can represent the Hamiltonian in the interaction picture as

The master equation of motion for the density operator in an arbitrary multilevel atomic system is usually written as

By expanding Eq. (4), we can easily arrive at the following dynamic equations

Considering that all three coherent fields are laser pulses, we also need the Maxwellian wave equations to numerically examine the dynamic evolution of atomic population and atomic coherence, which finally turn into

3. Numerical results and discussions

3.1. Population transfer

In this subsection, we explore the possibility of achieving the maximal population transfer from the ground state |1〉 to the Rydberg state |4〉 by combining the STIRAP technique and the π pulse technique. The time sequence of three coherent light pulses is shown in Fig. 2(a) while atomic populations ρ ₁₁, ρ ₃₃, and ρ ₄₄ are plotted against time t in Fig. 2(b). As we can see, the the pump pulse is applied after the Stokes pulse with a time delay τ_p in the counter-intuitive order and the full population in state |1〉 is assumed before the pump pulse arrives to trigger the atom-field interaction. When the pump pulse gradually enters the medium, atomic population is adiabatically transferred from state |1〉 to state |3〉 without populating state |2〉. This adiabatic process requires two basic conditions: the Stokes pulse should arrive before the pump pulse and extinguish first; both Stokes and pump pulses should vary slowly enough in time (i.e. have smooth rising and falling edges). The π pulse with a narrower time width (T_π < T_p = T_s) is applied, when the pump and Stokes pulses approximately have the same Rabi frequency [Ω_p(t) ≈ Ω_s(t)], to further transfer atomic population from state |3〉 to state |4〉. It is clear that about 98% (88%) atomic population is finally transferred into state |4〉 for an ideal (realistic) four-level ladder system and the population in state |4〉 decays little in 200 nanoseconds because the lifetime of a high Rydberg state is several tens of microseconds.

 

Fig. 2 (Color online) (a) Time sequence of the Stokes, pump, and π pulses. The pump and π pulses are delayed from the Stokes pulse by τ_p and τ_π, respectively. (b) Time evolution of atomic populations ρ ₁₁, ρ ₃₃, and ρ ₄₄. Dashed curves are plotted for an ideal four-level ladder system with Γ₃₂ = Γ₄₁ = Γ₄₃ = 0; solid curves are plotted for a realistic four-level ladder system with Γ₃₂ = 200 kHz, Γ₄₁ = 10 kHz, and Γ₄₃ = 50 kHz. Other parameters are set as Δ₂ = Δ₃ = 0, Γ₂₁ = 38 MHz, Ω_{p 0} = Ω_{s 0} = 400 MHz, Ω_{π 0} = 40 MHz, T_s = T_p = 80 ns, T_π = 22.15 ns, τ_p = 94 ns, and τ_π = 77 ns.

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The time width T_π and the delay time τ_π of the π pulse are two critical parameters to attain the maximal population transfer. In Fig. 3(a), we show atomic population ρ ₄₄ as a function of the delay time τ_π for an ideal four-level system where spontaneous emissions from state |3〉 and state |4〉 are neglected. The dashed, solid, and dash-dotted curves correspond to different peak Rabi frequencies Ω_{π 0} = 4 MHz, 40 MHz, 400 MHz and therefore different time widths T_π = 221.5 ns, 22.15 ns, 2.15 ns of the π pulse. For a smaller peak Rabi frequency Ω_{π 0} = 4 MHz, atomic population ρ ₄₄ changes much slower when the delay time τ_π is gradually increased and the maximal population ρ ₄₄ ≈ 96% can only be attained with τ_π > 300 ns. This is because a wider π pulse with smaller peak Rabi frequency interacts with the ideal four-level ladder system in a larger time interval and it has to be applied after state |3〉 has accumulated enough population. For the larger peak Rabi frequency (Ω_{π 0} = 40 MHz or Ω_{π 0} = 400 MHz), atomic population ρ ₄₄ is more sensitive to the delay time τ_π before the maximal population ρ ₄₄ ≈ 98% is attained after τ_π > 100 ns. That is, a small adjustment of the delay time τ_π may result in a large change in atomic population ρ ₄₄. In Fig. 3(b), we can find very similar dynamic behaviors for atomic population ρ ₄₄ in a realistic four-level ladder system where spontaneous emissions from state |3〉 and state |4〉 are suitably considered. The only difference lies in that the maximal population ρ ₄₄ is much smaller for Ω_{π 0} = 4 MHz because a part of atomic population ρ ₃₃ returns to state |1〉 due to the nonzero decay rates Γ₃₂, Γ₄₁, and Γ₄₃. Thus to guarantee that the maximal population in state |4〉 is larger than 90%, we have to choose a strong enough π pulse with a suitable delay time τ_π, e.g. Ω_{π 0} = 40 MHz and τ_π = 100 ns.

 

Fig. 3 (Color online) Atomic population ρ ₄₄ as a function of the delay time τ_π. for an ideal four-level ladder system in (a); for a realistic four-level ladder system in (b). Dashed, solid, and dash-dotted curves correspond to Ω_{π 0} = 4 MHz, 40 MHz, and 400 MHz, respectively. Other parameters are the same as in Fig. 2.

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3.2. Coherence generation

In this subsection, we explore the possibility of achieving the maximal atomic coherence between state |1〉 and state |4〉 by combining the fractional STIRAP technique and the π pulse technique. The time sequence of three coherent light pulses is shown in Fig. 4(a) while atomic population and coherence ρ ₁₁, ρ ₃₃, ρ ₄₄, and ρ ₁₄ are plotted against time t in Fig. 4(b). Once again the pump pulse is applied after the Stokes pulse in the counter-intuitive order but they have the same falling edge and extinguish simultaneously instead. This is necessary to transfer only a part of atomic population out of state |1〉 and finally create large enough atomic coherence between state |1〉 and state |4〉. Figure 4(b) shows that, for an ideal four-level ladder system, 50% atomic population is adiabatically transferred into state |3〉 from state |1〉 via the fractional STIRAP technique and then is pumped into state |4〉 by a π pulse with a suitable delay time τ_π. Accordingly, the maximal coherence ρ ₁₄ ≈ 0.5 is generated between state |1〉 and state |4〉. It is worth noting that both rising edge and falling edge of the Stokes pulse are Gaussian, which are however separated by a constant peak Rabi frequency extending from t = −400 ns to t = 0.0 ns [see Fig. 4(a)].

 

Fig. 4 (Color online) (a) Time sequence of the Stokes, pump, and π pulses. The pump and Stokes pulses have the same falling edge while the π pulse is delayed from the pump pulse by τ_π. (b) Time evolution of atomic population and coherence ρ ₁₁, ρ ₃₃, ρ ₄₄, and ρ ₁₄ for an ideal four-level ladder system with Γ₃₂ = Γ₄₁ = Γ₄₃ = 0. Other parameters are set as Δ₂ = Δ₃ = 0, Γ₂₁ = 38 MHz, Ω_{p 0} = Ω_{s 0} = 400 MHz, Ω_{π 0} = 40 MHz, T_s = T_p = 80 ns, T_π = 22.15 ns, τ_p = 0, and τ_π = 125 ns.

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Similar to the dynamic behaviors of atomic population ρ ₄₄ in Fig. 3(a), for a smaller peak Rabi frequency Ω_{π 0} = 4 MHz, atomic coherence ρ ₁₄ in Fig. 5(a) changes very slowly when the delay time τ_π is gradually increased and the maximal coherence ρ ₁₄ ≈ 0.5 can only be attained with a large enough delay time of τ_π > 200 ns. This is also because, when the peak Rabi frequency Ω_{π 0} is smaller, the π pulse will interact with the ideal four-level ladder system in a larger time interval so that its application has to be suitably postponed. For a π pulse with larger peak Rabi frequency (Ω_{π 0} = 40 MHz or Ω_{π 0} = 400 MHz), atomic coherence ρ ₁₄ is more sensitive to the delay time τ_π before the maximal coherence ρ ₁₄ ≈ 0.5 is attained after τ_π > 100. Note, in particular, that atomic coherence ρ ₁₄ oscillates drastically with the delay time τ_π between −100 ns < τ_π < 100 ns if the peak Rabi frequency Ω_{π 0} is very large. This peculiar phenomenon should be a result of the fast rising and falling edges of a narrower π pulse, which can remarkably affect the coherent population transfer between state |1〉 and state |3〉 in the case of fractional STIRAP. In Fig. 5(b), we consider instead a realistic four-level ladder system with Γ₃₂ = 200 kHz, Γ₄₁ = 10 kHz, and Γ₄₃ = 50 kHz. As we can see, the maximal coherence ρ ₁₄ is only about 0.43 and corresponds to an optimal delay time τ_π ≈ 100 ns for the realistic four-level ladder system.

 

Fig. 5 (Color online) Atomic coherence ρ ₁₄ as a function of the delay time τ_π for an ideal four-level ladder system in (a); for a realistic four-level ladder system in (b). Dashed, solid, and dash-dotted curves correspond to Ω_{π 0} = 4 MHz, 40 MHz, and 400 MHz, respectively. Other parameters are the same as in Fig. 4.

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4. Conclusions

In summary, we have demonstrated by numerical calculations a new scheme to optimize the population transfer (coherence generation) between the ground state and the highest state in a four-level ladder system with the STIRAP (fractional STIRAP) technique and the π pulse technique. This scheme of efficient population transfer and coherence generation are quite robust in the time scale of several hundreds of nanoseconds when the highest state refers to a long-lived Rydberg state, which however requires an appropriate selection of the time width and the delay time of the π pulse. Our results may be extended to realize short wavelength lasing between the highest state and the ground state. For example, when hyperfine states of rubidium atoms are chosen to construct the four-level ladder system, the transition wavelength between the highest state and the ground state is about 200 nm – 300 nm [18].