1. Introduction

Producing samples of atoms or molecules whose population resides almost entirely in a single desired quantum state is the goal of many important applications of the quantum chemistry, quantum optics, mechanical manipulation and cooling of atoms or molecules by laser radiation. On the other hand, the coherent superpositions of quantum states also play an important role in the above mentioned fields of science with very interesting and useful applications. The population trapping [1] and laser cooling below the recoil limit [2], construction of atomic beam-splitters based on the dark states [3], using the dressed state with zero quasienergy for complete transfer of the atomic population from one ground state into the other one without populating of the intermediate excited state in the scheme of the stimulated Raman adiabatic passage (STIRAP) [4], enhancement of the refractive index of a resonant medium using the appropriate coherent superposition of the ground states [5] is a list of important applications of the coherent superposition quantum states.

A coherent superposition of quantum states may be created, in principle using a radio frequency field coupling these states. Alternatively, the STIRAP technique in a three-level Λ-atom may be modified by maintaining a fixed ratio of Stokes and pump pulse amplitudes at the end of the interaction to create a definite superposition of the ground states as suggested in Refs. [6,7]. This method, however requires an accurate control of the relative strength of Stokes and pump pulses and, for example inhomogeneous transverse intensity distribution of the laser beams may cause problems. A method of generation of coherent superposition of long lived quantum states robust against small variations of the time and space parameters of the laser pulses was suggested in Ref [8]. This method is based on the usual three-level STIRAP scheme with coupling of the intermediate excited state to a forth metastable state by a third (control) pulse. The final coherent superposition of the ground states in this tripod-linked system can be governed by adjusting of the relative delay of the control pulse. The relative phase of the components of the superposition is determined by the relative phase of the Stokes and pump pulses.

We propose in this paper an alternative and more simple scheme for generation of arbitrary coherent superposition of two ground states of a three-level • • atom by using frequency-chirped short bichromatic laser pulses (BLP) each being superposition of two pulses of a same shape with different carrier (and, in general different Rabi) frequencies being in Raman resonance with the atom under the conditions of the adiabatic passage (AP) regime of interaction [9]. Each laser pulse of the BLP couples corresponding ground states of the • • atom to a common excited state, as it is shown in Fig. 1. Due to the identical shape and chirp of the pulses and the assumed condition of the Raman resonance, the system under consideration is equivalent to a two-state system consisting of a “bright” superposition of the two ground states which is coupled to the excited state by the laser field and of a decoupled “dark” superposition of the ground states which doesn’t interact with the laser field. As is well known, a frequency-chirped laser pulse produces complete transfer of populations between the states of a two-state atom in AP regime of interaction [9]. In our case, the frequency-chirped BLP produces complete population transfer from the “bright” superposition to the excited state in a time short compared to the relaxation times of the atom leaving unchanged the population of the "dark" superposition state. Since the “bright” and “dark” superposition states depend on the relative phase and relative strength of the pulses forming the BLP, this dependence will be transferred to the population of the excited state. In the same time an arbitrary coherent superposition of the ground states (an arbitrary Raman or Zeeman coherence) can be generated in the “dark” superposition of the ground states. The population of the excited state in a second step has to be transferred in a fast way into a nondecaying storage level (level ∣ 4⟩ in Fig. 1) by a subsequent frequency-chirped laser pulse to avoid the decoherence induced by the relaxation processes. It is important to note that the nondecaying storage level ∣ 4⟩ in our scheme is being coupled to the excited state ∣ 2⟩ only after the action of the BLP and doesn’t play any role during the interaction of BLP with the three-level • • atom. So, our scheme differs significantly from the tripod-linked scheme of Ref. [8].

Fast and robust writing, storage and reading of optical information is a task to be solved nowadays, especially when short laser pulses are used as the carriers of the optical information. The sensitivity of the population transfer in a multilevel quantum system to the relative phases of exciting laser fields considered above (see also [6–8,10,11]) may be used to solve this problem.

We propose in this paper to use the suggested above scheme of generation of superposition states which are sensitive to the relative phase and relative strength of pulses forming BLP for coherent writing, storage and reading of the phase or intensity optical information. The information which is assumed to be contained in the relative phase or relative amplitude of the pulses of the BLP is being written in the population of the excited state when the population of the “bright” superposition of the ground states is being completely transferred into the excited state by the frequency-chirped BLP. The subsequent frequency-chirped short laser pulse is used to transfer the population of the excited state (and so, the information written into the population of this state) into the storage nondecaying level (level ∣ 4⟩ in Fig. 1). The same laser pulse can be used for reading out the stored information by exciting the atom from level ∣ 4⟩. It is worth noting that the information writing and reading processes are fast and robust in the proposed scheme due to the shortness of the applied laser pulses and the robustness of the population transfer produced by the frequency-chirped pulses in the AP regime of interaction.

As was mentioned above, the three-level problem under consideration reduces to the interaction of a single laser pulse with an equivalent two-level system. There exists a number of exact analytic solutions to the problem of interaction of short laser pulses (with chirped frequencies as well) with two-level systems for certain pulse shapes and modulations in time [11,12]. We use here the simpler approximate solutions corresponding to the AP regime [9,13].

2. The mathematical formalism

We consider interaction of BLP consisting of two linearly polarised laser pulses having same shape f(t) with a three-level Λ-atom having two ground states ∣ 1⟩ and ∣ 3⟩ and excited state ∣ 2⟩, as it is shown in Fig. 1, where also is depicted a nondecaying (metastable) state ∣ 4⟩ which is assumed to be not coupled to the λ -system by BLP. This state will be used only for transfer of the population of the excited state (for information storage, see below) at a second step, by applying a short frequency-chirped laser pulse coupling the states ∣ 2⟩ and ∣ 4⟩ after the interaction with the BLP.

 

Fig.1. The scheme of the atomic system.

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The laser pulses forming the BLP have complex amplitudes A _1,2(t) = f(t)$A_{1,2}^{\left(0\right)}$; $A_{1,2}^{\left(0\right)}$ = $A_{1,2}^{\left(0\right)}$ exp[iΦ_1,2] and time-dependent frequencies ω _L1(t) and ω _L2(t). The pulse A ₁(t) couples the states ∣1⟩ and ∣2⟩, and the pulse A ₂(t) couples the states ∣3⟩ and ∣2⟩. The carrier frequencies of both pulses of the BLP are assumed to be chirped in time in the same way and the BLP duration is assumed to be much shorter than all relaxation times of the atom. It allows as to deal with the Schrödinger equation for the amplitudes a_j , j = 1,2,3 of the states of the • • atom in Fig. 1.

It is useful (for reduction of the three-state problem to a two-state one) to introduce a vector C, whose components c_i ,i = 1,2,3 are proportional to the amplitudes a_i ,i = 1,2,3 of the atomic states:

where the Rabi frequencies 2Ω₁₂(t) and 2Ω₃₂(t) with complex amplitudes Ω₁ and Ω₂ are introduced: ${\Omega }_{12}\left(t\right)=f\left(t\right){\Omega }_1=f\left(t\right)\frac{d_{12}}{2ħ}{A}_1^{\left(0\right)}$ and ${\Omega }_{32}\left(t\right)=f\left(t\right){\Omega }_2=f\left(t\right)\frac{d_{32}}{2ħ}{A}_2^{\left(0\right)}$ with d_ij being the dipole moment matrix element for the laser induced transition from state ∣ j⟩ to state ∣i⟩, (i,j = 1,2,3). ε ₂₁(t) = ω _L1(t)-ω ₂₁, ε ₂₃(t) = ω _L2(t)-ω ₂₃ are the detunings from the one-photon resonances with ω ₂₁ and ω ₂₃ being the resonant transition frequencies between the corresponding states. We assume, that the condition of the Raman resonance has been fulfilled: ε ₂₁(t) = ε ₂₃(t) = ε(t). In what follows, we assume the same linear temporal chirp of the carrier frequencies of the pulses forming BLP: ω_Lj (t) = ${\omega }_{\mathit{\text{Lj}}}^{\left(0\right)}$ +2βt, j = 1,2, where ${\omega }_{\mathit{\text{Lj}}}^{\left(0\right)}$ are the central frequencies and 2β is the speed of the chirp, see Fig.1.

We obtain the following equation for the vector C from the Schrödinger equation for the probability amplitudes a_j , j = 1,2,3, [12]:

The Hamiltonian Ĥ in the rotating wave approximation is:

One can introduce the following g ⁽⁺⁾ and g ^(-) amplitudes of the coupled to the excited state by the BLP “bright” and uncoupled “dark” superpositions of the ground states having amplitudes a ₁ (c ₁) and a ₃ (c ₃):

and the excited state amplitude e = c ₂ .

We obtain the following set of equations for the new state amplitudes using Eq.(1):

where $F\left(t\right)=f\left(t\right)\sqrt{{\mid {\Omega }_1\mid }^2+{\mid {\Omega }_2\mid }^2}.$

The first two equations in Eq.(2) describe an equivalent two-level system with the “bright” ground state amplitude g ⁽⁺⁾ being coupled to the excited state with amplitude e by the BLP. The “dark” ground state with amplitude g ^(-) does not interact with the BLP, as it follows from the third equation in Eq.(2).

It is well known that complete transfer of the populations of two-level atom takes place as a result of interaction with a frequency-chirped laser pulse in the AP regime [9]. It means that the population of the “bright” state will be completely transferred to the excited state and we have for the final amplitude $g_{\mathit{\text{fin}}}^{(+)}$ at the end of the laser pulse:

if the atom initially was in the ground state: e_in =c _2in =0. The subscripts in and fin stand for the initial (t- > -∞) and the final (t- > ∞) values.

We obtain for the probability amplitudes a _1fin and a _3fin of the ground states of our original three-level Λ-atom at the end of the laser pulse using Eq.(3):

In the simplest case of the same Rabi frequencies (∣Ω₁∣ = ∣Ω₂∣) of the laser pulses, we obtain for the final populations n_ifin and the phases ϕ_ifin (i=1,3) of the ground states:

where the populations and phases of the states are introduced as follows:

a_j = √n_j exp[iϕ_j ], j = 1,2,3 and ΔΦ₁₂ = Φ₁ -Φ₂ is the relative (difference) phase of the laser pulses.

We obtain for the final population n _2fin of the excited state after interaction with the BLP (in the case of ∣Ω₁∣ = ∣Ω₂∣):

As it follows from Eq.(6), the probability of excitation of the Λ- atom by frequency-chirped BLP in AP regime strongly depends on the relative phases Δϕ _13in and ΔΦ₁₂, when the initial state of the Λ-atom is a superposition of the ground states ∣1⟩ and ∣3⟩.

The dependence of the final population n _2fin on the initial phase Φ_in = Δϕ _13in + ΔΦ₁₂ is depicted in Fig. 2.

 

Fig.2. Dependence of the excited state population n _2fin on the phase Φ_in for different values of population n _1in of the ground state ∣1⟩: n _1in = 1 (1); .8 (2); .6 (3); .5 (4).

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There is complete transfer of the populations of the Λ-atom from the ground states to the excited state: n _2fin = 1, when the initial phase Φ_in = 0 and • there is no excitation at all, n _2fin = n _2in = 0 (dark state) at Φ_in =π in the case of equal initial population of the ground states, when n _1in = n _3in = 1/2.

The population of the excited state does not depend on phases when the atom is in a single ground state initially (n _1in = 1): n _2fin - 1/2 and n _1fin = n _3fin = 1/4.

As it follows from Eq.(4), we have in the case of the different Rabi frequencies (∣Ω₁∣ ≠ ∣Ω₂∣) and the atom in the ground state ∣1⟩ initially, n _1in = 1:

which means that the final population of the excited state is a function of the relative strengths of the pulses forming the BLP.

3. Results of the numerical simulations

We have solved numerically the set of Eqs.(1) for BLP having durations much shorter than the relaxation times of the system to verify the above obtained conclusions.

The envelope of the BLP has been taken as a Gaussian function: f(t) = exp[-t ²/2${\tau }_L^2$] (with t_L being the duration of the BLP), and the same linear chirp has been assumed for the pulses forming the BLP.

The dependence of the population of the excited state on the phase Φ_in = Δϕ _13in + ΔΦ₁₂ is clearly seen in Figs.3 for the Λ-atom being initially in the superposition of the ground states: There is effective excitation of the atom when Φ_in = 0 (Fig. 3a) and the excitation of the atom is suppressed when Φ_in =π (Fig. 3b). Note that in the latter case we have a “dark” state in case of equal initial populations of the two ground states (for ∣Ω₁∣ = ∣Ω₂∣).

 

Fig.3. Time dependence of the populations for n _1in =.7,n _3in =.3 at: (a) Φ_in =0, (b) Φ_in =π and ∣Ω₁∣ = ∣Ω₂∣ = Ω. The parameters applied are: Ωτ_L = 5, β ${\tau }_L^2$ = 5 ; green-n ₁(t), blue-n ₃(t), red-n ₂(t).

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4. Writing/reading of phase and intensity information

The atomic system under consideration may be used for construction of arbitrary coherent superpositions of the ground states and for writing/reading and storage of phase information. As it follows from Eq.(5), the relative phase of the probability amplitudes of the ground states is equal to Δϕ _13fin = π + ΔΦ₁₂, at the end of the BLP. So, an arbitrary phase difference Δϕ _13in (arbitrary value of the Raman, or Zeeman coherence) may be generated by controlling the phase difference ΔΦ₁₂ of the pulses forming the BLP.

As it follows from Eq.(6), the population n _2fin of the excited state is a function of the phase ΔΦ₁₂. So, the phase information contained in the BLP may be written in the population n _2fin of the excited state if the atoms are prepared initially in the same coherent superposition of the ground states, for e.g., in a “dark” state (with the same initial parameters n _1in, n _3in and Δϕ _13in). This information however will be distorted due to the spontaneous decay from the excited state during the decay time of this state. One of the way to preserve this information is to transfer the population of the excited state ∣2⟩ (with the information written therein) into the additional nondecaying metastable state ∣4⟩, see Fig. 1, by acting with a subsequent frequency-chirped short laser pulse in the AP regime of interaction. Reading of the phase information stored in this nondecaying state may be produced by acting by the same chirped laser pulse which transforms the phase information into the population of the excited state. The latter may be detected for example, by analysing the spontaneous or stimulated emission from this state.

According to Eq.(7), where the atoms are assumed to be optically pumped into the ground state ∣1⟩ initially, the population of the excited state resulting from the interaction with the frequency-chirped BLP is a function of the relative strength ∣Ω₂∣²/ ∣Ω₁∣². So, optical intensity information (image) contained in the relative intensity of the pulses forming the BLP may be written into the population n _2fin of the excited state. One have to store this information (image) to preserve it from distortion due to the spontaneous decay by transfer the population of the excited state into the nondecaying state (for e.g., into the state ∣4⟩ in Fig. 1). This may be done just like to the case of the considered above phase information storage using a subsequent frequency-chirped short laser pulse. The reading of the stored information may be produced also in the same way as that one of the phase information reading.

5. Conclusions

In conclusion, the results of analysis of the interaction of short frequency-chirped BLP with three-level A-atom has been presented. The dependence of the excitation probability on the relative phase and relative strength of the pulses forming the BLP has been proposed in this paper to be used for the generation of coherent superpositions of the ground states with arbitrary controllable values of Raman or Zeeman coherences, as well as for writing and reading of phase or intensity optical information. The physics of the proposed technique is as follows. The frequency-chirped short BLP in the AP regime of interaction produces complete transfer of population of the “bright” superposition of the ground states into the excited state leaving unchanged the population of the “dark” superposition state. Since the “bright” and “dark” states depend on the relative phase and relative strength of the BLP’s pulses, the phase and intensity information contained in the BLP are transferred to the excited state with the population of the “bright” state or are written in the population of the “dark” superposition in the ground state. This information writing is produced by BPL with duration much shorter compared to the atomic relaxation times.

It has been shown, that the atoms have to be prepared in the coherent superposition of the ground states initially before using them for writing the phase information. They have to be prepared in one of the ground state to be used for the intensity information writing. The storage of information in both cases is produced in a second step by a subsequent frequency-chirped short laser pulse which in a fast way transfers the population of the excited state (and hence, the information written therein) into one of the nondecaying state of the atom. The reading of the stored information is produced by excitation of atoms by the same frequency-chirped short laser pulse from this state.

It is worth noting that the information writing and reading are fast and robust in the proposed scheme. These processes have time scales equal to those of the laser pulses, which durations may be chosen to be very short. The restriction on the duration of the laser pulses is mainly connected with the conditions for the AP regime of interaction [9,13]. The robustness of these processes is coming from the robustness of the population transfer in quantum systems produced by the frequency-chirped laser pulses in the AP regime of interaction: It is well known that the effectiveness of this transfer may be near to 100% and is insensitive to the shapes and the transverse intensity distributions of the laser pulses, as well as to exact resonance conditions.