Stimulated Raman Adiabatic Passage in a Dense Medium

Abstract

We have considered a coherent population transfer to a higher excited singlet state (𝑆₂) of molecules with anomalous fluorescence in molecular assemblies (e.g., a dense medium). A direct excitation to 𝑆₂ requires light in the UV region. Because of this, the transition is conveniently realized by a two-step (two-photon) process: 𝑆₀→𝑆₁→𝑆₂, where transitions 𝑆₀→𝑆₁ and 𝑆₁→𝑆₂ correspond to the optical region. We have shown that efficient stimulated Raman adiabatic passage (STIRAP) in the ladder configuration can be realized in this case, using suitably chirped pulses, to compensate a change of the two-photon transition frequency in time, induced by the pulses themselves, due to near dipole-dipole interactions. We have provided a reduced state formulation of the optical control process. Chirping the “pump” pulse that excites transition 𝑆₀→𝑆₁ is nonequivalent to chirping the “Stokes” pulse that excites transition 𝑆₁→𝑆₂, with respect to the population of the intermediate state (𝑆₁) in the pulse nonadiabatic regime. We have also shown that with suitably chirped pulses, efficient STIRAP still persists even for a rather large decay of the intermediate state.

1. Introduction

Population transfer to a higher singlet state 𝑆_𝑛 (𝑛>1) of complex organic molecules makes no sense, since more often than not 𝑆_𝑛 (𝑛>1) rapidly relaxes back to the first excited singlet 𝑆₁ [1–4], though excited-state absorption 𝑆₁→𝑆_𝑛 (𝑛>1) has a profound effect on coherent population transfer 𝑆₀→𝑆₁ with shaped laser pulses [5]. The situation reverses for molecules with anomalous fluorescence like azulene, the molecule known as the textbook exception to Kasha's rule [6], and its derivatives. For such molecules, the first excited singlet state 𝑆₁ is very short lived (~1 ps) due to a conical intersection [7] and, as a consequence, a strong nonradiative decay of the transition 𝑆₁→𝑆₀. However, a higher excited singlet state 𝑆₂ is long lived (~1 ns) displaying a strong fluorescence band that corresponds to transition 𝑆₂→𝑆₀ [8–11]. Because of this, there are also reasons to consider a coherent population transfer to higher excited singlet states (𝑆₂) of the molecules under discussion. However, a direct excitation to a higher electronic state 𝑆₀→𝑆₂ requires light in the UV region where pulse shaping presents difficulties. Because of this, a coherent population transfer to 𝑆₂ is conveniently realized by a two-step (two-photon) process, 𝑆₀→𝑆₁→𝑆₂ where transitions 𝑆₀→𝑆₁ and 𝑆₁→𝑆₂ correspond to the optical region. Due to the absence of the center of symmetry in azulene (belongs to the 𝐶_𝑠 point group) and its derivatives, all three transitions 𝑆₀→𝑆₂, 𝑆₀→𝑆₁ and 𝑆₁→𝑆₂ are allowed. In this regard, using stimulated Raman adiabatic passage (STIRAP) [12–16] in the ladder configuration for the population transfer from 𝑆₀ to 𝑆₂ seems rather attractive. Indeed, STIRAP is a simple and powerful technique for complete and robust population transfer in three-state quantum systems. Using this technique, the population is transferred adiabatically from an initially populated state |1⟩ (𝑆₀) to a target state |3⟩ (𝑆₂), which are coupled via an intermediate state |2⟩ (𝑆₁) by two pulsed fields. A unique and very useful feature of STIRAP is that the intermediate state, whose presence is crucial for providing two strongly coupled single-photon transitions, never gets populated, not even transiently. The reason is that throughout the adiabatic evolution of the system, the population remains trapped in an adiabatic dark state, which is a superposition of states |1⟩ and |3⟩ only and does not involve the intermediate state |2⟩ that can be a strongly decaying state (𝑆₁). Such a dark state is formed by maintaining a two-photon resonance between the target |3⟩ and initial |1⟩ states during the interaction.

Azulene is a molecular multilevel system. In this relation, it is worthy to note the STIRAP application of controlling non-Franck-Condon transitions in Na₂ [17] and adiabatic passage by light-induced potentials in polyatomic molecules [18]. The effect of background states on STIRAP was considered by Kurkal and Rice [19] and Cheng et al. [20]. Nakajima et al. [21] discussed population transfer through continuum intermediate states.

In this study, we will concentrate on another aspect of STIRAP in molecular systems. The point is that many important molecular systems are made out of assemblies of coupled localized chromophores. Examples are molecular crystals, organic nanostructures, supramolecular structures, and so forth, which are of interest from the point of view of the design of new optical materials with desirable characteristics. Therefore, in this paper, we aim to apply STIRAP rather to the assembly of interacting azulene molecules (or its derivatives) than to noninteracting molecules. In general, calculating optical properties of interacting molecules becomes an intractable many-body problem even for a reduced state formulation of the optical control process in an isolated molecule which is used in this work (see below). The local field approximation (see, e.g., [22]) provides a simple phenomenological way of relating the optical characteristics of isolated molecules to those of molecular assemblies. In the local field approximation the corresponding equations of motion, describing the system evolution, become nonlinear thus displaying the dependence of the transition frequencies on optical excitation [23, 24]. Here, we will consider a dense medium of the molecules with the dependence of the transition frequencies on optical excitation due to near dipole-dipole (DD) interactions [25]. In the following, we show that efficient STIRAP in the ladder configuration can be achieved in a dense collection of molecules exhibiting anomalous fluorescence, but only if a change of the two-photon transition frequency, due to near DD interactions, is taken into account.

A two-photon resonance between the initial and final states during the interaction can not be achieved since its frequency depends on the population of these states (many-body effect) (see (13) and (20) below). Our goal is to get an efficient STIRAP in this system. We show that one can use pulses that are suitably chirped (time-dependent carrier frequency) to compensate for a change of frequency of the two-photon transition in time induced by the pulses themselves (a similar idea was proposed in studies of a two-state system in relation to Rabi oscillations in inter-subband transitions in quantum wells [26].)

The outline of the paper is as follows. In Section 2, we introduce our model and derive a closed set of equations for the expectation values of one-particle variables in the local field approximation. In Section 3, we show that suitably chirped pulses that compensate a change of frequency of the two-photon transition in time induced by the pulses themselves enable us to achieve a “dark” state and, hence, complete population transfer in the interacting media. We provide numerical solutions of the coupled equations for the expectation values of one-particle variables in Section 4. We summarize our results in Section 5.

2. Model and Basic Equations

Consider a dense collection of molecules, with densities such that there are many molecules within a cubic molecular resonance wavelength, with three electronic states with energies 𝐸₁<𝐸₂<𝐸₃ under the action of two (phase modulated) pulses 𝐄₁(𝑡) and 𝐄₂(𝑡)𝐄(𝐫,𝑡)=_𝑖=1,2𝐄_𝑖(𝑡)=12_𝑖=1,2𝐞_𝑖ℰ_𝑖(𝑡)exp−𝑖𝜔_𝑖𝑡+𝑖𝜑_𝑖(𝑡)+c.c. (1) The frequencies of which are close to those of the transitions 1→2 and 1→3, respectively. Here, ℰ_𝑖(𝑡) and 𝜑_𝑖(𝑡) describe the change of the pulse amplitude and phase in time, 𝐞_𝑖 are unit polarization vectors, and the instantaneous pulse frequencies are 𝜔_𝑖(𝑡)=𝜔_𝑖−𝑑𝜑_𝑖(𝑡)/𝑑𝑡. Three states, 1, 2, and 3 correspond to singlet states 𝑆₀, 𝑆₁, and 𝑆₂, of the azulene molecule, respectively (Figure 2). The potential-energy surfaces of polyatomic molecules are multidimensional, and the number of states related to a transition between initial and final states increases with the number of degrees of freedom of a molecule. For that reason, it is desirable to have a reduced space or a reduced state formulation of the optical control process. We will consider azulene in a Shpol'skii matrix for low temperatures; the 𝑆₂→𝑆₁ fluorescence spectrum shows very narrow lines [10] (see Figure 1). We assume that field 𝐄₂(𝑡) connects states 2 and 3 corresponding to 𝑆₁, with the excitation of one vibrational quantum of the vibration 1558 cm⁻¹ and the vibrationless state of 𝑆₂, respectively (see the second intensive line near 𝜆=810 nm in Figure 1). Of course, there is one more (less intensive) line corresponding to the vibration 1756 cm⁻¹ near the line related to 1558 cm⁻¹. However, we will consider only one state 2, representing both vibrationally excited states under discussion in 𝑆₁.

Figure 1 

𝑆2→𝑆1 fluorescence of azulene in 𝑛-hexane ad 77 K [10].

Figure 2 

A reduced states formulation of STIRAP in azulene. Field 𝐄2(𝑡) connects states 2 and 3 corresponding to 𝑆1 with the excitation of one vibrational quantum, and the vibrationless state of 𝑆2, respectively. State 1 represents the vibrationless state of a singlet 𝑆0 and field 𝐄1(𝑡) excites transition 1→2, so that the sum of frequencies 𝜔1(𝑡)+𝜔2(𝑡) is close to the frequency of purely electronic transition 𝑆0→𝑆2.

We assume that state 1 represents the vibrationless state of a singlet 𝑆₀ and field 𝐄₁(𝑡) excites transition 1→2 (see Figure 2), so that the sum of frequencies 𝜔₁(𝑡)+𝜔₂(𝑡) is close to the frequency of purely electronic transition 𝑆₀→𝑆₂. In the proposed scheme, the frequencies of transitions 1→2 and 2→3 are well separated in order to realize STIRAP (in spite of close energies of purely electronic transitions 𝑆₀→𝑆₁ and 𝑆₁→𝑆₂ [7, 11]), though complete and robust population transfer is also possible when each of the two laser pulses interacts with each of the pair of states [27].

The interaction of the dense collection of such systems with electromagnetic fields can be written as 𝐻_int=−_𝑛𝐏_𝑛⋅𝐄_𝑙,𝑛, (2) where 𝐄_𝑙,𝑛 is the electric field at the location of the 𝑛th molecule (local electric field), and 𝐏_𝑛 is its optical polarization operator. The local (microscopic) field is related to the Maxwell (macroscopic) field 𝐄(𝐫,𝑡), (1), by [22, 25]𝐄_𝑙,𝑛=𝐄(𝐫,𝑡)+4𝜋3^𝑚≠𝑛𝐏_𝑚, (3) where 𝐏_𝑚 denotes the expectation value of 𝐏_𝑚. The second term on the right-hand side of (3) represents the electrostatic longitudinal field created by all other particles 𝑚 (≠𝑛), that is, the instantanous dipole-dipole interactions between the molecules.

Let us introduce excitonic operators 𝐴_𝑛,12≡|𝑛1⟩⟨𝑛2|, 𝐴_𝑛,23≡|𝑛2⟩⟨𝑛3| and 𝐴_𝑛,13≡|𝑛1⟩⟨𝑛3|, where |𝑛𝑘⟩ denotes the 𝑘 state of molecule 𝑛. Then, the optical polarization operator of the 𝑛th molecule is given by𝐏_𝑛=𝐃_𝑛,12𝐴⁺_𝑛,12+𝐃_𝑛,23𝐴⁺_𝑛,23+𝐃_𝑛,13𝐴⁺_𝑛,13+H.c., (4) where 𝐃_{𝑛,𝑘𝑘′} is the transition dipole moment connecting states 𝑘 and 𝑘^′ of molecule 𝑛, and H.c. denotes Hermitian conjugate. We also introduce operators 𝑛_𝑚𝑘=|𝑚𝑘⟩⟨𝑚𝑘| describing the population of states |𝑚𝑘⟩. It is obvious that𝑛_𝑚1+𝑛_𝑚2+𝑛_𝑚3=𝐼 (5) due to the completeness relation for the 3-level electronic states space of the 𝑚th molecule. Here 𝐼is the unit operator. Operators 𝐴_{𝑛,𝑘𝑘′} obey the following commutation relation: 𝐴_{𝑚,𝑘𝑘′},𝐴⁺_{𝑚,𝑘𝑘}^′=𝑛_𝑚𝑘−𝑛_𝑚𝑘′, (6) where 𝑘=1,2, 𝑘^′=2,3, and 𝑘<𝑘^′. Then, the Hamiltonian of the dense collection of three-state systems under consideration can be written as 𝐻=𝐻₀+𝐻_int, (7) where𝐻₀=_{𝑚𝑘=1,2,3}𝐸_𝑚𝑘𝑛_𝑚𝑘. (8)𝐸_𝑚𝑘 is the energy of state 𝑘 of a molecule 𝑚.

Using the Heisenberg equations of motion, one obtains the equation for the expectation value of any operator 𝐹𝑑⟨𝑖𝑑𝑡𝐹⟩=ℏ𝐻₀+𝐻_int,𝐹≡𝑖ℏTr𝐻₀+𝐻_int,𝐹𝜌, (9) where 𝜌 is the density matrix. Straightforward operator algebra manipulations, using commutation relations, (6), yield equations for 𝑛_𝑚𝑘≡⟨𝑛_𝑚𝑘⟩ and 𝐴_{𝑚,𝑘𝑘′}≡⟨𝐴_{𝑚,𝑘𝑘′}⟩ in the rotating wave approximation (RWA). In the derivation of these equations, we kept only single-particle variables, using a factorization into population variables 𝑛_𝑚𝑘 and polarizations 𝐴_{𝑚,𝑘𝑘′} in accordance with the local field approximation [22]. Then, switching to the system that rotates with instantaneous frequency, 𝑎_𝑚,12=𝐴_𝑚,12exp𝑖𝜔₁𝑡−𝑖𝜑₁(𝑡,𝑎_𝑚,23)=𝐴_𝑚,23exp𝑖𝜔₂𝑡−𝑖𝜑₂(𝑡,𝑎_𝑚,13)=𝐴_𝑚,13exp𝑖𝜔₁+𝜔₂𝑡−𝑖𝜑₁(𝑡)+𝜑₂(𝑡), (10) we obtain equations for the quantities that vary slowly with time during the period of a light wave. Furthermore, considering a homogeneous excitation of an assembly of identical molecules (𝐃_{𝑛,𝑘𝑘′}=𝐃_𝑘𝑘′,𝐸_𝑚𝑘=𝐸_𝑘,𝑛_𝑚𝑘=𝑛_𝑘,𝑎_{𝑚,𝑘𝑘′}=𝑎_𝑘𝑘′), we get 𝑑𝑎₁₂𝜔𝑑𝑡=−𝑖₂₁−𝜔₁(𝑡)+𝑝₁₂𝑛₂−𝑛₁𝑎₁₂+Ω₁(𝑡)2×𝑛₂−𝑛₁−Ω₂(𝑡)2𝑎₁₃+𝑝₁₃−𝑝₂₃𝑎^∗₂₃𝑎₁₃,𝑑𝑎₂₃𝜔𝑑𝑡=−𝑖₃₂−𝜔₂(𝑡)+𝑝₂₃𝑛₃−𝑛₂𝑎₂₃+Ω₁(𝑡)2𝑎₁₃+Ω₂(𝑡)2𝑛₃−𝑛₂+𝑝₁₂−𝑝₁₃𝑎^∗₁₂𝑎₁₃,𝑑𝑎₁₃𝜔𝑑𝑡=−𝑖₃₁−𝜔₁(𝑡)−𝜔₂(𝑡)+𝑝₁₃𝑛₃−𝑛₁𝑎₁₃+Ω₁(𝑡)2𝑎₂₃−Ω₂(𝑡)2𝑎₁₂+𝑝₁₂−𝑝₂₃𝑎₂₃𝑎₁₂,𝑑𝑛𝑑𝑡𝑛¹₃=−Ω₁(𝑡)Im𝑎₁₂Ω₂(𝑡)Im𝑎₂₃𝑛₁,+𝑛₂+𝑛₃=1, (11)(12)(13)(14)(15) where 𝜔_𝑘𝑘′=(𝐸_𝑘−𝐸_𝑘′)/ℏ and Ω_𝑖(𝑡)=(𝐃_𝑖,𝑖+1⋅𝐞_𝑖)ℰ_𝑖(𝑡)/ℏ is the Rabi frequency corresponding to the 𝑖th pulse. The strength of the near DD interaction is given by terms 𝑝_𝑘𝑘′=4𝜋𝐷3ℏ^𝑘𝑘′2𝑁, (16) where 𝑁 is the density of molecules [25]. In writing (11)–(14), we have neglected the population-decay time of state 2 (𝑆₁). Extending this consideration to a decaying intermediate state is made in Section 4.2.

One can easily see from (13) that the two-photon resonance cannot be achieved with the pulses of fixed frequency, at least due to the presence of the “local field” term 𝑝₁₃(𝑛₃−𝑛₁) that depends on the state populations. However, suitably chirped pulses can compensate the “local field” detuning 𝑝₁₃(𝑛₃−𝑛₁), and the first term on the right-hand side of (13) may be zero during the interaction. We will address the question of whether the chirping allows the realisation of a “dark” state and thus allows complete population transfer in the interacting media in the next section.

3. “Dark” State in the Interacting Media

The expectation values 𝑎_{𝑚,𝑘𝑘′} in the system that rotates with instantaneous frequency, (10), can be expressed through the density matrix. Indeed, 𝑎_{𝑚,𝑘𝑘′}=Tr𝑎_{𝑚,𝑘𝑘′}𝜌=Tr|𝑚𝑘⟩𝑚𝑘^′_𝑗𝑗′𝑚𝑗𝜌_{𝑚,𝑗𝑗′}𝑚𝑗^′=𝜌_{𝑚,𝑘′𝑘}. (17) In particular, for pure states that are realized in the absence of relaxation, the density matrix can be represented as 𝜌_𝑘′𝑘=𝑐^∗_𝑘𝑐_𝑘′ where 𝑐_𝑘 are the amplitudes of states, and we consider a homogeneous excitation of an assembly of identical molecules. This enables us to express 𝑎_𝑘𝑘′ by the amplitudes of states 𝑐_𝑘. Putting 𝑎_𝑘𝑘′=𝑐^∗_𝑘𝑐_𝑘′ in equations (11)–(15), we get the following equations for the magnitudes 𝑐_𝑘:𝑑𝑐₁Ω𝑑𝑡=𝑖¹(𝑡)2𝑐₂,𝑑𝑐₂𝜔𝑑𝑡=−𝑖₂₁−𝜔₁(𝑡)+𝑝₁₂𝑐₂²−𝑐₁²+𝑝₁₃−𝑝₂₃𝑐₃²𝑐₂Ω+𝑖¹(𝑡)2𝑐₁Ω+𝑖²(𝑡)2𝑐₃,𝑑𝑐₃𝜔𝑑𝑡=−𝑖₃₁−𝜔₁(𝑡)−𝜔₂(𝑡)+𝑝₁₃𝑐₃²−𝑐₁²+𝑝₁₂−𝑝₂₃𝑐₂²𝑐₃Ω+𝑖²(𝑡)2𝑐₂,𝑐₁²+𝑐₂²+𝑐₃²=1. (18)(19)(20)(21)

Let us assume that pulses 𝐄₁(𝑡) and 𝐄₂(𝑡) are suitably chirped to fulfill the condition of the two-photon resonance during the interaction (the first term on the right-hand side of (20) is equal to zero). Then, one can easily see that under the adiabatic conditions (𝑑𝑐_𝑘/𝑑𝑡=0), the solutions of (18)–(21) are given by 𝑐₂=0,𝑐₃=−[Ω₁(𝑡)/Ω₂(𝑡)]𝑐₁ and 𝑐₁=Ω₂(𝑡)Ω²₁(𝑡)+Ω²₂(𝑡). (22) This enables us to obtain the adiabatic “dark” state Ψ_ad=𝑐₁|1⟩+𝑐₃|3⟩ as a superposition of states |1⟩ and |3⟩ only, similar to the case of usual noninteracting media without local field effects [12]. Complete population transfer occurs if lim_{𝑡→−∞}[Ω₁(𝑡)/Ω₂(𝑡)]=0 and lim_𝑡→+∞[Ω₂(𝑡)/Ω₁(𝑡)]=0, where 𝑡→−∞ and 𝑡→+∞ correspond to times before and after the interaction with optical pulses, respectively.

4. Numerical Calculations

4.1. Calculations without Decay

To check these ideas, we provided numerical solutions of coupled equations (11)–(15). For this purpose, it is convenient to switch to equations for the components of the 3-state coherence vector [28]𝑣_𝑘𝑘′=𝑖𝑎_𝑘𝑘′−𝑎_𝑘′𝑘,𝑢_𝑘𝑘′=𝑎_𝑘𝑘′+𝑎_𝑘′𝑘,𝑤₁=𝑛₂−𝑛₁,𝑤₂=−13𝑛₂+𝑛₁−2𝑛₃, (23) where 𝑘,𝑘^′=1,2,3 and 𝑘<𝑘^′. These components satisfy the following equations:𝑑𝑢₁₂𝑑𝑡=−Δ₂₁𝑣₁₂+Ω₂2𝑣₁₃,𝑑𝑢₂₃𝑑𝑡=−Δ₃₂𝑣₂₃−Ω₁2𝑣₁₃,𝑑𝑢₁₃𝑑𝑡=−Δ₃₁𝑣₁₃−Ω₁2𝑣₂₃+Ω₂2𝑣₁₂,𝑑𝑣₁₂𝑑𝑡=Δ₂₁𝑢₁₂+Ω₁𝑤₁−Ω₂2𝑢₁₃,𝑑𝑣₂₃𝑑𝑡=Δ₃₂𝑢₂₃+Ω₁2𝑢₁₃+Ω₂2√3𝑤₂−𝑤₁,𝑑𝑣₁₃𝑑𝑡=Δ₃₁𝑢₁₃+Ω₁2𝑢₂₃−Ω₂2𝑢₁₂,𝑑𝑤₁𝑑𝑡=−Ω₁𝑣₁₂+Ω₂2𝑣₂₃,𝑑𝑤₂√𝑑𝑡=−3Ω₂𝑣₂₃, (24) where detunings with respect to the renormalized optical transitions are given by Δ₂₁=𝜔₂₁−𝜔₁(𝑡)+𝑝₁₂𝑤₁+𝑝₁₃−𝑝₂₃√33𝑤₂+13Δ₃₂,=𝜔₃₂−𝜔₂(𝑡)+√32𝑤₂𝑝₂₃−13𝑝₁₂−𝑝₁₃−12𝑤₁𝑝₂₃+𝑝₁₂−𝑝₁₃+13𝑝₁₂−𝑝₁₃,Δ₃₁=𝜔₃₁−𝜔₁(𝑡)−𝜔₂(𝑡)+√32𝑤₂𝑝₁₃−13𝑝₁₂−𝑝₂₃+12𝑤₁𝑝₁₃+𝑝₁₂−𝑝₂₃+13𝑝₁₂−𝑝₂₃. (25)(26)(27) In deriving equations (24), we have used the relation 𝑎_𝑘𝑘′=𝑐^∗_𝑘𝑐_𝑘′ for pure states.

We considered the excitation of a dense collection of three-state systems with parameters 𝑝_𝑘𝑘′, (16), corresponding to those of azulene with two Gaussian pulses of the same duration, ℰ_𝑖(𝑡)=ℰ_0𝑖exp[−(𝑡−𝑡_𝑖)²/(2𝜏²)], and the amplitudes giving the same maximum Rabi frequencies for transitions 1→2 and 2→3, ordered counterintuitively, 𝐄₂(𝑡) before 𝐄₁(𝑡). Parameters 𝑝_𝑘𝑘′, (16), can be evaluated through radiation decay times 𝜏_rad of transitions 𝑆_𝑖→𝑆_𝑗 by 𝑝_𝑘𝑘′=𝜘_𝑘𝑘′𝑁, where 𝜘_𝑘𝑘′≡1/(8𝜋²𝜔³_𝑘′𝑘𝜏_rad), and 𝜏_rad were taken from [8]: 𝜏_rad(𝑆₂→𝑆₀)=5⋅10⁻⁹ s, 𝜏_rad(𝑆₁→𝑆₀)=8⋅10⁻⁷s and 𝜏_rad(𝑆₂→𝑆₁)=2⋅10⁻⁶s. This gives 𝜘₁₃=1.06⋅10⁻⁷[cm³/s],𝑝₁₃≡𝑝,𝑝₁₂=0.05𝑝,𝑝₂₃=0.02𝑝. The time delay between the pulses was 𝑡₁−𝑡₂=2𝜏, and the pulse areas 𝑆=∫Ω_1,2(𝑡)𝑑𝑡=√2𝜋Ω_max𝜏 ranged from 2 to 122. The product of the strength of the near DD interaction, 𝑝, and parameter 𝜏 characterizing the pulse duration, was equal to 𝑝𝜏=20. This value can be realized, for example, for the density of the active molecules 𝑁=3⋅10¹⁹ cm⁻³ and 𝜏=6.3ps (this value of 𝜏 exceeds the decay time of the intermediate state that will be taken into account in Section 4.2). It is worthy to note that 𝑁=3⋅10¹⁹cm⁻³ corresponds to the mean distance between active molecules of 3.2nm justifying the use of the dipole approximation for the near DD interaction.

One can see from (20) that the frequency of transition 1→3 shifts to the blue side during the population transfer 1→3, since 𝑝₁₃>0 and 𝑝₁₃≫𝑝₁₂,𝑝₂₃ for azulene. This denotes that the chirp preserving the two-photon resonance will not bring other levels into play close to the purely electronic transition 𝑆₀→𝑆₂.

Figure 3 shows the state populations, after the completion of the pulses action as functions of the ratio between the Rabi frequency at the pulse maximum and parameter 𝑝 for 𝜔₂₁−𝜔₁=0. Dotted lines correspond to the pulses with fixed frequencies when 𝜔₃₁−(𝜔₁+𝜔₂)=0. Other lines describe the population transfer under the two-photon resonance conditions, achieved by chirping the pulse 𝐄₁(𝑡) (solid lines) and 𝐄₂(𝑡) (dashed lines). Note that pulse chirping enables us to realize the two-photon resonance during the interaction, giving rise to efficient and robust population transfer by STIRAP in interacting media, in contrast to the excitation with pulses of fixed frequencies. Pronounced oscillations of the populations in the latter case (dotted lines) can be related to bistable (or multistable) behavior of the solution of (24), due to the dependence of the resonance frequency on the population difference [14, 23]. The bistability effect consists of a sudden switching of the population from a low level to a higher level with increasing pump intensity.

(a)

(b)

Figure 3 

Populations of the final (a) and intermediate (b) states after the completion of the pulses action as functions of Ωmax/𝑝. Solid lines: chirping the pulse 𝐄1(𝑡), dashed: chirping 𝐄2(𝑡), dots: pulses with fixed frequencies for 𝜔31−(𝜔1+𝜔2)=0.

Chirping the pulse 𝐄₂(𝑡) gives rise to a much larger population of intermediate state 2 for small values of Ω_max/𝑝, than chirping the pulse 𝐄₁(𝑡). This can be explained as follows. Let us assume that the chirped pulse is 𝐄₁(𝑡). To obey the two-photon resonance condition (see (20) and (27)), frequency 𝜔₁(𝑡) must be equal to 𝜔₁(𝑡)≈𝜔₃₁−𝜔₂+𝑝𝑛₃−𝑛₁=𝜔₁+𝑝𝑛₃−𝑛₁, (28) since for azulene 𝑝₁₃≫𝑝₁₂,𝑝₂₃. Substituting 𝜔₁(𝑡) from (28) into (25) for the detuning with respect to the renormalized frequency of transition 2→1 (see also (19)), we get 𝜔₂₁−𝜔₁(𝑡)+𝑝𝑛₃=𝜔₂₁−𝜔₁+𝑝𝑛₁=𝑝𝑛₁. In other words, detuning of transition 2→1 with respect to 𝜔₁(𝑡) comprises 𝑝𝑛₁ which prevents populating state 2 at the beginning of the process when 𝑛₁ is close to 1. In contrast, when one chirps the pulse 𝐄₂(𝑡), detuning of transition 2→1 with respect to 𝜔₁ comprises 𝜔₂₁−𝜔₁+𝑝𝑛₃=𝑝𝑛₃ which can be close to zero at the beginning of the process. Therefore, better conditions for populating state 2 are realized when the chirped pulse is 𝐄₂(𝑡).

Figure 4 depicts populations 𝑛₃ and 𝑛₂ for the pulses with fixed frequencies when 𝜔₃₁−(𝜔₁+𝜔₂)−𝑝₁₃=0 corresponding to the two-photon resonance condition for the initial populations of states (𝑛₁(−∞)=1, see (28)). Note again that only the pulse chirping gives rise to efficient and robust population transfer by STIRAP in interacting media. As to the excitation with pulses of fixed frequencies, pronounced oscillations remain for both populations 𝑛₂ and 𝑛₃, though 𝑛₃ is essentially smaller than that in Figure 3.

(a)

(b)

Figure 4 

Populations of the final (a) and intermediate (b) states after the completion of the pulses action as functions of Ωmax/𝑝. Solid lines: chirping the pulse 𝐄1(𝑡), dashed: chirping 𝐄2(𝑡), dots: pulses with fixed frequencies for 𝜔31−(𝜔1+𝜔2)−𝑝13=0.

4.2. Influence of the Decay of Intermediate State

In Section 4.1, we have shown that efficient STIRAP for population transfer to higher electronic states in interacting media of molecules with anomalous fluorescence can be realized in the absence of decay, using suitably chirped pulses. However, an intermediate state in this process belongs to the first singlet 𝑆₁ which decays very fast into 𝑆₀. In this section, we will extend the above consideration to a decaying intermediate state. We will use (18)–(21) for the magnitudes 𝑐_𝑘 where (19) is replaced by (29) 𝑑𝑐₂1𝑑𝑡=−2Γ𝑐₂−𝑖𝜔₂₁−𝜔₁(𝑡)+𝑝₁₂𝑐₂²−𝑐₁²+𝑝₁₃−𝑝₂₃𝑐₃²𝑐₂Ω+𝑖¹(𝑡)2𝑐₁Ω+𝑖²(𝑡)2𝑐₃ (29) taking into account a decaying intermediate state (the first term on the right-hand side of (29)).

We provided numerical solutions of coupled equations (18), (20), (21), and (29). Figures 5 and 6 show that with suitably chirped pulses, the efficient STIRAP still persists even for rather large decay 𝜏Γ=6 corresponding to 𝜏=6 ps (see Section 4.1) and 1/Γ=1 ps. However, for the excitation with pulses of fixed frequencies, the oscillations of 𝑛₃ are smoothed at a large decay which may be related to vanishing the bistable (or multistable) behavior for this condition.

Figure 5 

Populations of the final state 𝑛3 with the decaying intermediate state (𝜏Γ=0.6) after the completion of the pulses action as functions of Ωmax/𝑝. Solid line: chirping the pulse 𝐄1(𝑡), dashed: chirping 𝐄2(𝑡), dots: pulses with fixed frequencies for 𝜔31−(𝜔1+𝜔2)=0. Other parameters are identical to those of Figure 3.

Figure 6 

The same as in Figure 5 only for 𝜏Γ=6.

5. Conclusion

In this work, we have considered a coherent population transfer to a higher excited singlet state (𝑆₂) in molecular assemblies (e.g., a dense medium) of molecules with anomalous fluorescence. A direct excitation to 𝑆₂ requires light in the UV region. Because of this, the transition is conveniently realized by a two-step (two-photon) process: 𝑆₀→𝑆₁→𝑆₂ where transitions 𝑆₀→𝑆₁ and 𝑆₁→𝑆₂ correspond to the optical region. Since for such molecules, the first excited singlet state 𝑆₁ is very short living, using STIRAP in the ladder configuration for the population transfer 𝑆₀→𝑆₂ seems rather attractive, since a unique and very useful feature of STIRAP is that the intermediate state (𝑆₁) never gets populated, not even transiently. This is provided by maintaining a two-photon resonance between the target (𝑆₂) and initial (𝑆₀) states during the interaction.

The potential energy surfaces of polyatomic molecules are multidimensional, and the number of states related to a transition between initial and final states increases with the number of degrees of freedom of a molecule. We have provided a reduced states formulation of the optical control process, using azulene in a Shpol'skii matrix for low temperatures as an example.

Unfortunately, in molecular assemblies (a dense medium of molecules) a change of the frequency of the two-photon transition exists due to near DD interactions. We have shown that in spite of the frequency change, efficient STIRAP can be realized in this case as well, using suitably chirped pulses to compensate a change of the two-photon transition frequency in time induced by the pulses themselves. Chirping the “pump” pulse that excites transition 𝑆₀→𝑆₁ and the “Stokes” pulse that excites transition 𝑆₁→𝑆₂ are nonequivalent with respect to the population of the intermediate state (𝑆₁) in the pulse nonadiabatic regime. We have also shown that with suitably chirped pulses, the efficient STIRAP still persists, even for a rather large decay of the intermediate state.

It is worthy to note that the method for the compensation of two-photon detuning, proposed in our work, can be used not only in the case of violating the two-photon resonance due to near DD interactions, but also due to the off-resonant Stark shift [29]. In general, the pulse chirp must compensate the two-photon detuning stemming from both cases.

Acknowledgment

This work was supported by a grant from the Russia-Israel Scientific Research Cooperation.