Generalised EIT for 4 level systems

Consider a four-level ladder system driven by three coherent fields, denoted as probe, dressing and control, shown on Fig. 3.4(a). Their respective intensities are given by the coupling Rabi frequencies Ωp, Ωdand Ωc. With respective field detunings ∆p,

∆dand ∆c, the coherent dynamics of the system is described in the {|1〉, |2〉, |3〉, |4〉}

basis with Hamiltonian (ħh = 1)

Figure 3.4: Dressed state electromagnetic- ally induced transparency in Doppler-free (uniform-phase spin-wave) configuration. (a) Bare-states level diagram of the system driven by three coherent fields. (b) Levels in semi-dressed picture. With three driving fields oriented in plane as in (c), Doppler-free condition is ful- filled, and collective excitation of the ensemble of such four-level systems into state |4〉 will form uniform-phase spin-wave. Simultaneously, for

∆p= −∆c= Ωd/2 highly dispersive EIT window

opens for probe light (d), theoretically calculated here for Γ1= Γ2 ≡ Γ , Γ3= 0, (Ωd, Ωp, Ωc)/Γ =

(8, 0.1, 0.5) and (∆d, ∆c)/Γ = (0, −4) for a sta-

tionary four-level system.

Ωp, k1 Ωd, k2 Ωc, k3 _Ωc Ωp Γ2 Γ1 (a) (b) |1〉 |2〉 |3〉 |4〉 k₁ k3 Ωd P kl= 0 k2 Γ3 |+〉 |−〉

Probe detuning from the bare-state resonace, ∆p/Γ

Re (χ p ) ,− Im (χ p ) (c) (d) _{|−〉 |+〉} ∆c ∆d ∆p semidressed basis

H =      0 Ωp/2 0 0 Ωp/2 −∆p Ωd/2 0 0 Ωd/2 −∆p− ∆d Ωc/2 0 0 Ωc/2 −∆p− ∆d− ∆c     . (3.2)

In addition to the coherent driving, dissipation affecting the system is described by the Lindblad superoperator L[. . .] acting on the system’s density matrix ˆρ as L[ ˆρ] = P

α Lkρ Lˆ †α−

1

2L†αLαρˆ−12ρ Lˆ †αLα

. Spontaneous decays with rates Γi, i = 1, 2, 3 are included with Li=pΓi|i〉〈i + 1|. Overall, the system’s dynamics is governed by the master equation d

dtρ = −i[H , ˆˆ ρ] + L[ ˆρ]. As discussed in Sec. 2.4.3, solving this

in the case when all three beams are resonant will give rise to electromagnetically induced absorption, instead of transparency. We have to identify parameters for EIT to occur.

In the following, we focus our attention to the parameter regime where the middle driving field, resonant with the unperturbed transition |2〉 → |3〉, ∆d= 0, introduces

strong dressing Ωd Ωp, Ωc of the two intermediate states. The probe field will

then see an Autler-Townes split resonance [Fig. 3.4(d)], corresponding to the two states |+〉 and |−〉, that appear in the semi-dressed basis [Fig. 3.4(b) and Sec. 2.4.3]. Consider the situation where the probe and control fields are both detuned from

the bare-state resonances ∆p= −∆c = Ωd/2, so that they are resonant with one

of the semi-dressed states |+〉 or |−〉. This engineered dressed-state resonance can then be used in combination with control Ω_c and probe light Ω_p to open a narrow transparency window [Fig. 3.4(d)]. Typically, state |4〉 would be a long-lived Rydberg state, whose decay (Sec. 2.3.2) is much weaker compared to that of the two intermediate states Γ3 Γ1≈ Γ2. To a very good approximation a dark state |D〉 is

formed, which can be obtained by diagonalising the system Hamiltonian (Eq. 3.2). In the limiting worst case Ω_{p= Ωc}, when atoms are in an equal-weighted superposition of the ground |1〉 and excited state |4〉, we can obtain a clean expression for the dark state |D〉 = (|1〉 − ξ|2〉 − ξ|3〉 + |4〉)/N, (3.3) ξ ≡ −Ωd+ q Ω2_c+ Ω2d Ωc , (3.4)

where N is a normalization factor, and ξ characterises the admixture of the bright (radiatively coupled) states |2〉 and |3〉. In the limit of strong dressing the contribution of the bright states 2ξ ≈ Ωc/Ωd 1 is negligible. This is similar to the double-dark

resonance schemes explored in 4-level Λ-like systems [221]. The benefit of using the engineered state for excitation becomes apparent if one considers momentum-kick free, Doppler-free excitation. In contrast to typical two-photon driving schemes to highly excited states that, as discussed in the introduction, cannot fulfil the Doppler- free condition, this can be achieved with three fields arranged in a plane [Fig. 3.4(c)]. Additional advantages will be discussed in the following sections.

Intuitively, how EIT arises in this situation can be seen in a similar way as for the usual three-level EIT scheme discussed in Sec. 2.4.3. One can expect that one of the eigenstates of H (Eq. 3.2) has dominant composition of a1|1〉 + a2|4〉, so that in a

Figure 3.5: Generalisation of dressed-state EIT for N-middle levels. Example configuration for the five- (b) and six-level (c) system, shown in bare and semi-dressed basis. With all dress- ing beams resonant (yellow) ∆d,i= 0, and much

stronger then probe and control fields Ωd,i 

Ωc, Ωpnarrow transparency window opens when

probe and control are resonant with one of the states in the semi-dressed basis. For example for Ωd,i = 8Γ , (Ωp, Ωc)/Γ = (0.1, 0.5), narrow

transparency window opens for for ∆p= −∆c=

5.65 Γ for five-level system (a) and ∆p= −∆c=

6.45 Γ in six-level system (d). Arrows (a,d) high-

light the transparency window. _{Probe detuning from the bare-state resonance, ∆p/Γ} Probe detuning from the bare-state resonance, ∆p/Γ

Ωp Ωc Ωc Ωp Re (χ p ) ,− Im (χ p ) Re (χ p ) ,− Im (χ p ) (b) (c) (a) (d)

of the middle two states |2〉 and |3〉. If the coherent driving Ωdbetween these two

µagˆ"1 µbgˆ"2 µagˆ"1 µbgˆ"2 Ωc Ωc (a) (b) |g〉 |e〉 |a〉 |b〉 |e〉 |c〉 |g〉 |a〉 |b〉 |c〉

Figure 3.6: Extension of the dressed- state EIT scheme for coupling between bi- chromatic polaritons. (a) A uniform spin- wave between states |e〉 and |g〉, from light ex- citation ˆ"1stored over |g〉 → |a〉 → |c〉 → |e〉

levels, can be phase-matched for retrieval as field ˆ"2of different frequency, by retrieving over

|e〉 → |c〉 → |b〉 → |g〉. This modified diamond schemes is also interesting for exploring the influence of EIT on the efficiency of the four- wave mixing process. For continuous coupling of two propagating bi-chromatic fields ˆ"1and

ˆ"2the scheme shown on (b) is particularly in-

teresting although the spin-wave between |g〉 and |e〉 is not uniform-phase. That is because phase matching of spin waves formed over transitions |g〉 → |a〉 → |c〉 → |b〉 → |e〉 and |g〉 → |b〉 → |e〉 can be done only by adjusting the propagation direction of the two dressing fields on transitions |a〉 → |c〉 and |c〉 → |b〉, al- lowing co-propagation of the two fields ˆ"1and

ˆ"2. Using Rydberg state for |e〉 would addition-

ally open the possibility for non-trivial coup- lings between the two bi-chromatic quantum fields.

states is strong enough to coherently mix these two contributions, there has to exist a combination of the amplitudes a₁and a₂that will in this mixing destructively interfere in the amplitude for excitation of the middle manifold. Following this effective image, one would expect that if we have a middle manifold consisting of a ladder of N states |m1〉

Ωd,1

−−→ |m2〉

Ωd,2

−−→ . . .−−−→ |mΩd,N−1 N〉, all of them coupled with strong

dressing fields Ωd,i resonant with the unperturbed transitions, EIT would again appear if we tune probe and control laser to one of the dressed states. That is indeed the case, as we show on Fig. 3.5, where the calculation is performed with three and four middle states, amounting to a total of five and six states respectively. Finally, we note that these multi-level schemes can also be interesting because they open up interesting possibilities for coupling multiple weak (quantum) fields (Fig. 3.6), where a total system’s polariton that forms would have two quantum EM field modes with very different frequencies. Bi-chromatic quantum field interfacing can be interesting in several contexts, as different energies can be used as a frequency encoded qubits [222] or for interfacing and entanglement of heterogenous quantum systems with different resonant frequencies. Some possibilities will be discussed in Sec. 3.3.5.