Amplification without inversion and high refractive index in heterogeneous molecules

Optics Communications 328 (2014) 77–86

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Amplification without inversion and high refractive index in heterogeneous molecules O. Budriga Laser Department, National Institute for Laser, Plasma and Radiation Physics, P.O. Box MG-36, 077125 Magurele, Romania

art ic l e i nf o

a b s t r a c t

Article history: Received 23 December 2013 Received in revised form 24 April 2014 Accepted 26 April 2014 Available online 9 May 2014

The system of interest is a closed three-level V-type system with two excited near degenerated levels in a weak probe field and a strong coupling field, in the presence of an incoherent pumping field. We investigate analytically the conditions to have amplification without population inversion or high index of refraction without absorption due to spontaneously generated coherence and incoherent pumping. The perturbation solution of the density matrix equation in the steady state case is derived. We find that the refractive index and absorption coefficient have a periodic dependence on the relative phase of the probe and coupling fields. The relative phase values for which the system exhibits a high refractive index without absorption are obtained. The conditions on incoherent pumping rate related to that for the angle between the two transition dipole moments and spontaneous emission rates of the two excited states to obtain the probe field gain without population inversion are derived. We propose a three-level V-type system with spontaneously generated coherence from LiH molecule which can achieve quasi-high index of refraction without absorption and amplification without population inversion. & 2014 Elsevier B.V. All rights reserved.

Keywords: Refractive index without absorption Amplification without population inversion V-type system Spontaneously generated coherence Incoherent pumping

1. Introduction The spontaneously generated coherence appears as a result of quantum interferences produced by spontaneous decay. Javanien showed for the first time the possibility for a Λ system with neardegenerated levels to achieve a spontaneously generated coherence (SGC), as a superposition of two receiving states of the spontaneous emission from a single excited state [1]. Effects of the SGC on the electromagnetically induced transparency (EIT) and coherent population trapping were shown for the Λ system in the absence of an incoherent pump field [2]. It is possible to control the amplification without population inversion by the relative phase of the probe and coupling fields in the presence of the incoherent pumping for a Λ system with SGC [3]. Conditions to obtain the lasing without inversion (LWI) are modified by SGC for a V-type system [4]. The transient response of a three-level V-type system without incoherent pumping and SGC influences the LWI [5,6] and large refractive index without absorption [7]. The pumping with an incoherent light was studied for Λ atomic system [8] and V-type atomic system [9]. We showed in [10] that for a V-type system with SGC in the time-dependent regimen the use of an incoherent pumping leads to the increase of the transient gain without population inversion, but until a value of the incoherent pumping rate. Several Λ-type atomic systems

E-mail address: olimpia.budriga@inflpr.ro http://dx.doi.org/10.1016/j.optcom.2014.04.064 0030-4018/& 2014 Elsevier B.V. All rights reserved.

with SGC, without incoherent pumping which can achieve a high refractive index without absorption were proposed [11–14]. The experimental measurements on absorption and dispersion coefficients were done for a V-type cesium atom [15]. The enhancement of the refractive index was evidenced in various experiments in the low density atomic Rubidium (Rb) gas [16] and in a medium of ultra-cold 87Rb atoms [17]. Amplification without population inversion can be obtained in diatomic molecules H2 and Li2 [18], in the vacuum ultraviolet region and far-red region, respectively. LiH molecules [19] achieve a high gain without population inversion in the violet region wavelength and depend on the choice of the lasing schemes like V, ladder and Λ, same as for the abovementioned molecules. In all above-mentioned works the three-level systems (V, ladder and Λ) were studied as independent atomic or molecular systems which do not interact one another. The effect of the relative phase of the two fields, probe and coupling fields, on the collective populations dynamics of a N three-level V- and Λ atoms system was relieved in recent papers [20]. Macovei et al. have shown that the change of the relative phase between two strong fields, in the presence of two incoherent pumping fields and a weak probe field, can lead us to a convenient manipulation of the absorption properties of a degenerate three-level V or Λ atomic sample. For resonantly driven system of two-identical two-level atoms the adjusting of the laser phase at the positions of the atoms can affect the spectral features of cooperative emission significantly and generate strong entanglement in the system [21]. Semiconductor

78

O. Budriga / Optics Communications 328 (2014) 77–86

quantum dots were mentioned as the possible systems to observe the interference effects. Another interesting application of SGC (or vacuum-induced coherence VIC) is in the quantum information domain. Das et al. have shown that the quantum interference can lead to preservation of bipartite entanglement of the two initially entangled qubits formed out of two atomic system in V configuration [22]. In this paper we study the probe field gain and refractive index for a closed three-level V-type system with spontaneously generated coherence in the presence of an incoherent pumping field. We have seen in the prior work [23] that in this system, which is driven by two coherent fields without constraints and an incoherent pumping, we can control the high refractive index without absorption by variating the relative phase. In this general case, because the analytical expressions of the refractive index and the gain coefficient are complicated, we could not predict the behaviour of these quantities. We obtain a simple expression for the index of refraction and the gain in the case of a weak probe field and a strong coupling field. Besides our previous paper [23] we find the conditions whereby we predict the high refractive index without absorption and amplification without population inversion. We consider the case of a weak probe field and a strong coupling field, and we use the lower perturbation theory (LOPT) to derive the analytical expressions of the density matrix elements. Additionally, we derive analytically the constraints for the parameters of this system such as, in the steady state regimen, probe field amplification without population inversion can be detected. We apply density matrix formalism to a real V-type system in the heteronuclear LiH molecule, build with the external fields. We consider the general case in which the three fields, probe, coupling and incoherent pumping, act on both transitions, which was not studied before. In Section 2 we describe the system and its dynamics in the density matrix formalism. The analytical and numerical results are presented in Section 3, which has two subsections. We give the general analytical solution of the density matrix equations system in Section 3.1 and particular analytical solution of it in Section 3.2 along with the numerical results. Section 4 is devoted to an example of a real three-level V-type system in the LiH molecule. We conclude with some remarks in Section 5.

2. The system and density matrix equations The studied system is a closed three-level V-type system with two closely lying excited states j1〉 and j2〉 and a ground state j3〉, as shown in Fig. 1(a). The rates of spontaneous emission from levels j1〉 and j2〉 to ground level j3〉 are denoted by 2γ1 and 2γ2,

respectively. The transition j2〉2j3〉 with frequency ω23 is driven ! ! by a coupling coherent field ( E 2 ¼ ϵ 2 e  iω2 t þc:c:) with the Rabi ! ! ! frequency 2Gc ¼ 2 ϵ 2  d 23 =ℏ, where d 23 is the transition dipole moment. Between the levels j1〉 and j3〉 is applied a probe field ! ! ! ! ( E 1 ¼ ϵ 1 e  iω1 t þ c:c:) with the Rabi frequency 2g p ¼ 2 ϵ 1  d 13 =ℏ. The transition j1〉2j3〉 with the frequency ω13 and dipole moment ! d 13 is pumped with a rate 2Λ by an incoherent field. The detunings of the probe field and the coupling field are Δ1 ¼ ω13  ω1 and Δ2 ¼ ω23  ω2 , respectively. The semiclassical Hamiltonian of this system in the interaction picture, in a rotating-wave frame, under the rotating-wave approximation and dipole approximation, is written as H ¼  ℏðg p eiΔ1 t j1〉〈3j þ Gc eiΔ2 t j2〉〈3jþ H:c:Þ:

ð1Þ

The density matrix equations in the Markoff approximation with the phenomenological inclusion of the unidirectional incoherent pump terms and the spontaneous dumping terms are

ρ_ 11 ¼  2γ 1 ρ11 þ2Λρ33  ηðρ12 þ ρ21 Þ þ igp ρ31  ignp ρ13 ρ_ 22 ¼  2γ 2 ρ22  ηðρ12 þ ρ21 Þ þ iGc ρ32  iGnc ρ23 ρ_ 13 ¼  ðγ 1 þ Λ þ iΔ1 Þρ13  ηρ23 þ ig p ðρ33  ρ11 Þ  iGc ρ12 ρ_ 12 ¼  ½γ 1 þ γ 2 þ iðΔ1  Δ2 Þρ12 þ igp ρ32  iGnc ρ13  ηðρ11 þ ρ22 Þ ρ_ 23 ¼  ðγ 2 þ Λ  iΔ2 Þρ23  ηρ13  ig p ρ21 þ iGc ðρ33  ρ22 Þ:

ð2Þ

The diagonal density matrix elements satisfy the trace condition

ρ11 þ ρ22 þ ρ33 ¼ 1. The parameter η from (2) describes the quantum

interference between spontaneous emission from excited levels j1〉 and j2〉 to ground level j3〉 and depends on θ, the angle between the ! ! pffiffiffiffiffiffiffiffiffiffi two dipole moments d 13 and d 23 , η ¼ η0 γ 1 γ 2 cos θ, where 3=2 η0  ðω23 =ω13 Þ [24]. The SGC effects are important only for small energy spacing between the two excited levels [2], and for high energy spacing such effects will disappear [25]. As the excited levels j1〉 and j2〉 are near-degenerated, then ω23  ω13 and η0  1. The condition to have spontaneously generated coherence is η a 0, which means that we must have a nonorthogonal dipole moment of the two transitions, that is, θ a π =2. Therefore we choose the dipole ! ! moment so that one field acts on one transition ( d 13 ? ϵ 2 , ! ! d 23 ? ϵ 1 ) as it can be shown in Fig. 1(b). Rabi frequencies are ! ! connected to the angle θ by the relations 2g p ¼ 2j ϵ 1 jj d 13 j sin θ=ℏ ! ! and 2Gc ¼ 2j ϵ 2 jj d 23 j sin θ=ℏ. The nonorthogonality of the dipolar moments can be achieved from the mixing of the levels arising from internal [26] or external fields [27–31]. We shall use the method proposed by Ficek and

Fig. 1. (a) The three-level V-type system with two near-degenerated excited states j1〉, j2〉 and a ground level j3〉. (b) The electric dipole transition moments are chosen so that one field acts on only one transition.

O. Budriga / Optics Communications 328 (2014) 77–86

Swain to create a Vee-type system with antiparallel dipole moments, which lies in applying a strong laser field to one of the two transitions in a Lambda-type system [32]. The real threelevel Λ system is from LiH molecule and the data was chosen from the paper of Bhattacharjee et al. [19]. As noted before the two excited levels, j1〉 and j2〉 must be very close. Consequently the probe field absorption and dispersion depend on the probe and coupling field phases and the existence of the spontaneously generated coherence is related to the complex Rabi frequencies 2gp and 2Gc. We write the Rabi frequencies in the form 2g p ¼ 2geiϕp and 2Gc ¼ 2Geiϕc , where we denoted ϕp and ϕc the phases of the probe field and the coupling field, respectively. The quantities 2g and 2G are considered real. We do the variable changes, ρ~ 13 ¼ ρ13 e  iϕp , ρ~ 23 ¼ ρ23 e  iϕc , ρ~ 12 ¼ ρ12 e  iϕ , ρ~ ii ¼ ρii , i ¼ 1  3 in the density matrix equations system, where ϕ ¼ ϕp  ϕc is the relative phase of the two coherent driving fields. In the steady state conditions the equations system (2) becomes 2γ 1 ρ~ 11 þ2Λρ~ 33  η eiϕ ρ~ 12  η e  iϕ ρ~ 21 þ ig ρ~ 31  ig ρ~ 13 ¼ 0 2γ 2 ρ~ 22  η eiϕ ρ~ 12  η e  iϕ ρ~ 21 þ iGρ~ 32  iGρ~ 23 ¼ 0

ρ~ ð0Þ 22 ¼

aþb cd 1 D



Λ  η2 G2 Re  γ 1 þ Λ þ η2 Re Λ þ η2 Re

 X ρ~ ð0Þ 11 D

Y D

 iϕ ρ~ ð0Þ ðγ 1 þ Λ þ iΔ1 Þ 13 ¼ iGη e fðΛ  γ 1  iΔ1 Þρ~ ð0Þ 11 =x

iG D

ð3Þ

Our previous paper contains the analytical solution of this system in the general case without restrictions on the probe and coupling field, in all order of g and G [23]. In the present paper we shall obtain an approximative solution for a weak probe field and a strong coupling field using a perturbative method. Our system is different from that studied in the work of Xu et al. [7] by the introduction of the incoherent pumping field. The originality arises from the fact that the perturbative solution allows us to predict the values of the parameters of the system in three particulary cases for which one obtain high index of refraction without absorption. Also, we derived the constraints on angle between dipole transition moments, spontaneous emission rates of two excited states and the incoherent pumping rate to achieve amplification without population inversion.

ρ~ ð0Þ 21 ¼ 

η eiϕ ½Dðγ 1 þ Λ  iΔ1 Þ  G2 Zρ~ ð0Þ 11 D

y

ð0Þ ½Dðγ 1 þ Λ  iΔ1 Þ  G2 Wρ~ 22  G2 y þ y

ð6Þ

where Re means real part of a complex number and    1 Y W γ 2 þ η2 Re þ G2 Re a ¼ Λ  η2 G2 Re D D D !   2 Y 1 x  G b ¼ G2 Λ þ η2 Re G2 Re þ Re D D D    X Y W γ 2 þ η2 Re þ G2 Re c ¼ γ 1 þ Λ þ η2 Re D D D    Y X Z d ¼ Λ þ η2 Re η2 Re þ G2 Re D D D x ¼ G2 þ ðγ 1 þ ΛÞðγ 1 þ γ 2 Þ  Δ1 ðΔ1  Δ2 Þ þ i½Δ1 ð2γ 1 þ γ 2 þ ΛÞ  Δ2 ðγ 1 þ ΛÞ

3. Results If we assume that the probe field is weak then the interaction of the V-type system with the probe field will be weak, too. Under this consideration we can use the lower perturbation theory (LOPT) to solve the equations system (3) (see [33]). The interaction hamiltonian between the system and the weak probe field is proportional with the Rabi frequency of the probe field g. Hence we develop the solutions in the lowest power of g: where i; j ¼ 1  3:

ð4Þ

3.1. General solution In the zeroth order of g and all order of G the equations system (3) reduces to iϕ ~ ð0Þ  iϕ ~ ð0Þ ~ ð0Þ ~ ð0Þ 1 11 þ 2 33  ðe 12 þ e 21 Þ ¼ 0 ð0Þ ð0Þ i ϕ  i ϕ ~ ð0Þ ~ ð0Þ 2 2 ~ 22  ðe ~ 12 þ e Þ þ iGð ~ ð0Þ 21 32  23 Þ ¼ 0  iϕ ~ ð0Þ ~ ð0Þ ð 1 þ þ i 1 Þ ~ ð0Þ 13  e 23  iG 12 ¼ 0 ð0Þ ð0Þ ð0Þ þ ~ ð0Þ ½ 1 þ 2 þ ið 1  2 Þ ~ 12  iG ~ 13  e  iϕ ð ~ 11 22 Þ ¼ 0 ð0Þ ð0Þ ð0Þ ð0Þ ð 2 þ  i 2 Þ ~ 23  eiϕ ~ 13 þiGð ~ 33  ~ 22 Þ ¼ 0:

2γ ρ

ρ~ ð0Þ 11 ¼

~ ð0Þ ðZ ρ~ ð0Þ ρ~ ð0Þ 23 ¼  11 þ W ρ 22 þxÞ

 η e  iϕ ðρ~ 11 þ ρ~ 22 Þ ¼ 0

~ ð1Þ ρ~ ij ¼ ρ~ ð0Þ ij þ g ρ ij ;

In order to obtain the solutions of equations system (5) we use the elimination method. First of all, from the third equation we obtain the ~ ð0Þ ~ ð0Þ expression of ρ~ ð0Þ 13 dependent on ρ 23 and ρ 12 . Then we replace this expression in the fourth and the fifth equations of (5) and after that ð0Þ eliminate the variable ρ~ 23 . We shall obtain a formula for ρ~ ð0Þ 12 ~ ð0Þ depending on the populations ρ~ ð0Þ 11 and ρ 22 attending to the trace ð0Þ condition. Replacing this last expression in the formula of ρ~ 23 and then in the two first equations of the system (5) we have a common two ~ ð0Þ equation system with two variables ρ~ ð0Þ 11 and ρ 22 , which is straightforward to solve. By consecutive substitutions the solutions of (5) are

ð0Þ =xg þ ½Λ  2γ 1  γ 2 þ iðΔ2  2Δ1 Þρ~ 22

ðγ 1 þ Λ þ iΔ1 Þρ~ 13  η e  iϕ ρ~ 23 þ igðρ~ 33  ρ~ 11 Þ  iGρ~ 12 ¼ 0 ½γ 1 þ γ 2 þ iðΔ1  Δ2 Þρ~ 12 þ ig ρ~ 32  iGρ~ 13

ðγ 2 þ Λ  iΔ2 Þρ~ 23  η eiϕ ρ~ 13 ig ρ~ 21 þ iGðρ~ 33  ρ~ 22 Þ ¼ 0:

79

Λρ η ρ γ ρ η ρ γ Λ Δ ρ η γ γ Δ Δ ρ γ Λ Δ ρ η

ρ

ρ ρ

ρ

ρ

ρ ρ η ρ

ρ

D ¼ G2 ðγ 2 þ ΛÞ þ ðγ 1 þ γ 2 Þ½ðγ 1 þ ΛÞðγ 2 þ ΛÞ þ Δ1 Δ2  η2 

 ðΔ1  Δ2 Þ½Δ1 ðγ 2 þ ΛÞ  Δ2 ðγ 1 þ ΛÞ

þ ifðΔ1  Δ2 Þ½ðγ 1 þ ΛÞðγ 2 þ ΛÞ þ Δ1 Δ2  η2  þ ðγ 1 þ γ 2 Þ½Δ1 ðγ 2 þ ΛÞ  Δ2 ðγ 1 þ ΛÞ  Δ2 G2 g X ¼ η2 þ G2 ðγ 1 þ ΛÞðγ 2 þ ΛÞ  Δ1 Δ2  i½Δ1 ðγ 2 þ ΛÞ  Δ2 ðγ 1 þ ΛÞ Y ¼ X þ G2 Z ¼ η2 G2  ðγ 1 þ ΛÞðγ 1 þ γ 2 Þ þ Δ1 ðΔ1  Δ2 Þ  i½Δ1 ðγ 1 þ γ 2 Þ þ ðΔ1  Δ2 Þðγ 1 þ ΛÞ

ρ

ρ

y ¼ G2 þðγ 1 þ ΛÞðγ 1 þ γ 2 Þ  Δ1 ðΔ1  Δ2 Þ  i½Δ1 ð2γ 1 þ γ 2 þ ΛÞ  Δ2 ðγ 1 þ ΛÞ

ρ

ð5Þ

  W ¼ η2  2G2  2 ðγ 1 þ ΛÞðγ 1 þ γ 2 Þ  Δ1 ðΔ1  Δ2 Þ    2i Δ1 ðγ 1 þ γ 2 Þ þ ðΔ1  Δ2 Þðγ 1 þ ΛÞ

ð7Þ

80

O. Budriga / Optics Communications 328 (2014) 77–86

are parameters which appear in the density matrix elements (5). In the first order of perturbation theory, what means in the first order of Rabi probe field frequency g the density matrix equations system becomes ð1Þ ð1Þ  iϕ ~ ð1Þ  ηð eiϕ ρ~ ð1Þ ρ 21 Þ þ iGðρ~ 32  ρ~ ð1Þ  2γ 2 ρ~ 22 12 þ e 23 Þ ¼ 0

~ ð0Þ ~ ð1Þ  ½γ 1 þ γ 2 þ iðΔ1  Δ2 Þρ~ ð1Þ 12 þ ig ρ 32  iGρ 13 iϕ ~ ð1Þ ~ ð0Þ  ðγ 2 þ Λ  iΔ2 Þρ~ ð1Þ 23  η e ρ 13  ig ρ 21

ð8Þ

Following the same steps as in the zeroth order of the perturbation theory we can derive the expressions for the density matrix elements ρ~ ð1Þ ij , where i; j ¼ 1  3. The gain coefficient for the probe field which acts on the transition j1〉2j3〉 is proportional ð1Þ with imaginary part of ρ~ 31 density matrix element, Im ρ~ ð1Þ 31 . The ð1Þ probe laser will be amplified if the condition Im ρ~ 31 4 0 is met [34]. The refractive index of the probe field is proportional with ð1Þ the real part of ρ~ 31 , Re ρ~ ð1Þ 31 . To study the amplification and dispersion properties of the probe field we derived the ρ~ ð1Þ 31 density matrix element. The density matrix element ρ~ ð1Þ 31 depending on the level populations in the first order of g and the solutions of density matrix elements in the zero order of g is ð9Þ

with the numerator M having the form M ¼  igfiG½η2 ð1  e  iϕ Þ þ e  iϕ ðγ 2 þ Λ þ iΔ2 Þ  ðγ 1 þ Λ  iΔ1 Þðγ 1 þ Λ iΔ1 Þρ~ ð0Þ 23 ð0Þ 2 2  iϕ þ ðρ~ 33  ρ~ ð0Þ Þ þ e  iϕ ðγ 2 þ Λ þiΔ2 Þ 11 Þfη G ð1  e

 ðγ 1 þ Λ  iΔ1 ÞfG2 þ ðγ 1 þ Λ iΔ1 Þ 2  ½γ 1 þ γ 2  iðΔ1  Δ2 Þgg þ ρ~ ð0Þ 12 ηðγ 1 þ Λ  iΔ1 Þ

 ½γ 1 þ γ 2  iðΔ1  Δ2 Þg  iηGðγ 1 þ Λ  iΔ1 Þ

Polarizations

N ¼ ðγ 1 þ Λ  iΔ1 Þfη2 G2  e  iϕ ½η2  ðγ 2 þ Λ þ iΔ2 Þ  ðγ 1 þ Λ iΔ1 ÞfG2 þ ðγ 1 þ Λ  iΔ1 Þ The formulas for ρ

~ ð1Þ 23

and ρ

ð11Þ

0 -0.01

-0.03 0.03

As you have seen in the previous section the general expression of the ρ~ ð1Þ 31 density matrix element (9) is sophisticated. Hence we will discuss the cases of three particular values for detunings of the probe and coupling lasers.

0.01

Polarizations

0.02

ð12Þ

A2

0.01

3.2. Particular solutions

i eiϕ η γ 1 þ 2Λ ρ~ ð1Þ 31 ¼  G 2γ 1 þ 3Λ

A1

-0.02

are written in Appendix A.

3.2.1. Resonant probe and coupling lasers The first case that we consider is that of the resonant weak probe field and the resonant strong coupling field, i.e. Δ1 ¼ 0, Δ2 ¼ 0 and G⪢γ 1 ; γ 2 ; Λ⪢g. In these conditions, the solution for the density matrix element ρ~ ð1Þ 31 reduces to

ð14Þ

0.02

ð10Þ

and the denominator N is

~ ð1Þ 12

ð13Þ

η γ 1 þ 2Λ cos ϕ: G 2γ 1 þ 3Λ

0.03

þ eiϕ ½η2 ð1  e  iϕ Þ þ e  iϕ ðγ 2 þ Λ þ iΔ2 Þ

 ½γ 1 þ γ 2 iðΔ1  Δ2 Þgg

γ1 þ Λ : 2γ 1 þ 3Λ

In the absence of the incoherent pumping, i.e. Λ ¼ 0 from relation (12) we obtain the population of the state j1〉, ρ~ ð1Þ 11  0. Because the probe field is weak the probability for atoms to be excited to state j1〉 is very small. The steady state values for the populations of j2〉 and j3〉 must be equal with 0.5, the V-type system having the same probability to be excited to these levels. ~ ð1Þ Indeed, if we put Λ ¼ 0 in the expressions of the ρ~ ð1Þ 22 and ρ 11 from (12) we get the same value 0.5. From the relations (13) we obtain the values for ϕ to have high index of refraction (Re ρ~ ð1Þ 31 ¼ max) with zero absorption (Im ρ~ ð1Þ ¼ 0) for sin ϕ ¼ 1 and cos ϕ ¼ 0, 31 which implies that ϕ ¼ π =2 þ 2kπ , with k integer number, same result being obtained from numerical calculations by Xu et al. [7]. The use of an incoherent pumping laser to drive the transition j1〉2j3〉 leads to some modifications in the absorption and dispersion properties of the probe field [23]. Using our analytical expressions from [23] for the real and imaginary part of the ρ~ 31 density matrix element for the resonant weak probe and strong coupling fields and the parameters Δ1 ¼ Δ2 ¼ 0, γ 2 ¼ 2γ 1 , G ¼ 20γ 1 sin θ, g ¼ 0:01γ 1 sin θ, Λ ¼ 0:5γ 1 and θ ¼ π =3 we plotted the results in Fig. 2. Plotting the values from the relations (13) (Fig. 2(b)) we obtained identical curves with those which were found using the analytical formulas from our previous paper [23] (Fig. 2(a)). A straightforward calculation of the high value of Re ρ~ ð1Þ 31 give us the value 0.0236 almost the same with the numerical value of the exact expression of Re ρ~ 31 , pert 0.0237, the ordinates of the points A1, A2 and Apert 1 , A2 , respectively.

~ ð1Þ  fðγ 1 þ Λ  iΔ1 Þ½γ 1 þ γ 2  iðΔ1  Δ2 Þðρ~ ð1Þ 22  ρ 33 Þ ~ ð1Þ  ðγ 1 þ Λ  iΔ1 Þðρ~ ð1Þ 11 þ ρ 22 Þg

~ ð1Þ ρ~ ð1Þ 22 ¼ ρ 33 ¼

;

η γ 1 þ 2Λ sin ϕ; G 2γ 1 þ3Λ

Im ρ~ ð1Þ 31 ¼ 

~ ð1Þ  η e  iϕ ðρ~ ð1Þ 11 þ ρ 22 Þ ¼ 0

M N

Λ

2γ 1 þ 3Λ

ð1Þ ¼ Re ρ~ 31

ð0Þ  iϕ ~ ð1Þ ~ ð0Þ ρ 23 þ igðρ~ 33  ρ~ ð0Þ  ðγ 1 þ Λ þ iΔ1 Þρ~ ð1Þ 13  η e 11 Þ  iGρ 12 ¼ 0

ρ~ ð1Þ 31 ¼

ρ~ ð1Þ 11 ¼

~ ð1Þ ~ ð1Þ The real and imaginary parts of ρ~ ð1Þ 31 , Re ρ 31 and Im ρ 31 , respectively, will be

ð1Þ iϕ ~ ð1Þ  iϕ ~ ð1Þ ~ ð0Þ 2γ 1 ρ~ 11 þ 2Λρ~ ð1Þ ρ 21 Þ þ igðρ~ ð0Þ 33  ηð e ρ 12 þ e 31  ρ 13 Þ ¼ 0

ð1Þ þ iGðρ~ 33  ρ~ ð1Þ 22 Þ ¼ 0:

and the populations of the levels are

A1

pert

pert

A2

0 -0.01 -0.02 -0.03

-6

-4

-2

0

φ

2

4

6

Fig. 2. Re ρ~ 31 (solid line) and Im ρ~ 31 (dashed line) versus relative phase ϕ of the probe and coupling fields using: (a) exact analytical formulas and (b) perturbative formulas. The parameters used are Δ1 ¼ 0, Δ2 ¼ 0, γ 1 ¼ 1, γ 2 ¼ 2γ 1 , G ¼ 20γ 1 sin θ, g ¼ 0:01γ 1 sin θ, Λ ¼ 0:5γ 1 and θ ¼ π=3.

O. Budriga / Optics Communications 328 (2014) 77–86

3.2.2. Resonant coupling laser and the eigenvalues corresponding to the dressed states The states j2〉 and j3〉 are dressed by the strong coupling field. The dressed states are written as j þ 〉 ¼ cos αj2〉 þ sin αj3〉 j  〉 ¼  sin αj2〉 þ cos αj3〉;

ð15Þ

where 0 r 2α o π , tgð2αÞ ¼  2G=Δ2 . The eigenvalues corresponding to the two dressed states have the expressions ℏλ þ ¼ ℏðΔ2 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δ22 þ 4G2 Þ=2 and ℏλ  ¼ ðΔ2  Δ22 þ 4G2 Þ=2, respectively. We have seen in the previous paper [23] that the high indexes of refraction without absorption corresponds to a probe field detuning Δ1 ¼ λ 7 , when the other parameters mentioned before are constant. For this reason we are interested to study the two cases for strong resonant coupling laser (Δ2 ¼ 0), and correspondingly the probe laser detuning Δ1 ¼ 7 G. Assuming that the probe field is weak and the coupling laser is strong, i.e. G⪢γ 1 ; γ 2 ; Λ⪢g, the solutions of the density matrix element ρ~ ð1Þ 31 from (8) for Δ2 ¼ 0 and Δ1 ¼ 7 G are pffiffiffiffiffiffiffiffiffiffi ð1Þ γ 1 γ 2 cos θ eiϕ ðγ 2 þ ΛÞð1  ρ~ 33 Þ ð1Þ ρ~ 31 ¼7 ; ð16Þ ðγ 2 þ ΛÞð2γ 1 þ γ 2 þ ΛÞ  γ 1 γ 2 cos 2 θ which depends on the population of the ground state j3〉, ð1Þ ρ~ 33 ¼ ρ~ ð1Þ 22 with the same expression for both values of the probe field detuning Δ1 ¼ 7 G ð1Þ ρ~ 33 ¼

P Q

ð17Þ

where P ¼ γ 1 ðγ 2 þ ΛÞð2γ 1 þ γ 2 þ ΛÞ  ðγ 1 þ γ 2 þ ΛÞγ 1 γ 2 cos 2 θ Q ¼ ð2γ 1 þ ΛÞðγ 2 þ ΛÞð2γ 1 þ γ 2 þ ΛÞ  γ 1 γ 2 cos 2 θð2γ 1 þ γ 2 þ2ΛÞ: The quantities of interest are the real and the imaginary parts of ρ~ ð1Þ 31 , which can be written as pffiffiffiffiffiffiffiffiffiffi γ 1 γ 2 cos θ cos ϕðγ 2 þ ΛÞð1  ρ~ ð1Þ 33 Þ Þ ¼ 7 ; Reðρ~ ð1Þ 31 ðγ 2 þ ΛÞð2γ 1 þ γ 2 þ ΛÞ  γ 1 γ 2 cos 2 θ pffiffiffiffiffiffiffiffiffiffi γ 1 γ 2 cos θ sin ϕðγ 2 þ ΛÞð1  ρ~ ð1Þ 33 Þ : ð18Þ Imðρ~ ð1Þ 31 Þ ¼ 7 ðγ 2 þ ΛÞð2γ 1 þ γ 2 þ ΛÞ  γ 1 γ 2 cos 2 θ The dependence on the relative phase ϕ is done by the periodic functions cos ϕ and sin ϕ existing in the expressions (18). From the relation (17) one can notice that the populations of the states do not depend on the relative phase ϕ. We remark that for 0 r θ r π the refractive index is maximum for Δ1 ¼ G if cos ϕ ¼ 7 1, what means that ϕ ¼ 2kπ or ϕ ¼ ð2k þ 1Þπ , with k integer number. As result we have sin ϕ ¼ 0, and consequently Im ρ~ ð1Þ 31 ¼ 0, namely the absorption of the probe field vanishes. We obtain the same behaviour using the general formulas from our paper [23] as can be seen in Fig. 3. The other parameters of the system are γ 2 ¼ 2γ 1 , G ¼ 20γ 1 sin θ, g ¼ 0:01γ 1 sin θ, Λ ¼ 0:5γ 1 and θ ¼ π =3. The highest value of ð1Þ Re ρ~ 31 , the ordinate of B1 point from Fig. 3, is 0.102 and is calculated ð1Þ in the general case. The perturbatively highest value of Re ρ~ 31 , the

B1

Polarizations

0.1 0.05 0 -0.05 -0.1

pert

B1

0.1

Polarizations

The abscissae of A1 and Apert points are identical equal with  1.576 1 and that for A2 and Apert points are 1.576. 2 The relations (12) and (13) show us that for all incoherent ð1Þ ~ ð1Þ pumping rate Λ, there is no inversion population, ρ~ 11 o ρ~ ð1Þ 22 ; ρ 33 , and the probe field is absorbed and amplified periodically in relative phase ϕ due to the existence in the quantity Im ρ~ ð1Þ 31 of the cosinus function with ϕ as variable. ~ ð1Þ The dependence of the Re ρ~ ð1Þ 31 and Im ρ 31 on the angle θ between dipolar moments of the transitions j1〉-j3〉 and j2〉-j3〉 is included in the quantity η, directly proportional with cos θ. Same variation with θ has been obtained in [35].

81

0.05 0 -0.05 -0.1 -6

-4

-2

0

2

4

6

φ Fig. 3. Reðρ~ 31 Þ (solid line) and Im ρ~ 31 (dashed line) versus relative phase ϕ of the probe and coupling fields derived in different ways: (a) in all order of g and G and (b) in the first order of g, and strong coupling field, i.e. G⪢γ 1 ; γ 2 ; Λ⪢g. The parameters used are Δ1 ¼ G, Δ2 ¼ 0, γ 1 ¼ 1, γ 2 ¼ 2γ 1 , G ¼ 20γ 1 sin θ, g ¼ 0:01γ 1 sin θ, Λ ¼ 0:5γ 1 and θ ¼ π=3.

ordinate of Bpert point from Fig. 3, is 0.103, very close value with that 1 obtained with analytical formulas from [23]. For Δ1 ¼  G the highest value of the index of refraction appears, only for cos ϕ ¼ 8 1 which implies that ϕ ¼ ð2k þ 1Þπ , or ϕ ¼ 2kπ with k integer number. As well as the above situation, the absorption coefficient is zero, because Im ρ~ ð1Þ 31 is zero. We predicted the possibility to obtain higher refractive indexes without absorption in the presence of the incoherent pumping laser at the same values for the relative phase that in the absence of it [7]. In Fig. 4 we plotted Re ρ~ 31 and Im ρ~ 31 with the same parameters as in Fig. 3. Only the probe field detuning is different, Δ1 ¼  G. In Fig. 4(a) we plotted the quantities Re ρ~ 31 and Im ρ~ 31 in all order of g. It can be seen that the points C1 and C2 correspond to the high index of refraction without absorption the maximum value 0.102. Values almost identical are obtained in the first order of g being equal with 0.103, the ordinates of the points Cpert and Cpert 1 2 . We proved that the values obtained in the general case and in the perturbative regimen, for weak probe field and strong coupling field, are almost identical. In order to have gain of the probe field, and consequently amplification without inversion of population, it must fulfill two ð1Þ ~ ð1Þ ~ ð1Þ conditions: Im ρ~ 31 4 0 and ρ~ ð1Þ 33 ¼ ρ 22 4 ρ 11 . The populations of the ð1Þ states are all smaller than 1, then 1  ρ~ 33 4 0. In this type of system, the two dipole moments must not be orthogonal, so the value of θ is given as 0 r θ r π and θ a π =2. Then there are two situations. First, when 0 r θ o π =2 which implies cos θ 4 0. Consequently, for Δ2 ¼ 0 and Δ1 ¼ G, Im ρ~ ð1Þ 31 4 0 if and only if sin ϕ 4 0, equivalently with 2kπ o ϕ o ð2k þ 1Þπ with k integer number, and the second condition implies that 3ρ~ ð1Þ 33  1 40. The last inequality assesses for the parameters of the V-type system in the following constraints:

π

0rθo ; 2

ð19Þ

82

O. Budriga / Optics Communications 328 (2014) 77–86

C1

Polarizations

0.1

C2

0.05 0 -0.05 -0.1 pert

0.1

Polarizations

pert

C1

C2

0.05 0 Fig. 5. Three-dimensional plot of the quantity proportional with the probe field gain, Imðρ~ 31 Þ (vertical axis) versus relative phase ϕ of the probe and coupling fields (horizontal axis across the page) and angle between the two transition dipole moments θ (horizontal axis into the page) with parameters γ 1 ¼ 1, γ 2 ¼ 2γ 1 , G ¼ 20γ 1 sin θ, g ¼ 0:01γ 1 sin θ, Δ1 ¼ G, Δ2 ¼ 0, Λ ¼ 0:5γ 1 .

-0.05 -0.1 -6

-4

-2

0

φ

2

4

6

Fig. 4. Reðρ~ 31 Þ (solid line) and Imðρ~ 31 Þ (dashed line) versus relative phase ϕ of the probe and coupling fields derived in different ways: (a) in all order of g and G and (b) in the first order of g, and strong coupling field, i.e. G⪢γ 1 ; γ 2 ; Λ⪢g. The parameters used are Δ1 ¼  G, Δ2 ¼ 0, γ 1 ¼ 1, γ 2 ¼ 2γ 1 , G ¼ 20γ 1 sin θ, g ¼ 0:01γ 1 sin θ, Λ ¼ 0:5γ 1 and θ ¼ π=3. 0

0oΛoΛ ;

ð20Þ

0

where Λ is the real positive solution of the third order in incoherent pumping field rate Λ equation

Λ3 þ Λ2 ðγ 1 þ 2γ 2 Þ þ Λðγ 22 þ γ 1 γ 2 cos 2 θ 2γ 21 Þ  γ 1 γ 2 ½ð2  cos 2 θÞγ 1 þ ð1  2 cos 2 θÞγ 2  ¼ 0

ð21Þ

and additional relations between the rates of spontaneous emission from levels j1〉 and j2〉 to the ground level j3〉, 2γ1 and 2γ2, respectively, and θ, the angle between the two transition dipole ! ! moments d 13 and d 23 , which are detailed in Appendix B. Second case, when π =2 o θ r π , then Im ρ~ ð1Þ 31 4 0 if and only if sin ϕ o 0, that means ð2k  1Þπ o ϕ o 2kπ with k integer number. The condition 3ρ~ ð1Þ 33  1 4 0 implies that the relations (20) and (21) and (B.1)–(B.4) to be accomplished too. In Fig. 5 we represented the values of Im ρ~ 31 , which are proportional with the gain coefficient using the exact analytical formulas from [23]. The parameters of the three-level V-type system are chosen to satisfy relations (19)–(21) and (B.1)–(B.4). Their values are γ 2 ¼ 2γ 1 , G ¼ 20γ 1 sin θ, g ¼ 0:01γ 1 sin θ, Δ1 ¼ G, Δ2 ¼ 0, θ ¼ π =3, Λ ¼ 0:5γ 1 . As we predicted in the perturbation theory, if the relative phase ϕ is, on a period, between zero and π =2, there is gain in the probe field (Im ρ~ 31 4 0), and for π =2 o ϕ o π we have absorption of the probe field (Im ρ~ 31 o 0). The other case of interest is for the resonant coupling field and Δ1 ¼ G. As we proved, the expressions of the density matrix element ρ~ ð1Þ 31 for the last two cases studied are opposite in sign and the populations of the three levels are unchanged. Therefore the relative phase ϕ will be ð2k 1Þπ o ϕ o 2kπ , with k integer number for 0 r θ o π =2 or 2kπ o ϕ oð2k þ 1Þπ for π =2 o θ r π and the system parameters will fulfil the same relations as in the previous case, (19)–(21) and the others from Appendix B. This result coincides with that obtained using our analytical formulas from [23] as can be seen in Fig. 6, with parameters γ 2 ¼ 2γ 1 ,

Fig. 6. Three-dimensional plot of the quantity proportional with the probe field gain, Imðρ~ 31 Þ (vertical axis) versus relative phase ϕ of the probe and coupling fields (horizontal axis across the page) and angle between the two transition dipole moments θ (horizontal axis into the page) with parameters γ 1 ¼ 1, γ 2 ¼ 2γ 1 , G ¼ 20γ 1 sin θ, g ¼ 0:01γ 1 sin θ, Δ1 ¼  G, Δ2 ¼ 0, Λ ¼ 0:5γ 1 .

G ¼ 20γ 1 sin θ, g ¼ 0:01γ 1 sin θ, Δ1 ¼ G, Δ2 ¼ 0, θ ¼ π =3, Λ ¼ 0:5γ 1 . If we compare Fig. 5 with Fig. 6 we can observe that whenever the absorption (gain) probe field coefficient proportional with Im ρ~ 31 is positive for Δ1 ¼ G, for the same values this quantity is negative for Δ1 ¼  G, and viceversa. We have chosen the parameters to the conditions from psatisfy ffiffiffi (19) to (21) and (B.4). For γ 2 ¼ 2γ 1 4 2γ 1 and   1 2γ þ γ 4 θ ¼ π =3 ð cos θÞ2 ¼ o 1 2 ¼ 4 γ 1 þ 2γ 2 5 the incoherent pumping rate Λ must be smaller than the root of the third order equation 2Λ þ 10γ 1 Λ þ 5γ 21 Λ  11γ 31 ¼ 0: 3

2

ð22Þ 0

There is only one real solution of this equation Λ ¼ 0:783γ 1 . The 0 incoherent pumping rate Λ ¼ 0:5γ 1 is smaller than Λ . Therefore,

O. Budriga / Optics Communications 328 (2014) 77–86

the condition (20) and the other constraints to have amplification without population inversion are fulfilled.

4. V-type system with spontaneously generated coherence in LiH molecule For a real system the nonorthogonal transitions dipole moments are very hard to come by in nature. Applying external fields to build a mixing of atomic or molecular states leads to a system with parallel or antiparallel transitions dipole moments [32]. A way to create a Vee-type system with transition dipole moments antiparallel is to use a strong laser field to couple one of the two transitions in a Λ-type atom or molecule [32]. We shall obtain such V-type system in LiH molecule which has electronic transitions in the violet energy range. The LiH molecule is a good example of a system for studying the dependence of the gain (absorption) coefficient and refractive index on the choice of the ro-vibrational levels, as the hyperfine splitting (1 MHz) is much smaller than the spacing between two rotational ( 102 GHz) or two vibrational levels ( 104 GHz). We choose the Λ system built with the excited þ vibrational state A1 Σ ðv ¼ 1; j ¼ 1Þ and the vibrational states of þ 1 þ the ground level X Σ ðv ¼ 1; j ¼ 0Þ and X 1 Σ ðv ¼ 0; j ¼ 0Þ. When a strong field with the frequency ωL is applied between þ þ A1 Σ ðv ¼ 1; j ¼ 1Þ and X 1 Σ ðv ¼ 1; j ¼ 0Þ states of the LiH molecule, as we have shown in 3.2.2 it produces the dressed states ja〉 ¼ sin ψ j2〉 þ cos ψ j1〉 jb〉 ¼ cos ψ j2〉  sin ψ j1〉:

ð23Þ

with 0 r2ψ o π , tgð2ψ Þ ¼  2GL =ΔL , where GL is the real strong coupling field Rabi frequency and ΔL the detuning of the strong laser frequency from the molecular transition j1〉-j2〉 [36]. The separation energy between the two dressed states ja〉 and jb〉 are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 equal with ℏΩ, where Ω ¼ ΔL þ 4G2L . The energies of the two dressed states are the eigenvalues of the atomic system Hamiltonian and they are associated to the eigenstates ja〉 and jb〉, ℏλ 7 ¼ ℏðΔL 7 ΩÞ=2 (see Section 3.2.2). The energies of the dressed states when measured with respect to state j3〉 are W a3 ¼ ℏω13 þ ℏλ þ W b3 ¼ ℏω13  ℏλ þ :

ð24Þ

We build a V-type system with the upper dressed states ja〉, jb〉 þ and the ground level X 1 Σ ðv ¼ 0; j ¼ 0Þ. From relations (23) we obtain the transition dipole matrix elements ! ! ! d a3 ¼ d 23 sin ψ þ d 13 cos ψ ! ! ! d b3 ¼ d 23 cos ψ  d 13 sin ψ : ð25Þ The transition j2〉-j3〉 has the rate 2γ2 very small, 43.1 Hz [37], and it can be approximated with zero [19]. Therefore its dipole

83

! transition moment d 23 is null and the expressions of the two dipole moments from (25) become ! ! d a3 ¼ d 13 cos ψ ! ! d b3 ¼  d 13 sin ψ : ð26Þ We can see from the relation (26) the two transitions from the dressed states to the ground level have antiparallel dipole ! moments. We consider that the weak probe laser E p , with the angular frequency ωp and intensity I1 drives both dressed states ja〉 ! and jb〉 and the ground state j3〉 with Rabi frequencies 2g 1 ¼ 2 E p  ! ! ! 0 d a3 =ℏ and 2g 1 ¼ 2 E p  d b3 =ℏ, respectively. The strong coherent ! pump laser E c , with the angular frequency ωc and the intensity I2, acts on the transitions between both dressed states and the ground state too. The Rabi frequencies of the coupling field applied ! ! on the transitions ja〉-j3〉 and jb〉-j3〉 are 2G1 ¼ 2 E c  d b3 =ℏ and ! ! 0 2G1 ¼ 2 E c  d a3 =ℏ, respectively. The detunings of the probe laser frequency and the coupling laser frequency from the transitions ja〉-j3〉 and jb〉-j3〉 are denoted with Δp ¼ ωa3  ωp and Δc ¼ ωb3  ωc , respectively. An incoherent pumping laser is used in the transition ja〉-j3〉. This field can act in the transition between the levels jb〉 and j3〉 too. We assume that the incoherent pumping rate will have the same rate, 2Λ for both transitions. In Fig. 7 the diagrams with the Λ-type system (Fig. 7(a)) and V-type system (Fig. 7(b)) in LiH molecule are represented. Using the same formalism like in our previous work [23] and assuming that the two laser angular frequencies have almost the same values, i.e. ωp  ωc , we obtain the density matrix equation system  2γ 1 ρ~ aa þ 2Λρ~ 33  η eiϕ ρ~ ab  η e  iϕ ρ~ ba þ iðg þ G0 Þρ~ 3a  iðg þ G0 Þρ~ a3 ¼ 0  2γ 2 ρ~ bb þ 2Λρ~ 33  η eiϕ ρ~ ab  η e  iϕ ρ~ ba þ iðG þ g 0 Þρ~ 3b  iðG þ g 0 Þρ~ b3 ¼ 0  ðγ 1 þ 2Λ þ iΔp Þρ~ a3  η e  iϕ ρ~ b3 þ iðg þ G0 Þðρ~ 33  ρ~ aa Þ  iðG þ g 0 Þρ~ ab ¼ 0  ðγ 2 þ 2Λ  iΔc Þρ~ b3  η eiϕ ρ~ a3  iðg þ G0 Þρ~ ba þ iðG þ g 0 Þðρ~ 33  ρ~ bb Þ ¼ 0  ½γ 1 þ γ 2 þ iðΔp  Δc Þρ~ ab  η e  iϕ ðρ~ aa þ ρ~ bb Þ þ iðg þG0 Þρ~ 3b  iðG þ g 0 Þρ~ a3 ¼ 0

ð27Þ

0

with g; g 0 ; G; G the real Rabi frequencies corresponding to the complex Rabi frequencies g 1 ; g 01 ; G1 and G01 , respectively. This system of linear equations differs from the system of linear equations (3) by the appearance of the term 2Λρ~ 33 in the second equation. The other equations have the same terms as (3), where the Λ, g and G are replaced by 2Λ, g þ G0 and G þ g 0 , respectively. We solve the system of equations (27) numerically. The fact that the probe and coupling fields drive and couple both transitions ja〉-j3〉 and jb〉-j3〉 implies that there are no restrictions over the field polarizations and allow us to choose the electric polarizations ! ! ϵ p parallel with the dipole transition moment! d a3 and ! ϵc parallel with the dipole transition moment d b3 . Therefore the real Rabi frequencies are 2g ¼ 2Ep da3 =ℏ, 2g 0 ¼  2Ep db3 =ℏ, 2G ¼ 2Ec db3 =ℏ and 2G0 ¼  2Ec da3 =ℏ. The probe and coupling field

Fig. 7. (a) LiH molecule Λ system consisted of the excited level A1 Σ þ ðv ¼ 1; j ¼ 1Þ and vibrational states X 1 Σ þ ðv ¼ 1; j ¼ 0Þ and X 1 Σ þ ðv ¼ 0; j ¼ 0Þ of the ground level. An external strong laser field couples the two excited levels named j1〉 and j2〉. (b) The subsystem with the upper dressed states ja〉 and jb〉 and the ground level X 1 Σ þ ðv ¼ 0; j ¼ 0Þ ! ! form a three-level Vee-type system with the dipole moments of the two transitions, d a3 and d b3 , antiparallel.

O. Budriga / Optics Communications 328 (2014) 77–86

n ¼ 1þ

Nda3

ϵ0 ℏg

ðRe ρ~ a3 da3 Re ρ~ b3 db3 Þ;

-1000

ð28Þ

where ϵ0 is the vacuum electric permittivity, c is the speed of light in vacuum and ℏ is the reduced Planck constant. A similar formula can be deduced for the spontaneous emission rate of the state jb〉 2 to the ground level j3〉, 2γ b ¼ ω3b3 db3 =ð3π ℏϵ0 c3 Þ, with the frequency of the transition jb〉-j3〉, ωb3 ¼ W b3 =ℏ. The wave lengths of the þ transitions between the level A1 Σ ðv ¼ 1; j ¼ 1Þ and the levels þ þ X 1 Σ ðv ¼ 1; j ¼ 0Þ and X 1 Σ ðv ¼ 0; j ¼ 0Þ are λ12 ¼ 406:12 nm and λ13 ¼ 385:02 nm, respectively [19]. We derived the electric dipole moment transition d13 from the Fermi's golden rule, i.e. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d13 ¼ λ13 3ℏϵ0 γ 13 λ13 , with γ 13 ¼ 33 kHz [19]. The rate of the þ þ transition between the states A1 Σ ðv ¼ 1; j ¼ 1Þ and X 1 Σ ðv ¼ 1; j ¼ 0Þ is γ 12 ¼ 0:148 MHz [19]. The absorption coefficient α and the refractive index n of the probe field in SI units are 2N ωp da3 ðIm ρ~ a3 da3 Im ρ~ b3 db3 Þ; α¼ ϵ0 ℏcg

-500

1.002

-4

-2

0

2

4

6

2

4

6

M

0.998 0.996 0.994 0.992 -6

0.7

-4

-2

0

0.6

ð29Þ

0.5

ρ33

0.4

ρaa

0.3 0.2

ð30Þ

where N is the density of LiH molecules and g is the real Rabi frequency of the probe field. The system exhibits probe field absorption if α 4 0, gain of the probe field for α o0 and no probe absorption if α ¼ 0. We consider a resonant strong coupling field applied in the Vee-type system of interest. Hence, the strong field detuning Δc is zero. In Fig. 8 we represented the absorption coefficient α in units of m  1 and the refractive index n of the probe field for the probe detuning Δp ¼ G using numerical solutions of (27) and the expressions (29). The intensity of the probe field is I 1 ¼ 10  8 W=cm2 , the intensity of the strong coupling field is I 2 ¼ 1:5 W=cm2 and the other parameters of the system are γ a ¼ 2:475  104 Hz, γ b ¼ 0:33γ a , g ¼ 0:033γ a , g 0 ¼  0:019γ a , G ¼ 235:35γ a , G0 ¼  407:64γ a , Λ ¼ 0:5γ a , cos ψ ¼ 0:5, N ¼ 1012 molecules=cm3 . We choose the probe laser frequency ωp ¼ ωa3 G. From Fig. 8(a) one can see that the three level V-type system achieves periodically in the relative phase ϕ absorption and amplification (gain) of the probe field, when the absorption coefficient is positive and negative, respectively. The graphics show a quasi-high refractive index without absorption. The point M from Fig. 8(b) corresponds to zero absorption, a refractive index n ¼0.99979 for a relative phase ϕ ¼ 0:274, close by the high index of refraction value 1.00035 at ϕ ¼ 0. This behaviour is the consequence of the fact that all the fields act on both the transitions, evidenced by the presence of the imaginary and real parts of both density matrix elements ρ~ a3 and ρ~ b3 in the formulas of the absorption coefficient α and refractive index n, respectively. In Fig. 8(c) it can be seen that for some values of the relative phase ϕ the population of the highest excited levels of the system, ρ~ aa is lower than the population of the ground state, ρ~ 33 and the other excited level, ρ~ bb . At the same time the absorption coefficient α is negative. Hence, we have amplification without population inversion. In the light of the results from Section 3.2.2 we solved Eq. (21) with γ 1 ¼ γ a , γ 2 ¼ γ b and θ ¼ π . The solution of this 0 equation is Λ ¼ 0:764γ a . Numerical calculations of the absorption coefficient with the conditions α o 0, ρ~ aa o ρ~ bb and ρ~ aa o ρ~ 33 show us that we have amplification without population inversion for

-6

1

n

ω3a3 d2a3 ; 3π ℏϵ0 c 3

0

Populations

2γ a ¼

500

α (m-1)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi strengthsp depend onffi the intensities of the fields as Ep ¼ 2I 1 =ðcϵ0 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and Ec ¼ 2I 2 =ðcϵ0 Þ, respectively. The spontaneous emission rate from the dressed state ja〉 to the ground level j3〉, 2γa, can be expressed in terms of the dipole transition moment and the frequency of the transition ja〉-j3〉, ωa3 ¼ W a3 =ℏ in International System of Units by the Fermi's golden rule

ρbb

0.1 0

-6

-4

-2

0

2

4

6

φ Fig. 8. (a) The absorption coefficient α of the probe field, (b) the refractive index n of the probe field and (c) the populations of the V-type system levels versus relative phase ϕ with parameters γ a ¼ 2:475  104 Hz, γ b ¼ 0:33γ a , Δp ¼ G, g ¼ 0:033γ a , g 0 ¼  0:019γ a , Δc ¼ 0, G ¼ 235:35γ a , G0 ¼  407:64γ a , Λ ¼ 0:5γ a , cos ψ ¼ 0:5, N ¼ 1012 molecules=cm3 .

1.0×105

φ=π φ = π/3 φ = 3π/2 φ = 2π

8.0×104

α (m-1)

84

6.0×104 4.0×104 2.0×104 0.0 0

10

20

30

40

50

60

Λ/γa Fig. 9. The absorption coefficient α versus incoherent pumping rate Λ=γ a . The other parameters are the same as in Fig. 8. 0

Λ o 0:7γ a . The value 0:7γ a is very close to Λ . Therefore, the lower perturbation theory applied in the previous sections is appropriate for the three-level V-type system from the LiH molecule. The absorption coefficient α versus the incoherent pumping rate Λ for different relative phase ϕ is drawn in Fig. 9. It can be seen that for any ϕ the absorption coefficient α is negative for Λ o 0:763γ a and it increases with the increase of the incoherent pumping rate Λ. The greatest values of absorption coefficient are achieved for the relative phase ϕ ¼ π .

O. Budriga / Optics Communications 328 (2014) 77–86

1.01

5. Conclusions

1 0.99

φ=π φ = 3π/2 φ = −2π φ = π/3

n

0.98 0.97 0.96 0.95 0.94

0

10

20

30

40

50

60

Λ/γa Fig. 10. The index of refraction n versus incoherent pumping rate Λ=γ a . The other parameters are the same as in Fig. 8.

1

2000

α m

85

1

0 0.8 0.7 0.6 0.5

2000 4000 6 4

0.4

2 φ

γa

The three-level V-type system with spontaneously generated coherence and incoherent pumping was studied for a weak probe field and a strong coupling field. The perturbative method is good for this V-type system to solve the density matrix equations system. In this approximation we derived analytic explicit formulas for the gain probe field coefficient and refractive index, direct proportional with the periodic function in the relative phase ϕ. Thus we predicted the results obtained using the complicated formulas from [23]. For the first time we deduced the constraints on the incoherent pumping rate Λ, the angle θ, the spontaneous emission rates γ1 and γ2 to have amplification without population inversion. We found a real system of the LiH heteronuclear molecule as an example of a three-level V-type system with spontaneously generated coherence and incoherent pumping which achieves amplification without population inversion and quasi-high index of refraction without absorption with a proper choice of the relative phase of the probe and coupling fields and the incoherent pumping rate. The behaviour of the absorption coefficient is little different from the case of the theoretical case, because for the real LiH molecule system we consider that the three fields, probe, coupling and incoherent pumping fields, act on both transitions, and not only on one transition. In the future work we will study the subluminal or superluminal light propagation in the same three-level V-type system from LiH molecule.

Acknowledgements The author thanks the support of Ministry of Education and Research, Romania (program Laplas 3, PN 09 39). The author wishes to express acknowledgments to Madalina Boca for her advice on the numerical calculations.

0.3

0

0.2

2

0.1

4 6

Fig. 11. Three-dimensional plot of the absorption coefficient α versus relative phase ϕ (horizontal axis across the page) and incoherent pumping rate Λ=γ a (horizontal axis into the page). The other parameters are the same as in Fig. 8.

Appendix A. Off-diagonal density matrix elements The other off-diagonal density matrix elements, the solutions of equations system (8) are

ρ~ ð1Þ 12 ¼

M1 N1

ðA:1Þ

with We plotted in Fig. 10 the dependence of the refractive index n on the incoherent pumping rate for some relative phases. We can see that the refractive index n increases when the incoherent pumping rate increases until a critical value which differs for each relative phase ϕ. For an incoherent pumping rate greater than the critical value the refractive index decreases to a value which remains constant when Λ increases. The highest value of the refractive index n is obtained for ϕ ¼ π , at Λ ¼ 4:2γ a and is 1.0045871. The periodically absorption (α 4 0) and amplification (α o 0) of the probe field in the relative phase ϕ are emphasized in the Fig. 11 for the incoherent pumping rate between 0 and γa. Fig. 11 shows that for Λ 4 0:7γ a there is only absorption, because for each relative phase ϕ the absorption coefficient is positive. The three-level Vee-type system with spontaneously generated coherence and incoherent pumping built by applying the external fields in a Λ-system from a LiH molecule is a real system which can be modeled with the density matrix formalism. The quasi-high refractive index without absorption and amplification without population inversion can be obtained with the variation of the relative phase between the weak probe field and the strong coupling field ϕ and of the incoherent pumping rate Λ.

M 1 ¼ ηðγ 1 þ Λ þ iΔ1 Þf½η2  ðγ 2 þ Λ  iΔ2 Þ 2 ~ ð1Þ ~ ð1Þ ~ ð1Þ ðγ 1 þ Λ þ iΔ1 Þðρ~ ð1Þ 11 þ ρ 22 Þ  G ðρ 33  ρ 22 Þ ð0Þ ð0Þ iϕ þ gGρ~ 21 g þ η2 Ggeiϕ ðρ~ 33  ρ~ ð0Þ 11 Þ  ige ðγ 1 þ Λ þ iΔ1 Þ

½η2  ðγ 2 þ Λ  iΔ2 Þðγ 1 þ Λ þ iΔ1 Þρ~ ð0Þ 32

ðA:2Þ

N1 ¼ η2 G2  eiϕ ½η2  ðγ 2 þ Λ  iΔ2 Þðγ 1 þ Λ þ iΔ1 Þ fG2 þ ðγ 1 þ Λ þ iΔ1 Þ½γ 1 þ γ 2 þ iðΔ1  Δ2 Þg

ðA:3Þ

and

ρ~ ð1Þ 23 ¼

M2 N2

with M 2 ¼ iGðγ 1 þ Λ þ iΔ1 ÞffG2 þ ðγ 1 þ Λ þiΔ1 Þ ð1Þ ½γ 1 þ γ 2 þ iðΔ1  Δ2 Þgðρ~ 33  ρ~ ð1Þ 22 Þ ð1Þ ~ ð0Þ  η2 e  iϕ ðρ~ 11 þ ρ~ ð1Þ 22 Þg  ηgGðγ 1 þ Λ þ iΔ1 Þρ 32

 igffG2 þ ðγ 1 þ Λ þ iΔ1 Þ½γ 1 þ γ 2 þiðΔ1  Δ2 Þg iϕ ~ ð0Þ ~ ð0Þ ðγ 1 þ Λ þ iΔ1 Þρ~ ð0Þ 21 þ η e ðρ 33  ρ 11 Þg

ðA:4Þ

86

O. Budriga / Optics Communications 328 (2014) 77–86

  N 2 ¼ η2  ðγ 2 þ Λ  iΔ2 Þðγ 1 þ Λ þ iΔ1 Þ fG2 þ ðγ 1 þ Λ þ iΔ1 Þ½γ 1 þ γ 2 þ iðΔ1  Δ2 Þg  G2 η2 e  iϕ :

ðA:5Þ

The expressions of the level populations in the zeroth order of the perturbation theory are given by the relation (5). The explicit formulas for the first order perturbation theory population are too complicated, and are not deduced. Appendix B. Constraints on the parameters of the V-type system

γ 2 ¼ γ 1 and 0 o θ o π

ðB:1Þ

γ 1 4 γ 2 and 0 r θ r π

ðB:2Þ

γ1γ2

or

2γ 21  γ 22

γ1 γ2

o ð cos θÞ2 o

2γ 1 þ γ 2 γ 1 þ2γ 2

pffiffiffi

2γ 1 þ γ 2 : γ 1 þ 2γ 2

References [1] J. Javanainen, Europhys. Lett. 17 (1992) 407. [2] S. Menon, G.S. Agarwal, Phys. Rev. A 57 (1998) 4014.

[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

 ðB:3Þ

γ 2 4 2γ 1 and ð cos θÞ2 o

[9] [10] [11] [12] [13] [14]

To have amplification without population inversion of the probe field in a closed three-level V-type system, the rates of spontaneous emission from levels j1〉 and j2〉 to ground level j3〉, 2γ1 and 2γ2, respectively, and θ (θ a π =2), the angle between the ! ! two transition dipole moments d 13 and d 23 , must fulfil one of the conditions

pffiffiffi γ 1 o γ 2 o 2γ 1 and  2γ 2  γ 22 ð cos θÞ2 o 1

[3] [4] [5] [6] [7] [8]

ðB:4Þ

[29] [30] [31] [32] [33] [34] [35] [36] [37]

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