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Phase conjugate emission enhancement via incoherent processes
Optics Communications 226 (2003) 393–404 www.elsevier.com/locate/optcom
Phase conjugate emission enhancement via incoherent processes Suneel Singh *, Suresh Sripada School of Physics, University of Hyderabad, Hyderabad 500 046, India Received 29 April 2003; received in revised form 31 July 2003; accepted 4 August 2003
Abstract Optical phase conjugation by backward nondegenerate four-wave mixing in four-level systems is studied. It is found that incoherent processes like incoherent excitation of intermediate levels and (dephasing) collisions can create large atomic coherences by modifying the interference effects and lead to large enhancement of phase conjugate emission. Ó 2003 Published by Elsevier B.V.
1. Introduction Optical phase conjugation by degenerate and nondegenerate four-wave mixing in either homogeneously or Doppler broadened multilevel systems has been of great interest in the past [1–4] as well as in the recent years with the inclusion of electromagnetic (field) induced transparency (EIT) and coherent population trapping (CPT) effects [5]. Four-wave mixing in general denotes the interaction of four light waves with different frequencies and propagation directions. The interaction is due to the induced thirdorder nonlinear polarization that is proportional to the atomic coherences excited (by the incident fields) in the material. It is thus desirable to excite large coherences in the medium in order to obtain enhanced signal emission. In multilevel systems however signal generation efficiency critically depends upon interference between different quantum mechanical pathways (involving intermediate states) that lead to the excitation of atomic coherence pertaining to the four wave mixing. Enhanced (extra) resonances in four-wave mixing in multilevel systems [6] have been observed where dephasing collisions were utilized to remove (destructive) interference effects between quantum mechanical paths. These studies of collision induced extra resonances dealt with excitation of atomic coherence between either close lying (same parity) levels within the same electronic manifold [7] or different (parity) electronic levels [8]. More recently the
*
Corresponding author. E-mail address: [email protected] (S. Singh).
0030-4018/$ - see front matter Ó 2003 Published by Elsevier B.V. doi:10.1016/j.optcom.2003.08.029
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role of coherent and incoherent excitation (pumping) on Doppler broadened transitions has also been explored in the context of four-wave mixing without [9] and with electromagnetically induced transparency [10]. In this work we present a study of optical phase conjugation by nearly degenerate four-wave mixing in a homogenously broadened four-level system subjected to, in addition to the incident (coherent) fields, an external (incoherent) excitation mechanism. The latter may also account for (even in the absence of any external mechanism) the possibility of any excitation of finite populations (negligible compared to the ground state) in the intermediate levels by the weak incident fields in real experimental situations. In third-order perturbation theory where it is assumed that almost all population remains in the ground state, this effect may be incorporated phenomenologically as weak incoherent pumping. The form of third-order coherence pertaining to phase conjugate emission by near resonant four-wave mixing is obtained. Collisional (dephasing) effects of a perturber gas introduced in the medium are also analyzed. More specifically we show how these incoherent processes can create large atomic coherences by modifying the inherent interference effects and lead to large enhancement of coherent signal emission. A near resonant interaction scheme is considered in which the applied field frequencies are tuned far (large detuning) from the atomic resonance frequencies. As the one-photon detunings are very large compared to the applied field amplitudes (Rabi frequencies) we apply third-order perturbation theory for calculation of signals. Hence no CPT or EIT effects are included in the present study as these generally involve the application of rather strong coherent (pump) fields. The organization of the paper is as follows: Section 2 deals with formulation of the problem. Resonant interaction of a model four-level system with three coherent electromagnetic (laser) fields leading to generation of a fourth coherent field is treated. Considering the time evolution of density matrix in interaction picture, equations of motion of density matrix element of the four-level system are derived. The atomic (population) and collisional (coherence) relaxation as well as incoherent excitation (pumping) processes are incorporated phenomenologically in the equations of motion. In Section 3 we solve the set of density matrix equations (obtained in Section 2) using third-order perturbation theory. The form of third-order coherence pertaining to phase conjugate emission by near resonant four-wave mixing is obtained. Phase conjugate signal line shapes are evaluated numerically as a function of probe detuning for large (fixed) pump detunings. These are discussed with the aid of analytical calculations for different schemes of incoherent pumping of the intermediate levels. Contact is made with an earlier work [9] in which three-photon resonance was studied in the limiting case of only one of the intermediate level being pumped externally. Finally, in Section 4 we present a summary of results and conclusions.
2. Four-level system: density matrix equations 2.1. Formulation In the present work we have used the backward phase conjugate geometry to study the four-wave mixing. In the backward pump geometry, two counter propagating pump beams are labelled as the forward pump ~ Ef (at frequency xf , wave vector ~ kf ) and the backward pump ~ Eb (at frequency xb , wave vector ~ kb ) as shown in Fig. 1(a). A third beam, indicated as probe beam ~ Ep (at frequency xp , wave vector ~ kp ), is incident at small angle to the direction of the forward beam. Interaction between fields and the medium creates, among several others, one particular Fourier component of nonlinear optical polarization at frequency xs ¼ xf þ xb xp . This component generates a signal field oscillating at frequency xs and propagating in a direction given by the phase matching condition ~ ks ¼ ~ kf þ ~ kb ~ kp ’ ~ kp , i.e., in almost opposite direction to the probe field ~ Ep .
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Fig. 1. (a) Phase conjugation geometry of four-wave mixing. (b) Energy levels and interaction scheme of fields with various transitions.
The resonance behavior of the medium is modelled in terms of a four-level system. The relevant energylevel structure and near resonant interaction of the fields with the system is as depicted in Fig. 1(b). j1i is assumed to be the ground state. The spontaneous (natural) decay rates of the upper level j4i to the intermediate levels j3i and j2i are c43 and c42 , respectively. The spontaneous decay rates of the intermediate levels j3i and j2i to the ground level are c31 and c21 , respectively. Incoherent excitation (pumping) between the ground and intermediate levels is described by the parameter Kij ði; j ¼ 1; 2; 3Þ. The three laser field ~ Ef , ~ Ep and ~ Eb couple to dipole transitions j1i j3i, j1i j2i and j3i j4i, respectively. ~ Es is the field corresponding to the frequency xs ¼ xf xb þ xb generated via the process of four-wave mixing. The electric fields can be written as ~ Ej ðtÞ ¼ ~ Aj eixj t þ c:c:;
j ¼ f ; b; p;
ð1Þ
Aj ¼ ~ j expðikj zÞ are frequency and amplitude, respectively of the field ~ Ej ðj ¼ f ; b; pÞ. where xj and ~ The Hamiltonian for the combined atom and field system is H ¼ H0 þ V ;
ð2Þ
where H0 is the atomic Hamiltonian and the field–atom interaction Hamiltonian V is X ~ Ej ðj ¼ f ; b; pÞ: V ¼ ~ l~ E ¼ ~ l
ð3Þ
j
The Hamiltonian V in the interaction picture, under rotating-wave approximation, has the form V int ¼ h½af eiD31 t j3ih1j þ ap eiD21 t j2ih1j þ ab eiD43 t j4ih3j þ H:c::
ð4Þ
d ~ Aj Þ= h ðj ¼ f ; b; pÞ are the Rabi frequencies and jlihkj ðl; k ¼ 1; 2; 3; 4Þ are the atomic raising Here aj ¼ ð~ or lowering operators. The dipole operator matrix elements are ~ d ¼ hkj~ ljli ðl; k ¼ 1; 2; 3; 4Þ. The detuning of field frequencies from atomic resonance frequencies are denoted as D31 ¼ x31 xf ;
D21 ¼ x21 xp
and
where the atomic resonance frequencies are
D43 ¼ x43 xb ;
ð5Þ
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Wk Wl h with Wk ðWl Þ being the energy of the level jki ðjliÞ. xkl ¼
ð6Þ
2.2. Density matrix equations The time evolution of ijth element of the density matrix of the system in interaction picture is q_ ij ¼
i X ðqik Vkjint Vikint qkj Þ þ ðq_ ij Þrel þ Kij ; h k
ð7Þ
where int ¼ hmjV int jni Vmn
ð8Þ
can be calculated from Eq. (4). The phenomenologically incorporated second and the third terms in Eq. (7) describe incoherent processes such as the relaxation due to spontaneous emission, (dephasing) collisions and external excitation (pumping) of energy levels, respectively. From Eqs. (7), (8) and (4) and with the inclusion of the spontaneous emission, collisional relaxation and incoherent excitation (pumping) processes phenomenologically, we get the time evolution of the slowly varying components of the density matrix elements as follows: q~_ 12 ¼ ½iD21 C12 ~ q12 þ iap ðq22 q11 Þ þ iaf q~32 ;
ð9Þ
q~_ 13 ¼ ½iD31 C13 ~ q13 þ iaf ðq33 q11 Þ þ iap q~23 iab q~14 ;
ð10Þ
q34 þ iaf q~14 þ iab ðq44 q33 Þ; q~_ 34 ¼ ½iD43 C34 ~
ð11Þ
q~_ 14 ¼ ½iðD31 þ D43 Þ C14 ~ q14 þ iap q~24 þ iaf q~34 iab q~13 ;
ð12Þ
q~_ 23 ¼ ½iðD31 D21 Þ C23 ~ q23 þ iap q~13
ð13Þ
iaf q~21
iab q~24 ;
q~_ 24 ¼ ½iðD31 þ D43 D21 Þ C24 ~ q24 þ iap q~14 iab q~23 ;
ð14Þ
q_ 22 ¼ i½ap q~12 c:c: ðc21 þ K21 Þq22 þ K12 q11 þ c42 q44 ;
ð15Þ
q_ 33 ¼ i½af q~13 c:c: i½ab q~34 c:c: ðc31 þ K31 Þq33 þ K13 q11 þ c43 q44 ;
ð16Þ
q_ 44 ¼ i½ab q~34 c:c: ðc43 þ c42 Þq44 ;
ð17Þ
q_ 11 ¼ q_ 22 q_ 33 q_ 44 :
ð18Þ
The last equation follows from the population conservation condition: q11 ¼ 1 q22 q33 q44
ð19Þ
for a closed four-level system. In the above equations the off-diagonal (coherence) relaxation rates due to spontaneous emission and collisional decay are 1 1 C12 ¼ ðK12 þ K13 Þ þ ðc21 þ K21 Þ þ cph ; 2 2 1 1 C13 ¼ ðK12 þ K13 Þ þ ðc31 þ K31 Þ þ cph ; 2 2 1 1 C34 ¼ ðc31 þ K31 Þ þ ðc43 þ c42 Þ þ cph ; 2 2
ð20Þ ð21Þ ð22Þ
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1 1 C14 ¼ ðc43 þ c42 Þ þ ðK12 þ K13 Þ þ cph ; 2 2 1 1 C24 ¼ ðc21 þ K21 Þ þ ðc43 þ c42 Þ þ cph ; 2 2 1 1 C23 ¼ ðc21 þ K21 Þ þ ðc31 þ K31 Þ þ cph : 2 2
397
ð23Þ ð24Þ ð25Þ
3. Phase conjugation by near resonant four-wave mixing 3.1. Steady-state nonlinear (third-order) coherence Interaction between fields and the medium creates, among several others, one particular Fourier component of nonlinear optical polarization at frequency xs ¼ xf þ xb xp . This component generates a fourwave mixing signal field oscillating at frequency xs and propagating in a direction given by the phase matching condition ~ ks ¼ ~ kf þ ~ kb ~ kp ’ ~ kp , i.e., in almost opposite direction to the probe field ~ Ep . From the equations of motion of the atomic density matrix elements set up in Section 2, we observe that the induced (macroscopic) nonlinear polarization density at frequency xs ¼ xf þ xb xp pertaining to fourwave mixing is related to the slowly varying component q~24 of the off-diagonal matrix element oscillating at frequency xs and with wave vector ~ ks . In the limit of weak (nonsaturating) intensity pumps and probe fields, we make third-order expansion of density matrix to derive the third-order component of the off-diagonal ð3Þ ð3Þ matrix element q~24 oscillating at frequency xs . Using this perturbation scheme the calculation of q~24 under steady-state conditions is as follows. In the absence of the input field, initial population exist in ground and intermediate states and are given by q033 ¼
K13 ðc21 þ K21 Þ ; ðc31 þ K31 þ K13 Þðc21 þ K21 Þ þ K12 ðc31 þ K31 Þ
ð26Þ
q022 ¼
K12 ðc31 þ K31 Þ ; ðc31 þ K31 þ K13 Þðc21 þ K21 Þ þ K12 ðc31 þ K31 Þ
ð27Þ
q011 ¼
ðc31 þ K31 Þðc21 þ K21 Þ : ðc31 þ K31 þ K13 Þðc21 þ K21 Þ þ K12 ðc31 þ K31 Þ
ð28Þ
It should be noted that in absence of external excitation (pumping) ðK13 ¼ K31 ¼ K12 ¼ K21 ¼ 0Þ, the intermediate levels are unpopulated and the entire population lies in the ground state, i.e., q022 ¼ q033 ¼ 0, q011 ¼ 1. The action of radiation fields, to first order in the applied fields, generates one-photon coherence q011 q033 ; C13 þ iD31 q0 q022 ðpÞ q21 ðxp Þ ¼ iap 11 ; C12 þ iD21 q033 ðbÞ q43 ðxb Þ ¼ iab C34 þ iD43 ðfÞ
q31 ðxf Þ ¼ iaf
ð29Þ ð30Þ ð31Þ
between the transition connected by them. In the second-order approximation the action of forward ðf;bÞ and backward pumps generate a two-photon coherence q~14 between the highest excited state j4i and the ground state j1i, oscillating at the sum frequency xf þ xb . Similarly the forward pump and probe ðf;pÞ fields generate a coherence q~23 between the close lying components j3i and j2i of the intermediate state, oscillating at the difference frequency xf xp , given by
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ðf;bÞ q~14
ðf;pÞ q~23
" # ð0Þ ð0Þ ð0Þ ð0Þ af ab q44 q33 q33 q11 ¼ ; ½ðD31 þ D43 Þ þ iC14 D43 þ iC34 D31 þ iC13 " # ð0Þ ð0Þ ð0Þ ð0Þ ap af q33 q11 q22 q11 ¼ þ : ½ðD31 D21 Þ þ iC23 D31 þ iC13 D21 þ iC21
ð32Þ
ð33Þ
ðf;bÞ ðf;pÞ The coupling of probe wave ap to q~14 and the backward pump ab to q~23 generates the steady-state four-wave mixing coherence n o 1 ðf;b;pÞ ðf;bÞ ðf;pÞ ¼ ap q~14 ab q~23 : q~24 ð34Þ D43 þ D31 D21 þ iC24
It is evident from the above expression that the four-wave mixing polarization in the present case arises ðf;bÞ from pure (second-order) two-photon coherences as the terms in the curly brackets are proportional to q~14 ðf;pÞ and q~23 . It does not involve (to second order in fields), the excitation of population in the intermediate states j2i and j3i. We now assume all spontaneous decay rates equal c ð¼ c21 ¼ c31 ¼ c4 ¼ c42 þ c43 Þ and put ðf;b;pÞ K2 ð¼ K12 ¼ K21 Þ, K3 ð¼ K31 ¼ K13 Þ. The full form of the third-order coherence q~24 now can be written as ap ab af c2 ðc14 þ c23 Þ; D43 þ D31 D21 þ iC24 ðc þ 2K3 Þðc þ K2 Þ þ K2 ðc þ K3 Þ " # ðKc2 Þ ð1 þ Kc2 ÞðKc3 Þ 1 1 ; c14 ¼ þ D43 þ iC34 D31 þ D43 þ iC14 D31 þ iC13 ðD31 þ iC13 ÞðD43 þ D31 þ iC14 Þ " # ðKc2 Þ ðKc3 Þ ið1:9cph þ K2 þ K3 Þ 1 c23 ¼ þ þ D31 D21 þ iC23 ðD31 þ iC13 ÞðD21 þ iC12 Þ D31 þ iC13 ðD21 þ iC12 Þ
ðf;b;pÞ q~24 ¼
þ
1 : ðD31 þ iC13 ÞðD21 þ iC12 Þ
ð35aÞ ð35bÞ
ð35cÞ
ðf;pÞ ðf;bÞ Here c23 and c14 are contributions of coherences q~23 and q~14 , respectively. The phase conjugate signal generated via near resonant wave-mixing in an optically thin medium is proportional to S given by q~ðf;b;pÞ 2 24 S¼ ð36Þ : ap ab af
3.2. Numerical and analytical results and discussion Expression (35a)–(35c) has the usual resonances at Dkj þ iCjk ðk; j ¼ 1–4Þ when the input field frequencies coincide with the atomic resonance frequencies. We are however interested in a near resonant case in which the input field frequencies are tuned far from the atomic resonance frequencies. Such that the detuning Dkj ðj; k ¼ 1; 4Þ cj ðj ¼ 2; 3; 4Þ, Cjk ðj; k ¼ 1–4Þ and Doppler width cD , etc. Also the large forward ðD31 Þ and backward ðD43 Þ pump detunings are usually held constant and the behavior of the signal is studied as a function of the probe detuning D21 . Therefore holding the pumping detunings fixed and by varying the probe detuning we expect to observe the following resonances in the signal lineshape:
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(i) At D21 ¼ D31 þ D43 with a width C24 . This resonance condition describes coherent signal emission when the combination of the incident field frequencies xf þ xb xp coincides with the frequency separation x42 of the atomic transition j4i $ j2i. (ii) At D21 ¼ D31 with a width C23 . This is a Raman type resonance originating from the term c23 , that is the ðf;pÞ contribution of (two-photon) coherence q~23 . This describes signal emission when the difference of the forward pump and probe frequencies xf xp is equal to the separation between the intermediate state frequencies x32 . The signal S is numerically computed using Eqs. (35a)–(35c) and (36) for typical experimental parameters [6,7], is plotted as a function of probe detuning D21 in Figs. 2–7. The forward and backward pump detunings are equal and fixed to a large (compared to the Doppler width, cD value, i.e., D ¼ D31 ¼ D43 ¼ 10cD . All other parameters are also scaled in units of the Doppler width, i.e., cD ¼ 1. We now derive analytical results from Eq. (35a)–(35c) to explain the numerical results. In a purely radiatively broadened medium, in which there is no collisions and no (incoherent) excitation of levels, both the resonances described above will be absent. This can be seen by putting K2 ¼ K3 ¼ cph ¼ 0 in Eq. (35a)–(35c). We first consider the resonance at D21 þ D43 . In this case we observe that only the last term of c23 (see Eq. (35b)) and c14 (see Eq. (35c)) survive, which then combine to cancel the first term in the denominator of Eq. (35a), because C24 ¼ C12 þ C14 ¼ ð1=2Þðc2 þ c4 Þ. The cancellation of the resonance at D21 ¼ D31 þ D43 may thus be interpreted as due to destructive interference between two (two-photon) coðf;pÞ ðf;bÞ herences q~23 and q~14 that form, the two distinct channel for excitation of the third-order coherence pertaining to coherent signal emission. As a consequence of this destructive interference, the generated ðf;b;pÞ third-order coherence q~24 is vanishingly small. Moreover, this is true even in the presence of dephasing ðf;b;pÞ collisions with a perturber gas introduced in the medium. This is apparent from the form of q~24 obtained from Eq. (35a)–(35c) in the presence of dephasing collisions cph 6¼ 0 as ap ab af 1 ðf;b;pÞ ¼ q~24 ðD31 þ iC13 ÞðD21 þ iC12 Þ D43 þ D31 þ iC14 icph 1 1:9 þ þ : ð37Þ D43 þ D31 D21 þ iC24 D43 þ D31 þ iC14 D31 D21 þ iC23
Fig. 2. Phase conjugate signal S as a function of the probe detuning D21 and incoherent excitation rate K2 (indicated on the right-hand side of the curves) for fixed (large) values of the forward and backward pump detunings, D43 ¼ D31 ¼ 10. (a) No collisions, cph ¼ 0. This particular resonance at D21 ¼ D31 þ D43 does not occur in the absence of incoherent excitation. (b) Effect of dephasing collisions, cph ¼ 10c, on the resonance shown in (a). Note the large enhancement of the signal in the presence of incoherent excitation. The resonances exhibit collisional broadening with a full line width 2cph at half maximum. The spontaneous decay rates in this and subsequent figures are taken as c ¼ 0:01 (in units of Doppler width cD ) and the value of dephasing rate for C23 is 0.1cph .
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Fig. 3. Phase conjugate signal S as a function of the probe detuning D21 and incoherent excitation rate K3 (indicated on the right-hand side of the curves) for fixed (large) values of the forward and backward pump detunings, D43 ¼ D31 ¼ 10. (a) No collisions, cph ¼ 0. (b) Effect of dephasing collisions, cph ¼ 10c, on the resonance shown in (a). The resonances exhibit collisional broadening with a full line width 2cph at half maximum.
Fig. 4. Phase conjugate signal S as a function of the probe detuning D21 and equal incoherent excitation rate K2 ¼ K3 (indicated on the right-hand side of the curves) for fixed (large) values of the forward and backward pump detunings, D43 ¼ D31 ¼ 10. (a) No collisions, cph ¼ 0. (b) Effect of dephasing collisions, cph ¼ 10c, on the resonance shown in (a). The resonances exhibit collisional broadening with a full line width 2cph at half maximum.
Fig. 5. Phase conjugate signal S as a function of the probe detuning D21 and incoherent excitation rate K2 (indicated on the right-hand side of the curves) for fixed (large) values of the forward and backward pump detunings, D43 ¼ D31 ¼ 10. (a) No collisions, cph ¼ 0. This Raman type resonance at D21 ¼ D31 does not occur in the absence of incoherent excitation. (b) Effect of dephasing collisions, cph ¼ 10c, on the resonance shown in (a). Note the large enhancement of the signal in the presence of incoherent excitation.
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Fig. 6. Phase conjugate signal S as a function of the probe detuning D21 and incoherent excitation rate K3 for fixed (large) values of the forward and backward pump detunings, D43 ¼ D31 ¼ 10. (a) No collisions, cph ¼ 0. This Raman type resonance at D21 ¼ D31 does not occur in the absence of incoherent excitation. (b) Effect of dephasing collisions, cph ¼ 10c, on the resonance shown in (a). The signal behavior is same as that observed in the previous Fig. 5.
Fig. 7. Phase conjugate signal S as a function of the probe detuning D21 and equal incoherent excitation rate K2 ¼ K3 for fixed (large) values of the forward and backward pump detunings, D43 ¼ D31 ¼ 10. (a) No collisions, cph ¼ 0. (b) Effect of dephasing collisions, cph ¼ 10c, on the resonance shown in (a). The signal is very weak in this case.
Because D43 ¼ D31 ¼ D cph ; ck ; Cjk in the near resonant case under consideration we find that at the resonance condition D21 ¼ D43 þ D31 ¼ 2D, the above expression reduces to " # 1 1:4cph ðf;b;pÞ 1 q~24 ’ ap ab af 3 ; ð38Þ 2D 2 c þ cph which shows that even in the presence of large dephasing collisions, i.e., cph c the signal given by Eq. (36) that goes as Sdep ¼ 0:15D6 is very weak. If now any one of the intermediate levels (say level j2i) is subjected to incoherent excitation, we obtain from (35a)–(35c) by putting K3 ¼ cph ¼ 0 and neglecting nonresonant terms K2 1 1 1 1 ðf;b;pÞ ~ q24 ’ af ab ap þ ; c þ 2K2 ðD43 þ D31 D21 þ iC24 Þ ðD31 þ iC13 Þ D43 þ D31 þ iC14 D31 D21 þ iC23 ð39Þ
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which shows the dependence of the resonance at D21 ¼ D43 þ D21 on (incoherent) pumping parameter K2 . The value of the signal around D21 ¼ D31 þ D43 goes as ðK2 c DÞ Sinc ¼
K c þ 2K2
2
1 1 : 4c2 D4
The ratio Sinc =Sdep reveals that the emitted signal is enhanced by many orders of magnitude in the presence of incoherent excitation. This is due to the fact that when incoherent excitation is applied to one of (or both) the intermediate level(s), the (second-order) coherences created between the levels involved are modified (see Eqs. (32) and (33)) through change in level populations. It is clear that the contributions of ðf;b;pÞ the two-photon coherences to the third-order coherence q~24 (the two terms in curly brackets of Eq. (34) are no longer equal and therefore the destructive interference is not complete). Consequently, the thirdorder coherence is large and we observe a highly enhanced coherence at D21 ¼ D31 þ D43 whose intensity increases with increasing incoherent excitation rate. This behavior is illustrated in Fig. 2. Dephasing collisions tend to broaden the resonance and the signal magnitude is reduced considerably (Fig. 2(b)) but is still very large compared to case when only dephasing collisions are present. These are in agreement with the numerical results presented earlier [9] for this particular case in a slightly different interaction scheme. Fig. 3 show the phase conjugate signal emission when only level j3i is incoherently pumped, i.e., K3 6¼ 0 and K2 ¼ 0. The emitted signal intensity in this case is found to be very weak. An expansion the terms c14 and c23 in the limit of Cij Dij yields c 1 ðK3 =cÞ ðf;b;pÞ ’ af ab ap q~24 c þ 2K3 ðD43 þ D31 D21 þ iC24 Þ ðD31 þ D43 ÞD43 iC34 iC14 ðK3 =cÞ iC23 iC12 1þ þ þ 1þ þ þ þ D21 ðD21 D31 Þ D43 ðD31 þ D43 Þ ðD21 D31 Þ D21 iK3 ðD43 þ D31 D21 þ iC24 Þ þ þ ; ð40Þ þ ðD21 ÞD31 ðD31 þ D43 Þ D21 ðD21 D31 ÞðD31 þ D43 Þ which shows that around the three-photon resonance when D21 ¼ D31 þ D43 the leading terms (proportional to K3 =c) cancel out and only the higher-order terms (proportional to K3 Cij ) remain, giving rise to a very weak signal. Moreover, when K3 c the antiresonance structure at the line center is due to the destructive interference between the resonant and non-resonant (last term) contributions. As K3 increases further the resonant term dominates and a peak appears at the line center whose magnitude continues to increase and does saturate only when K3 ’ Dij . Fig. 4 illustrates the case when both the intermediate states j2i and j3i are incoherently excited at the same rate K2 ¼ K3 . Except for overall reduction of magnitude, the signal behavior in this case is qualitatively similar to that observed in Fig. 2. Similarly the Raman type (two-photon) resonance (described by the c23 in Eq. (35a)–(35c)) that would occur when the probe detuning D21 ¼ D31 is absent in a purely radiatively broadened system when cph ¼ K2 ¼ K3 ¼ 0. Physically this is due to fact that the two contributions coming from two channels ðf;pÞ beading to creation of q~23 (see Eq. (33)) interfere destructively when Cij are governed only by spontaneous decay rates. In the presence of incoherent excitation (which modifies the ground and intermediate energy level populations) the contributions of the two channels is rendered unequal, thereby leading to a finite ðf;b;pÞ value of coherence q~24 . These type of Raman resonance between close lying excited (ground) states of a V (K) type three level system have been previously observed in collisional studies of nearly degenerate four wave mixing in Na vapor [6,7]. These were termed pressure induced extra resonances (PIER4) since they were found to occur only in the presence of dephasing collisions (with a buffer gas) which removes the
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destructive interference between coherent quantum mechanical pathways. The numerically computed signals in Figs. 5–7 show the occurrence of extra resonance in the region where D21 ¼ D31 . We find that the signal intensity obtained from an incoherently pumped system is highly enhanced compared to that in the presence of dephasing collisions. However, we find in Fig. 7 that the signals are very weak when both the level are pumped at equal rate, i.e.,K2 ¼ K3 consistent with the fact that Raman type resonances between equally populated levels vanish. This behavior can be understood from Eq. (35a)–(35c) if we ignore the nonresonant term coming from c14 . Only the term c23 (Eq. (35c)) contributes to the coherence; the magnitude of which around the resonance D21 ¼ D31 is found to be invariant under interchange of the pumping rates K2 with K3 . When K2 ¼ K3 , the leading terms of the order of K2 =c and K3 =c (the second and the third term in the square bracket of Eq. (35c)) combine to cancel the resonance at D21 ¼ D31 .
4. Summary and conclusions In this work we have studied phase conjugate signal emission by near resonant four-wave mixing in an incoherently pumped four-level system. Three possible schemes involving incoherent pumping of the intermediate levels j2i and j3i are considered, viz., (i) when only level j2i is pumped, K3 ¼ 0, (ii) when only level j3i is pumped, K2 ¼ 0 and (iii) when both the levels j2i and j3i are pumped at the same rate, K2 ¼ K3 . Phase conjugate signal emission is studied as a function of the probe detuning for fixed large values of the forward and backward pump detunings. Effects of dephasing collisions that may occur when a foreign (perturber) gas is added to the medium are also incorporated in the theory. It is found that even very weak incoherent pumping of the intermediate levels can induce (extra) resonances whose intensity is enhanced by several orders of magnitude compared to those induced by dephasing collisions alone. In reality such a situation may arise in experiments where the weak incident fields may excite a negligibly small yet finite amount of population in intermediate levels, that could act like a weak incoherent pumping. For the resonance at D21 ¼ D31 þ D43 , the emitted PC signals saturate fast (K2 of the order of spontaneous decay rates c) in the case when only level j2i is pumped incoherently. In contrast to this behavior, we find that due to cancellation of contributions from the terms c14 and c23 the signal is very weak in the case when only level j3i is (incoherently) pumped. This resonance still exists, though with somewhat reduced intensity, when both the intermediate levels are incoherently pumped at the same rate (K2 ¼ K3 ). Whereas at resonance condition D21 ¼ D31 the phase conjugate signal depends symmetrically on the level pumping parameter, i.e., signal intensity remains the same if K2 and K3 are interchanged. This signal is very weak when both the intermediate levels are pumped at the same rate. Effect of dephasing collisions is to broaden the resonance lineshape and to decrease the magnitude of the signals. The resonances observed here provide yet another example of generation of a coherent phenomenon by an incoherent process. It must be emphasized however that the mechanism involved in generation here is different from that involving dephasing collisions. Two distinct channels are involved in the excitation of the coherence pertaining to phase conjugate emission. The third-order coherence here has contributions from a couple of two-photon coherences, arising out of the two different channels of excitation that are opposite in nature. In a purely radiatively broadened system the two contributions are equal (in magnitude) and therefore destructive interference between them destroys the third-order coherence. Incoherent pumping of population in intermediate level(s) causes (through modification of level populations), an imbalance between the excitation channels, as a consequence of which the destructive interference between them is not complete. In our study we have neglected the effect of atomic motion. The result for a Doppler broadened medium can be obtained from Eq. (35a)–(35c) by replacing the detunings Dij with Dmn þ~ v ~ kj , m; n ¼ 1–4, j ¼ f ; b; p. Since in the present study one-photon detunings are very large compared to the
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Doppler width, the additional Doppler broadening factor occurring in the one-photon detuning terms can be neglected. Only the terms D21 D31 D43 and D21 D31 will then have additional factors of ð~ kp ~ kf ~ kb Þ ~ v and ð~ kp ~ kf Þ ~ v. In that case the resonance at D21 ¼ D31 D43 will appear Doppler broadened (typically of the order of 2 GHz) with peak intensity somewhat reduced. Owing to the residual Doppler broadening the two-photon Raman resonance at D21 ¼ D31 then would have additional width that is of the order of 50–100 MHz [6,7] in typical experimental situations. Finally, it should be pointed out that the magnitude of interfering contributions depends upon the energy level populations which in turn also depend on relaxation processes and intensity of applied coherent radiation [10] apart from incoherent excitation mechanism. Hence the observed nonlinear optical phenomena in multilevel systems that involve generation of nonlinear coherence could also be modified by a variation in these parameters. Such studies [11] are of great interest in the recent times.
References [1] G.P. Agarwal, Phys. Rev. A 28 (1983) 2286; D. Bloch, M. Ducloy, J. Opt. Soc. Am. 73 (1983) 635; D.G. Steel, J.F. Lam, Opt. Commun. 40 (1981) 77; D.G. Steel, R.C. Lind, J.F. Lam, Phys. Rev. A 23 (1981) 2531; J.P. Woerdmann, M.F.H. Schuurmans, Opt. Lett. 6 (1981) 239; G.P. Agarwal, Opt. Commun. 39 (1981) 272. [2] M.T. Araujo, S.S. Vianna, G. Grynberg, Opt. Commun. 80 (1990) 79; G. Gryneberg, M. Pinard, P. Verkerk, Opt. Commun. 50 (1984) 261; M. Pinard, P. Verkerk, G. Grynberg, Opt. Lett. 9 (1984) 399; M. Pinard, B. Kleinmann, G. Grynberg, Opt. Commun. 51 (1984) 182. [3] D. Bloch, M. Ducloy, J. Opt. Soc. Am. 73 (1983) 1844(E); M. Ducloy, F.A.M. de Olivervia, D. Bloch, Phys. Rev. A 32 (1984) 1614; S. Le Boiteux, P. Simoneau, D. Bloch, F.A.M. de Oliveria, M. Ducloy, IEEE J. Quantum Electron. QE-22 (1986) 1229. [4] A.M. Levine, N. Chencinski, W.M. Schreiber, A.N. Weiszmann, Y. Prior, Phys. Rev. A 35 (1987) 2250; F.A.M. de Oliveria, Cid B. de Araujo, Jose R. Rios Leite, Phys. Rev. A 38 (1988) 5688. [5] P.R. Hemmer, D.P. Katz, J. Donoghue, M. Cronin-Golomb, M. Shahriar, P. Kumar, Opt. Lett. 20 (1995) 982; T.T. Grove, M. Shahriar, P.R. Hemmer, P. Kumar, V.S. Sudarshanam, M. Cronin-Golomb, Opt. Lett. 22 (1997) 769. [6] Review article by L. Rothberg, in: E. Wolf (Ed.), Progress in Optics, Elsevier, Amsterdam, 1987. [7] Y. Prior, A.R. Bogdan, M. Dagenais, N. Bloembergen, Phys. Rev. Lett. 46 (1981) 111; A.R. Bogdan, M. Downer, N. Bloembergen, Opt. Lett. 6 (1981) 348. [8] E. Giacobino, P.R. Berman, Phys. Rev. Lett. 58 (1987) 21. [9] Suneel Singh, Opt. Lett. 27 (2002) 539. [10] A.K. Popov, Proc. SPIE 3485 (1998) 252. [11] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge, 1997, pp. 220–245, and references therein.
Phase conjugate emission enhancement via incoherent processes Suneel Singh *, Suresh Sripada School of Physics, University of Hyderabad, Hyderabad 500 046, India Received 29 April 2003; received in revised form 31 July 2003; accepted 4 August 2003
Abstract Optical phase conjugation by backward nondegenerate four-wave mixing in four-level systems is studied. It is found that incoherent processes like incoherent excitation of intermediate levels and (dephasing) collisions can create large atomic coherences by modifying the interference effects and lead to large enhancement of phase conjugate emission. Ó 2003 Published by Elsevier B.V.
1. Introduction Optical phase conjugation by degenerate and nondegenerate four-wave mixing in either homogeneously or Doppler broadened multilevel systems has been of great interest in the past [1–4] as well as in the recent years with the inclusion of electromagnetic (field) induced transparency (EIT) and coherent population trapping (CPT) effects [5]. Four-wave mixing in general denotes the interaction of four light waves with different frequencies and propagation directions. The interaction is due to the induced thirdorder nonlinear polarization that is proportional to the atomic coherences excited (by the incident fields) in the material. It is thus desirable to excite large coherences in the medium in order to obtain enhanced signal emission. In multilevel systems however signal generation efficiency critically depends upon interference between different quantum mechanical pathways (involving intermediate states) that lead to the excitation of atomic coherence pertaining to the four wave mixing. Enhanced (extra) resonances in four-wave mixing in multilevel systems [6] have been observed where dephasing collisions were utilized to remove (destructive) interference effects between quantum mechanical paths. These studies of collision induced extra resonances dealt with excitation of atomic coherence between either close lying (same parity) levels within the same electronic manifold [7] or different (parity) electronic levels [8]. More recently the
*
Corresponding author. E-mail address: [email protected] (S. Singh).
0030-4018/$ - see front matter Ó 2003 Published by Elsevier B.V. doi:10.1016/j.optcom.2003.08.029
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role of coherent and incoherent excitation (pumping) on Doppler broadened transitions has also been explored in the context of four-wave mixing without [9] and with electromagnetically induced transparency [10]. In this work we present a study of optical phase conjugation by nearly degenerate four-wave mixing in a homogenously broadened four-level system subjected to, in addition to the incident (coherent) fields, an external (incoherent) excitation mechanism. The latter may also account for (even in the absence of any external mechanism) the possibility of any excitation of finite populations (negligible compared to the ground state) in the intermediate levels by the weak incident fields in real experimental situations. In third-order perturbation theory where it is assumed that almost all population remains in the ground state, this effect may be incorporated phenomenologically as weak incoherent pumping. The form of third-order coherence pertaining to phase conjugate emission by near resonant four-wave mixing is obtained. Collisional (dephasing) effects of a perturber gas introduced in the medium are also analyzed. More specifically we show how these incoherent processes can create large atomic coherences by modifying the inherent interference effects and lead to large enhancement of coherent signal emission. A near resonant interaction scheme is considered in which the applied field frequencies are tuned far (large detuning) from the atomic resonance frequencies. As the one-photon detunings are very large compared to the applied field amplitudes (Rabi frequencies) we apply third-order perturbation theory for calculation of signals. Hence no CPT or EIT effects are included in the present study as these generally involve the application of rather strong coherent (pump) fields. The organization of the paper is as follows: Section 2 deals with formulation of the problem. Resonant interaction of a model four-level system with three coherent electromagnetic (laser) fields leading to generation of a fourth coherent field is treated. Considering the time evolution of density matrix in interaction picture, equations of motion of density matrix element of the four-level system are derived. The atomic (population) and collisional (coherence) relaxation as well as incoherent excitation (pumping) processes are incorporated phenomenologically in the equations of motion. In Section 3 we solve the set of density matrix equations (obtained in Section 2) using third-order perturbation theory. The form of third-order coherence pertaining to phase conjugate emission by near resonant four-wave mixing is obtained. Phase conjugate signal line shapes are evaluated numerically as a function of probe detuning for large (fixed) pump detunings. These are discussed with the aid of analytical calculations for different schemes of incoherent pumping of the intermediate levels. Contact is made with an earlier work [9] in which three-photon resonance was studied in the limiting case of only one of the intermediate level being pumped externally. Finally, in Section 4 we present a summary of results and conclusions.
2. Four-level system: density matrix equations 2.1. Formulation In the present work we have used the backward phase conjugate geometry to study the four-wave mixing. In the backward pump geometry, two counter propagating pump beams are labelled as the forward pump ~ Ef (at frequency xf , wave vector ~ kf ) and the backward pump ~ Eb (at frequency xb , wave vector ~ kb ) as shown in Fig. 1(a). A third beam, indicated as probe beam ~ Ep (at frequency xp , wave vector ~ kp ), is incident at small angle to the direction of the forward beam. Interaction between fields and the medium creates, among several others, one particular Fourier component of nonlinear optical polarization at frequency xs ¼ xf þ xb xp . This component generates a signal field oscillating at frequency xs and propagating in a direction given by the phase matching condition ~ ks ¼ ~ kf þ ~ kb ~ kp ’ ~ kp , i.e., in almost opposite direction to the probe field ~ Ep .
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395
Fig. 1. (a) Phase conjugation geometry of four-wave mixing. (b) Energy levels and interaction scheme of fields with various transitions.
The resonance behavior of the medium is modelled in terms of a four-level system. The relevant energylevel structure and near resonant interaction of the fields with the system is as depicted in Fig. 1(b). j1i is assumed to be the ground state. The spontaneous (natural) decay rates of the upper level j4i to the intermediate levels j3i and j2i are c43 and c42 , respectively. The spontaneous decay rates of the intermediate levels j3i and j2i to the ground level are c31 and c21 , respectively. Incoherent excitation (pumping) between the ground and intermediate levels is described by the parameter Kij ði; j ¼ 1; 2; 3Þ. The three laser field ~ Ef , ~ Ep and ~ Eb couple to dipole transitions j1i j3i, j1i j2i and j3i j4i, respectively. ~ Es is the field corresponding to the frequency xs ¼ xf xb þ xb generated via the process of four-wave mixing. The electric fields can be written as ~ Ej ðtÞ ¼ ~ Aj eixj t þ c:c:;
j ¼ f ; b; p;
ð1Þ
Aj ¼ ~ j expðikj zÞ are frequency and amplitude, respectively of the field ~ Ej ðj ¼ f ; b; pÞ. where xj and ~ The Hamiltonian for the combined atom and field system is H ¼ H0 þ V ;
ð2Þ
where H0 is the atomic Hamiltonian and the field–atom interaction Hamiltonian V is X ~ Ej ðj ¼ f ; b; pÞ: V ¼ ~ l~ E ¼ ~ l
ð3Þ
j
The Hamiltonian V in the interaction picture, under rotating-wave approximation, has the form V int ¼ h½af eiD31 t j3ih1j þ ap eiD21 t j2ih1j þ ab eiD43 t j4ih3j þ H:c::
ð4Þ
d ~ Aj Þ= h ðj ¼ f ; b; pÞ are the Rabi frequencies and jlihkj ðl; k ¼ 1; 2; 3; 4Þ are the atomic raising Here aj ¼ ð~ or lowering operators. The dipole operator matrix elements are ~ d ¼ hkj~ ljli ðl; k ¼ 1; 2; 3; 4Þ. The detuning of field frequencies from atomic resonance frequencies are denoted as D31 ¼ x31 xf ;
D21 ¼ x21 xp
and
where the atomic resonance frequencies are
D43 ¼ x43 xb ;
ð5Þ
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Wk Wl h with Wk ðWl Þ being the energy of the level jki ðjliÞ. xkl ¼
ð6Þ
2.2. Density matrix equations The time evolution of ijth element of the density matrix of the system in interaction picture is q_ ij ¼
i X ðqik Vkjint Vikint qkj Þ þ ðq_ ij Þrel þ Kij ; h k
ð7Þ
where int ¼ hmjV int jni Vmn
ð8Þ
can be calculated from Eq. (4). The phenomenologically incorporated second and the third terms in Eq. (7) describe incoherent processes such as the relaxation due to spontaneous emission, (dephasing) collisions and external excitation (pumping) of energy levels, respectively. From Eqs. (7), (8) and (4) and with the inclusion of the spontaneous emission, collisional relaxation and incoherent excitation (pumping) processes phenomenologically, we get the time evolution of the slowly varying components of the density matrix elements as follows: q~_ 12 ¼ ½iD21 C12 ~ q12 þ iap ðq22 q11 Þ þ iaf q~32 ;
ð9Þ
q~_ 13 ¼ ½iD31 C13 ~ q13 þ iaf ðq33 q11 Þ þ iap q~23 iab q~14 ;
ð10Þ
q34 þ iaf q~14 þ iab ðq44 q33 Þ; q~_ 34 ¼ ½iD43 C34 ~
ð11Þ
q~_ 14 ¼ ½iðD31 þ D43 Þ C14 ~ q14 þ iap q~24 þ iaf q~34 iab q~13 ;
ð12Þ
q~_ 23 ¼ ½iðD31 D21 Þ C23 ~ q23 þ iap q~13
ð13Þ
iaf q~21
iab q~24 ;
q~_ 24 ¼ ½iðD31 þ D43 D21 Þ C24 ~ q24 þ iap q~14 iab q~23 ;
ð14Þ
q_ 22 ¼ i½ap q~12 c:c: ðc21 þ K21 Þq22 þ K12 q11 þ c42 q44 ;
ð15Þ
q_ 33 ¼ i½af q~13 c:c: i½ab q~34 c:c: ðc31 þ K31 Þq33 þ K13 q11 þ c43 q44 ;
ð16Þ
q_ 44 ¼ i½ab q~34 c:c: ðc43 þ c42 Þq44 ;
ð17Þ
q_ 11 ¼ q_ 22 q_ 33 q_ 44 :
ð18Þ
The last equation follows from the population conservation condition: q11 ¼ 1 q22 q33 q44
ð19Þ
for a closed four-level system. In the above equations the off-diagonal (coherence) relaxation rates due to spontaneous emission and collisional decay are 1 1 C12 ¼ ðK12 þ K13 Þ þ ðc21 þ K21 Þ þ cph ; 2 2 1 1 C13 ¼ ðK12 þ K13 Þ þ ðc31 þ K31 Þ þ cph ; 2 2 1 1 C34 ¼ ðc31 þ K31 Þ þ ðc43 þ c42 Þ þ cph ; 2 2
ð20Þ ð21Þ ð22Þ
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1 1 C14 ¼ ðc43 þ c42 Þ þ ðK12 þ K13 Þ þ cph ; 2 2 1 1 C24 ¼ ðc21 þ K21 Þ þ ðc43 þ c42 Þ þ cph ; 2 2 1 1 C23 ¼ ðc21 þ K21 Þ þ ðc31 þ K31 Þ þ cph : 2 2
397
ð23Þ ð24Þ ð25Þ
3. Phase conjugation by near resonant four-wave mixing 3.1. Steady-state nonlinear (third-order) coherence Interaction between fields and the medium creates, among several others, one particular Fourier component of nonlinear optical polarization at frequency xs ¼ xf þ xb xp . This component generates a fourwave mixing signal field oscillating at frequency xs and propagating in a direction given by the phase matching condition ~ ks ¼ ~ kf þ ~ kb ~ kp ’ ~ kp , i.e., in almost opposite direction to the probe field ~ Ep . From the equations of motion of the atomic density matrix elements set up in Section 2, we observe that the induced (macroscopic) nonlinear polarization density at frequency xs ¼ xf þ xb xp pertaining to fourwave mixing is related to the slowly varying component q~24 of the off-diagonal matrix element oscillating at frequency xs and with wave vector ~ ks . In the limit of weak (nonsaturating) intensity pumps and probe fields, we make third-order expansion of density matrix to derive the third-order component of the off-diagonal ð3Þ ð3Þ matrix element q~24 oscillating at frequency xs . Using this perturbation scheme the calculation of q~24 under steady-state conditions is as follows. In the absence of the input field, initial population exist in ground and intermediate states and are given by q033 ¼
K13 ðc21 þ K21 Þ ; ðc31 þ K31 þ K13 Þðc21 þ K21 Þ þ K12 ðc31 þ K31 Þ
ð26Þ
q022 ¼
K12 ðc31 þ K31 Þ ; ðc31 þ K31 þ K13 Þðc21 þ K21 Þ þ K12 ðc31 þ K31 Þ
ð27Þ
q011 ¼
ðc31 þ K31 Þðc21 þ K21 Þ : ðc31 þ K31 þ K13 Þðc21 þ K21 Þ þ K12 ðc31 þ K31 Þ
ð28Þ
It should be noted that in absence of external excitation (pumping) ðK13 ¼ K31 ¼ K12 ¼ K21 ¼ 0Þ, the intermediate levels are unpopulated and the entire population lies in the ground state, i.e., q022 ¼ q033 ¼ 0, q011 ¼ 1. The action of radiation fields, to first order in the applied fields, generates one-photon coherence q011 q033 ; C13 þ iD31 q0 q022 ðpÞ q21 ðxp Þ ¼ iap 11 ; C12 þ iD21 q033 ðbÞ q43 ðxb Þ ¼ iab C34 þ iD43 ðfÞ
q31 ðxf Þ ¼ iaf
ð29Þ ð30Þ ð31Þ
between the transition connected by them. In the second-order approximation the action of forward ðf;bÞ and backward pumps generate a two-photon coherence q~14 between the highest excited state j4i and the ground state j1i, oscillating at the sum frequency xf þ xb . Similarly the forward pump and probe ðf;pÞ fields generate a coherence q~23 between the close lying components j3i and j2i of the intermediate state, oscillating at the difference frequency xf xp , given by
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ðf;bÞ q~14
ðf;pÞ q~23
" # ð0Þ ð0Þ ð0Þ ð0Þ af ab q44 q33 q33 q11 ¼ ; ½ðD31 þ D43 Þ þ iC14 D43 þ iC34 D31 þ iC13 " # ð0Þ ð0Þ ð0Þ ð0Þ ap af q33 q11 q22 q11 ¼ þ : ½ðD31 D21 Þ þ iC23 D31 þ iC13 D21 þ iC21
ð32Þ
ð33Þ
ðf;bÞ ðf;pÞ The coupling of probe wave ap to q~14 and the backward pump ab to q~23 generates the steady-state four-wave mixing coherence n o 1 ðf;b;pÞ ðf;bÞ ðf;pÞ ¼ ap q~14 ab q~23 : q~24 ð34Þ D43 þ D31 D21 þ iC24
It is evident from the above expression that the four-wave mixing polarization in the present case arises ðf;bÞ from pure (second-order) two-photon coherences as the terms in the curly brackets are proportional to q~14 ðf;pÞ and q~23 . It does not involve (to second order in fields), the excitation of population in the intermediate states j2i and j3i. We now assume all spontaneous decay rates equal c ð¼ c21 ¼ c31 ¼ c4 ¼ c42 þ c43 Þ and put ðf;b;pÞ K2 ð¼ K12 ¼ K21 Þ, K3 ð¼ K31 ¼ K13 Þ. The full form of the third-order coherence q~24 now can be written as ap ab af c2 ðc14 þ c23 Þ; D43 þ D31 D21 þ iC24 ðc þ 2K3 Þðc þ K2 Þ þ K2 ðc þ K3 Þ " # ðKc2 Þ ð1 þ Kc2 ÞðKc3 Þ 1 1 ; c14 ¼ þ D43 þ iC34 D31 þ D43 þ iC14 D31 þ iC13 ðD31 þ iC13 ÞðD43 þ D31 þ iC14 Þ " # ðKc2 Þ ðKc3 Þ ið1:9cph þ K2 þ K3 Þ 1 c23 ¼ þ þ D31 D21 þ iC23 ðD31 þ iC13 ÞðD21 þ iC12 Þ D31 þ iC13 ðD21 þ iC12 Þ
ðf;b;pÞ q~24 ¼
þ
1 : ðD31 þ iC13 ÞðD21 þ iC12 Þ
ð35aÞ ð35bÞ
ð35cÞ
ðf;pÞ ðf;bÞ Here c23 and c14 are contributions of coherences q~23 and q~14 , respectively. The phase conjugate signal generated via near resonant wave-mixing in an optically thin medium is proportional to S given by q~ðf;b;pÞ 2 24 S¼ ð36Þ : ap ab af
3.2. Numerical and analytical results and discussion Expression (35a)–(35c) has the usual resonances at Dkj þ iCjk ðk; j ¼ 1–4Þ when the input field frequencies coincide with the atomic resonance frequencies. We are however interested in a near resonant case in which the input field frequencies are tuned far from the atomic resonance frequencies. Such that the detuning Dkj ðj; k ¼ 1; 4Þ cj ðj ¼ 2; 3; 4Þ, Cjk ðj; k ¼ 1–4Þ and Doppler width cD , etc. Also the large forward ðD31 Þ and backward ðD43 Þ pump detunings are usually held constant and the behavior of the signal is studied as a function of the probe detuning D21 . Therefore holding the pumping detunings fixed and by varying the probe detuning we expect to observe the following resonances in the signal lineshape:
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399
(i) At D21 ¼ D31 þ D43 with a width C24 . This resonance condition describes coherent signal emission when the combination of the incident field frequencies xf þ xb xp coincides with the frequency separation x42 of the atomic transition j4i $ j2i. (ii) At D21 ¼ D31 with a width C23 . This is a Raman type resonance originating from the term c23 , that is the ðf;pÞ contribution of (two-photon) coherence q~23 . This describes signal emission when the difference of the forward pump and probe frequencies xf xp is equal to the separation between the intermediate state frequencies x32 . The signal S is numerically computed using Eqs. (35a)–(35c) and (36) for typical experimental parameters [6,7], is plotted as a function of probe detuning D21 in Figs. 2–7. The forward and backward pump detunings are equal and fixed to a large (compared to the Doppler width, cD value, i.e., D ¼ D31 ¼ D43 ¼ 10cD . All other parameters are also scaled in units of the Doppler width, i.e., cD ¼ 1. We now derive analytical results from Eq. (35a)–(35c) to explain the numerical results. In a purely radiatively broadened medium, in which there is no collisions and no (incoherent) excitation of levels, both the resonances described above will be absent. This can be seen by putting K2 ¼ K3 ¼ cph ¼ 0 in Eq. (35a)–(35c). We first consider the resonance at D21 þ D43 . In this case we observe that only the last term of c23 (see Eq. (35b)) and c14 (see Eq. (35c)) survive, which then combine to cancel the first term in the denominator of Eq. (35a), because C24 ¼ C12 þ C14 ¼ ð1=2Þðc2 þ c4 Þ. The cancellation of the resonance at D21 ¼ D31 þ D43 may thus be interpreted as due to destructive interference between two (two-photon) coðf;pÞ ðf;bÞ herences q~23 and q~14 that form, the two distinct channel for excitation of the third-order coherence pertaining to coherent signal emission. As a consequence of this destructive interference, the generated ðf;b;pÞ third-order coherence q~24 is vanishingly small. Moreover, this is true even in the presence of dephasing ðf;b;pÞ collisions with a perturber gas introduced in the medium. This is apparent from the form of q~24 obtained from Eq. (35a)–(35c) in the presence of dephasing collisions cph 6¼ 0 as ap ab af 1 ðf;b;pÞ ¼ q~24 ðD31 þ iC13 ÞðD21 þ iC12 Þ D43 þ D31 þ iC14 icph 1 1:9 þ þ : ð37Þ D43 þ D31 D21 þ iC24 D43 þ D31 þ iC14 D31 D21 þ iC23
Fig. 2. Phase conjugate signal S as a function of the probe detuning D21 and incoherent excitation rate K2 (indicated on the right-hand side of the curves) for fixed (large) values of the forward and backward pump detunings, D43 ¼ D31 ¼ 10. (a) No collisions, cph ¼ 0. This particular resonance at D21 ¼ D31 þ D43 does not occur in the absence of incoherent excitation. (b) Effect of dephasing collisions, cph ¼ 10c, on the resonance shown in (a). Note the large enhancement of the signal in the presence of incoherent excitation. The resonances exhibit collisional broadening with a full line width 2cph at half maximum. The spontaneous decay rates in this and subsequent figures are taken as c ¼ 0:01 (in units of Doppler width cD ) and the value of dephasing rate for C23 is 0.1cph .
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Fig. 3. Phase conjugate signal S as a function of the probe detuning D21 and incoherent excitation rate K3 (indicated on the right-hand side of the curves) for fixed (large) values of the forward and backward pump detunings, D43 ¼ D31 ¼ 10. (a) No collisions, cph ¼ 0. (b) Effect of dephasing collisions, cph ¼ 10c, on the resonance shown in (a). The resonances exhibit collisional broadening with a full line width 2cph at half maximum.
Fig. 4. Phase conjugate signal S as a function of the probe detuning D21 and equal incoherent excitation rate K2 ¼ K3 (indicated on the right-hand side of the curves) for fixed (large) values of the forward and backward pump detunings, D43 ¼ D31 ¼ 10. (a) No collisions, cph ¼ 0. (b) Effect of dephasing collisions, cph ¼ 10c, on the resonance shown in (a). The resonances exhibit collisional broadening with a full line width 2cph at half maximum.
Fig. 5. Phase conjugate signal S as a function of the probe detuning D21 and incoherent excitation rate K2 (indicated on the right-hand side of the curves) for fixed (large) values of the forward and backward pump detunings, D43 ¼ D31 ¼ 10. (a) No collisions, cph ¼ 0. This Raman type resonance at D21 ¼ D31 does not occur in the absence of incoherent excitation. (b) Effect of dephasing collisions, cph ¼ 10c, on the resonance shown in (a). Note the large enhancement of the signal in the presence of incoherent excitation.
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Fig. 6. Phase conjugate signal S as a function of the probe detuning D21 and incoherent excitation rate K3 for fixed (large) values of the forward and backward pump detunings, D43 ¼ D31 ¼ 10. (a) No collisions, cph ¼ 0. This Raman type resonance at D21 ¼ D31 does not occur in the absence of incoherent excitation. (b) Effect of dephasing collisions, cph ¼ 10c, on the resonance shown in (a). The signal behavior is same as that observed in the previous Fig. 5.
Fig. 7. Phase conjugate signal S as a function of the probe detuning D21 and equal incoherent excitation rate K2 ¼ K3 for fixed (large) values of the forward and backward pump detunings, D43 ¼ D31 ¼ 10. (a) No collisions, cph ¼ 0. (b) Effect of dephasing collisions, cph ¼ 10c, on the resonance shown in (a). The signal is very weak in this case.
Because D43 ¼ D31 ¼ D cph ; ck ; Cjk in the near resonant case under consideration we find that at the resonance condition D21 ¼ D43 þ D31 ¼ 2D, the above expression reduces to " # 1 1:4cph ðf;b;pÞ 1 q~24 ’ ap ab af 3 ; ð38Þ 2D 2 c þ cph which shows that even in the presence of large dephasing collisions, i.e., cph c the signal given by Eq. (36) that goes as Sdep ¼ 0:15D6 is very weak. If now any one of the intermediate levels (say level j2i) is subjected to incoherent excitation, we obtain from (35a)–(35c) by putting K3 ¼ cph ¼ 0 and neglecting nonresonant terms K2 1 1 1 1 ðf;b;pÞ ~ q24 ’ af ab ap þ ; c þ 2K2 ðD43 þ D31 D21 þ iC24 Þ ðD31 þ iC13 Þ D43 þ D31 þ iC14 D31 D21 þ iC23 ð39Þ
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which shows the dependence of the resonance at D21 ¼ D43 þ D21 on (incoherent) pumping parameter K2 . The value of the signal around D21 ¼ D31 þ D43 goes as ðK2 c DÞ Sinc ¼
K c þ 2K2
2
1 1 : 4c2 D4
The ratio Sinc =Sdep reveals that the emitted signal is enhanced by many orders of magnitude in the presence of incoherent excitation. This is due to the fact that when incoherent excitation is applied to one of (or both) the intermediate level(s), the (second-order) coherences created between the levels involved are modified (see Eqs. (32) and (33)) through change in level populations. It is clear that the contributions of ðf;b;pÞ the two-photon coherences to the third-order coherence q~24 (the two terms in curly brackets of Eq. (34) are no longer equal and therefore the destructive interference is not complete). Consequently, the thirdorder coherence is large and we observe a highly enhanced coherence at D21 ¼ D31 þ D43 whose intensity increases with increasing incoherent excitation rate. This behavior is illustrated in Fig. 2. Dephasing collisions tend to broaden the resonance and the signal magnitude is reduced considerably (Fig. 2(b)) but is still very large compared to case when only dephasing collisions are present. These are in agreement with the numerical results presented earlier [9] for this particular case in a slightly different interaction scheme. Fig. 3 show the phase conjugate signal emission when only level j3i is incoherently pumped, i.e., K3 6¼ 0 and K2 ¼ 0. The emitted signal intensity in this case is found to be very weak. An expansion the terms c14 and c23 in the limit of Cij Dij yields c 1 ðK3 =cÞ ðf;b;pÞ ’ af ab ap q~24 c þ 2K3 ðD43 þ D31 D21 þ iC24 Þ ðD31 þ D43 ÞD43 iC34 iC14 ðK3 =cÞ iC23 iC12 1þ þ þ 1þ þ þ þ D21 ðD21 D31 Þ D43 ðD31 þ D43 Þ ðD21 D31 Þ D21 iK3 ðD43 þ D31 D21 þ iC24 Þ þ þ ; ð40Þ þ ðD21 ÞD31 ðD31 þ D43 Þ D21 ðD21 D31 ÞðD31 þ D43 Þ which shows that around the three-photon resonance when D21 ¼ D31 þ D43 the leading terms (proportional to K3 =c) cancel out and only the higher-order terms (proportional to K3 Cij ) remain, giving rise to a very weak signal. Moreover, when K3 c the antiresonance structure at the line center is due to the destructive interference between the resonant and non-resonant (last term) contributions. As K3 increases further the resonant term dominates and a peak appears at the line center whose magnitude continues to increase and does saturate only when K3 ’ Dij . Fig. 4 illustrates the case when both the intermediate states j2i and j3i are incoherently excited at the same rate K2 ¼ K3 . Except for overall reduction of magnitude, the signal behavior in this case is qualitatively similar to that observed in Fig. 2. Similarly the Raman type (two-photon) resonance (described by the c23 in Eq. (35a)–(35c)) that would occur when the probe detuning D21 ¼ D31 is absent in a purely radiatively broadened system when cph ¼ K2 ¼ K3 ¼ 0. Physically this is due to fact that the two contributions coming from two channels ðf;pÞ beading to creation of q~23 (see Eq. (33)) interfere destructively when Cij are governed only by spontaneous decay rates. In the presence of incoherent excitation (which modifies the ground and intermediate energy level populations) the contributions of the two channels is rendered unequal, thereby leading to a finite ðf;b;pÞ value of coherence q~24 . These type of Raman resonance between close lying excited (ground) states of a V (K) type three level system have been previously observed in collisional studies of nearly degenerate four wave mixing in Na vapor [6,7]. These were termed pressure induced extra resonances (PIER4) since they were found to occur only in the presence of dephasing collisions (with a buffer gas) which removes the
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destructive interference between coherent quantum mechanical pathways. The numerically computed signals in Figs. 5–7 show the occurrence of extra resonance in the region where D21 ¼ D31 . We find that the signal intensity obtained from an incoherently pumped system is highly enhanced compared to that in the presence of dephasing collisions. However, we find in Fig. 7 that the signals are very weak when both the level are pumped at equal rate, i.e.,K2 ¼ K3 consistent with the fact that Raman type resonances between equally populated levels vanish. This behavior can be understood from Eq. (35a)–(35c) if we ignore the nonresonant term coming from c14 . Only the term c23 (Eq. (35c)) contributes to the coherence; the magnitude of which around the resonance D21 ¼ D31 is found to be invariant under interchange of the pumping rates K2 with K3 . When K2 ¼ K3 , the leading terms of the order of K2 =c and K3 =c (the second and the third term in the square bracket of Eq. (35c)) combine to cancel the resonance at D21 ¼ D31 .
4. Summary and conclusions In this work we have studied phase conjugate signal emission by near resonant four-wave mixing in an incoherently pumped four-level system. Three possible schemes involving incoherent pumping of the intermediate levels j2i and j3i are considered, viz., (i) when only level j2i is pumped, K3 ¼ 0, (ii) when only level j3i is pumped, K2 ¼ 0 and (iii) when both the levels j2i and j3i are pumped at the same rate, K2 ¼ K3 . Phase conjugate signal emission is studied as a function of the probe detuning for fixed large values of the forward and backward pump detunings. Effects of dephasing collisions that may occur when a foreign (perturber) gas is added to the medium are also incorporated in the theory. It is found that even very weak incoherent pumping of the intermediate levels can induce (extra) resonances whose intensity is enhanced by several orders of magnitude compared to those induced by dephasing collisions alone. In reality such a situation may arise in experiments where the weak incident fields may excite a negligibly small yet finite amount of population in intermediate levels, that could act like a weak incoherent pumping. For the resonance at D21 ¼ D31 þ D43 , the emitted PC signals saturate fast (K2 of the order of spontaneous decay rates c) in the case when only level j2i is pumped incoherently. In contrast to this behavior, we find that due to cancellation of contributions from the terms c14 and c23 the signal is very weak in the case when only level j3i is (incoherently) pumped. This resonance still exists, though with somewhat reduced intensity, when both the intermediate levels are incoherently pumped at the same rate (K2 ¼ K3 ). Whereas at resonance condition D21 ¼ D31 the phase conjugate signal depends symmetrically on the level pumping parameter, i.e., signal intensity remains the same if K2 and K3 are interchanged. This signal is very weak when both the intermediate levels are pumped at the same rate. Effect of dephasing collisions is to broaden the resonance lineshape and to decrease the magnitude of the signals. The resonances observed here provide yet another example of generation of a coherent phenomenon by an incoherent process. It must be emphasized however that the mechanism involved in generation here is different from that involving dephasing collisions. Two distinct channels are involved in the excitation of the coherence pertaining to phase conjugate emission. The third-order coherence here has contributions from a couple of two-photon coherences, arising out of the two different channels of excitation that are opposite in nature. In a purely radiatively broadened system the two contributions are equal (in magnitude) and therefore destructive interference between them destroys the third-order coherence. Incoherent pumping of population in intermediate level(s) causes (through modification of level populations), an imbalance between the excitation channels, as a consequence of which the destructive interference between them is not complete. In our study we have neglected the effect of atomic motion. The result for a Doppler broadened medium can be obtained from Eq. (35a)–(35c) by replacing the detunings Dij with Dmn þ~ v ~ kj , m; n ¼ 1–4, j ¼ f ; b; p. Since in the present study one-photon detunings are very large compared to the
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Doppler width, the additional Doppler broadening factor occurring in the one-photon detuning terms can be neglected. Only the terms D21 D31 D43 and D21 D31 will then have additional factors of ð~ kp ~ kf ~ kb Þ ~ v and ð~ kp ~ kf Þ ~ v. In that case the resonance at D21 ¼ D31 D43 will appear Doppler broadened (typically of the order of 2 GHz) with peak intensity somewhat reduced. Owing to the residual Doppler broadening the two-photon Raman resonance at D21 ¼ D31 then would have additional width that is of the order of 50–100 MHz [6,7] in typical experimental situations. Finally, it should be pointed out that the magnitude of interfering contributions depends upon the energy level populations which in turn also depend on relaxation processes and intensity of applied coherent radiation [10] apart from incoherent excitation mechanism. Hence the observed nonlinear optical phenomena in multilevel systems that involve generation of nonlinear coherence could also be modified by a variation in these parameters. Such studies [11] are of great interest in the recent times.
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