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Modifying spontaneous emission via interferences from incoherent pump fields
Physics Letters A 307 (2003) 8–12 www.elsevier.com/locate/pla
Modifying spontaneous emission via interferences from incoherent pump fields David Bullock, Jörg Evers, Christoph H. Keitel ∗ Theoretische Quantendynamik, Fakultät für Mathematik und Physik, University of Freiburg, Hermann-Herder Str. 3, 79104 Freiburg, Germany Received 10 July 2002; received in revised form 9 September 2002; accepted 13 September 2002 Communicated by P.R. Holland
Abstract The spontaneous emission spectrum is evaluated from the doublet of a 4-level atomic system. Excitation from the ground state of the closed system is supplied solely by two incoherent pump fields with one involving the doublet states; i.e., no laser fields are involved at any stage of the problem. We find that the interference which arises due to the incoherent pump field induces substantial modifications of the spontaneous emission spectrum. In particular, the occurrence of a narrow spectral feature is shown to be quite sensitive to the incoherent pump-induced interference. 2002 Elsevier Science B.V. All rights reserved. PACS: 42.50.Lc; 42.50.Ct; 42.50.Hz
Laser field induced quantum interferences have been proven to be seminal in controlling the optical properties of an atomic medium. Both stimulated processes such as light absorption and refraction [1] and also spontaneous processes were shown to be altered almost at will due to laser-induced, i.e., stimulated extra indistinguishable quantum mechanical pathways [2–4]. In addition incoherent processes such as spontaneous emission [3,4] and incoherent pumping [5] turned out to induce interferences in particular circumstances and thus to modify stimulated processes like absorption and refraction. What thus remains to be shown is that incoherently induced interferences are in the position to modify spontaneous
* Corresponding author.
E-mail address: [email protected] (C.H. Keitel).
processes as well. In this Letter we consider spontaneous emission from doublet states of an incoherently pumped 4-level system and point out that interferences due to the incoherent field may indeed notably modify the spontaneous emission spectrum without any laser field. Special emphasis will be placed on the role of the incoherently induced interferences on narrow spectral features. The 4-level system of interest involves a Λ configuration driven by a linearly polarised incoherent laser field, with a spectral width which may be greater than the separation of the two closely lying doublet states. The level configuration and notation, including that of the various spontaneous emission and pumping rates may be viewed from Fig. 1. The spontaneous emission spectrum of the doublet states shall be calculated in dependence in particular of the involved pumping processes (we have assumed
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(02)01667-5
D. Bullock et al. / Physics Letters A 307 (2003) 8–12
9
1 ρ˙a a = − iωa a + (γ1 + γ2 + r1 + r2 ) ρa a 2 √ 1 √ − Ks γ1 γ2 + Kp r1 r2 (ρaa + ρa a ) 2 √ + Kp r1 r2 ρcc . (1)
Fig. 1. The 4-level system of interest with the corresponding level notation and showing all employed decay rates γ1 and γ2 and indirect pump r and direct pump processes r1 and r2 . Emphasis is placed here on the role of the interferences arising from direct pump processes r1 and r2 on the spontaneous emission from the doublet with level spacing 2δ.
the decay rates of level |c to be negligible for simplicity). The indirect pump process r via some unspecified auxiliary state is employed to supply steady state population in the excited states. The direct pumping however is known to induce interferences [5]. Following [3–6], the equations of motion for the relevant matrix elements of the density operator ρ can be cast in the form: ρ˙aa = −(γ1 + r1 )ρaa + r1 ρcc √ 1 √ − Ks γ1 γ2 + Kp r1 r2 (ρa a + ρaa ), 2 ρ˙a a = −(γ2 + r2 )ρa a + r2 ρcc √ 1 √ − Ks γ1 γ2 + Kp r1 r2 (ρa a + ρaa ), 2 ρ˙bb = −rρbb + γ1 ρaa + γ2 ρa a √ + Ks γ1 γ2 (ρa a + ρaa ), ρ˙cc = −(r1 + r2 )ρcc + r1 ρaa + r2 ρa a + rρbb √ + Kp r1 r2 (ρa a + ρaa ), 1 ρ˙ab = − iωab + (γ1 + r1 + r) ρab 2 √ 1 √ − Ks γ1 γ2 + Kp r1 r2 ρa b , 2 1 ρ˙a b = − iωa b + (γ2 + r2 + r) ρa b 2 √ 1 √ − Ks γ1 γ2 + Kp r1 r2 ρab , 2
Kp and Ks are measures of the interferences between the two direct incoherent pump processes and the two processes of spontaneous emission, respectively. In both cases Kp/s = 0 corresponds to no interference while Kp/s = 1 means maximum interference. These parameters were introduced to compare the situations with and without interference. Using ρaa + ρa a + ρbb + ρcc = 1 for a closed system, the equations of motion can be transformed into matrix
Here ρ =
form: ρ˙ = Bρ + A. σ with denoting the quantum mechanical expectation value and σ = (|a a|, |a a |, |b b|, |a b|, |b a|, |a b|, |b a |, |a a|, |a a |)T . The precise form of the time dependent matrix B and of the constant vector A
follows from Eqs. (1). In order to obtain the spontaneous emission spectrum, we need calculate the Fourier transform of the two-time correlation function, D (+) (t)D (−) (t ) where D (+) (t) = µab σ4 (t) + µa b σ6 (t) and D (−) (t) = (D (+) (t))† . For the dipole moments on the transitions a ↔ b and a ↔ b we have √ |µab |2 ∝ γ1 , |µa b |2 ∝ γ2 and µab µa b ∝ Ks γ1 γ2 and σi is the ith elements of σ for i ∈ {1, . . . , 9}. Thus we obtain √ S = Re K5 γ1 R 4 + Ks γ1 γ2 R 6 √ + K7 γ2 R 6 + Ks γ1 γ2 R 4 , (2) where Re denotes the real part. Here K = (iωI − B)−1 with Ki being its ith row. Furthermore, we employed R i = δσ (0)δσi (0)ss with σi = σi + δσi for i ∈ {1, . . . , 9}, which can be transformed to read R i = σ ss σi ss . The superindices “ss” refer to
σ σi ss −
the steady state values of the corresponding variable. By integration we then obtain the total intensity √ ss ss + γ2 ρass a + Ks γ1 γ2 ρass a + ρaa Itotal = π γ1 ρaa . (3) In what follows we evaluate and discuss the spectrum S as a function of the frequency ω − ω0 , where ω0 is defined to be the energy gap between level |b and the middle between the states |a and |a . The general features of a typical spectrum in Fig. 2 are as expected. There are two Lorentzian peaks
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D. Bullock et al. / Physics Letters A 307 (2003) 8–12
Fig. 2. Inelastic spectral intensity S via Eq. (2) as a function of frequency ω. For all four lines γ1 = γ2 = 1, r1 = r2 = 0.2, r = 0.01 and δ = 0.7. The dotted and dot-dashed lines denote the situation with no interferences due to spontaneous emission Ks = 0, with rather small dependence on Kp (dotted: Kp = 1, dot-dashed: Kp = 0). For the dashed line: Ks = Kp = 1 and for the solid line: Ks = 1, Kp = 0. The vertical grey solid lines indicate the frequency of the transitions |a ↔ |b and |a ↔ |b.
centred on the transition frequencies from levels a and a to the ground level b, because we are calculating the light emitted due to the spontaneous emission from the doublet. When the levels are brought closer (2δ < γ1 , γ2 ), the peaks are found to merge into one. In Fig. 2 the role of interferences due to incoherent pumping is found to be small without the interference via spontaneous emission; i.e., for Ks = 0 setting Kp to 0 or 1 is found to make little difference on the scale of the figure. Thus we find for these parameters that pump interference terms only show a significant effect when we also have decay interference. Mathematically this can be associated with the fact that many of the interference terms in S in Eq. (2) appear as a function of the product of Ks and Kp . In the case of all interference terms being present (Ks = 1, Kp = 1), we find destructive interference, predominantly at the centre, whereas in the case of there only being decay interference we have higher peaks and no large trough in the middle of the spectrum. When we compare the maxima of the peaks to the transition frequencies shown in the figure, we see that only the plot with just the decay interference terms has its maxima off these frequencies, actually nearer the centre. Thus, there is clear evidence that interference due to incoherent pumping substantially changes the spontaneous emission spectrum. Fig. 3, involving smaller values of the pump rates, displays the effect of the pump root terms, i.e., the
pump interference terms, more clearly. Only the case of Ks = 1 has been drawn to accentuate the difference between the two plots, showing the effect of the pump root terms. As in Fig. 2, the solid graph has peaks nearer the centre than the plot with both root terms. We see in the solid graph that when varying the doublet separation, at some point we observe a high peak at the centre of the spectrum, presumably an effect of constructive interference between the decay paths, namely, spectral narrowing [7–10]. We could expect this feature to be seen somewhere in the spectra, but it is interesting that when the pump interference terms are included (dashed plot), the narrowing effect is lost. At the extreme cases of large and small doublet separation we see that the interference effects play a lessening role. The plots merge into each other. Finally, in Fig. 4 we note that interference effects can also be seen for larger values of the pump rates. All four possibilities of including or excluding the interference terms of the pump and decay processes are shown and are well distinguishable. There are no spectral narrowing effects here to be seen. The only large variation is the height of the peaks and the minimum at the centre of the spectrum. In both diagrams the dot-dashed plot, corresponding to neither root terms being included, is highest in the central region and the dashed plot, corresponding to all the terms being present, has both the lowest peaks and minimum. Thus the action of the terms in this example is destructive interference. Fig. 4(b) shows the spectrum for the asymmetric case of one of the pump rates being larger. Noticeable is that the plot of no interference terms is even more clearly above the others. The main result of this figure though is, that interference due to incoherent pumping substantially changes spontaneous emission when there is no interference due to spontaneous emission. A qualitative understanding of pump interferences and coherences arises from the fact that the width of the pump field covers both transitions from the auxiliary level c to the doublet states. This way we obtain a cross coupling of the coherences to the ground state and a substantial coherence between the doublet states themselves. The pump interference terms thus modify the equations of motion similarly to the well studied decay interference terms. As a consequence the pump interference terms modify the doublet populations and all coherences involving the doublet states
D. Bullock et al. / Physics Letters A 307 (2003) 8–12
11
Fig. 3. Inelastic spectral intensities S via Eq. (2) for varying δ. For both graphs in all figures, γ1 = γ2 = 1, r1 = r2 = 0.01, r = 0.1 and Ks = 1. In (a) δ = 1.5, (b) δ = 0.7, (c) δ = 0.1, (d) δ = 0.03. The dashed line considers the situation Kp = 1 and the solid line Kp = 0, i.e., we note the significance of interferences via incoherent pumping.
Fig. 4. Inelastic spectral intensity S via Eq. (2) for large pump rates. For all lines in both figures, γ1 = γ2 = δ = r = 1 and furthermore in (a) r1 = r2 = 1 and in (b) r1 = 1, r2 = 3. The solid line corresponds to Ks = 1, Kp = 0, the dashed to Ks = Kp = 1, the dotted to Ks = 0, Kp = 1 and the dot-dashed to Ks = Kp = 0. Interference due to incoherent pumping is significant even without the presence of interference via spontaneous emission.
and thus also the spontaneous emission spectrum. The most sensitive area in the spectra with respect to pump interferences is as usual the central frequency between the transitions from each of the doublet states. We note further that quantitative interpretations are demanding due to the complexity of the system and the equation
for the spectrum in Eq. (2). We find in this equation that two of the four terms fully disappear without decay interference and decay and pump interferences often appear in sums in the equations of motion (1). Since the various matrix elements in the dynamical equations are coupled we expect the spectral response
12
D. Bullock et al. / Physics Letters A 307 (2003) 8–12
to involve higher order mixed expressions of decay and pump interference terms. Thus it is understandable that pump interferences may strongly depend on the existence of decay interferences such as in Fig. 2. Concluding, interferences via fully incoherent pumping processes have been proven to have a substantial effect on the spontaneous emission spectrum, even though to a somewhat lesser extent than via coherent fields. In particular, narrow spectral features in the area half way between two transitions turned out to be remarkably sensitive to this incoherently induced interference.
[4]
[5]
Acknowledgements Funding by Erasmus Exchange Program and Deutsche Forschungsgemeinschaft (Nachwuchsgruppe within SFB 276) is gratefully acknowledged. We thank Martin Haas for valuable advice.
[6] [7]
References [8] [1] S.E. Harris, Phys. Today 50 (1997) 36; J.P. Marangos, J. Mod. Opt. 45 (1998) 471; See also some of the early articles by G. Alzetta, A. Gozzini, L. Moi, G. Orriols, Nuovo Cimento B 36 (1976) 5; E. Arimondo, G. Orriols, Lett. Nuovo Cimento 17 (1976) 333; H.R. Gray, R.M. Whitley, C.R. Stroud Jr., Opt. Lett. 6 (1978) 218; P.M. Radmore, P.L. Knight, J. Phys. B 15 (1982) 561. [2] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge Univ. Press, Cambridge, 2001; Z. Ficek, Quantum Interference in Atomic and Molecular Systems, in: Advances in Chemical Physics, Vol. 119, Wiley, New York, 2001, part 1, and references therein; S. Swain, Z. Ficek, Quantum Interference (special issue), J. Mod. Opt. 49 (2002) 1, and the articles in this issue. [3] P.L. Knight, J. Phys. B 12 (1979) 3297; D.A. Cardimona, M.G. Raymer, C.R. Stroud, J. Phys. B 15 (1982) 55; A. Imamoˇglu, Phys. Rev. A 40 (1989) R2835; M. Fleischhauer, et al., Opt. Commun. 94 (1994) 599;
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J. Javanainen, Europhys. Lett. 17 (1992) 407; X.-M. Hu, J.-S. Peng, J. Phys. B 33 (2000) 921; J. Evers, D. Bullock, C.H. Keitel, Opt. Commun. 209 (2002) 173. S.-Y. Zhu, M.O. Scully, Phys. Rev. Lett. 76 (1996) 388; H.R. Xia, C.Y. Ye, S.-Y. Zhu, Phys. Rev. Lett. 77 (1996) 1032; H. Huang, S.-Y. Zhu, M.S. Zubairy, Phys. Rev. A 55 (1997) 744; G.S. Agarwal, Phys. Rev. A 55 (1997) 2457; M.O. Scully, S.Y. Zhu, Science 281 (1998) 1973; Y.P. Yang, S.-Y. Zhu, M.S. Zubairy, Opt. Commun. 166 (1999) 79; S.-Y. Zhu, L.M. Narducci, M.O. Scully, Phys. Rev. A 52 (1995) 4791; S.-Y. Zhu, R.C.F. Chan, C.P. Lee, Phys. Rev. A 52 (1995) 1050. M. Fleischhauer, C.H. Keitel, M.O. Scully, C. Su, Opt. Commun. 87 (1992) 109. C.H. Keitel, Phys. Rev. Lett. 83 (1999) 1307; C.H. Keitel, S.Y. Zhu, P.L. Knight, Las. Phys. 9 (1999) 826. L.M. Narducci, et al., Phys. Rev. A 42 (1990) 1630; L.M. Narducci, G.L. Oppo, M.O. Scully, Opt. Commun. 75 (1990) 111; D.J. Gauthier, Y. Zhu, T.W. Mossberg, Phys. Rev. Lett. 66 (1991) 2460; C.H. Keitel, L.M. Narducci, M.O. Scully, Appl. Phys. B 60 (1995) 153; G.S. Agarwal, Phys. Rev. A 54 (1996) R3734. P. Zhou, S. Swain, Phys. Rev. Lett. 77 (1996) 3995; P. Zhou, S. Swain, Phys. Rev. A 56 (1997) 3011; P. Zhou, S. Swain, Z. Ficek, Opt. Commun. 148 (1998) 159; P. Zhou, S. Swain, Phys. Rev. A 59 (1999) 841; B.J. Dalton, Z. Ficek, S. Swain, J. Mod. Opt. 46 (1999) 379; U. Akram, Z. Ficek, S. Swain, J. Mod. Opt. 48 (2001) 1059; Z. Ficek, S. Swain, J. Mod. Opt. 49 (2002) 3. E. Paspalakis, P.L. Knight, Phys. Rev. Lett. 81 (1998) 293; E. Paspalakis, C.H. Keitel, P.L. Knight, Phys. Rev. A 58 (1998) 4868; S.Q. Gong, E. Paspalakis, P.L. Knight, J. Mod. Opt. 45 (1998) 2433; E. Paspalakis, S.-Q. Gong, P.L. Knight, Opt. Commun. 152 (1998) 293; M.A.G. Martinez, et al., Phys. Rev. A 55 (1997) 4483; E. Paspalakis, P.L. Knight, J. Mod. Opt. 47 (2000) 1025. G.C. Hegerfeldt, M.B. Plenio, Phys. Rev. A 52 (1995) 3333; B.M. Garraway, M.S. Kim, P.L. Knight, Opt. Commun. 117 (1995) 550; V. Bühner, C. Tamm, Phys. Rev. A 61 (2000) 061801(R); J. Evers, C.H. Keitel, Phys. Rev. A 65 (2002) 033813.
Modifying spontaneous emission via interferences from incoherent pump fields David Bullock, Jörg Evers, Christoph H. Keitel ∗ Theoretische Quantendynamik, Fakultät für Mathematik und Physik, University of Freiburg, Hermann-Herder Str. 3, 79104 Freiburg, Germany Received 10 July 2002; received in revised form 9 September 2002; accepted 13 September 2002 Communicated by P.R. Holland
Abstract The spontaneous emission spectrum is evaluated from the doublet of a 4-level atomic system. Excitation from the ground state of the closed system is supplied solely by two incoherent pump fields with one involving the doublet states; i.e., no laser fields are involved at any stage of the problem. We find that the interference which arises due to the incoherent pump field induces substantial modifications of the spontaneous emission spectrum. In particular, the occurrence of a narrow spectral feature is shown to be quite sensitive to the incoherent pump-induced interference. 2002 Elsevier Science B.V. All rights reserved. PACS: 42.50.Lc; 42.50.Ct; 42.50.Hz
Laser field induced quantum interferences have been proven to be seminal in controlling the optical properties of an atomic medium. Both stimulated processes such as light absorption and refraction [1] and also spontaneous processes were shown to be altered almost at will due to laser-induced, i.e., stimulated extra indistinguishable quantum mechanical pathways [2–4]. In addition incoherent processes such as spontaneous emission [3,4] and incoherent pumping [5] turned out to induce interferences in particular circumstances and thus to modify stimulated processes like absorption and refraction. What thus remains to be shown is that incoherently induced interferences are in the position to modify spontaneous
* Corresponding author.
E-mail address: [email protected] (C.H. Keitel).
processes as well. In this Letter we consider spontaneous emission from doublet states of an incoherently pumped 4-level system and point out that interferences due to the incoherent field may indeed notably modify the spontaneous emission spectrum without any laser field. Special emphasis will be placed on the role of the incoherently induced interferences on narrow spectral features. The 4-level system of interest involves a Λ configuration driven by a linearly polarised incoherent laser field, with a spectral width which may be greater than the separation of the two closely lying doublet states. The level configuration and notation, including that of the various spontaneous emission and pumping rates may be viewed from Fig. 1. The spontaneous emission spectrum of the doublet states shall be calculated in dependence in particular of the involved pumping processes (we have assumed
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(02)01667-5
D. Bullock et al. / Physics Letters A 307 (2003) 8–12
9
1 ρ˙a a = − iωa a + (γ1 + γ2 + r1 + r2 ) ρa a 2 √ 1 √ − Ks γ1 γ2 + Kp r1 r2 (ρaa + ρa a ) 2 √ + Kp r1 r2 ρcc . (1)
Fig. 1. The 4-level system of interest with the corresponding level notation and showing all employed decay rates γ1 and γ2 and indirect pump r and direct pump processes r1 and r2 . Emphasis is placed here on the role of the interferences arising from direct pump processes r1 and r2 on the spontaneous emission from the doublet with level spacing 2δ.
the decay rates of level |c to be negligible for simplicity). The indirect pump process r via some unspecified auxiliary state is employed to supply steady state population in the excited states. The direct pumping however is known to induce interferences [5]. Following [3–6], the equations of motion for the relevant matrix elements of the density operator ρ can be cast in the form: ρ˙aa = −(γ1 + r1 )ρaa + r1 ρcc √ 1 √ − Ks γ1 γ2 + Kp r1 r2 (ρa a + ρaa ), 2 ρ˙a a = −(γ2 + r2 )ρa a + r2 ρcc √ 1 √ − Ks γ1 γ2 + Kp r1 r2 (ρa a + ρaa ), 2 ρ˙bb = −rρbb + γ1 ρaa + γ2 ρa a √ + Ks γ1 γ2 (ρa a + ρaa ), ρ˙cc = −(r1 + r2 )ρcc + r1 ρaa + r2 ρa a + rρbb √ + Kp r1 r2 (ρa a + ρaa ), 1 ρ˙ab = − iωab + (γ1 + r1 + r) ρab 2 √ 1 √ − Ks γ1 γ2 + Kp r1 r2 ρa b , 2 1 ρ˙a b = − iωa b + (γ2 + r2 + r) ρa b 2 √ 1 √ − Ks γ1 γ2 + Kp r1 r2 ρab , 2
Kp and Ks are measures of the interferences between the two direct incoherent pump processes and the two processes of spontaneous emission, respectively. In both cases Kp/s = 0 corresponds to no interference while Kp/s = 1 means maximum interference. These parameters were introduced to compare the situations with and without interference. Using ρaa + ρa a + ρbb + ρcc = 1 for a closed system, the equations of motion can be transformed into matrix
Here ρ =
form: ρ˙ = Bρ + A. σ with denoting the quantum mechanical expectation value and σ = (|a a|, |a a |, |b b|, |a b|, |b a|, |a b|, |b a |, |a a|, |a a |)T . The precise form of the time dependent matrix B and of the constant vector A
follows from Eqs. (1). In order to obtain the spontaneous emission spectrum, we need calculate the Fourier transform of the two-time correlation function, D (+) (t)D (−) (t ) where D (+) (t) = µab σ4 (t) + µa b σ6 (t) and D (−) (t) = (D (+) (t))† . For the dipole moments on the transitions a ↔ b and a ↔ b we have √ |µab |2 ∝ γ1 , |µa b |2 ∝ γ2 and µab µa b ∝ Ks γ1 γ2 and σi is the ith elements of σ for i ∈ {1, . . . , 9}. Thus we obtain √ S = Re K5 γ1 R 4 + Ks γ1 γ2 R 6 √ + K7 γ2 R 6 + Ks γ1 γ2 R 4 , (2) where Re denotes the real part. Here K = (iωI − B)−1 with Ki being its ith row. Furthermore, we employed R i = δσ (0)δσi (0)ss with σi = σi + δσi for i ∈ {1, . . . , 9}, which can be transformed to read R i = σ ss σi ss . The superindices “ss” refer to
σ σi ss −
the steady state values of the corresponding variable. By integration we then obtain the total intensity √ ss ss + γ2 ρass a + Ks γ1 γ2 ρass a + ρaa Itotal = π γ1 ρaa . (3) In what follows we evaluate and discuss the spectrum S as a function of the frequency ω − ω0 , where ω0 is defined to be the energy gap between level |b and the middle between the states |a and |a . The general features of a typical spectrum in Fig. 2 are as expected. There are two Lorentzian peaks
10
D. Bullock et al. / Physics Letters A 307 (2003) 8–12
Fig. 2. Inelastic spectral intensity S via Eq. (2) as a function of frequency ω. For all four lines γ1 = γ2 = 1, r1 = r2 = 0.2, r = 0.01 and δ = 0.7. The dotted and dot-dashed lines denote the situation with no interferences due to spontaneous emission Ks = 0, with rather small dependence on Kp (dotted: Kp = 1, dot-dashed: Kp = 0). For the dashed line: Ks = Kp = 1 and for the solid line: Ks = 1, Kp = 0. The vertical grey solid lines indicate the frequency of the transitions |a ↔ |b and |a ↔ |b.
centred on the transition frequencies from levels a and a to the ground level b, because we are calculating the light emitted due to the spontaneous emission from the doublet. When the levels are brought closer (2δ < γ1 , γ2 ), the peaks are found to merge into one. In Fig. 2 the role of interferences due to incoherent pumping is found to be small without the interference via spontaneous emission; i.e., for Ks = 0 setting Kp to 0 or 1 is found to make little difference on the scale of the figure. Thus we find for these parameters that pump interference terms only show a significant effect when we also have decay interference. Mathematically this can be associated with the fact that many of the interference terms in S in Eq. (2) appear as a function of the product of Ks and Kp . In the case of all interference terms being present (Ks = 1, Kp = 1), we find destructive interference, predominantly at the centre, whereas in the case of there only being decay interference we have higher peaks and no large trough in the middle of the spectrum. When we compare the maxima of the peaks to the transition frequencies shown in the figure, we see that only the plot with just the decay interference terms has its maxima off these frequencies, actually nearer the centre. Thus, there is clear evidence that interference due to incoherent pumping substantially changes the spontaneous emission spectrum. Fig. 3, involving smaller values of the pump rates, displays the effect of the pump root terms, i.e., the
pump interference terms, more clearly. Only the case of Ks = 1 has been drawn to accentuate the difference between the two plots, showing the effect of the pump root terms. As in Fig. 2, the solid graph has peaks nearer the centre than the plot with both root terms. We see in the solid graph that when varying the doublet separation, at some point we observe a high peak at the centre of the spectrum, presumably an effect of constructive interference between the decay paths, namely, spectral narrowing [7–10]. We could expect this feature to be seen somewhere in the spectra, but it is interesting that when the pump interference terms are included (dashed plot), the narrowing effect is lost. At the extreme cases of large and small doublet separation we see that the interference effects play a lessening role. The plots merge into each other. Finally, in Fig. 4 we note that interference effects can also be seen for larger values of the pump rates. All four possibilities of including or excluding the interference terms of the pump and decay processes are shown and are well distinguishable. There are no spectral narrowing effects here to be seen. The only large variation is the height of the peaks and the minimum at the centre of the spectrum. In both diagrams the dot-dashed plot, corresponding to neither root terms being included, is highest in the central region and the dashed plot, corresponding to all the terms being present, has both the lowest peaks and minimum. Thus the action of the terms in this example is destructive interference. Fig. 4(b) shows the spectrum for the asymmetric case of one of the pump rates being larger. Noticeable is that the plot of no interference terms is even more clearly above the others. The main result of this figure though is, that interference due to incoherent pumping substantially changes spontaneous emission when there is no interference due to spontaneous emission. A qualitative understanding of pump interferences and coherences arises from the fact that the width of the pump field covers both transitions from the auxiliary level c to the doublet states. This way we obtain a cross coupling of the coherences to the ground state and a substantial coherence between the doublet states themselves. The pump interference terms thus modify the equations of motion similarly to the well studied decay interference terms. As a consequence the pump interference terms modify the doublet populations and all coherences involving the doublet states
D. Bullock et al. / Physics Letters A 307 (2003) 8–12
11
Fig. 3. Inelastic spectral intensities S via Eq. (2) for varying δ. For both graphs in all figures, γ1 = γ2 = 1, r1 = r2 = 0.01, r = 0.1 and Ks = 1. In (a) δ = 1.5, (b) δ = 0.7, (c) δ = 0.1, (d) δ = 0.03. The dashed line considers the situation Kp = 1 and the solid line Kp = 0, i.e., we note the significance of interferences via incoherent pumping.
Fig. 4. Inelastic spectral intensity S via Eq. (2) for large pump rates. For all lines in both figures, γ1 = γ2 = δ = r = 1 and furthermore in (a) r1 = r2 = 1 and in (b) r1 = 1, r2 = 3. The solid line corresponds to Ks = 1, Kp = 0, the dashed to Ks = Kp = 1, the dotted to Ks = 0, Kp = 1 and the dot-dashed to Ks = Kp = 0. Interference due to incoherent pumping is significant even without the presence of interference via spontaneous emission.
and thus also the spontaneous emission spectrum. The most sensitive area in the spectra with respect to pump interferences is as usual the central frequency between the transitions from each of the doublet states. We note further that quantitative interpretations are demanding due to the complexity of the system and the equation
for the spectrum in Eq. (2). We find in this equation that two of the four terms fully disappear without decay interference and decay and pump interferences often appear in sums in the equations of motion (1). Since the various matrix elements in the dynamical equations are coupled we expect the spectral response
12
D. Bullock et al. / Physics Letters A 307 (2003) 8–12
to involve higher order mixed expressions of decay and pump interference terms. Thus it is understandable that pump interferences may strongly depend on the existence of decay interferences such as in Fig. 2. Concluding, interferences via fully incoherent pumping processes have been proven to have a substantial effect on the spontaneous emission spectrum, even though to a somewhat lesser extent than via coherent fields. In particular, narrow spectral features in the area half way between two transitions turned out to be remarkably sensitive to this incoherently induced interference.
[4]
[5]
Acknowledgements Funding by Erasmus Exchange Program and Deutsche Forschungsgemeinschaft (Nachwuchsgruppe within SFB 276) is gratefully acknowledged. We thank Martin Haas for valuable advice.
[6] [7]
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