- Email: [email protected]
Spontaneous emission in a V-type three-level atom driven by a classical field
6 December 1999
Physics Letters A 263 Ž1999. 250–256 www.elsevier.nlrlocaterphysleta
Spontaneous emission in a V-type three-level atom driven by a classical field Gao-xiang Li ) , Klaas Allaart, Christa Hooijer, Daan Lenstra Department of Physics and Astronomy, Vrije UniÕersiteit Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands Received 10 September 1999; accepted 10 October 1999 Communicated by P.R. Holland
Abstract We have studied the spectrum of spontaneous emission of a V-configuration three-level atom, whose two upper levels are linked by a classical driving field and their energy spacing is much larger than the spontaneous emission widths. It is shown that the spectrum can be controlled by the driving field and peaks with subnatural linewidth can be produced. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Introduction Recently the effects of quantum interference among multiple atomic transition pathways have been extensively studied because of applications in a number of different contexts such as lasing without population inversion w1–3x, electromagnetically induced transparency w4–6x, and enhancement of the index of refraction without absorption w7,8x. Usually this interference is reduced by spontaneous emission. However, it is now well known that the spontaneous decay of a system of two closely lying atomic levels induced by interaction with a common bath may create spontaneously generated interference w9–16x. This quantum interference can lead to the appearance of ultra narrow spectral lines and fluorescence quenching in the spontaneous emission spectrum w9– )
Corresponding author. Permanent address: Department of Physics, Huazhong Normal University, Wuhan 430079, PR China. E-mail: [email protected]
16x, and to atomic population trapping in excited levels w14–17x. However, it should be noted that only for a small energy spacing between the two closely lying levels the effects of the spontaneously generated interference are important. For large energy spacing the rapid oscillation between these two levels will average out such effects w18,19x. So it is interesting to study whether the effects of this quantum interference, such as line narrowing in spontaneous emission spectrum, may still be preserved for the case of a large energy spacing. In this Letter, we study the spontaneous emission spectrum of a V-type three-level atom in a vacuum bath, of which the two excited levels are coupled by a classical microwave field while the energy spacing between them is much larger than their spontaneous emission widths. In Section 2, we present the model and derive the time-dependent state function of the atom-field coupled system. Section 3 is devoted to a discussion of the effects of the driving microwave field on the spontaneous emission spectrum. It will
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 7 1 8 - 5
G.-x. Li et al.r Physics Letters A 263 (1999) 250–256
be shown that the spontaneous emission spectrum can be changed drastically by adjusting the intensity, the frequency and the phase of the microwave field. A similar line narrowing phenomenon as found in Refs. w9–16x, which results from the spontaneously generated interference, appears when the classical field is chosen appropriately. Finally our conclusions are summarized in Section 4. 2. Model and equations of motion Consider a V-configuration three-level atom, illustrated in Fig. 1a, whose two upper levels < i : Ži s 1,2. are linked to its ground state <0: by the same vacuum bath. The two upper levels with energy spacing v 21 s v 2 y v 1 are strongly coupled by a classical source of microwave radiation with frequency vc and phase f . In the rotating-wave approximation, the interaction Hamiltonian of the system in the interaction representation is of the form HI Ž t . s H A q V Ž t . , Ž 1.
hilation and creation operators for the k th mode of the electromagnetic field with frequency v k . The coupling constants g 1 k and g 2 k are respectively associated with the atomic transition <2: <0: and <1: <0: driven by the k th mode of the vacuum field. The parameter V 0 represents the Rabi frequency and is proportional to the intensity of the classical field. And d 0 and d k are frequency detuning parameters, which are defined as d 0 s vc y v 21 and d k s v k y Ž v 1 y v 0 .. A convenient basis for the atomic Hilbert space can be constructed by the unperturbed ground state <0: and the dressed states < " : originating from the interaction of the atom with the classical field. The dressed states can be easily derived from the eigenvalue equation
™
™
HA < " : s l " < " : ,
HA s yd 0 <2:²2 < q i Ž V 0 e i f <2:²1 < y V 0 eyi f <1:²2 < . ,
Ž " s 1. Ž 2. yi d k t † < :² < i d k t < :² < V Ž t . s i Ý g1 k Ž ak 1 0 e y ak 0 1 e . k
d0
q
2
(ž
d0 2
2
q V 02 ,
/
< q : s sin u <1: q ie i f cos u <2: ;
lys y
q i Ý g 2 k Ž a k <2:²0
Ž 4.
with eigenvalues and corresponding eigenstates
lqs y
with
251
d0 2
y
(ž
d0 2
2
/
q V 02 ,
k
=²2
Ž 3.
Here the frame rotating with the frequency v c has been adopted. The symbols a k and a†k are the anni-
< y : s cos u <1: y ie i f sin u <2: ,
Ž 5.
where sin u s
V0
(l
2 2 q qV0
,
cos u s
lq
(l
2 2 q qV0
.
Ž 6.
Therefore, the state vector of the atom-field coupled system at any time t can be expanded as
Ž 7.
k
Substituting Eqs. Ž1. – Ž3. and Ž7. into the Schrodi¨ nger equation in the interaction representation and using the Weisskopf–Wigner theory w9–13x, the
G.-x. Li et al.r Physics Letters A 263 (1999) 250–256
252
equations of motion of the probability amplitudes Cq Ž t ., Cy Ž t . and C0 k Ž t . are given by d dt
Cys yi ly Cyy g 1cos 2u q g 2 sin2u yg sin Ž 2 u . sin Ž vc t y f . Cy y Ž g 1 y g 2 . sin Ž 2 u . r2
d
qig Ž cos 2u eyi Ž v c ty f . qsin2u e iŽ v c ty f . . Cq , d dt
Ž 8.
dt
dt
Cqs yi lq Cqy g 1 sin u qg 2 cos 2u q g sin Ž 2 u . sin Ž vc t y f . Cq
d
y Ž g 1 y g 2 . sin Ž 2 u . r2
dt
C0 k s yg 1 k Ž sin u Cqq cos u Cy . e i d k t y ie i f g 2 k Ž cos u Cqy sin u Cy . e iŽ d kyv c .t . Ž 10 .
Here 2g 1 s 2 p Ž g 1 . 2 D Ž v 1 y v 0 . and 2g 2 s 2p Ž g 2 . 2 DŽ v 2 y v 0 . are the rates of spontaneous emissions from the upper levels <1: and <2: to the ground level <0:, DŽ v . is the mode density and g s p g 1 g 2 Žy1 F p F 1. represents the quantum interference resulting from the cross coupling between the transitions <1: <0: and <2: <0:. It reflects the fact that as the atom decays from the upper level <1: it drives the other excited level <2: and vice versa. This spontaneously generated interference is sensitive to the orientations of the atomic dipole polarizations w9–17x. If the matrix element of the atomic dipole moment of the transition from <0: to <1: is parallel Žor antiparallel. to that from <0: to 2:, then p s 1 Žor p s y1. and the quantum interference is strongest, while if these two matrix elements are perpendicular to each other, then p s 0 and this interference vanishes. Here we are interested in the properties of the spontaneous emission spectrum in the atom-field coupled system, so we only need to be concerned with the asymptotic behavior of the probability am-
(
l
C˜ys yig cos 2u exp yi Ž vc q lqy ly . t q i f qsin2u exp i Ž vc y lqq ly . t y i f 4 C˜q , Ž 11 .
2
yig Ž cos 2u e iŽ v c ty f . q sin2u eyiŽ v c ty f . . Cy , Ž 9. d
plitudes at times which are much longer than 1rg 1 and 1rg 2 . If we assume that the energy spacing v 21 of the two upper levels is much larger than g 1 and g 2 , then it is reasonable to neglect for long times Ž t 4 1rg 1 ,1rg 2 . the contribution of the terms containing the rapidly oscillating function sinŽ vc t y f . in Eqs. Ž8. and Ž9.. Then Eqs. Ž8. and Ž9. reduce to
l
C˜qs ig sin2u exp yi Ž vc y lqq ly . t q i f qcos 2u exp i Ž vc q lqy ly . t y i f 4 C˜y , Ž 12 .
where C˜"s C " expwŽ i l "q g 1 . t x and we have also assumed g 1 s g 2 in order to simplify the further calculations. If g s 0, which means that there is no quantum interference, Eqs. Ž11. and Ž12. indicate that the dressed levels < " : decay independently in the course of time. However, when g / 0, the spontaneously generated quantum interference leads to a coupling between the dressed levels. If we adjust properly the frequency vc and the intensity V 0 of the classical radiation field which is used to drive the two upper levels <2: and <1: so that lqy lyf v c , then the factors expw"iŽ vc q lqy ly . t x are rapidly varying in comparison with expw"iŽ v c q lyy lq . t x. Therefore, it is appealing to introduce a further, effective, rotating-wave approximation w20x, dropping the terms containing expw"iŽ vc q lyy lq . t x in Eqs. Ž11. and Ž12.. Then, the time-dependence of the probability amplitudes C " Ž t . can be expressed as Cq Ž t . s A1eŽ i v ayi lqyg 1 .t q A 2 eŽ i v byi lqyg 1 .t , Ž 13 . Cy Ž t . s
1
g sin2u e i f
A1 v a eyŽ i v bqi ly q g 1 .t
qv b A 2 eyŽ i v aqi ly q g 1 .t ,
Ž 14 .
G.-x. Li et al.r Physics Letters A 263 (1999) 250–256
with
va s
(ž
lqy lyy vc
A 2 sin u X d k y a q ig 1
y g sin u ,
qig 2 k cos u e i f
2
lqy lyy vc 2
1
q
2
/
2
4
y g sin u ,
Ž 15 .
v b Cq Ž 0 . y g sin2u e i f Cy Ž 0 . ,
v b y va
as
1
v a Cq Ž 0 . y g sin2u e i f Cy Ž 0 . ,
va y v b
Now we discuss the properties of the spontaneous emission spectrum of the V-type three-level atom. The spontaneous emission spectrum SŽ v ., in the continuum limit, is proportional to the Fourier transform of the field correlation function w9–13x ² Ey Ž t q t . Eq Ž t . :t ™ `
s ²C Ž t . < Ý a†k a k X e i v k Ž tq t . eyi v kX t
`
Hy`d v D Ž v . < C
0k
Ž ` . < 2 e i vt .
Ž 16 .
The above equation shows that the spontaneous emission spectrum SŽ v . is proportional to < C0 k Ž`.< 2 . Substituting Eqs. Ž13. and Ž14. into Eq. Ž10., we obtain the steady-state value of C0 k Ž`. as C0 k Ž ` . s yi
q
½
g 1 k cos u 2
g sin u e
if
ž
A1 v a X d k q vc q a q ig 1
A2 v b X d k q vc y a q ig 1
g1 k y
g1 k y
ig 2 k v a
g sin2u ig 2 k v b
g sin2u
/ /
A1 X d k y vc q a q ig 1
A2 X d k y vc y a q ig 1
/
(
Ž lqy lyy vc .
d kX s d k y
3. Spontaneous emission spectra
k,k
ž
ž ž
/5
Ž 17 .
with
where C " Ž0. are just the initial probability amplitudes of the atom in the dressed levels < " :.
s
q 4
lqy lyy vc
(ž
A2 s
2
/
2
y
A1 s
A1 sin u X d k q a q ig 1
2 q
vb s
q
lqy lyy vc 2
253
4
d0 2
q
vc 2
2
y g 2 sin4u ,
.
Ž 18 .
Evidently, if g s 0, which means that there is no quantum interference arising from the cross coupling between the transitions <1: <0: and <2: <0: under the interaction of vacuum field, then the spontaneous emission spectrum exhibits four peaks, centered at v k s v 1 y v 0 q vc q l " and v k s v 1 y v 0 q l " with the same widths 2g 1. Because the upper two levels of the atom are strongly driven by the classical microwave field, this induces that the system has two dressed-level doublets with energies v 1 q l " and v 1 q vc q l " as shown in Fig. 1b. The spontaneous decay of the atom from these four dressed levels to the ground state <0: is responsible for these four peaks. If the frequency vc and the intensity V 0 of the microwave field satisfy lqy ly y v c ; g 1 , then the two peaks centered at v k s v 1 y v 0 q lq and at v k s v 1 y v 0 q vc q ly overlap, but there is no interference between these two peaks because the two dipole moments of the transitions from <1: to <0: and from <2: to <0: are perpendicular to each other. This also leads to the spontaneous emission spectrum being independent of the phase f of the microwave field. When lqy lys vc , the dressed levels with energies v 1 q vc q ly and v 1 q lq become degenerate. Then there are only three peaks left. However, if g / 0 and Ž lqy lyy vc . 2 ) 4g 2 sin4u , Eq. Ž17. shows that the spontaneous emis-
l
l
254
G.-x. Li et al.r Physics Letters A 263 (1999) 250–256
sion spectrum has six peaks which center at d kX s yvc " a, d kX s "a, and d kX s vc " a, all with the same widths 2g 1. Eqs. Ž11. and Ž12. indicate that the dressed levels < q : and < y : are coupled due to the existence of the spontaneously generated quantum interference. So the dressed levels < " : receive a second dressing through the effect of this quantum interference. In the effective rotating-wave approximation we only consider the coupling between the two dressed levels < " : with energies v 1 q vc q ly and v 1 q lq and neglect the effect of the coupling between the other two dressed levels as shown in Fig. 1b ŽThe effects of the coupling between the dressed levels within every doublet are cancelled when g 1 s g 2 as predicted in Eqs. Ž8. and Ž9... The four dressed levels are then split into eight seconddressed levels. This situation is similar to the case of a two-level atom driven by a strong pumping field and a weak probing field with different frequencies w21x. The energies of these eight second-dressed levels are v 1 q vc q lqy v˜ , v 1 q vc q lyq v˜ , v 1 q lqy v˜ and v 1 q lyq v˜ as shown in Fig. 1c, where v˜ denotes v a or v b . It is easy to verify that the four middle levels are two-by-two degenerate, so the spontaneous decay of the atom from these eight second-dressed levels to its ground state <0: only exhibits six peaks in SŽ v .. On the other hand, Eq. Ž17. also shows that the spontaneous emission spectrum depends on the phase f of the classical field. This is because the atom-field coupled system forms a closed one w22–25x when the spontaneously generated quantum interference is included Žg / 0.. So after a finite time, the probability amplitudes of the atom in the second-dressed levels depend on the phase f , which also gives rise to SŽ v . to be phase dependent. Therefore, by adjusting the phase f one can change the properties of SŽ v . as shown in Fig. 2. In Fig. 2 we have assumed that the atom is initially in the bare state <1:, g s g 1 and d 0 s 0. As we see, SŽ v . has six peaks, but when two peaks are close in energy they are asymmetric due to the interference between them. For different phases, the properties of SŽ v . can change evidently. Next, we examine the case of g / 0 and Ž lqy lyy vc . 2 - 4g 2 sin4u . In this case, Eq. Ž17. shows that the centers of two neighbouring peaks coincide, but their widths are different. One width equals 2Žg 1 q < a <. which is broader than the natural linewidth
Fig. 2. The spontaneous emission spectrum SŽ v . as a function of X the detuning d k for g 1 sg s1, vc s v 21 s10, V 0 s 7, and different values of the phase f . Solid line: f s 3p r2; long-dashed line: f s 0; short-dashed line: f sp r2.
2g 1 and the other width is 2Žg 1 y < a <. which is narrower than the natural one. These phenomena originate from the coupling between the dressed levels < " : which arises from the quantum interference. This coupling leads to a splitting of each dressed level into two ones with complex eigenvalues. Because the parameter < a < is related to the Rabi frequency V 0 and to the frequency vc , not only the centers of the peaks but also the widths in SŽ v . can be controlled by the driving microwave field. Since every narrow peak is superposed on a broad peak, it is non-trivial to distinguish these narrow peaks from the broad ones. Fortunately, the probability amplitudes of the atom in the dressed levels are strongly dependent on the phase f , so the height of every peak can be controlled by changing f . By a suitable choice of f , the broad peaks can be significantly reduced, while the narrow peaks can be enhanced. Fig. 3 displays SŽ v . for different f , assuming that the atom is initially in <1:, g s g 1 , d 0 s 0 and vc s 2 V 0 . In this case, the parameters A1 and A 2 become A1 s A2 s
1
'2 1
'2
cos
cos
ž ž
p
f y
4
2
p
f q
4
2
p
f
/ ž / / ž / exp yi
y
4
p
exp i
2
f
q
4
,
2
,
Ž 19 .
G.-x. Li et al.r Physics Letters A 263 (1999) 250–256
255
4. Conclusions
X Fig. 3. The spectrum SŽ v . as a function of the detuning d k for different values of the phase f , g 1 sg s1, vc s v 21 s10, and V 0 s 5. Short-dashed line: f s1.3p ; long-dashed line: f s1.4p ; solid line: f s1.5p ; dotted line: f s 0.5p .
We have studied the spectrum of spontaneous emission of a V-configuration three-level atom, whose two upper levels are coupled by a classical field and their energy spacing is much larger than the spontaneous emission widths. The spontaneously generated interference can induce the spectrum to exhibit six peaks and depend on the phase of the classical field. These six peaks result from the spontaneous decay of six second-dressed levels, in which two of them are twofold degenerate. If specific values of the parameters such as frequency, intensity and phase of the driving field are chosen, then the peaks with supernatural linewidth can vanish and only the ones with subnatural linewidth appear in the spontaneous emission spectrum evidently even in the case of large energy spacing.
and the probability amplitude C0 k Ž`. is C0 k Ž ` . s yi
q
q
q
g1 k
'2
ž
iA1eyi f
Acknowledgements
d kX q 2 V 0 q i3g 1r2
yiA 2 eyi f
d kX q 2 V 0 q ig 1r2
q
2 A1 X d k q i3g 1r2
iA1e i f X d k y 2 V 0 q i3g 1r2 ie i fA 2
d kX y 2 V 0 q ig 1r2
/
.
Ž 20 .
Evidently, the narrow peak centered at d kX s 0 disappears because of the completely destructive interference between the spontaneous emission processes of the atom in the twofold degenerate dressed levels with complex energies v 1 q vcr2 y ig 1r2. If f s pr2, then A 2 s 0. In that case all the narrow peaks vanish and there are three broad peaks left. However, if p - f - 2p , the heights of the narrow peaks are higher than those of the broad peaks. When f s 3pr2, A1 s 0, so all the broad peaks disappear and there are only two narrow peaks left in SŽ v . as shown in Fig. 3. So we have demonstrated that even in the case of large energy spacing between the upper two levels <1: and <2:, the line narrowing phenomenon resulting from the spontaneously generated quantum interference can still appear clearly in the spontaneous emission spectrum.
G.X.L. and C.H. are supported by the ‘Stichting voor Fundamenteel Onderzoek der Materie ŽFOM.’, which is financially supported by the ‘Nederlandse Orgranisatie voor Wetenschappelijk Onderzoek ŽNWO.’.
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w14x S.Y. Zhu, R.C.F. Chan, C.P. Lee, Phys. Rev. A 52 Ž1995. 710. w15x P. Zhou, S. Swain, Phys. Rev. Lett. 77 Ž1996. 3995. w16x P. Zhou, S. Swain, Phys. Rev. A 56 Ž1997. 3011. w17x G.X. Li, J.S. Peng, Phys. Lett. A 189 Ž1994. 449. w18x S. Menon, G.S. Agarwal, Phys. Rev. A 57 Ž1998. 4014. w19x G.S. Agarwal, Phys. Rev. A 58 Ž1998. 687. w20x J.A. Bergou, M. Orszag, M.O. Scully, Phys. Rev. A 38 Ž1988. 754.
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Physics Letters A 263 Ž1999. 250–256 www.elsevier.nlrlocaterphysleta
Spontaneous emission in a V-type three-level atom driven by a classical field Gao-xiang Li ) , Klaas Allaart, Christa Hooijer, Daan Lenstra Department of Physics and Astronomy, Vrije UniÕersiteit Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands Received 10 September 1999; accepted 10 October 1999 Communicated by P.R. Holland
Abstract We have studied the spectrum of spontaneous emission of a V-configuration three-level atom, whose two upper levels are linked by a classical driving field and their energy spacing is much larger than the spontaneous emission widths. It is shown that the spectrum can be controlled by the driving field and peaks with subnatural linewidth can be produced. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Introduction Recently the effects of quantum interference among multiple atomic transition pathways have been extensively studied because of applications in a number of different contexts such as lasing without population inversion w1–3x, electromagnetically induced transparency w4–6x, and enhancement of the index of refraction without absorption w7,8x. Usually this interference is reduced by spontaneous emission. However, it is now well known that the spontaneous decay of a system of two closely lying atomic levels induced by interaction with a common bath may create spontaneously generated interference w9–16x. This quantum interference can lead to the appearance of ultra narrow spectral lines and fluorescence quenching in the spontaneous emission spectrum w9– )
Corresponding author. Permanent address: Department of Physics, Huazhong Normal University, Wuhan 430079, PR China. E-mail: [email protected]
16x, and to atomic population trapping in excited levels w14–17x. However, it should be noted that only for a small energy spacing between the two closely lying levels the effects of the spontaneously generated interference are important. For large energy spacing the rapid oscillation between these two levels will average out such effects w18,19x. So it is interesting to study whether the effects of this quantum interference, such as line narrowing in spontaneous emission spectrum, may still be preserved for the case of a large energy spacing. In this Letter, we study the spontaneous emission spectrum of a V-type three-level atom in a vacuum bath, of which the two excited levels are coupled by a classical microwave field while the energy spacing between them is much larger than their spontaneous emission widths. In Section 2, we present the model and derive the time-dependent state function of the atom-field coupled system. Section 3 is devoted to a discussion of the effects of the driving microwave field on the spontaneous emission spectrum. It will
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 7 1 8 - 5
G.-x. Li et al.r Physics Letters A 263 (1999) 250–256
be shown that the spontaneous emission spectrum can be changed drastically by adjusting the intensity, the frequency and the phase of the microwave field. A similar line narrowing phenomenon as found in Refs. w9–16x, which results from the spontaneously generated interference, appears when the classical field is chosen appropriately. Finally our conclusions are summarized in Section 4. 2. Model and equations of motion Consider a V-configuration three-level atom, illustrated in Fig. 1a, whose two upper levels < i : Ži s 1,2. are linked to its ground state <0: by the same vacuum bath. The two upper levels with energy spacing v 21 s v 2 y v 1 are strongly coupled by a classical source of microwave radiation with frequency vc and phase f . In the rotating-wave approximation, the interaction Hamiltonian of the system in the interaction representation is of the form HI Ž t . s H A q V Ž t . , Ž 1.
hilation and creation operators for the k th mode of the electromagnetic field with frequency v k . The coupling constants g 1 k and g 2 k are respectively associated with the atomic transition <2: <0: and <1: <0: driven by the k th mode of the vacuum field. The parameter V 0 represents the Rabi frequency and is proportional to the intensity of the classical field. And d 0 and d k are frequency detuning parameters, which are defined as d 0 s vc y v 21 and d k s v k y Ž v 1 y v 0 .. A convenient basis for the atomic Hilbert space can be constructed by the unperturbed ground state <0: and the dressed states < " : originating from the interaction of the atom with the classical field. The dressed states can be easily derived from the eigenvalue equation
™
™
HA < " : s l " < " : ,
HA s yd 0 <2:²2 < q i Ž V 0 e i f <2:²1 < y V 0 eyi f <1:²2 < . ,
Ž " s 1. Ž 2. yi d k t † < :² < i d k t < :² < V Ž t . s i Ý g1 k Ž ak 1 0 e y ak 0 1 e . k
d0
q
2
(ž
d0 2
2
q V 02 ,
/
< q : s sin u <1: q ie i f cos u <2: ;
lys y
q i Ý g 2 k Ž a k <2:²0
Ž 4.
with eigenvalues and corresponding eigenstates
lqs y
with
251
d0 2
y
(ž
d0 2
2
/
q V 02 ,
k
=²2
Ž 3.
Here the frame rotating with the frequency v c has been adopted. The symbols a k and a†k are the anni-
< y : s cos u <1: y ie i f sin u <2: ,
Ž 5.
where sin u s
V0
(l
2 2 q qV0
,
cos u s
lq
(l
2 2 q qV0
.
Ž 6.
Therefore, the state vector of the atom-field coupled system at any time t can be expanded as
Ž 7.
k
Substituting Eqs. Ž1. – Ž3. and Ž7. into the Schrodi¨ nger equation in the interaction representation and using the Weisskopf–Wigner theory w9–13x, the
G.-x. Li et al.r Physics Letters A 263 (1999) 250–256
252
equations of motion of the probability amplitudes Cq Ž t ., Cy Ž t . and C0 k Ž t . are given by d dt
Cys yi ly Cyy g 1cos 2u q g 2 sin2u yg sin Ž 2 u . sin Ž vc t y f . Cy y Ž g 1 y g 2 . sin Ž 2 u . r2
d
qig Ž cos 2u eyi Ž v c ty f . qsin2u e iŽ v c ty f . . Cq , d dt
Ž 8.
dt
dt
Cqs yi lq Cqy g 1 sin u qg 2 cos 2u q g sin Ž 2 u . sin Ž vc t y f . Cq
d
y Ž g 1 y g 2 . sin Ž 2 u . r2
dt
C0 k s yg 1 k Ž sin u Cqq cos u Cy . e i d k t y ie i f g 2 k Ž cos u Cqy sin u Cy . e iŽ d kyv c .t . Ž 10 .
Here 2g 1 s 2 p Ž g 1 . 2 D Ž v 1 y v 0 . and 2g 2 s 2p Ž g 2 . 2 DŽ v 2 y v 0 . are the rates of spontaneous emissions from the upper levels <1: and <2: to the ground level <0:, DŽ v . is the mode density and g s p g 1 g 2 Žy1 F p F 1. represents the quantum interference resulting from the cross coupling between the transitions <1: <0: and <2: <0:. It reflects the fact that as the atom decays from the upper level <1: it drives the other excited level <2: and vice versa. This spontaneously generated interference is sensitive to the orientations of the atomic dipole polarizations w9–17x. If the matrix element of the atomic dipole moment of the transition from <0: to <1: is parallel Žor antiparallel. to that from <0: to 2:, then p s 1 Žor p s y1. and the quantum interference is strongest, while if these two matrix elements are perpendicular to each other, then p s 0 and this interference vanishes. Here we are interested in the properties of the spontaneous emission spectrum in the atom-field coupled system, so we only need to be concerned with the asymptotic behavior of the probability am-
(
l
C˜ys yig cos 2u exp yi Ž vc q lqy ly . t q i f qsin2u exp i Ž vc y lqq ly . t y i f 4 C˜q , Ž 11 .
2
yig Ž cos 2u e iŽ v c ty f . q sin2u eyiŽ v c ty f . . Cy , Ž 9. d
plitudes at times which are much longer than 1rg 1 and 1rg 2 . If we assume that the energy spacing v 21 of the two upper levels is much larger than g 1 and g 2 , then it is reasonable to neglect for long times Ž t 4 1rg 1 ,1rg 2 . the contribution of the terms containing the rapidly oscillating function sinŽ vc t y f . in Eqs. Ž8. and Ž9.. Then Eqs. Ž8. and Ž9. reduce to
l
C˜qs ig sin2u exp yi Ž vc y lqq ly . t q i f qcos 2u exp i Ž vc q lqy ly . t y i f 4 C˜y , Ž 12 .
where C˜"s C " expwŽ i l "q g 1 . t x and we have also assumed g 1 s g 2 in order to simplify the further calculations. If g s 0, which means that there is no quantum interference, Eqs. Ž11. and Ž12. indicate that the dressed levels < " : decay independently in the course of time. However, when g / 0, the spontaneously generated quantum interference leads to a coupling between the dressed levels. If we adjust properly the frequency vc and the intensity V 0 of the classical radiation field which is used to drive the two upper levels <2: and <1: so that lqy lyf v c , then the factors expw"iŽ vc q lqy ly . t x are rapidly varying in comparison with expw"iŽ v c q lyy lq . t x. Therefore, it is appealing to introduce a further, effective, rotating-wave approximation w20x, dropping the terms containing expw"iŽ vc q lyy lq . t x in Eqs. Ž11. and Ž12.. Then, the time-dependence of the probability amplitudes C " Ž t . can be expressed as Cq Ž t . s A1eŽ i v ayi lqyg 1 .t q A 2 eŽ i v byi lqyg 1 .t , Ž 13 . Cy Ž t . s
1
g sin2u e i f
A1 v a eyŽ i v bqi ly q g 1 .t
qv b A 2 eyŽ i v aqi ly q g 1 .t ,
Ž 14 .
G.-x. Li et al.r Physics Letters A 263 (1999) 250–256
with
va s
(ž
lqy lyy vc
A 2 sin u X d k y a q ig 1
y g sin u ,
qig 2 k cos u e i f
2
lqy lyy vc 2
1
q
2
/
2
4
y g sin u ,
Ž 15 .
v b Cq Ž 0 . y g sin2u e i f Cy Ž 0 . ,
v b y va
as
1
v a Cq Ž 0 . y g sin2u e i f Cy Ž 0 . ,
va y v b
Now we discuss the properties of the spontaneous emission spectrum of the V-type three-level atom. The spontaneous emission spectrum SŽ v ., in the continuum limit, is proportional to the Fourier transform of the field correlation function w9–13x ² Ey Ž t q t . Eq Ž t . :t ™ `
s ²C Ž t . < Ý a†k a k X e i v k Ž tq t . eyi v kX t
`
Hy`d v D Ž v . < C
0k
Ž ` . < 2 e i vt .
Ž 16 .
The above equation shows that the spontaneous emission spectrum SŽ v . is proportional to < C0 k Ž`.< 2 . Substituting Eqs. Ž13. and Ž14. into Eq. Ž10., we obtain the steady-state value of C0 k Ž`. as C0 k Ž ` . s yi
q
½
g 1 k cos u 2
g sin u e
if
ž
A1 v a X d k q vc q a q ig 1
A2 v b X d k q vc y a q ig 1
g1 k y
g1 k y
ig 2 k v a
g sin2u ig 2 k v b
g sin2u
/ /
A1 X d k y vc q a q ig 1
A2 X d k y vc y a q ig 1
/
(
Ž lqy lyy vc .
d kX s d k y
3. Spontaneous emission spectra
k,k
ž
ž ž
/5
Ž 17 .
with
where C " Ž0. are just the initial probability amplitudes of the atom in the dressed levels < " :.
s
q 4
lqy lyy vc
(ž
A2 s
2
/
2
y
A1 s
A1 sin u X d k q a q ig 1
2 q
vb s
q
lqy lyy vc 2
253
4
d0 2
q
vc 2
2
y g 2 sin4u ,
.
Ž 18 .
Evidently, if g s 0, which means that there is no quantum interference arising from the cross coupling between the transitions <1: <0: and <2: <0: under the interaction of vacuum field, then the spontaneous emission spectrum exhibits four peaks, centered at v k s v 1 y v 0 q vc q l " and v k s v 1 y v 0 q l " with the same widths 2g 1. Because the upper two levels of the atom are strongly driven by the classical microwave field, this induces that the system has two dressed-level doublets with energies v 1 q l " and v 1 q vc q l " as shown in Fig. 1b. The spontaneous decay of the atom from these four dressed levels to the ground state <0: is responsible for these four peaks. If the frequency vc and the intensity V 0 of the microwave field satisfy lqy ly y v c ; g 1 , then the two peaks centered at v k s v 1 y v 0 q lq and at v k s v 1 y v 0 q vc q ly overlap, but there is no interference between these two peaks because the two dipole moments of the transitions from <1: to <0: and from <2: to <0: are perpendicular to each other. This also leads to the spontaneous emission spectrum being independent of the phase f of the microwave field. When lqy lys vc , the dressed levels with energies v 1 q vc q ly and v 1 q lq become degenerate. Then there are only three peaks left. However, if g / 0 and Ž lqy lyy vc . 2 ) 4g 2 sin4u , Eq. Ž17. shows that the spontaneous emis-
l
l
254
G.-x. Li et al.r Physics Letters A 263 (1999) 250–256
sion spectrum has six peaks which center at d kX s yvc " a, d kX s "a, and d kX s vc " a, all with the same widths 2g 1. Eqs. Ž11. and Ž12. indicate that the dressed levels < q : and < y : are coupled due to the existence of the spontaneously generated quantum interference. So the dressed levels < " : receive a second dressing through the effect of this quantum interference. In the effective rotating-wave approximation we only consider the coupling between the two dressed levels < " : with energies v 1 q vc q ly and v 1 q lq and neglect the effect of the coupling between the other two dressed levels as shown in Fig. 1b ŽThe effects of the coupling between the dressed levels within every doublet are cancelled when g 1 s g 2 as predicted in Eqs. Ž8. and Ž9... The four dressed levels are then split into eight seconddressed levels. This situation is similar to the case of a two-level atom driven by a strong pumping field and a weak probing field with different frequencies w21x. The energies of these eight second-dressed levels are v 1 q vc q lqy v˜ , v 1 q vc q lyq v˜ , v 1 q lqy v˜ and v 1 q lyq v˜ as shown in Fig. 1c, where v˜ denotes v a or v b . It is easy to verify that the four middle levels are two-by-two degenerate, so the spontaneous decay of the atom from these eight second-dressed levels to its ground state <0: only exhibits six peaks in SŽ v .. On the other hand, Eq. Ž17. also shows that the spontaneous emission spectrum depends on the phase f of the classical field. This is because the atom-field coupled system forms a closed one w22–25x when the spontaneously generated quantum interference is included Žg / 0.. So after a finite time, the probability amplitudes of the atom in the second-dressed levels depend on the phase f , which also gives rise to SŽ v . to be phase dependent. Therefore, by adjusting the phase f one can change the properties of SŽ v . as shown in Fig. 2. In Fig. 2 we have assumed that the atom is initially in the bare state <1:, g s g 1 and d 0 s 0. As we see, SŽ v . has six peaks, but when two peaks are close in energy they are asymmetric due to the interference between them. For different phases, the properties of SŽ v . can change evidently. Next, we examine the case of g / 0 and Ž lqy lyy vc . 2 - 4g 2 sin4u . In this case, Eq. Ž17. shows that the centers of two neighbouring peaks coincide, but their widths are different. One width equals 2Žg 1 q < a <. which is broader than the natural linewidth
Fig. 2. The spontaneous emission spectrum SŽ v . as a function of X the detuning d k for g 1 sg s1, vc s v 21 s10, V 0 s 7, and different values of the phase f . Solid line: f s 3p r2; long-dashed line: f s 0; short-dashed line: f sp r2.
2g 1 and the other width is 2Žg 1 y < a <. which is narrower than the natural one. These phenomena originate from the coupling between the dressed levels < " : which arises from the quantum interference. This coupling leads to a splitting of each dressed level into two ones with complex eigenvalues. Because the parameter < a < is related to the Rabi frequency V 0 and to the frequency vc , not only the centers of the peaks but also the widths in SŽ v . can be controlled by the driving microwave field. Since every narrow peak is superposed on a broad peak, it is non-trivial to distinguish these narrow peaks from the broad ones. Fortunately, the probability amplitudes of the atom in the dressed levels are strongly dependent on the phase f , so the height of every peak can be controlled by changing f . By a suitable choice of f , the broad peaks can be significantly reduced, while the narrow peaks can be enhanced. Fig. 3 displays SŽ v . for different f , assuming that the atom is initially in <1:, g s g 1 , d 0 s 0 and vc s 2 V 0 . In this case, the parameters A1 and A 2 become A1 s A2 s
1
'2 1
'2
cos
cos
ž ž
p
f y
4
2
p
f q
4
2
p
f
/ ž / / ž / exp yi
y
4
p
exp i
2
f
q
4
,
2
,
Ž 19 .
G.-x. Li et al.r Physics Letters A 263 (1999) 250–256
255
4. Conclusions
X Fig. 3. The spectrum SŽ v . as a function of the detuning d k for different values of the phase f , g 1 sg s1, vc s v 21 s10, and V 0 s 5. Short-dashed line: f s1.3p ; long-dashed line: f s1.4p ; solid line: f s1.5p ; dotted line: f s 0.5p .
We have studied the spectrum of spontaneous emission of a V-configuration three-level atom, whose two upper levels are coupled by a classical field and their energy spacing is much larger than the spontaneous emission widths. The spontaneously generated interference can induce the spectrum to exhibit six peaks and depend on the phase of the classical field. These six peaks result from the spontaneous decay of six second-dressed levels, in which two of them are twofold degenerate. If specific values of the parameters such as frequency, intensity and phase of the driving field are chosen, then the peaks with supernatural linewidth can vanish and only the ones with subnatural linewidth appear in the spontaneous emission spectrum evidently even in the case of large energy spacing.
and the probability amplitude C0 k Ž`. is C0 k Ž ` . s yi
q
q
q
g1 k
'2
ž
iA1eyi f
Acknowledgements
d kX q 2 V 0 q i3g 1r2
yiA 2 eyi f
d kX q 2 V 0 q ig 1r2
q
2 A1 X d k q i3g 1r2
iA1e i f X d k y 2 V 0 q i3g 1r2 ie i fA 2
d kX y 2 V 0 q ig 1r2
/
.
Ž 20 .
Evidently, the narrow peak centered at d kX s 0 disappears because of the completely destructive interference between the spontaneous emission processes of the atom in the twofold degenerate dressed levels with complex energies v 1 q vcr2 y ig 1r2. If f s pr2, then A 2 s 0. In that case all the narrow peaks vanish and there are three broad peaks left. However, if p - f - 2p , the heights of the narrow peaks are higher than those of the broad peaks. When f s 3pr2, A1 s 0, so all the broad peaks disappear and there are only two narrow peaks left in SŽ v . as shown in Fig. 3. So we have demonstrated that even in the case of large energy spacing between the upper two levels <1: and <2:, the line narrowing phenomenon resulting from the spontaneously generated quantum interference can still appear clearly in the spontaneous emission spectrum.
G.X.L. and C.H. are supported by the ‘Stichting voor Fundamenteel Onderzoek der Materie ŽFOM.’, which is financially supported by the ‘Nederlandse Orgranisatie voor Wetenschappelijk Onderzoek ŽNWO.’.
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