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Effects of quantum interference on coherent population trapping states of a four-level atom interacting with coherent fields
1 April 1999
Optics Communications 162 Ž1999. 155–161
Full length article
Effects of quantum interference on coherent population trapping states of a four-level atom interacting with coherent fields Fu-li Li
a,b
, Shi-Yao Zhu
a
a
b
Department of Physics, Hong Kong Baptist UniÕersity, Hong Kong, China Department of Applied Physics, Xian Jiaotong UniÕersity, Xian 710049, China
Received 21 October 1998; received in revised form 12 January 1999; accepted 19 January 1999
Abstract We find that a four-level atom–two-driving field coupled system has four coherent population trapping states. We show that quantum interference between different spontaneous emission pathways from the upper to the lower levels can lead to coherent population trapping states stable against the radiative decay of the upper levels. By adjusting the detuning of the driving fields, we can control the population trapped in the upper levels via quantum interference. We also find that quantum interference can much largely delay the relaxation of the atom to the steady state. We also notice that the quantum interference results in the dependence of the atomic steady state on the initial condition although the interaction of the atom with the vacuum modes is considered. q 1999 Elsevier Science B.V. All rights reserved.
As it is well known, spontaneous emission from excited atoms is one of the most important noise sources in a laser system. On the other hand, pumping enough population to highly excited levels of atoms to generate an ultra-high frequency laser is problematic since the spontaneous emission rate is typically proportional to the transition frequency cubed Ž v 3 .. Therefore, depression of spontaneous emission has been highly desired. Recent studies have shown that spontaneous emission from excited atomic levels can be depressed or even totally cancelled via quantum interference between different decay pathways in a multi-level atom w1–6x. In Ref. w1x, Scully, Zhu and Gavrielides studied the interaction between a single mode of the signal field and three-level atoms, and showed the cancellation of emission from the two nondegenerate upper levels into the signal mode. In Ref. w2x, Zhu, Chan and Lee investigated the spontaneous emission from a three-level atom in which two upper levels are coupled to the ground state by the same vacuum modes, and showed that quantum interference can result in a dark line in the spontaneous emission spectrum. In Ref. w3x, Zhu and Scully studied the spontaneous emission spectrum of a four-level atom in which two upper levels are coupled to one of the lower levels by
a coherent field and to another one by the same vacuum modes, and showed that quantum interference can lead to the depression of not only the transition mode but also all the modes of spontaneous emission. The theoretical expectation has been demonstrated in an experiment w4,5x. The cancellation of spontaneous emission originally results from atomic coherence between the two excited levels, which leads to quantum interference between different decay pathways of excited levels. The atomic coherence can be created by applying a coherent field between the two excited levels and the lower single level or a lower auxiliary level w3–9x. Considering that from the dressed states, for these driving schemes, there is one Žonly one. dressed state with zero eigenvalue and it does not evolve in time under the governing Hamiltonian, the transition from the dressed state to the lowly lying single state is composed of two pathways. These pathways can have a completely destructive interference, which results in cancellation of the spontaneous emission from the dressed state w3x and the depression of resonance fluorescence w7–9x. In this way, the dressed state becomes nondecaying and a portion of the populations is trapped in the upper levels. This effect is useful, in principle, for generation of high-frequency and high power lasers.
0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 0 2 2 - X
F. Li, S.-Y. Zhu r Optics Communications 162 (1999) 155–161
156
The dressed states which have zero eigenvalue and hence are stationary states of the Hamiltonian are called coherent population trapping states Žhereafter abbreviated as CTP states. w10x. To our knowledge, most atom–driving field coupled systems which have been investigated have only one CPT state. These states might become unstable or still remain nonevolving in the presence of the radiative relaxation of the atom depending on the dipole polarizations of the atom and its composition. In this Letter, we shall consider a double V-configuration of a four-level atom interacting with two coherent fields, in contrast with the V-configuration of a three-level atom, in which the upper level is replaced by two closely spaced levels. We shall show that via quantum interference the system can have four CPT states which are stable against the radiative decay of the upper levels. It is more interesting because quantum interference can lead to the dependence of the steady state of the atom on the initial condition although the interaction of the atom with the vacuum modes is considered, and the steady-state upperlevel population can be controlled by adjusting the detuning of the driving fields. The atomic level scheme under consideration is shown in Fig. 1. The upper levels < a1 : and < a2 : are coupled by the same vacuum modes to the lower level < b : or < c :. We assume that the vacuum modes coupling < a1 ) and < a2 : to < b : are totally different from those coupling < a1 : and < a 2 : to < c :. The transitions from < a1 : and < a 2 : to < b : and < c : are simultaneously driven by two coherent fields at frequencies v 1 and v 2 , respectively. The dipole transition between < b : and < c : is forbidden because of equal parity. With the rotation-wave approximation, in the rotated frame defined by the transformation expŽ i"y1 Hˆ0 t . where Hˆ0 s " Ž v b q v 1 . Ž < a1 : ² a1 < q < a 2 : ² a 2 < . q " v b < b: ² b < q " Ž v b q v 1 y v 2 . < c: ² c < qÝ
" v q a†q a q q
q
Ý
Fig. 1. Level scheme of the model atom.
In the above expressions, " v a1, " v a2 , " v b , and " v c are the energies of the levels < a1 :, < a 2 :, < b : and < c :, D1,2 s v a1,2 y v b y v 1 , and D3,4 s v a1,2 y v c y v 2 , and a q Ž a†q . and b k Ž b†k . are the annihilation Žcreation. operators for the qth and k th vacuum modes which energies are " v q and " v k , respectively. In Eqs. Ž2. – Ž5., g qŽ1,2. Ž g kŽ3,4. . are the coupling constants between the qth Ž k th. vacuum mode and the atomic transitions from < a1 : and < a 2 : to < b : Ž< c :., and V 1,2 Ž V 3,4 . are the Rabi frequencies of the coherent field which drives the transitions. Here k and q stand for both the momentum and polarization of the vacuum modes. The dressed states of the atom–driving field coupled system are solutions of the eigenequation
Ž Hˆ q Vˆ
" v k b†k b k ,
s
1
(
2 V 12 q D s2
: s ´s < ws : ,
Ž7.
w yV 1 < a1 : q V 1 < a2 : q Ds < b : x . Ž8.
Žii. When D3 s D s but D1 / D s ,
the Hamiltonian of the system can be written as Hˆ s Hˆb q Vˆcb q Vˆcc q VˆR b q VˆRc ,
.
where < ws : s C1 s < a1 : q C2 s < a 2 : q C3 s < b : q C4s < c : with the integer s Žs 1,2,3,4. labelling the four eigenvectors. On the assumption that V 1 s V 2 and V 3 s V 4 , from Ž7., we find that the system under consideration has the four CPT states: Ži. When D1 s v 12r2 ' D s but D3 / D s , < w I: s
k
ˆ
cb q Vcc
b
Ž1.
where
1
< w II : s
(
2 V 32 q D s2
w yV 3 < a1 : q V 3 < a2 : q Ds < c : x . Ž9.
Žiii. When D1 s D3 / D s ,
Hˆb s " D1 < a1 : ² a1 < q " D2 < a 2 : ² a 2 < q " Ž D1 y D3 . < c : ² c < , Ž2. Vˆcb s " V 1 < a1 : ² b < q " V 2 < a 2 : ² b < q h.c.,
Ž3.
Vˆcc s " V 3 < a1 : ² c < q " V 4 < a2 : ² c < q h.c.,
Ž4.
VˆR b s " Ý e
iŽ v 1 y v q .t
< w III : s
w V 3 < b: y V 1 < c:x .
Ž 10 .
Živ. When D1 s D3 s D s , the zero eigenvalue becomes two-fold degenerate. The corresponding CPT states are
q
Ž5.
VˆRc s " Ý e iŽ v 2y v k .t g kŽ3. a k < a1 : ² c < q g kŽ4. a k < a1 : ² c < k
q h.c.
(
< w IV : s < w III : ,
g qŽ1. a q < a1 : ² b < q g qŽ2. a q < a1 : ² b <
q h.c.
1
V 12 q V 32
< w V:s
( y
Ž6.
2Ž
Ž 11 .
V 12 q V 32 V 12 q V 32 q D s2
.
Ds V 12 q V 32
< a1 : y < a 2 :
Ž V 1 < b: q V 3 < c:.
.
Ž 12.
F. Li, S.-Y. Zhu r Optics Communications 162 (1999) 155–161
Obviously, governed by the Hamiltonian Hˆb q Vˆcb q Vˆcc , these states do not evolve in time. Moreover, these CPT states can be stable even if the radiative decay of the upper levels is considered. The state < w III : or < w IV : is stable as long as < b : and < c : are stable states. In the following discussions, we shall see that quantum interference between different decay processes of the upper levels can make the CPT states < w I :, < w II : and < w V : stable against spontaneous emission from the upper levels. The decay rates from the dressed states < ws : to < ws X : by emitting a photon into the corresponding vacuum modes are given by w6x
gs X s Ž q . s
d
i
Ý dt
y
q
d dt
t
X
H d t ² ws <²1 " 0 X
i
Ý k
y
q
< VˆR b < Oq :< ws :
,
r˙ a1 a1 s y Ž g 1 q g 3 . r a1 a1
t
X
H0 d t ² ws <²1 < Vˆ k
Rc
< O k :< ws :
y 12 p 1 g 1g 2 q p 2 g 3g4
Ž '
'
.
Ž 14. Replacing the summations over k and q by integration and using the Weisskoff-Wigner approximation, we obtain
= e i v 12 rt a2 a1 q eyi v 12 rt a1 a2
Ž
q i V 3) eyi D 3 rt a1 c ,
y 12 p 1 g 1g 2 q p 2 g 3g4 e i v 12 t Ž r a1 a1 q r a 2 a 2 .
Ž '
'
Ž 15.
r˙ a1 b s y 12 Ž g 1 q g 3 . r a1 b
.
gs X s Ž k . s g 3 C12s q g4 C22s q 2 p 2 g 3 g4 C1 s C2 s C42s X ,
'
Ž
.
Ž 16. where g 1,2 Žg 3,4 . are the spontaneous emission rates from the two upper levels < a1 : and < a 2 : to the lower level < b : Ž< c :., and p 1 s m a b P m a br< m a b < < m a b < and p 2 s m a c P 1 2 1 2 1 m a2 cr< m a1 c < < m a2 c < with m a1 b , m a2 b , and m a1 c , m a2 c the dipole moments of the transitions referred by the subscripts. The C in Ž15. and Ž16. are the coefficients in the definitions of the dressed states of Ž7.. If the dipole moments of the two transitions are parallel or antiparallel, we have p 1,2 s "1, while for the orthogonal case we have p1,2 s 0. The crossing terms proportional to 2 p1 g 1g 2 and 2 p 2 g 3g4 in Ž15. and Ž16. lead to an interference between the decay processes from < a1 : to < b : Ž< c :. and < a 2 : to < b : Ž< c :.. From Ž15. and Ž16., we see that when p1,2 s "1, and the amplitudes of the components in < ws : satisfy the conditions C 1 s rC 2 s s . g 2 rg 1 or C 1 s rC 2 s s . g4rg 3 the interference becomes completely destructive and correspondingly the decay rate is zero. The CPT states < w I :, < w II : and < w V : have a common feature that the amplitudes of the upper levels are the same with opposite signs. This feature suggests that there is a completely destructive interference between the two spontaneous emission pathways Ž< a1 : ™ < b : or < c : and < a 2 : ™ < b : or < c :. in the transitions from these CPT states to any
'
'
'
'
.
y i V 1e i D 1 tr b a 2 y i V 3 e i D 3 rt c a 2 q i V 2) eyi D 2 rt a1 b q i V 4) eyi D 4 tr a1 c ,
'
Ž 17 .
r˙ a1 a2 s y 12 Ž g 1 q g 2 q g 3 q g4 . r a1 a 2
gs X s Ž q . s g 1C12s q g 2 C22s q 2 p1 g 1 g 2 C1 s C2 s C32s X ,
Ž
. .
y i V 1e i D 1 rt b a1 q i V 1) eyi D 1 rt a1 b y i V 3 e i D 3 rt c a1
2 X
"
dressed states < ws : if p 1 s p 2 s 1, g 1 s g 2 and g 3 s g4 . Therefore, under these conditions, the CPT states < w I :, < w II : and < w V : become stable against the radiative decay of the upper levels. When the atom is in one of these states the population in the upper levels holds and the resonance fluorescence is depressed. To study the time evolution of the atom to the CPT states, we need the equation of motion for the reduced atomic density matrix. According to the generalized reservoir theory with the Weisskoff-Wigner approximation, we can obtain w6x
2
Ž 13. gs X s Ž k . s
157
Ž 18.
y 12 p 1 g 1g 2 q p 2 g 3g4 e i v 12 rt a 2 b
Ž '
y i V 1e
i D1 t
'
.
Ž r b b y r a1 a1 .
q i V 2 e i D 2 rt a1 a 2 y i V 3 eyi D 3 rt c b ,
Ž 19.
r˙ a1 c s y 12 Ž g 1 q g 3 . r a1 c y 12 p 1 g 1g 2 q p 2 g 3g4 e i v 12 rt a 2 c
Ž '
y i V 3e
i D3 t
'
.
Ž 1 y r b b y ra
y 2 r a1 a1 .
2 a2
q i V 4 e i D 4 tr a1 a 2 y i V 1e i D 1 tr b c ,
Ž 20.
r˙ a2 a2 s y Ž g 2 q g4 . r a 2 a 2 y 12 p 1 g 1g 2 q p 2 g 3g4
Ž '
= e
Ž
i v 12 t
'
yi v 12 t
r a2 a1 q e
r a1 a2
. .
y i V 2 e i D 2 rt b a 2 q i V 2) eyi D 2 rt a 2 b y i V 4 e i D 4 rt c a 2 q i V 4) eyi D 4 rt a 2 c ,
Ž 21.
1 2
r˙ a2 b s y Ž g 2 q g4 . r a 2 b y 12 p 1 g 1g 2 q p 2 g 3g4 eyi v 12 tr a1 b
Ž '
y i V 2e
i D2 t
'
.
Ž r b b y ra2 a2 .
q i V 1e i D 1 rt a 2 a1 y i V 4 e i D 4 rt c b ,
Ž 22.
F. Li, S.-Y. Zhu r Optics Communications 162 (1999) 155–161
158
r˙ a2 c s y 12 Ž g 2 q g4 . r a 2 c y 12 p 1 g 1g 2 q p 2 g 3g4 eyi v 12 tr a1 c
Ž '
'
y i V4e
i D4 t
q i V 3e
i D3 t
.
Ž 1 y r b b y ra a y 2 ra a . 1 1
r a 2 a1 y i V 2 e
2
2
i D2 t
rbc ,
Ž 23 .
r˙ bb s g 1 r a1 a1 q g 2 r a 2 a 2 q p1 g 1g 2 eyi v 12 tr a1 a 2 q e i v 12 rt a 2 a1
'
Ž
.
q i V 1e i D 1 rt b a1 y i V 1) eyi D 1 rt a1 b q i V 2 e i D 2 rt b a 2 y i V 2) eyi D 2 tr a 2 b ,
Ž 24 .
r bc˙ s yi V 1) eyi D 1 rt a1 c y i V 2) eyi D 2 tr a 2 c q i V 3 e i D 3 rt b a1 q i V 4 e i D 4 rt b a 2 .
Ž 25.
'
In Eqs. Ž17. – Ž25., as pointed out above, the term p1 g 1g 2 Ž p 2 g 3g4 . represents the effect of quantum interference between the spontaneous emission pathways from < a1 : to < b : Ž< c :. and from < a 2 : to < b : Ž< c :.. It reflects the fact that as the atom decays from the excited sublevel < a1 : it drives the other excited sublevel < a2 : and vice versa because of atomic coherence between < a1 : and < a2 :, which is generated by the driving fields. We notice that in the above equations the usual decay terms proportional to g 1,2 and g 3,4 are always acompanied by decay interference terms proportional to p 1 g 1g 2 and p 2 g 3g4 , respectively, as long as p 1 / 0 and p 2 / 0. The interference terms might cancel the usual decay terms, which will result in spontaneous emission cancellation. Consequently, the atom takes one of the four CPT states as its stationary state. Substituting the transformations C 1 s eyi D 1 rt a1 b , C 2 s eyi D 2 rt a 2 b , C 3 s eyi D 3 rt a1 c , C4 s eyi D 4 rt a 2 c , C 5 s eyiD rt a1 a2 , C 14 s eyiŽ D 3yD 1 .tr b c , C6 s r a1 a1, C 7 s r a 2 a 2 , C 8 s r b b , and C 9 s C 1) , C 10 s C 2) , C 11 s C 3) , C 12 s C4) , C 13 s C5) , C 15 s C 14) into Ž17. – Ž25., we can write the above equations into a compact vector form
'
'
d dt
ˆ Cˆ s LˆCˆ q I,
'
photon resonance case in a three-level atom interacting with two fields w11,12x. For D1 s D3 s D s , V 1 s V 2 and V 3 s V 4 , the vector Cˆ determined by
Ž 26.
where Cˆ is a column vector whose ith component is Ci and the inhomogeneous term Iˆ is also a column vector with the non-zero components Iˆ3 s yi V 3 , Iˆ11 s Iˆ3) , Iˆ4 s yi V 4 , Iˆ12 s Iˆ4). In Ž26., Lˆ is a time-independent 15 = 15 matrix which explicit expression can easily be derived using Eqs. Ž17. – Ž25.. It can directly be verified that for p1 s p 2 s 1, g 1 s g 2 , g 3 s g4 , V 1 s V 2 and V 3 s V 4 the vectors Cˆ determined by the CPT states < w I :, < w II : and < w V : are the steady-state solutions of Ž26. if D1 s D s and D3 / D s , or, D1 / D s and D3 s D s , or, D1 s D3 s D s , respectively. However, for D1 s D3 / D s , V 1 s V 2 and V 3 s V 4 the vector Cˆ determined by the CPT state
Fig. 2. Steady-state populations in the upper levels versus D3 rg 3 with V 1 s V 2 s 2.0, V 3 s V 4 s 5.0, g 1 sg 2 s 0.5, g 3 sg4 s 1.0, v 12 s 2.0.
F. Li, S.-Y. Zhu r Optics Communications 162 (1999) 155–161
159
Fig. 3. Time evolution of the level populations. The initial state is < c : with V 1 s V 2 s 1.2, V 3 s V 4 s 1.5, g 1 s g 2 s 0.5, g 3 s g4 s 1.0, v 12 s 2.0, D1 s 1.0, D3 s 0.0. Ža. p1 s p 2 s 1; Žb. p1 s 0, p 2 s 1; Žc. p1 s p 2 s 0.
160
F. Li, S.-Y. Zhu r Optics Communications 162 (1999) 155–161
to switch the steady state population of the atom from the lower levels to the upper levels. Now let us see some numerical results. In Fig. 2 the steady-state populations in the upper levels are shown as
functions of D3rg 3 with fixed D1 s 0.5 and p 1 s p 2 s 1.0. The points A Ž D3rg 3 s 0.5. and B Ž D3rg 3 s 1.0. represent two special cases in which the atom is in the CPT states
Fig. 4. Time evolution of level populations. The initial state is: Ža. < c :; Žb. < a1 :; Žc. 0.5 r b b q 0.5 rc c . Other parameters are same as in Fig. 3 but p1 s p 2 s 1.
F. Li, S.-Y. Zhu r Optics Communications 162 (1999) 155–161
between the two limits the upper-level populations can drastically be changed from 0.0 to 2 V 32rŽ2 V 32 q D s2 . or vice versa by varying the detuning D3. By fixing D3 Ž/ D s . and varying the detuning D1 we have the same result. Using this effect, we can transfer the population trapped in < c : Ž< b :. to < b : Ž< c :.. For example, at the point B, the portions of population trapped in < c : and < b : are D s2rŽ2 V 32 q D s2 . and 0.0, respectively. We now change D3 from D s to D1 , and then fix D3 and increase D1 to D s . At the point D1 s D s and D3 / D1 , the atom gets into the CPT state
161
steady state on the initial conditions has been studied by Agarwal w14x. Here, we would like to emphasize that this effect originates from the quantum interference, which decouples the atom from the vacuum modes. In summary, we have found that the four-level atom interacting with two coherent fields has four coherent population trapping states. We have shown that the quantum interference between the spontaneous emission pathways from the upper levels to the lower levels can lead to these states stable against the radiative decay of the upper levels. The atom can take these states as its steady state. Quantum interference strongly delays the relaxation of the atom to these states. We have also noticed that quantum interference can lead to the dependence of the atomic steady state on the initial condition although the atom is coupled to the vacuum modes.
Acknowledgements This work was supported by the Baptise University of Hong Kong. F.L. Li also acknowledges support from the National Natural Science Foundation of China.
References w1x M.O. Scully, S.Y. Zhu, A. Gavrielides, Phys. Rev. Lett. 62 Ž1989. 2813. w2x S.Y. Zhu, R.C.F. Chan, C.P. Lee, Phys. Rev. A 52 Ž1995. 710. w3x S.Y. Zhu, M.O. Scully, Phys. Rev. Lett. 76 Ž1996. 388. w4x H.R. Xia, C.Y. Ye, S.Y. Zhu, Phys. Rev. Lett. 77 Ž1996. 1032. w5x G.S. Agarwal, Phys. Rev. A 55 Ž1997. 2457. w6x H. Lee, P. Polynkin, M.O. Scully, S.-Y. Zhu, Phys. Rev. A 55 Ž1997. 4454. w7x D.A. Cardimona, M.G. Raymer, C.R. Stroud Jr., J. Phys. B 15 Ž1982. 55. w8x P. Zhou, S. Swain, Phys. Rev. Lett. 77 Ž1996. 3995. w9x P. Zhou, S. Swain, Phys. Rev. A 56 Ž1997. 3011. w10x G.S. Agarwal, Phys. Rev. Lett. 71 Ž1993. 1351. w11x S. Swain, J. Phys. B 15 Ž1982. 3405. w12x P.M. Radmore, P.L. Knight, J. Phys. B 15 Ž1982. 561. w13x C.E. Carroll, F.T. Hioe, Phys. Rev. Lett. 68 Ž1992. 3523. w14x G.S. Agarwal, Quantum Optics, Springer Tracts in Modern Physics, vol. 70, Springer, Berlin, 1974, p. 95.
Optics Communications 162 Ž1999. 155–161
Full length article
Effects of quantum interference on coherent population trapping states of a four-level atom interacting with coherent fields Fu-li Li
a,b
, Shi-Yao Zhu
a
a
b
Department of Physics, Hong Kong Baptist UniÕersity, Hong Kong, China Department of Applied Physics, Xian Jiaotong UniÕersity, Xian 710049, China
Received 21 October 1998; received in revised form 12 January 1999; accepted 19 January 1999
Abstract We find that a four-level atom–two-driving field coupled system has four coherent population trapping states. We show that quantum interference between different spontaneous emission pathways from the upper to the lower levels can lead to coherent population trapping states stable against the radiative decay of the upper levels. By adjusting the detuning of the driving fields, we can control the population trapped in the upper levels via quantum interference. We also find that quantum interference can much largely delay the relaxation of the atom to the steady state. We also notice that the quantum interference results in the dependence of the atomic steady state on the initial condition although the interaction of the atom with the vacuum modes is considered. q 1999 Elsevier Science B.V. All rights reserved.
As it is well known, spontaneous emission from excited atoms is one of the most important noise sources in a laser system. On the other hand, pumping enough population to highly excited levels of atoms to generate an ultra-high frequency laser is problematic since the spontaneous emission rate is typically proportional to the transition frequency cubed Ž v 3 .. Therefore, depression of spontaneous emission has been highly desired. Recent studies have shown that spontaneous emission from excited atomic levels can be depressed or even totally cancelled via quantum interference between different decay pathways in a multi-level atom w1–6x. In Ref. w1x, Scully, Zhu and Gavrielides studied the interaction between a single mode of the signal field and three-level atoms, and showed the cancellation of emission from the two nondegenerate upper levels into the signal mode. In Ref. w2x, Zhu, Chan and Lee investigated the spontaneous emission from a three-level atom in which two upper levels are coupled to the ground state by the same vacuum modes, and showed that quantum interference can result in a dark line in the spontaneous emission spectrum. In Ref. w3x, Zhu and Scully studied the spontaneous emission spectrum of a four-level atom in which two upper levels are coupled to one of the lower levels by
a coherent field and to another one by the same vacuum modes, and showed that quantum interference can lead to the depression of not only the transition mode but also all the modes of spontaneous emission. The theoretical expectation has been demonstrated in an experiment w4,5x. The cancellation of spontaneous emission originally results from atomic coherence between the two excited levels, which leads to quantum interference between different decay pathways of excited levels. The atomic coherence can be created by applying a coherent field between the two excited levels and the lower single level or a lower auxiliary level w3–9x. Considering that from the dressed states, for these driving schemes, there is one Žonly one. dressed state with zero eigenvalue and it does not evolve in time under the governing Hamiltonian, the transition from the dressed state to the lowly lying single state is composed of two pathways. These pathways can have a completely destructive interference, which results in cancellation of the spontaneous emission from the dressed state w3x and the depression of resonance fluorescence w7–9x. In this way, the dressed state becomes nondecaying and a portion of the populations is trapped in the upper levels. This effect is useful, in principle, for generation of high-frequency and high power lasers.
0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 0 2 2 - X
F. Li, S.-Y. Zhu r Optics Communications 162 (1999) 155–161
156
The dressed states which have zero eigenvalue and hence are stationary states of the Hamiltonian are called coherent population trapping states Žhereafter abbreviated as CTP states. w10x. To our knowledge, most atom–driving field coupled systems which have been investigated have only one CPT state. These states might become unstable or still remain nonevolving in the presence of the radiative relaxation of the atom depending on the dipole polarizations of the atom and its composition. In this Letter, we shall consider a double V-configuration of a four-level atom interacting with two coherent fields, in contrast with the V-configuration of a three-level atom, in which the upper level is replaced by two closely spaced levels. We shall show that via quantum interference the system can have four CPT states which are stable against the radiative decay of the upper levels. It is more interesting because quantum interference can lead to the dependence of the steady state of the atom on the initial condition although the interaction of the atom with the vacuum modes is considered, and the steady-state upperlevel population can be controlled by adjusting the detuning of the driving fields. The atomic level scheme under consideration is shown in Fig. 1. The upper levels < a1 : and < a2 : are coupled by the same vacuum modes to the lower level < b : or < c :. We assume that the vacuum modes coupling < a1 ) and < a2 : to < b : are totally different from those coupling < a1 : and < a 2 : to < c :. The transitions from < a1 : and < a 2 : to < b : and < c : are simultaneously driven by two coherent fields at frequencies v 1 and v 2 , respectively. The dipole transition between < b : and < c : is forbidden because of equal parity. With the rotation-wave approximation, in the rotated frame defined by the transformation expŽ i"y1 Hˆ0 t . where Hˆ0 s " Ž v b q v 1 . Ž < a1 : ² a1 < q < a 2 : ² a 2 < . q " v b < b: ² b < q " Ž v b q v 1 y v 2 . < c: ² c < qÝ
" v q a†q a q q
q
Ý
Fig. 1. Level scheme of the model atom.
In the above expressions, " v a1, " v a2 , " v b , and " v c are the energies of the levels < a1 :, < a 2 :, < b : and < c :, D1,2 s v a1,2 y v b y v 1 , and D3,4 s v a1,2 y v c y v 2 , and a q Ž a†q . and b k Ž b†k . are the annihilation Žcreation. operators for the qth and k th vacuum modes which energies are " v q and " v k , respectively. In Eqs. Ž2. – Ž5., g qŽ1,2. Ž g kŽ3,4. . are the coupling constants between the qth Ž k th. vacuum mode and the atomic transitions from < a1 : and < a 2 : to < b : Ž< c :., and V 1,2 Ž V 3,4 . are the Rabi frequencies of the coherent field which drives the transitions. Here k and q stand for both the momentum and polarization of the vacuum modes. The dressed states of the atom–driving field coupled system are solutions of the eigenequation
Ž Hˆ q Vˆ
" v k b†k b k ,
s
1
(
2 V 12 q D s2
: s ´s < ws : ,
Ž7.
w yV 1 < a1 : q V 1 < a2 : q Ds < b : x . Ž8.
Žii. When D3 s D s but D1 / D s ,
the Hamiltonian of the system can be written as Hˆ s Hˆb q Vˆcb q Vˆcc q VˆR b q VˆRc ,
.
where < ws : s C1 s < a1 : q C2 s < a 2 : q C3 s < b : q C4s < c : with the integer s Žs 1,2,3,4. labelling the four eigenvectors. On the assumption that V 1 s V 2 and V 3 s V 4 , from Ž7., we find that the system under consideration has the four CPT states: Ži. When D1 s v 12r2 ' D s but D3 / D s , < w I: s
k
ˆ
cb q Vcc
b
Ž1.
where
1
< w II : s
(
2 V 32 q D s2
w yV 3 < a1 : q V 3 < a2 : q Ds < c : x . Ž9.
Žiii. When D1 s D3 / D s ,
Hˆb s " D1 < a1 : ² a1 < q " D2 < a 2 : ² a 2 < q " Ž D1 y D3 . < c : ² c < , Ž2. Vˆcb s " V 1 < a1 : ² b < q " V 2 < a 2 : ² b < q h.c.,
Ž3.
Vˆcc s " V 3 < a1 : ² c < q " V 4 < a2 : ² c < q h.c.,
Ž4.
VˆR b s " Ý e
iŽ v 1 y v q .t
< w III : s
w V 3 < b: y V 1 < c:x .
Ž 10 .
Živ. When D1 s D3 s D s , the zero eigenvalue becomes two-fold degenerate. The corresponding CPT states are
q
Ž5.
VˆRc s " Ý e iŽ v 2y v k .t g kŽ3. a k < a1 : ² c < q g kŽ4. a k < a1 : ² c < k
q h.c.
(
< w IV : s < w III : ,
g qŽ1. a q < a1 : ² b < q g qŽ2. a q < a1 : ² b <
q h.c.
1
V 12 q V 32
< w V:s
( y
Ž6.
2Ž
Ž 11 .
V 12 q V 32 V 12 q V 32 q D s2
.
Ds V 12 q V 32
< a1 : y < a 2 :
Ž V 1 < b: q V 3 < c:.
.
Ž 12.
F. Li, S.-Y. Zhu r Optics Communications 162 (1999) 155–161
Obviously, governed by the Hamiltonian Hˆb q Vˆcb q Vˆcc , these states do not evolve in time. Moreover, these CPT states can be stable even if the radiative decay of the upper levels is considered. The state < w III : or < w IV : is stable as long as < b : and < c : are stable states. In the following discussions, we shall see that quantum interference between different decay processes of the upper levels can make the CPT states < w I :, < w II : and < w V : stable against spontaneous emission from the upper levels. The decay rates from the dressed states < ws : to < ws X : by emitting a photon into the corresponding vacuum modes are given by w6x
gs X s Ž q . s
d
i
Ý dt
y
q
d dt
t
X
H d t ² ws <²1 " 0 X
i
Ý k
y
q
< VˆR b < Oq :< ws :
,
r˙ a1 a1 s y Ž g 1 q g 3 . r a1 a1
t
X
H0 d t ² ws <²1 < Vˆ k
Rc
< O k :< ws :
y 12 p 1 g 1g 2 q p 2 g 3g4
Ž '
'
.
Ž 14. Replacing the summations over k and q by integration and using the Weisskoff-Wigner approximation, we obtain
= e i v 12 rt a2 a1 q eyi v 12 rt a1 a2
Ž
q i V 3) eyi D 3 rt a1 c ,
y 12 p 1 g 1g 2 q p 2 g 3g4 e i v 12 t Ž r a1 a1 q r a 2 a 2 .
Ž '
'
Ž 15.
r˙ a1 b s y 12 Ž g 1 q g 3 . r a1 b
.
gs X s Ž k . s g 3 C12s q g4 C22s q 2 p 2 g 3 g4 C1 s C2 s C42s X ,
'
Ž
.
Ž 16. where g 1,2 Žg 3,4 . are the spontaneous emission rates from the two upper levels < a1 : and < a 2 : to the lower level < b : Ž< c :., and p 1 s m a b P m a br< m a b < < m a b < and p 2 s m a c P 1 2 1 2 1 m a2 cr< m a1 c < < m a2 c < with m a1 b , m a2 b , and m a1 c , m a2 c the dipole moments of the transitions referred by the subscripts. The C in Ž15. and Ž16. are the coefficients in the definitions of the dressed states of Ž7.. If the dipole moments of the two transitions are parallel or antiparallel, we have p 1,2 s "1, while for the orthogonal case we have p1,2 s 0. The crossing terms proportional to 2 p1 g 1g 2 and 2 p 2 g 3g4 in Ž15. and Ž16. lead to an interference between the decay processes from < a1 : to < b : Ž< c :. and < a 2 : to < b : Ž< c :.. From Ž15. and Ž16., we see that when p1,2 s "1, and the amplitudes of the components in < ws : satisfy the conditions C 1 s rC 2 s s . g 2 rg 1 or C 1 s rC 2 s s . g4rg 3 the interference becomes completely destructive and correspondingly the decay rate is zero. The CPT states < w I :, < w II : and < w V : have a common feature that the amplitudes of the upper levels are the same with opposite signs. This feature suggests that there is a completely destructive interference between the two spontaneous emission pathways Ž< a1 : ™ < b : or < c : and < a 2 : ™ < b : or < c :. in the transitions from these CPT states to any
'
'
'
'
.
y i V 1e i D 1 tr b a 2 y i V 3 e i D 3 rt c a 2 q i V 2) eyi D 2 rt a1 b q i V 4) eyi D 4 tr a1 c ,
'
Ž 17 .
r˙ a1 a2 s y 12 Ž g 1 q g 2 q g 3 q g4 . r a1 a 2
gs X s Ž q . s g 1C12s q g 2 C22s q 2 p1 g 1 g 2 C1 s C2 s C32s X ,
Ž
. .
y i V 1e i D 1 rt b a1 q i V 1) eyi D 1 rt a1 b y i V 3 e i D 3 rt c a1
2 X
"
dressed states < ws : if p 1 s p 2 s 1, g 1 s g 2 and g 3 s g4 . Therefore, under these conditions, the CPT states < w I :, < w II : and < w V : become stable against the radiative decay of the upper levels. When the atom is in one of these states the population in the upper levels holds and the resonance fluorescence is depressed. To study the time evolution of the atom to the CPT states, we need the equation of motion for the reduced atomic density matrix. According to the generalized reservoir theory with the Weisskoff-Wigner approximation, we can obtain w6x
2
Ž 13. gs X s Ž k . s
157
Ž 18.
y 12 p 1 g 1g 2 q p 2 g 3g4 e i v 12 rt a 2 b
Ž '
y i V 1e
i D1 t
'
.
Ž r b b y r a1 a1 .
q i V 2 e i D 2 rt a1 a 2 y i V 3 eyi D 3 rt c b ,
Ž 19.
r˙ a1 c s y 12 Ž g 1 q g 3 . r a1 c y 12 p 1 g 1g 2 q p 2 g 3g4 e i v 12 rt a 2 c
Ž '
y i V 3e
i D3 t
'
.
Ž 1 y r b b y ra
y 2 r a1 a1 .
2 a2
q i V 4 e i D 4 tr a1 a 2 y i V 1e i D 1 tr b c ,
Ž 20.
r˙ a2 a2 s y Ž g 2 q g4 . r a 2 a 2 y 12 p 1 g 1g 2 q p 2 g 3g4
Ž '
= e
Ž
i v 12 t
'
yi v 12 t
r a2 a1 q e
r a1 a2
. .
y i V 2 e i D 2 rt b a 2 q i V 2) eyi D 2 rt a 2 b y i V 4 e i D 4 rt c a 2 q i V 4) eyi D 4 rt a 2 c ,
Ž 21.
1 2
r˙ a2 b s y Ž g 2 q g4 . r a 2 b y 12 p 1 g 1g 2 q p 2 g 3g4 eyi v 12 tr a1 b
Ž '
y i V 2e
i D2 t
'
.
Ž r b b y ra2 a2 .
q i V 1e i D 1 rt a 2 a1 y i V 4 e i D 4 rt c b ,
Ž 22.
F. Li, S.-Y. Zhu r Optics Communications 162 (1999) 155–161
158
r˙ a2 c s y 12 Ž g 2 q g4 . r a 2 c y 12 p 1 g 1g 2 q p 2 g 3g4 eyi v 12 tr a1 c
Ž '
'
y i V4e
i D4 t
q i V 3e
i D3 t
.
Ž 1 y r b b y ra a y 2 ra a . 1 1
r a 2 a1 y i V 2 e
2
2
i D2 t
rbc ,
Ž 23 .
r˙ bb s g 1 r a1 a1 q g 2 r a 2 a 2 q p1 g 1g 2 eyi v 12 tr a1 a 2 q e i v 12 rt a 2 a1
'
Ž
.
q i V 1e i D 1 rt b a1 y i V 1) eyi D 1 rt a1 b q i V 2 e i D 2 rt b a 2 y i V 2) eyi D 2 tr a 2 b ,
Ž 24 .
r bc˙ s yi V 1) eyi D 1 rt a1 c y i V 2) eyi D 2 tr a 2 c q i V 3 e i D 3 rt b a1 q i V 4 e i D 4 rt b a 2 .
Ž 25.
'
In Eqs. Ž17. – Ž25., as pointed out above, the term p1 g 1g 2 Ž p 2 g 3g4 . represents the effect of quantum interference between the spontaneous emission pathways from < a1 : to < b : Ž< c :. and from < a 2 : to < b : Ž< c :.. It reflects the fact that as the atom decays from the excited sublevel < a1 : it drives the other excited sublevel < a2 : and vice versa because of atomic coherence between < a1 : and < a2 :, which is generated by the driving fields. We notice that in the above equations the usual decay terms proportional to g 1,2 and g 3,4 are always acompanied by decay interference terms proportional to p 1 g 1g 2 and p 2 g 3g4 , respectively, as long as p 1 / 0 and p 2 / 0. The interference terms might cancel the usual decay terms, which will result in spontaneous emission cancellation. Consequently, the atom takes one of the four CPT states as its stationary state. Substituting the transformations C 1 s eyi D 1 rt a1 b , C 2 s eyi D 2 rt a 2 b , C 3 s eyi D 3 rt a1 c , C4 s eyi D 4 rt a 2 c , C 5 s eyiD rt a1 a2 , C 14 s eyiŽ D 3yD 1 .tr b c , C6 s r a1 a1, C 7 s r a 2 a 2 , C 8 s r b b , and C 9 s C 1) , C 10 s C 2) , C 11 s C 3) , C 12 s C4) , C 13 s C5) , C 15 s C 14) into Ž17. – Ž25., we can write the above equations into a compact vector form
'
'
d dt
ˆ Cˆ s LˆCˆ q I,
'
photon resonance case in a three-level atom interacting with two fields w11,12x. For D1 s D3 s D s , V 1 s V 2 and V 3 s V 4 , the vector Cˆ determined by
Ž 26.
where Cˆ is a column vector whose ith component is Ci and the inhomogeneous term Iˆ is also a column vector with the non-zero components Iˆ3 s yi V 3 , Iˆ11 s Iˆ3) , Iˆ4 s yi V 4 , Iˆ12 s Iˆ4). In Ž26., Lˆ is a time-independent 15 = 15 matrix which explicit expression can easily be derived using Eqs. Ž17. – Ž25.. It can directly be verified that for p1 s p 2 s 1, g 1 s g 2 , g 3 s g4 , V 1 s V 2 and V 3 s V 4 the vectors Cˆ determined by the CPT states < w I :, < w II : and < w V : are the steady-state solutions of Ž26. if D1 s D s and D3 / D s , or, D1 / D s and D3 s D s , or, D1 s D3 s D s , respectively. However, for D1 s D3 / D s , V 1 s V 2 and V 3 s V 4 the vector Cˆ determined by the CPT state
Fig. 2. Steady-state populations in the upper levels versus D3 rg 3 with V 1 s V 2 s 2.0, V 3 s V 4 s 5.0, g 1 sg 2 s 0.5, g 3 sg4 s 1.0, v 12 s 2.0.
F. Li, S.-Y. Zhu r Optics Communications 162 (1999) 155–161
159
Fig. 3. Time evolution of the level populations. The initial state is < c : with V 1 s V 2 s 1.2, V 3 s V 4 s 1.5, g 1 s g 2 s 0.5, g 3 s g4 s 1.0, v 12 s 2.0, D1 s 1.0, D3 s 0.0. Ža. p1 s p 2 s 1; Žb. p1 s 0, p 2 s 1; Žc. p1 s p 2 s 0.
160
F. Li, S.-Y. Zhu r Optics Communications 162 (1999) 155–161
to switch the steady state population of the atom from the lower levels to the upper levels. Now let us see some numerical results. In Fig. 2 the steady-state populations in the upper levels are shown as
functions of D3rg 3 with fixed D1 s 0.5 and p 1 s p 2 s 1.0. The points A Ž D3rg 3 s 0.5. and B Ž D3rg 3 s 1.0. represent two special cases in which the atom is in the CPT states
Fig. 4. Time evolution of level populations. The initial state is: Ža. < c :; Žb. < a1 :; Žc. 0.5 r b b q 0.5 rc c . Other parameters are same as in Fig. 3 but p1 s p 2 s 1.
F. Li, S.-Y. Zhu r Optics Communications 162 (1999) 155–161
between the two limits the upper-level populations can drastically be changed from 0.0 to 2 V 32rŽ2 V 32 q D s2 . or vice versa by varying the detuning D3. By fixing D3 Ž/ D s . and varying the detuning D1 we have the same result. Using this effect, we can transfer the population trapped in < c : Ž< b :. to < b : Ž< c :.. For example, at the point B, the portions of population trapped in < c : and < b : are D s2rŽ2 V 32 q D s2 . and 0.0, respectively. We now change D3 from D s to D1 , and then fix D3 and increase D1 to D s . At the point D1 s D s and D3 / D1 , the atom gets into the CPT state
161
steady state on the initial conditions has been studied by Agarwal w14x. Here, we would like to emphasize that this effect originates from the quantum interference, which decouples the atom from the vacuum modes. In summary, we have found that the four-level atom interacting with two coherent fields has four coherent population trapping states. We have shown that the quantum interference between the spontaneous emission pathways from the upper levels to the lower levels can lead to these states stable against the radiative decay of the upper levels. The atom can take these states as its steady state. Quantum interference strongly delays the relaxation of the atom to these states. We have also noticed that quantum interference can lead to the dependence of the atomic steady state on the initial condition although the atom is coupled to the vacuum modes.
Acknowledgements This work was supported by the Baptise University of Hong Kong. F.L. Li also acknowledges support from the National Natural Science Foundation of China.
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