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Coherent population trapping in multilevel laser-induced continuum structure system
15 May 1997
OPTICS
COM~U~I~ATJO~S Optics Communications 138 (1997) 59-64
ELSEXIER
Coherent population trapping in multilevel laser-induced continuum structure system Gao-xiang Li, Jin-sheng Peng Department
of Physics, Hmzhong
Nomnl
Unit:er.~ity, Wuhff~ 430079, China
Received 16 July 1996; revised 25 November 1996; accepted 14 January 1997
Abstract We have investigated atomic coherent population trapping in a multilevel model of laser-induced continuum structure, in which a quasicontinuum of excited levels to be modeled as the Bixon-Jortner quasicontinuum is laser-embedded into a previously structureless atomic continuum. The condition leading to,atomic coherent population trapping and the coherent population trapping state are given explicitly. The effects of nondegenerate detuning, the Fano factors and the laser intensities on the amount of population trapped in the atomic bound states and the populations distributed in the qu~icontinuum are also examined.
As is known, a strong laser can embed a low-lying atomic bound state into a flat atomic continuum to produce a tunable resonance of adjustable width f 11. Such laser-induced continuum structure (LICS) behaves in many ways as an autoionizin~ state. Because the LICS system provides a suitable scheme for observation of the quantum mechanical interference between transition amplitudes involving the atomic ionization continuum, the properties of LICS systems have drawn considerable attention both theoretically fl,~] and experimeiltally [3,4]. If such a structured continuum is probed from another atomic bound state by a weak laser, then the familiar asymmetric resonance occurs in the ionizing rate against the laser detuning curve [ 1.21, which has been confirmed in recent experiments in atomic sodium and atomic calcium [3,4]. But for the strong-probing case, Knight and his co-workers found that [ 11, if the intensities of two lasers and the nondegenerate two-photon detuning satisfy an approp~ate cond~~on~ the atom can be ionized partially and atomic coherent population trapping occurs. This atomic coherent population trapping is proposed to realize coherent population transfer [5] and elec~omagnetically induced transparency with matched pulses 161. Especially, K~apana~ioti et al. [7] observed autoionization suppression resulting in coherent population trapping due to strong electromagnetic coupling of two autoionizing states in a magnesium atom_ ~30-40~8/97/$17.00
Recently, the LICS model was generalized by Parzynski et al. [8] to a multilevel model of laser-induced continu~lm structure, in which, instead of a single level, a quasicontinuum of excited levels is laser embedded into a previously structureless atomic continuuln and shown to give a richer ionization spectrum. However, the atomic coherent population trapping in the multilevel LICS system has been less considered. In this paper, we focus our attention on the atomic coherent population trapping in a multilevel LICS system. Here the excited levels, which are laser embedded into the structureless continuum, are modeled as the Bixon-Jortner quasicontinuum [&I. The condition resulting in atomic coherent population happing and the coherent population trapping state are given. The effects of the Fano factors, the nondegenerate Raman two-photon detuning and the laser intensities on the amount of population trapped in the atomic bound states and the populations redis~b~Ited in the excited levels are also discussed in the long time limit. The atomic model, sketched in Fig. 1, consists of a ground state 1,g > with frequency wg, a set of structureless continuum states {ir)}, and an’infinite quasi-continnnm of excited levels (ij)} with frequencies wi, which will be modeled as the Bixon-Jortner quasicontinuum [S]. The one-photon transitions Ig) ++ is) and ij) ++ 1~) are driven by the laser fields with frequencies oP and We, respec-
Copyright 0 1997 Elsevier Science B.V. All rights reserved.
PII s0030-4018(97)00052-7
60
Fig. I. Diagram of the atom-field coupling system.
tively. In the rotating-wave approximation, Hamiltonian of the atom-field coupling interaction picture is written as
the interaction system in the
continuum photoexcitation 1101 and Rydberg-atom stabilization in strong laser fields [ll,lZ]. It is an infinite sequence of eqLlidist~t levels, with energy spacing A, which are coupled to the continuum with the same strength independent of the quasicontinuum index j. In fact, for a highly-excited Rydberg atom, two assumptions such as (i) constant bound-free coupling matrix elements, V,, = lh~;~/‘, and (ii) approxima~o~ of equidistant levels, E, = ER, + (n - n&z;3 can be regarded as reasonable [ 121. In this case, the nondeg~nerate Raman two-photon detunings Agj can be expressed as Agj = 8, - jA, with j = 0, f I, f 3_, _. . , i. 05,here 6, refer to the Raman two-photon detuning related &o the transitions [g) --) {IS)} ---)IO), and the Fano asymmetric factors 4, and gjjT and the photoionization rate p; can be assumed to be independent of the state index j, i.e., qgj = q, qjjr = qo, rj = ro. Under these assumptions, solving Eqs. (3) and (4) gives
where A,, =E-Wg-Wp and Ajc=~-~j-~d are one-photon detunings, VgB and VjS twe the matrix elements of the atom-field interaction. If the atom is initially in its ground state Ig), then in the interaction picture, the state vector of the atom-field coupling system at time t develops into IV!“(t))
= C,lg)
+
I’
+ CCje-‘“ri”lj)
d& C, e-‘A~erlc),
=i(~/A)cot[(n,‘A)(is-6,)f.
w
where A,i = Aj, - A,, is just the nondegenerate Raman two-photon detuning. Using the Laplace form of the SchriSdinger equation in the interaction picture and the pole approximation, we find that the amplitudes of the discrete levels obey
Generally, Eq. (3) shows that the solutions of F(s) = 0 are complex, so the atom initially in the ground state lg> will be ionized completely under the interaction of two Iaser fields. In order to avoid the atom to be ionized completely, one root of F(s) = 0 must be reduced to pure . . Imaginary, I.e., s = iL;;‘.Since fliE) = -ig(E), here g(E) = X7= _-li I,/@ + Atj) is a real function, the equation F(iE) = 0 can be rewritten as px f g(QOrg(qo
- 2q) f J-%(E)~o = 0,
E+Eg(E)r,q,-q2r,rog(E)=0.
(7) (8)
The solutions of the above equations are given as
Were s is the Laplace variable, 4 and 4 are the Laplace transforms of C, and Ci, rg and rj represent the photoionization rates of the states 1g ) and lj) respectively, qSj and qjjP are the Fano asymmetric factors associated with the two-photon tr~sitions 18) cf (/s)} ++ ij) and /j) ++ {is)> c-, Ij’), respectively. For simplicity of the calculation, we model the quasicontinuum (Ij)} as a Rixon-Jortner quasicontinuum [9], which is a very popular structure particularly in the theories of in~amolecuIar processes 191, quasi-
Noticing the relation between that if yg. t-o and 6, satisfy % = (A/~)~ctan[(~/A)(~
f(iE)
- &PO]
and g(E),
- Y, $
we find
P)
then s = iqrs is just a pure imaginary root of PCS(S) = 0. In this case, the atom initially in its ground state / g > can only be ionized partially and atomic cohere& population trapping in the bound states occurs. Knight et al. [I] pointed
G. Li, J. Peng / Optics Communications
out that in the standard LICS system, in which the atom has only one excited level IO), the condition resulting in atomic coherent population trapping is 6, = q(r, - r,) which is different from Eq. (10). This is because the atomic excited states we consider here are an infinite family of equidistant states with the frequency spacing A and there are two types of Raman two-photon transition processes, one is the nondegenerate two-photon transition process / g) t-* (1E)} CJ{I j)}, which is characterized by the Fano factor q, and the other one is the degenerate two-photon transition process Ii) c-) {E)} ++ 1j), which is characterized by q,,. This leads to the condition 10 to be more complicated than that in the standard LICS system. Now we turn to the discussion of the coherent population trapping state and the amount of population trapping in the bound states. Because s = iqr8 is a pole of both cg and c,, the bound states of this model trap some amount of population, even on the long-time scale t 4 a, preventing the atom from being completely ionized. The long-time asymptotics of discrete-state population amplitudes, found by the method of residues, are
138 (19971 59-64 100
61
*
rP
0 80
1
0.61
-
Fig. 2. Population P, trapped in the atomic bound states versus r0 for different detunings 6,, 4 = 2 and q0 = 1. Solid line: 6, = 0, long dashed line: 6,/A = -0.005, short dashed line: 6, /A = - 0.05, and dotted line: 6, /A = - 0.5.
1 ~,(~)
= I +(rJra)[l
+(~z/A2)(q-qa)Zr,2]
' (11)
Therefore, the amount of population trapped in the coherent population trapping state Iqr,> that will not be ionized reads as 1
'jC30)
“=
+G&-40)
(15)
1+(r,/r0)[1+(n2/A2)(q-q0)2r~]
=(4~~
+
6, -jA) 1
'
’
1+(r,/ro)[l+(.ir2/A2)(q-q~)2~~]
(12) The above amplitudes allow us to construct the state vector of the atom, which survived after t + ~0, as
Certainly, here we only consider the nondegenerate twophoton Raman coupling between the ground state 1g > and the quasicontinuum (1j)} via the continuum {Is)}. If the two-photon Raman couplings between 1g) and (1j)} through other channels are included, these additional cou-
1 00
1 lW(a))
=
1 + ( rg/ra) [ 1 + (n2/A’)(q
0.80
- ~a)2’r;]
pt 1
,;,;,;~~“““~-,--~-‘.‘..~~
/d,
’
-
,
\
\
/’ //
1
\
\
1
\
14
\ \ I i /
(13)
0.60
/ /
-
I I /
In the view of the dressed-state analysis, the pole of s = iqr, corresponds to the coherent population trapping state decoupled from the continuum and thus unionized. This coherent population trapping state lqr,>, which survived after t + ~0, can be found as
/ / , i / i
\ jj
IT*) =
1
._ /
-4.0
1+(~~/ro)[l+(~2/A2)(q-q,)2r02]
(14)
I
,
-2.0
I
,
0.0
I
/
I,
I
2.0
I
/
,
4.0
Fig. 3. Pt versus r0 for different 4, 6, /A = -0.005 and q0 = 1. Solid line: 4 = 2, long dashed line: 4 = 1.1, short dashed line: 4 = 1.01, and dotted line: 4 = 1.001.
G. Li, J. Peng/
62
Optics Communications
plings would give rise to a reduction of the achievable coherent population trapping as pointed out by Dai and Lambropoulos [13]. But here for simplicity, we have neglected the effect of these couplings. Evidently, this population trapping results in reduction of the long-time ionization probability from 1 to 1 - Pt. Because the photoionization rates rS and ~a are correlated by Eq. (lo), the atomic population P, trapped in IT,) in the steady-state case is determined by the Fano asymmetric factors 4 and qO, the nondegenerate two-photon detuning I?,,, and the photoionization rate rr,. Fig. 2 illustrates P, versus r0 for different 6,. When 8, = 0, which means that the nondegenerate two-photon transition 18) t) {IS)} @ IO) is resonant, P,
0 25
(a)
I38 (1997) 59-64
decreases with increasing rO. But for the nonresonant case, with increasing rO, the amount of the population trapped in the bound states increases in the weak intensity region of the laser with frequency wd and decreases in the strong intensity region of the laser wd. Fig. 2 also shows that the nondegenerate two-photon detuning 6, has a reduction effect on the value of P,. Fig. 3 gives the illustration of P, for different q - qO. As we see, the decrease of q - q,, has an enhancing effect on P,. We find that when q - q0 is very small, the atom initially in its ground state / g ) can be nearly trapped in its bound state 1g ) and i/j)} not to be ionized in the strong intensity region of the laser od. According to Eqs. (11) and (12) the entire trapped
(b)
0 25
w i
Fig. 4. Populations Pj in the states lj> versus r,, for different detunings S,, q = 2 and q,, = 1. (a) 6,/A = 0. Solid line: j = 0, long dashed line: j = 1, short dashed line: j = - 1, and dotted line: j = 2. (b) Same as (a) except for 6,/A = -0.005. (c) Same as (a) except for 6,/A = - 0.5.
63
G. Li, .I. Peng/ Optics Communications 138 (1997) 59-64 population is distributed in the following form
over the bare atomic bound states
Fig. 4 shows the trapped population distributed in the bare states jO), 1Jr 1) and /Z> for different detunings 8,. As we see from Fig. 4, the trapped population occupied in the quasicontinuum {lj)} IS mainly distributed in the states IO), 1_t I) and 12) neighbouring the state IO). And the population PO is sufficiently larger than P, 1 and P2 when rO/A < 1. That is to say, the atomic population which is transferred in the quasicontinuum {Ij)} is dominantly distributed in the state IO). Fig. 4b (6,/A = - 0.005) shows that a small detuning 6, has an enhancing effect on the population trapped in the state 10) when ro/A is chosen as lo-’ < ro/A < 1. But large detuning (8,/A = -0.5) has a reduction effect on PO as shown in Fig. 4~. However Fig. 4c also indicates that for the large detuning case, the most population trapped in the coherent ~p~lation state 13Fr,) is transferred into the state IO) and the remaining population Pg is very small compared to PO. Fig. 5 illustrates P,,, P, 1 and P, for different 4 - q,,. Comparing Fig. 5 with Fig- 4b, we fiud that with decreasing 4 - qO, the population P, exhibits a distinct asymmetric two-peak distribution with increasing ~a. The smaller the value of q - q,,, the more distinct the two-peak distribution. Fig. 5 also indicates that with decreasing q - qo, the population P, in the state 11) increases significantly in the strong intensity region of the laser wd and can nearly be coincident with the right peak of Fe. That is to say, for small 9 - qo, the trapped population P, is redis~buted in both the state IO) and /I > equally when the laser field wd is very strong. In conclusion, we have studied the atomic coherent population trapping in a multilevel model of laser-induced continuum structure. We find that in order to prevent the atom intially in its ground state 18) from being ionized completely, the nondegenerate Raman two-photon detuning and the intensities of two laser fields must obey Eq. (10). The nondegenerate two-photon detuning has a reduction effect on the amount of population trapped in the atomic bound states. Small detuning can enhance the population in the state IO> explictly and for large detuning, the most population trapped in the coherent population state I’&) is distributed in lo>. We also find that decreasing the
(a)
Pi in the states Ij) versus r0 for different q, and q,,=l. (a) q=l.l. Solid line: j=O, long dashed line: j = 1, short dashed line: j = - 1, and dotted line: j= 2. (b) Same as (a) except for q = 1.011.. Fig. 5. Populations
&,/A=-0.005
value of q-q0 can increase the amount of population trapped in lYr,). For small q - qo, the population PO exhibits a distinct asymmetric two-peak ~s~bution against the photoionization late rO, and in the strong intensity region of the laser wd, the trapped population P, is almost distributed in both the states IO) and il} equally.
Acknowledgements One of the authors (J.S. Peng) would like to thank the International Atomic Energy Agency and UNISCO for hospitality at the International Center for Theoretical Physics, Trieste, Italy. G.X. Li wishes to thank Dr. I?. Zhou for his kind help. This work is supported by the National Natural Science Fo~dation of China and the National Natural Science Foundation of Hubei Province, China.
64
G. Li, J. Peng / Optics Co~~unicat~o~s 138 (199715944
References [ll P.E. Coleman, P.L. Knight and K. Burnett, Optics Comm. 42 (1982) 171; P.L. Knight, Comments At. Mol. Phys. 15 (1984) 193; P.L. Knight, M.A. Lauder and B.S. Dalton, Phys. Rep. 190 (1990) 1. [Z] J. Zhang and P. Lambropoulos, Phys. Rev. A 45 (1992) 489; T. Nakajima, P. Lambropoulos, S. Cavalieri and M. Matera, Phys. Rev. A 46 (1992) 7315; S. Cavalieri, R. Era,, R. Buffa and M. Matera, Pbys. Rev. A 51 (1995) 2974; G.X. Li and 3.S. Peng, Phys. Rev. A 52 (199.5) 465; Optics Comm. 123 (1996) 94; Phys. Lett. A 218 (1996) 41. [3] Y.L. Shao, D. Ch~a~ambidis, C. Fotakis, J. Zhang and P. Lambropoulos, Phys. Rev. Len. 67 (1991) 3669; 0. Faucher, D. Charahunbidis, C. Fotakis, J. Zhang, and P. Lambropou10s Phys. Rev. Lett. 70 (1993) 3004; 0. Faucher, Y.L. Shao, D. Charalambidis and C. Fotakis, Phys. Rev. A 50 (1994) 641. [4] S. Cavalieri, F.S. Pavone and M. Matera, Phys. Rev. Lett. 67 (1991) 3673; S. Cavalieri, R. Eramo and L. Fini, J. Phys. B 28 (1995) 1793; B 28 (1995) L637.
151 T. Nakajima, M. Elk, J. Zhang and P. Lambropouios, Pbys. Rev. A 50 (1994) R913. 161 S.J. van Enk, 5. Zhang and P. Lambropoulos, Phys. Rev. A 50 (1994) 2777. [?‘I N.E. Karapanagioti, 0. Faucher, Y.L. Shao, D. Charalambidis, H. Bachau and E. Cormier, Phys. Rev. Lett. 74 (1995) 2431. [81 R. Parzynski, J. Schmidt and A. Wojcik, J. Opt. Sot. Am. B 11 (1994) 644; R. Parzynski, A. Wojcik and J. Schmidt, Phys. Rev. A 50 (1994) 3285. [9] M. Bixon and J. Jortner, J. Chem. Phys. 48 (1968) 715. [lo] J.H. Eberly, J.J. Yeh and C.M. Bowden, Chem. Phys. Len. 86 (1982) 76; E. Kyrola and J.H. Eberly, f. Chem. Phys. 82 (1985) 1841; P.M. Radmore, J. Mod. Optics 42 (1995) 579. [ll] A. Wojcik and R. Parznski, Phys. Rev. A 50 (1994) 2475; J. Opt. Sot. Am. B 12 (1995) 369. [12] M.V. Fedrov and A.M. Movesian, J. Opt. Sot. Am. B 6 (1989) 928. [13] Bo-nian Dai and P. Lambropoulos, Phys. Rev. A 36 (1987) 5205.
OPTICS
COM~U~I~ATJO~S Optics Communications 138 (1997) 59-64
ELSEXIER
Coherent population trapping in multilevel laser-induced continuum structure system Gao-xiang Li, Jin-sheng Peng Department
of Physics, Hmzhong
Nomnl
Unit:er.~ity, Wuhff~ 430079, China
Received 16 July 1996; revised 25 November 1996; accepted 14 January 1997
Abstract We have investigated atomic coherent population trapping in a multilevel model of laser-induced continuum structure, in which a quasicontinuum of excited levels to be modeled as the Bixon-Jortner quasicontinuum is laser-embedded into a previously structureless atomic continuum. The condition leading to,atomic coherent population trapping and the coherent population trapping state are given explicitly. The effects of nondegenerate detuning, the Fano factors and the laser intensities on the amount of population trapped in the atomic bound states and the populations distributed in the qu~icontinuum are also examined.
As is known, a strong laser can embed a low-lying atomic bound state into a flat atomic continuum to produce a tunable resonance of adjustable width f 11. Such laser-induced continuum structure (LICS) behaves in many ways as an autoionizin~ state. Because the LICS system provides a suitable scheme for observation of the quantum mechanical interference between transition amplitudes involving the atomic ionization continuum, the properties of LICS systems have drawn considerable attention both theoretically fl,~] and experimeiltally [3,4]. If such a structured continuum is probed from another atomic bound state by a weak laser, then the familiar asymmetric resonance occurs in the ionizing rate against the laser detuning curve [ 1.21, which has been confirmed in recent experiments in atomic sodium and atomic calcium [3,4]. But for the strong-probing case, Knight and his co-workers found that [ 11, if the intensities of two lasers and the nondegenerate two-photon detuning satisfy an approp~ate cond~~on~ the atom can be ionized partially and atomic coherent population trapping occurs. This atomic coherent population trapping is proposed to realize coherent population transfer [5] and elec~omagnetically induced transparency with matched pulses 161. Especially, K~apana~ioti et al. [7] observed autoionization suppression resulting in coherent population trapping due to strong electromagnetic coupling of two autoionizing states in a magnesium atom_ ~30-40~8/97/$17.00
Recently, the LICS model was generalized by Parzynski et al. [8] to a multilevel model of laser-induced continu~lm structure, in which, instead of a single level, a quasicontinuum of excited levels is laser embedded into a previously structureless atomic continuuln and shown to give a richer ionization spectrum. However, the atomic coherent population trapping in the multilevel LICS system has been less considered. In this paper, we focus our attention on the atomic coherent population trapping in a multilevel LICS system. Here the excited levels, which are laser embedded into the structureless continuum, are modeled as the Bixon-Jortner quasicontinuum [&I. The condition resulting in atomic coherent population happing and the coherent population trapping state are given. The effects of the Fano factors, the nondegenerate Raman two-photon detuning and the laser intensities on the amount of population trapped in the atomic bound states and the populations redis~b~Ited in the excited levels are also discussed in the long time limit. The atomic model, sketched in Fig. 1, consists of a ground state 1,g > with frequency wg, a set of structureless continuum states {ir)}, and an’infinite quasi-continnnm of excited levels (ij)} with frequencies wi, which will be modeled as the Bixon-Jortner quasicontinuum [S]. The one-photon transitions Ig) ++ is) and ij) ++ 1~) are driven by the laser fields with frequencies oP and We, respec-
Copyright 0 1997 Elsevier Science B.V. All rights reserved.
PII s0030-4018(97)00052-7
60
Fig. I. Diagram of the atom-field coupling system.
tively. In the rotating-wave approximation, Hamiltonian of the atom-field coupling interaction picture is written as
the interaction system in the
continuum photoexcitation 1101 and Rydberg-atom stabilization in strong laser fields [ll,lZ]. It is an infinite sequence of eqLlidist~t levels, with energy spacing A, which are coupled to the continuum with the same strength independent of the quasicontinuum index j. In fact, for a highly-excited Rydberg atom, two assumptions such as (i) constant bound-free coupling matrix elements, V,, = lh~;~/‘, and (ii) approxima~o~ of equidistant levels, E, = ER, + (n - n&z;3 can be regarded as reasonable [ 121. In this case, the nondeg~nerate Raman two-photon detunings Agj can be expressed as Agj = 8, - jA, with j = 0, f I, f 3_, _. . , i. 05,here 6, refer to the Raman two-photon detuning related &o the transitions [g) --) {IS)} ---)IO), and the Fano asymmetric factors 4, and gjjT and the photoionization rate p; can be assumed to be independent of the state index j, i.e., qgj = q, qjjr = qo, rj = ro. Under these assumptions, solving Eqs. (3) and (4) gives
where A,, =E-Wg-Wp and Ajc=~-~j-~d are one-photon detunings, VgB and VjS twe the matrix elements of the atom-field interaction. If the atom is initially in its ground state Ig), then in the interaction picture, the state vector of the atom-field coupling system at time t develops into IV!“(t))
= C,lg)
+
I’
+ CCje-‘“ri”lj)
d& C, e-‘A~erlc),
=i(~/A)cot[(n,‘A)(is-6,)f.
w
where A,i = Aj, - A,, is just the nondegenerate Raman two-photon detuning. Using the Laplace form of the SchriSdinger equation in the interaction picture and the pole approximation, we find that the amplitudes of the discrete levels obey
Generally, Eq. (3) shows that the solutions of F(s) = 0 are complex, so the atom initially in the ground state lg> will be ionized completely under the interaction of two Iaser fields. In order to avoid the atom to be ionized completely, one root of F(s) = 0 must be reduced to pure . . Imaginary, I.e., s = iL;;‘.Since fliE) = -ig(E), here g(E) = X7= _-li I,/@ + Atj) is a real function, the equation F(iE) = 0 can be rewritten as px f g(QOrg(qo
- 2q) f J-%(E)~o = 0,
E+Eg(E)r,q,-q2r,rog(E)=0.
(7) (8)
The solutions of the above equations are given as
Were s is the Laplace variable, 4 and 4 are the Laplace transforms of C, and Ci, rg and rj represent the photoionization rates of the states 1g ) and lj) respectively, qSj and qjjP are the Fano asymmetric factors associated with the two-photon tr~sitions 18) cf (/s)} ++ ij) and /j) ++ {is)> c-, Ij’), respectively. For simplicity of the calculation, we model the quasicontinuum (Ij)} as a Rixon-Jortner quasicontinuum [9], which is a very popular structure particularly in the theories of in~amolecuIar processes 191, quasi-
Noticing the relation between that if yg. t-o and 6, satisfy % = (A/~)~ctan[(~/A)(~
f(iE)
- &PO]
and g(E),
- Y, $
we find
P)
then s = iqrs is just a pure imaginary root of PCS(S) = 0. In this case, the atom initially in its ground state / g > can only be ionized partially and atomic cohere& population trapping in the bound states occurs. Knight et al. [I] pointed
G. Li, J. Peng / Optics Communications
out that in the standard LICS system, in which the atom has only one excited level IO), the condition resulting in atomic coherent population trapping is 6, = q(r, - r,) which is different from Eq. (10). This is because the atomic excited states we consider here are an infinite family of equidistant states with the frequency spacing A and there are two types of Raman two-photon transition processes, one is the nondegenerate two-photon transition process / g) t-* (1E)} CJ{I j)}, which is characterized by the Fano factor q, and the other one is the degenerate two-photon transition process Ii) c-) {E)} ++ 1j), which is characterized by q,,. This leads to the condition 10 to be more complicated than that in the standard LICS system. Now we turn to the discussion of the coherent population trapping state and the amount of population trapping in the bound states. Because s = iqr8 is a pole of both cg and c,, the bound states of this model trap some amount of population, even on the long-time scale t 4 a, preventing the atom from being completely ionized. The long-time asymptotics of discrete-state population amplitudes, found by the method of residues, are
138 (19971 59-64 100
61
*
rP
0 80
1
0.61
-
Fig. 2. Population P, trapped in the atomic bound states versus r0 for different detunings 6,, 4 = 2 and q0 = 1. Solid line: 6, = 0, long dashed line: 6,/A = -0.005, short dashed line: 6, /A = - 0.05, and dotted line: 6, /A = - 0.5.
1 ~,(~)
= I +(rJra)[l
+(~z/A2)(q-qa)Zr,2]
' (11)
Therefore, the amount of population trapped in the coherent population trapping state Iqr,> that will not be ionized reads as 1
'jC30)
“=
+G&-40)
(15)
1+(r,/r0)[1+(n2/A2)(q-q0)2r~]
=(4~~
+
6, -jA) 1
'
’
1+(r,/ro)[l+(.ir2/A2)(q-q~)2~~]
(12) The above amplitudes allow us to construct the state vector of the atom, which survived after t + ~0, as
Certainly, here we only consider the nondegenerate twophoton Raman coupling between the ground state 1g > and the quasicontinuum (1j)} via the continuum {Is)}. If the two-photon Raman couplings between 1g) and (1j)} through other channels are included, these additional cou-
1 00
1 lW(a))
=
1 + ( rg/ra) [ 1 + (n2/A’)(q
0.80
- ~a)2’r;]
pt 1
,;,;,;~~“““~-,--~-‘.‘..~~
/d,
’
-
,
\
\
/’ //
1
\
\
1
\
14
\ \ I i /
(13)
0.60
/ /
-
I I /
In the view of the dressed-state analysis, the pole of s = iqr, corresponds to the coherent population trapping state decoupled from the continuum and thus unionized. This coherent population trapping state lqr,>, which survived after t + ~0, can be found as
/ / , i / i
\ jj
IT*) =
1
._ /
-4.0
1+(~~/ro)[l+(~2/A2)(q-q,)2r02]
(14)
I
,
-2.0
I
,
0.0
I
/
I,
I
2.0
I
/
,
4.0
Fig. 3. Pt versus r0 for different 4, 6, /A = -0.005 and q0 = 1. Solid line: 4 = 2, long dashed line: 4 = 1.1, short dashed line: 4 = 1.01, and dotted line: 4 = 1.001.
G. Li, J. Peng/
62
Optics Communications
plings would give rise to a reduction of the achievable coherent population trapping as pointed out by Dai and Lambropoulos [13]. But here for simplicity, we have neglected the effect of these couplings. Evidently, this population trapping results in reduction of the long-time ionization probability from 1 to 1 - Pt. Because the photoionization rates rS and ~a are correlated by Eq. (lo), the atomic population P, trapped in IT,) in the steady-state case is determined by the Fano asymmetric factors 4 and qO, the nondegenerate two-photon detuning I?,,, and the photoionization rate rr,. Fig. 2 illustrates P, versus r0 for different 6,. When 8, = 0, which means that the nondegenerate two-photon transition 18) t) {IS)} @ IO) is resonant, P,
0 25
(a)
I38 (1997) 59-64
decreases with increasing rO. But for the nonresonant case, with increasing rO, the amount of the population trapped in the bound states increases in the weak intensity region of the laser with frequency wd and decreases in the strong intensity region of the laser wd. Fig. 2 also shows that the nondegenerate two-photon detuning 6, has a reduction effect on the value of P,. Fig. 3 gives the illustration of P, for different q - qO. As we see, the decrease of q - q,, has an enhancing effect on P,. We find that when q - q0 is very small, the atom initially in its ground state / g ) can be nearly trapped in its bound state 1g ) and i/j)} not to be ionized in the strong intensity region of the laser od. According to Eqs. (11) and (12) the entire trapped
(b)
0 25
w i
Fig. 4. Populations Pj in the states lj> versus r,, for different detunings S,, q = 2 and q,, = 1. (a) 6,/A = 0. Solid line: j = 0, long dashed line: j = 1, short dashed line: j = - 1, and dotted line: j = 2. (b) Same as (a) except for 6,/A = -0.005. (c) Same as (a) except for 6,/A = - 0.5.
63
G. Li, .I. Peng/ Optics Communications 138 (1997) 59-64 population is distributed in the following form
over the bare atomic bound states
Fig. 4 shows the trapped population distributed in the bare states jO), 1Jr 1) and /Z> for different detunings 8,. As we see from Fig. 4, the trapped population occupied in the quasicontinuum {lj)} IS mainly distributed in the states IO), 1_t I) and 12) neighbouring the state IO). And the population PO is sufficiently larger than P, 1 and P2 when rO/A < 1. That is to say, the atomic population which is transferred in the quasicontinuum {Ij)} is dominantly distributed in the state IO). Fig. 4b (6,/A = - 0.005) shows that a small detuning 6, has an enhancing effect on the population trapped in the state 10) when ro/A is chosen as lo-’ < ro/A < 1. But large detuning (8,/A = -0.5) has a reduction effect on PO as shown in Fig. 4~. However Fig. 4c also indicates that for the large detuning case, the most population trapped in the coherent ~p~lation state 13Fr,) is transferred into the state IO) and the remaining population Pg is very small compared to PO. Fig. 5 illustrates P,,, P, 1 and P, for different 4 - q,,. Comparing Fig. 5 with Fig- 4b, we fiud that with decreasing 4 - qO, the population P, exhibits a distinct asymmetric two-peak distribution with increasing ~a. The smaller the value of q - q,,, the more distinct the two-peak distribution. Fig. 5 also indicates that with decreasing q - qo, the population P, in the state 11) increases significantly in the strong intensity region of the laser wd and can nearly be coincident with the right peak of Fe. That is to say, for small 9 - qo, the trapped population P, is redis~buted in both the state IO) and /I > equally when the laser field wd is very strong. In conclusion, we have studied the atomic coherent population trapping in a multilevel model of laser-induced continuum structure. We find that in order to prevent the atom intially in its ground state 18) from being ionized completely, the nondegenerate Raman two-photon detuning and the intensities of two laser fields must obey Eq. (10). The nondegenerate two-photon detuning has a reduction effect on the amount of population trapped in the atomic bound states. Small detuning can enhance the population in the state IO> explictly and for large detuning, the most population trapped in the coherent population state I’&) is distributed in lo>. We also find that decreasing the
(a)
Pi in the states Ij) versus r0 for different q, and q,,=l. (a) q=l.l. Solid line: j=O, long dashed line: j = 1, short dashed line: j = - 1, and dotted line: j= 2. (b) Same as (a) except for q = 1.011.. Fig. 5. Populations
&,/A=-0.005
value of q-q0 can increase the amount of population trapped in lYr,). For small q - qo, the population PO exhibits a distinct asymmetric two-peak ~s~bution against the photoionization late rO, and in the strong intensity region of the laser wd, the trapped population P, is almost distributed in both the states IO) and il} equally.
Acknowledgements One of the authors (J.S. Peng) would like to thank the International Atomic Energy Agency and UNISCO for hospitality at the International Center for Theoretical Physics, Trieste, Italy. G.X. Li wishes to thank Dr. I?. Zhou for his kind help. This work is supported by the National Natural Science Fo~dation of China and the National Natural Science Foundation of Hubei Province, China.
64
G. Li, J. Peng / Optics Co~~unicat~o~s 138 (199715944
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