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Exact results for a model of a three-level atom
Volume 107A, number 4
28 January 1985
PHYSICS LETTERS
EXACT RESULTS FOR A MODEL OF A THREE-LEVEL ATOM N.N. BOGOLUBOV Jr., FAM LE KIEN and A.S. SHUMOVSKY Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, P.O. Box 79, 101000, Moscow, USSR Received 7 November 1984
For a three-level atom interacting with two modes of the quantized radiation field the time dependence of transition operators, photon amplitudes, and of operators of level populations and photon numbers is examined rigorously. It is shown that the availability of a mode detuning leads to the splitting of the “soft branch” of frequencies of two-photon Rabi oscillations.
The problem of dynamics of three-level atoms interacting with the electromagnetic field has been the subject of gained active research for the last ten years (see refs. [l- 1l] and references therein). It is central to discussions of two-photon coherence [ 1,2], resonance Raman scattering and double-resonance processes [3], three-level superradiance [4], twomode lasers [S], three-level echoes [6], etc. A number of recent papers [5,7- 1l] has been dedicated to a careful consideration of the problem of the dynamics of a single three-level atom interacting with two resonant modes of the radiation field. The semiclassical formalism for the treatment of this problem has been discussed in refs. [7-91. In another series of articles [5,10,1 l] the fully quantized theory was studied. Exact Schrijdinger wavefunctions were obtained for some special initial states [5]. In ref. [lo] the explicit expression of the evolution operator in the interaction picture was derived. The rigorous examinations of the dynamical behaviour of level populations and photon numbers have been realized in the Heisenberg picture [ 1I]. In the present letter we shall derive the time dependence of atomic transition operators and photon amplitudes as an exact solution of the equations of motion, taking into account a detuning of near-resonant modes. We shall discuss also some interesting consequences of this solution. We consider now a three-level atom (see fig. 1) in which nonzero dipole moments exist only between
f t
-_
__ 3
w--m -L
*i
wz
2
I
Fig. 1. Energy-level and transition structure of the model considered.
levels 1 and 3, and 2 and 3. Let the atom be at rest in a lossless cavity and interact with two modes of the quantized radiation field. The model hamiltonian of the system under consideration is [ 10,l l] H=HA+HF+HAF.
(1)
Here HA, HF describe the free atom and free field, respectively, and HAP describes the atom-field interaction in the dipole and rotating wave approximations 3
2
(2) The operator Rji -= lj)
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28 January 1985
PHYSICS LETTERS
the atomic transition from level j to level i. The vectors of atomic eigenstates Ii>, j = 1,2,3, form the basis of the state space of the three-level atom
system [ 131. From (7) we obtain for Ra3 the following equation, { [d/dt + i(o,
- C - A)] [d/dt + i(w,
- C)]
3 H~Jj)=Aajlj),
Cili)=Sfj,
C
1.
li)
(3)
j=l
+ h; +g;}Ra3
= 0.
(9)
Its solution can be presented in the following form: The operators Rji (i, j = 1, 2, 3) obey the following relations [4,1 l] :
Ra3(t) = exp[i(C+ X {cos[t(X2
& Rii = 1 . j=l
Rii Rkl = Rilsjk 9
The parameters ga! are the constants coupling. It can be shown that the operators
of atom-mode
,
+ b:R&
- A(R,,
+R,,)
>
(5)
commute with the hamiltonian (1) and with each other. They are therefore constant operators. Let us introduce the following auxiliary operators: V, = g$,R,,
where h2 = Xi + &A2. From the equation for the photon amplitudes db,/dt
= -iw,b,
dl/,/dt
t i(w,
- C)Ra3 = -iv, - C - A)V,
,
combined with expression (10) one can obtain the explicit time dependence for b,(t). After a number of cumbersome transformations this dependence can be represented in the form {b,(O) t B_, t B,
- exp{-i[iA
- (A2 +g2)lj2] a
- exp{-i[:A
t (A2 +g2)‘12] t}B a } ’ 01
t}B_ OL
where for simplicity we have introduced
(11) the notation
B ra! = ga { [C t : A f (X2 + g;)1’2] [2(h2 + g;)1’2]}-’ X {[(X2 +g;)1’2
7 V,(O)} .
(12)
P2, V,l = -g:V, .
(13)
T :A] Ra3(0)
[~2,Ra31= -g:Ra3,
So, for an arbitrary operator function f(h2)Ro,3
+ i(w,
- ig,R,,
of motion
(6)
where&=2ifcr= 1 andE= lifcr=2.Using(4)one can obtain now the equations of motion for the operators R,, and V,, dR,,/dt
(10)
It should be noted that
3
+ g&R,,
Ra3(0) - i(A2 t gi)-‘/2
b,(t) = exp[i(C + A - o,)t]
2 C= &FL g,(b,R,,
+g$1/2]
Xsin[t(h2+g~)1/2][VQ(0)+fAR,3(O)]},
a3 - s1, - w1 = S13 - a2 - o2 = A.
- R,,
o,)t]
(4)
The operators bi, b, (a = 1,2) are the photon amplitudes (or so-called creation and annihilation operators) corresponding to the mode with the energy Ao, of the quantized radiation field. We shall consider the case of exact two-photon resonance [8]
Ma = b;b,
iA-
,
= -i(X$ + gz)Ra3
= &3f
f(h2) we have
(A2- & 7
f02Ya = Vaf02 -g:> * ,
(14)
(7) Then, expression (10) can be rewritten in the form
where ho is determined
by Ra3(t) = exp[i(C+
:A - o,)t]
(Ra3(0)
cos Xt
hi = x; + x,2 ) - i[V,(O) t fAR,3(0)]x-1 x, Eg&MT
o= 1,2.
(8)
The operators X, can be considered as the quantum expressions of the one-photon Rabi frequencies in the 174
sin At}.
(15)
Analogously, with the help of eq. (14) we can obtain from expressions (11) and (12)
b,(t) = exp[i(C + A - w,)t] - B_,
exp(ih_t)
28 January 1985
PHYSICS LETTERS
Volume 107A, number 4
-B,
[b,(O) + B_,
exp(-iX+t)]
/~+a= W)a{PMO) - R,,(O)l~:~;~~~~,)-’
+ B,
,
(16)
+glg2BM~f - 4>W$J1 f &C2@)
where B Ta =&-JR,3(0)&
T
y-#>lPMC+~*)1-' 3 (17)
and the operators h,, X_ are determined
by
P,, = GOa{&lA1(0) and
Expressions (15) and (16) represent our main results. They describe the time dependence of the atomic transition and photon operators. Using these results we can calculate various correlation functions and averages of photon numbers, level populations and atomic dipole moments. Below, we shall consider some.consequences. First, we note that the model of a two-level atom interacting with a single mode radiation field is a particular case of the model considered here. In this case we have either g2 = 0 or g1 = 0, and expressions (15), (16) can then be reduced to the results of ref. [12]. For the third-level population operator R,,(t), from (15) and the relationRg3 = RSaRor3, we get
D zi(blb2R12
2ht - 1) t LJsin 2ht] tRs3(0).(18)
Here u = I&(O)
+ &2(O)
- X&(O)
+gl@(O)
+~A[g1C1(0)+g2C2(0)1}(2X2)-1 , IJ=-klAl@) +~2~2031w-1
,
(19)
and C,=bLR,, B = b;b2R12
A, =i(bkRa3
+b,Rj,, + b,b;R,,
- b&R,,),
.
(20)
From eq. (16) we obtain for the photon-number operator N, E b! b, , N,(t)
= ~r,(cos h+t - 1) t j3, sin h+t
+ p_,(cos + u,(cos
A-t - 1) + /3_, sin X-t 2ht - 1) t ua! sin 2Xt t N,(O),
where u, = (&l@~
3
uol = (X,/X&,
7
- &2A2(W%$1 (22)
f g~g2Nw~~*)-11 ,
X,=h?;A.
RjS(f) = -[u(cos
- A&,c, @)Im&-9
(21)
- blblR21).
Using the conservation law (5): N,(t) = R&t) t Ma and the relation R,,R,, = R,,, from (21) we obtain: R&t)
= pJcos
+ p_,(cos
h+t - 1) t /I, sin X+t
h-t - 1) + p_, sin X-t
+ u,(cos 2ti - 1) t uol sin 2ti t R,,(O)
,
(23)
and also N:(t)
= M,” + [(Ma + l)m - M,m]R,,(t),
(24)
where 01= 1, 2 and m is an arbitrary natural number (m = 1,2, 3, . ..). Expression (24) is useful for the investigation of photon statistics (see ref. [ 151). It is easily seen that the operators X,, X_, and 2h define the frequencies of two-photon Rabi oscillations of level populations and photon numbers in the system [7-l I]. At one-photon resonance A = 0 expressions (18), (2 1) and (23) coincide with the results of ref. [ 111. In this case there are two branches of the twophoton Rabi frequencies defined by the operators X and 2X [ 111. It should be noted that the existence of the “soft branch” h is a characteristic feature of the three-level system. Such a kind of oscillation frequencies is absent in the two-level system [ 13,141. Our present results (18), (21) and (23) show that the detuning A in the case of two-photon resonance leads to the splitting of the “soft branch” h to two branches characterized by the frequency operators h, = h t f A and X_ = A - $A. This conclusion of the fully quantized theory is in accordance with the results of the semiclassical theory [ 91. Thus, in this paper we derived in the Heisenberg picture the explicit expressions of the time dependence for the atomic transition operators, photon amplitudes, and also for the level population and photon-number 175
Volume 107A, number 4
PHYSICS LETTERS
operators. The frequency operators A,, A_, 2A of twophoton Rabi oscillations have been defined. Our results can be generalized to the other types of threelevel atoms [8]. Further discussions will be made in a future publication. References [l] N. Tan-no, K. Yokoto and H. Inaba, J. Phys. B8 (1975) 339. [ 21 D. Grischkowsky, M.M.T. Loy and P.F. Liao, Phys. Rev. Al2 (1975) 2514. [3] R.M. Whitley and C.R. Stroud, Phys. Rev. Al4 (1976) 1498. [4] CM. Bowden and C.C. Sung, Phys. Rev. A18 (1978) 1588; A20 (1980) 2033. (51 Shi-Yao-Chu and DaChun-Su, Phys. Rev. A25 (1982) 3169.
176
28 January 1985
[6] T.W. Mossberg and S.R. Hartmann, Phys. Rev. A23 (1981) 1271. [7] J.N. EIgin, Phys. Lett. 80A (1980) 140. [8] F.T. Hioe and J.H. Eberly, Phys. Rev. A25 (1982) 2168. [9] L. Kancheva, D. Pushkarov and S. Rashev, J. Phys. B14 (1981) 573. [lo] X. Li and N. Bei, Phys. Lett. 1OlA (1984) 169. [ 111 N.N. Bogolubov Jr., Fam Le Kien and A.S. Schumovsky, Phys. Lett. 1OlA (1984) 201; JINR, E17-84-292 (Dubna, 1984). [ 121 J.R. Ackerhalt and K. Rzazewski, Phys. Rev. Al2 (1975) 2549. [13] L. Allen and J.H. Eberly, Optical resonance and twolevel atoms (Wiley, New York. 1975). [14] B. Buck and C.V. Sukumar, Phys. Lett. 81A (1981) 132; 83A (1981) 211. [ 151 N.N. Bogolubov Jr., Fam Le Kien and A.S. Shumovsky, to be published.
28 January 1985
PHYSICS LETTERS
EXACT RESULTS FOR A MODEL OF A THREE-LEVEL ATOM N.N. BOGOLUBOV Jr., FAM LE KIEN and A.S. SHUMOVSKY Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, P.O. Box 79, 101000, Moscow, USSR Received 7 November 1984
For a three-level atom interacting with two modes of the quantized radiation field the time dependence of transition operators, photon amplitudes, and of operators of level populations and photon numbers is examined rigorously. It is shown that the availability of a mode detuning leads to the splitting of the “soft branch” of frequencies of two-photon Rabi oscillations.
The problem of dynamics of three-level atoms interacting with the electromagnetic field has been the subject of gained active research for the last ten years (see refs. [l- 1l] and references therein). It is central to discussions of two-photon coherence [ 1,2], resonance Raman scattering and double-resonance processes [3], three-level superradiance [4], twomode lasers [S], three-level echoes [6], etc. A number of recent papers [5,7- 1l] has been dedicated to a careful consideration of the problem of the dynamics of a single three-level atom interacting with two resonant modes of the radiation field. The semiclassical formalism for the treatment of this problem has been discussed in refs. [7-91. In another series of articles [5,10,1 l] the fully quantized theory was studied. Exact Schrijdinger wavefunctions were obtained for some special initial states [5]. In ref. [lo] the explicit expression of the evolution operator in the interaction picture was derived. The rigorous examinations of the dynamical behaviour of level populations and photon numbers have been realized in the Heisenberg picture [ 1I]. In the present letter we shall derive the time dependence of atomic transition operators and photon amplitudes as an exact solution of the equations of motion, taking into account a detuning of near-resonant modes. We shall discuss also some interesting consequences of this solution. We consider now a three-level atom (see fig. 1) in which nonzero dipole moments exist only between
f t
-_
__ 3
w--m -L
*i
wz
2
I
Fig. 1. Energy-level and transition structure of the model considered.
levels 1 and 3, and 2 and 3. Let the atom be at rest in a lossless cavity and interact with two modes of the quantized radiation field. The model hamiltonian of the system under consideration is [ 10,l l] H=HA+HF+HAF.
(1)
Here HA, HF describe the free atom and free field, respectively, and HAP describes the atom-field interaction in the dipole and rotating wave approximations 3
2
(2) The operator Rji -= lj)
Volume 107A. number 4
28 January 1985
PHYSICS LETTERS
the atomic transition from level j to level i. The vectors of atomic eigenstates Ii>, j = 1,2,3, form the basis of the state space of the three-level atom
system [ 131. From (7) we obtain for Ra3 the following equation, { [d/dt + i(o,
- C - A)] [d/dt + i(w,
- C)]
3 H~Jj)=Aajlj),
Cili)=Sfj,
C
1.
li)
(3)
j=l
+ h; +g;}Ra3
= 0.
(9)
Its solution can be presented in the following form: The operators Rji (i, j = 1, 2, 3) obey the following relations [4,1 l] :
Ra3(t) = exp[i(C+ X {cos[t(X2
& Rii = 1 . j=l
Rii Rkl = Rilsjk 9
The parameters ga! are the constants coupling. It can be shown that the operators
of atom-mode
,
+ b:R&
- A(R,,
+R,,)
>
(5)
commute with the hamiltonian (1) and with each other. They are therefore constant operators. Let us introduce the following auxiliary operators: V, = g$,R,,
where h2 = Xi + &A2. From the equation for the photon amplitudes db,/dt
= -iw,b,
dl/,/dt
t i(w,
- C)Ra3 = -iv, - C - A)V,
,
combined with expression (10) one can obtain the explicit time dependence for b,(t). After a number of cumbersome transformations this dependence can be represented in the form {b,(O) t B_, t B,
- exp{-i[iA
- (A2 +g2)lj2] a
- exp{-i[:A
t (A2 +g2)‘12] t}B a } ’ 01
t}B_ OL
where for simplicity we have introduced
(11) the notation
B ra! = ga { [C t : A f (X2 + g;)1’2] [2(h2 + g;)1’2]}-’ X {[(X2 +g;)1’2
7 V,(O)} .
(12)
P2, V,l = -g:V, .
(13)
T :A] Ra3(0)
[~2,Ra31= -g:Ra3,
So, for an arbitrary operator function f(h2)Ro,3
+ i(w,
- ig,R,,
of motion
(6)
where&=2ifcr= 1 andE= lifcr=2.Using(4)one can obtain now the equations of motion for the operators R,, and V,, dR,,/dt
(10)
It should be noted that
3
+ g&R,,
Ra3(0) - i(A2 t gi)-‘/2
b,(t) = exp[i(C + A - o,)t]
2 C= &FL g,(b,R,,
+g$1/2]
Xsin[t(h2+g~)1/2][VQ(0)+fAR,3(O)]},
a3 - s1, - w1 = S13 - a2 - o2 = A.
- R,,
o,)t]
(4)
The operators bi, b, (a = 1,2) are the photon amplitudes (or so-called creation and annihilation operators) corresponding to the mode with the energy Ao, of the quantized radiation field. We shall consider the case of exact two-photon resonance [8]
Ma = b;b,
iA-
,
= -i(X$ + gz)Ra3
= &3f
f(h2) we have
(A2- & 7
f02Ya = Vaf02 -g:> * ,
(14)
(7) Then, expression (10) can be rewritten in the form
where ho is determined
by Ra3(t) = exp[i(C+
:A - o,)t]
(Ra3(0)
cos Xt
hi = x; + x,2 ) - i[V,(O) t fAR,3(0)]x-1 x, Eg&MT
o= 1,2.
(8)
The operators X, can be considered as the quantum expressions of the one-photon Rabi frequencies in the 174
sin At}.
(15)
Analogously, with the help of eq. (14) we can obtain from expressions (11) and (12)
b,(t) = exp[i(C + A - w,)t] - B_,
exp(ih_t)
28 January 1985
PHYSICS LETTERS
Volume 107A, number 4
-B,
[b,(O) + B_,
exp(-iX+t)]
/~+a= W)a{PMO) - R,,(O)l~:~;~~~~,)-’
+ B,
,
(16)
+glg2BM~f - 4>W$J1 f &C2@)
where B Ta =&-JR,3(0)&
T
y-#>lPMC+~*)1-' 3 (17)
and the operators h,, X_ are determined
by
P,, = GOa{&lA1(0) and
Expressions (15) and (16) represent our main results. They describe the time dependence of the atomic transition and photon operators. Using these results we can calculate various correlation functions and averages of photon numbers, level populations and atomic dipole moments. Below, we shall consider some.consequences. First, we note that the model of a two-level atom interacting with a single mode radiation field is a particular case of the model considered here. In this case we have either g2 = 0 or g1 = 0, and expressions (15), (16) can then be reduced to the results of ref. [12]. For the third-level population operator R,,(t), from (15) and the relationRg3 = RSaRor3, we get
D zi(blb2R12
2ht - 1) t LJsin 2ht] tRs3(0).(18)
Here u = I&(O)
+ &2(O)
- X&(O)
+gl@(O)
+~A[g1C1(0)+g2C2(0)1}(2X2)-1 , IJ=-klAl@) +~2~2031w-1
,
(19)
and C,=bLR,, B = b;b2R12
A, =i(bkRa3
+b,Rj,, + b,b;R,,
- b&R,,),
.
(20)
From eq. (16) we obtain for the photon-number operator N, E b! b, , N,(t)
= ~r,(cos h+t - 1) t j3, sin h+t
+ p_,(cos + u,(cos
A-t - 1) + /3_, sin X-t 2ht - 1) t ua! sin 2Xt t N,(O),
where u, = (&l@~
3
uol = (X,/X&,
7
- &2A2(W%$1 (22)
f g~g2Nw~~*)-11 ,
X,=h?;A.
RjS(f) = -[u(cos
- A&,c, @)Im&-9
(21)
- blblR21).
Using the conservation law (5): N,(t) = R&t) t Ma and the relation R,,R,, = R,,, from (21) we obtain: R&t)
= pJcos
+ p_,(cos
h+t - 1) t /I, sin X+t
h-t - 1) + p_, sin X-t
+ u,(cos 2ti - 1) t uol sin 2ti t R,,(O)
,
(23)
and also N:(t)
= M,” + [(Ma + l)m - M,m]R,,(t),
(24)
where 01= 1, 2 and m is an arbitrary natural number (m = 1,2, 3, . ..). Expression (24) is useful for the investigation of photon statistics (see ref. [ 151). It is easily seen that the operators X,, X_, and 2h define the frequencies of two-photon Rabi oscillations of level populations and photon numbers in the system [7-l I]. At one-photon resonance A = 0 expressions (18), (2 1) and (23) coincide with the results of ref. [ 111. In this case there are two branches of the twophoton Rabi frequencies defined by the operators X and 2X [ 111. It should be noted that the existence of the “soft branch” h is a characteristic feature of the three-level system. Such a kind of oscillation frequencies is absent in the two-level system [ 13,141. Our present results (18), (21) and (23) show that the detuning A in the case of two-photon resonance leads to the splitting of the “soft branch” h to two branches characterized by the frequency operators h, = h t f A and X_ = A - $A. This conclusion of the fully quantized theory is in accordance with the results of the semiclassical theory [ 91. Thus, in this paper we derived in the Heisenberg picture the explicit expressions of the time dependence for the atomic transition operators, photon amplitudes, and also for the level population and photon-number 175
Volume 107A, number 4
PHYSICS LETTERS
operators. The frequency operators A,, A_, 2A of twophoton Rabi oscillations have been defined. Our results can be generalized to the other types of threelevel atoms [8]. Further discussions will be made in a future publication. References [l] N. Tan-no, K. Yokoto and H. Inaba, J. Phys. B8 (1975) 339. [ 21 D. Grischkowsky, M.M.T. Loy and P.F. Liao, Phys. Rev. Al2 (1975) 2514. [3] R.M. Whitley and C.R. Stroud, Phys. Rev. Al4 (1976) 1498. [4] CM. Bowden and C.C. Sung, Phys. Rev. A18 (1978) 1588; A20 (1980) 2033. (51 Shi-Yao-Chu and DaChun-Su, Phys. Rev. A25 (1982) 3169.
176
28 January 1985
[6] T.W. Mossberg and S.R. Hartmann, Phys. Rev. A23 (1981) 1271. [7] J.N. EIgin, Phys. Lett. 80A (1980) 140. [8] F.T. Hioe and J.H. Eberly, Phys. Rev. A25 (1982) 2168. [9] L. Kancheva, D. Pushkarov and S. Rashev, J. Phys. B14 (1981) 573. [lo] X. Li and N. Bei, Phys. Lett. 1OlA (1984) 169. [ 111 N.N. Bogolubov Jr., Fam Le Kien and A.S. Schumovsky, Phys. Lett. 1OlA (1984) 201; JINR, E17-84-292 (Dubna, 1984). [ 121 J.R. Ackerhalt and K. Rzazewski, Phys. Rev. Al2 (1975) 2549. [13] L. Allen and J.H. Eberly, Optical resonance and twolevel atoms (Wiley, New York. 1975). [14] B. Buck and C.V. Sukumar, Phys. Lett. 81A (1981) 132; 83A (1981) 211. [ 151 N.N. Bogolubov Jr., Fam Le Kien and A.S. Shumovsky, to be published.