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Intensity dependent, two-photon Jaynes-Cummings model
Physica A 166 (1990) North-Holland
INTENSITY
347-360
DEPENDENT,
TWO-PHOTON
JAYNES-CUMMINGS
MODEL
V. BARTZIS Department of Mathematics, University of Manchester Institute of Science and Technology, P.O. Box 88, Manchester M60 lQD, UK and Department of Physics, University of Patras, Patras 26110, Greece
Received 14 September 1989 Revised manuscript received 3 January
1990
In this study we consider the intensity dependent two-photon generalized JaynesCummings model. As initial state of the radiation mode we consider a squeezed state, which, as is well known, is the most general Gaussian pure state. We study the time evolution of the mean value of the inversion operator of the atom, of the mean photon number and of the dispersions of the two quadrature components of the electric field of the electromagnetic wave. We also examine the existence of the quantum noise squeezing phenomenon in the quadrature components.
1. Introduction
The Jaynes-Cummings model [l-3] of a two-level atom interacting with a quantized single mode electromagnetic field is very important because it is the simplest solvable model that describes the essential physics of radiation-matter interaction. Its Hamiltonian after the rotating-wave approximation [3] can be expressed in terms of the inversion, raising and lowering operators denoted by 03, (++ , (T_ and the annihilation and creation operators a, u’ of the radiation mode as H =
giw,
CT3 +
hwa+a+ hh(cT+a
+
a-u+> .
(1.1)
Here o,, is the transition frequency of the atom and w is the mode frequency; A is a coupling strength for the radiation-atom interaction. The u’s are 2 x 2 Pauli matrices,
(
1
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0
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0 -1
) 0
>
m+=
( >
1990 - Elsevier
0
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Science
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(1.2)
348
V. Bartzis I Two-photon Jaynes-Cummings
The U’S and a’s obey the following
[a,,a,] = +20-, , =
1(1++),
In a series
of articles
u+u_
Jaynes-Cummings
[u+,u-1
relations:
=
a3
)
[a,ai]= 1 ) c; = 1.
Ur+=i(l-q), [4-61 Sukumar
models
with
the
model
and Buck
corresponding
considered interaction
(1.3) (1.4) two generalized Hamiltonians,
respectively, H,,, =
?iA(a+u~ + cl&% a’) ,
Hi,, = KA(u+a” +
u_utm).
(1.5) (1.6)
We note that in the first model (1.5) the coupling strength depends on the number operator II = a’u (or otherwise on the radiation intensity), whereas in the second model (1.6) the transmission of the atom from one level to the other is accompanied by the absorption or emission of m photons. The above authors studied the time evolution of the mean value of the inversion operator of the atom. Singh [7] also studied the photon number distribution and the time evolution of the mean photon number for the above models and for various initial states. The model (1.6) with m = 2 has been studied in ref. [8] to obtain the steady state photon statistics in a two-photon laser in which the decay of the lasing levels was taken into account. The two-photon lasing has been observed by Toschek et al. [9]. However, with the recent interest in Rydberg atoms, Haroche et al. [lo] have recently observed the two-photon laser emission in Rb. The model (1.6) has been widely studied to investigate the dynamics of the field and atomic variables with Rydberg atoms in mind [ll, 121. In this paper by continuing the study of the generalized models we consider one with an interaction Hi”, = hh (u+u ‘a
Hamiltonian
of the form
+ udzz ut2) .
(1.7)
We observe that the transmission of the atom from one level to the other is accompanied by absorption or emission of two photons. The other feature is that the coupling strength is, as in the model (1.5), intensity dependent. In the standard Jaynes-Cummings model it is considered, as is well known, constant. However, observing that the radiation intensity depends on the time (the system is not in a steady state as in ref. [8]) we can consider that the coupling strength depends on the intensity. The exact form of
V. Barks
I Two-photon Jaynes-Cummings
model
349
the dependence is very involved, so our model may be considered to be mathematical, showing the effect of the intensity dependent A. As the initial state of the radiation mode we consider a squeezed state [13-191 which is the most general Gaussian pure state. In theory, a squeezed state is defined as
1%2) = S(z)D(a) lo>>
(1.8)
where D(a) = exp(cya’ - (~*a) is the Weyl displacement z = r emi0 ,
S(z) = exp[ $ (za’ - ~*a+*)] , the squeeze operator. In the n-representation,
operator
and (1.9)
the squeezed state has the form [14]
(1.10) C, =
(~)“‘2Hn[a(2pv)m1’2] exp[ -tlal’ + 5 a*]
&
,
where p = cash r ,
v = eie sinh
r
(1.11)
.
The mean photon number for a squeezed state has the form n= l&l2 + /VI2 )
(1.12)
where ii=p*a-KY*.
The two quadrature
(1.13) components
are defined as
x, = ;(a + a+> )
(1.14)
x2 = i
(1.15)
(a - a’).
So, the electric field of the radiation E(t) =X1 cos wf + X2 sin ot
mode can be expressed as (1.16)
V. Barks
350
/ Two-photon Jaynes-Cummings
model
The dispersions of X, and X, for a squeezed state with 8 = 0 are ((A Xi)‘) = i em2r,
(1.17)
((A X,)‘)
(1.18)
= i e2r ,
which means that the quantum noise in one quadrature component is lower than that of the coherent state (((A Xi)‘) = a, i = 1, 2) and in the other higher. This is called the squeezing phenomenon. In the second section of this paper we study the equation of motion of the inversion operator in the Heisenberg picture and we plot its mean value as a function of time for various values of the field constants. In the third section we deduce the evolution operator for the field, the mean photon number, the quantum noise distribution in two quadrature components and we study the above results for various values of the field constants. Finally, we comment on the above findings and we come to some conclusions .
2. Time evolution
of the atom inversion
The Hamiltonian
operator
of our model is
H = ;nwOC$ + ~wa’a + hh(cT+a2 6%
+ &Zz
at2) .
(2.1)
First of all we define the operators
(2.2)
c = u+a + u3 ) B = hh(a+a* V%z - u_ I&
a+‘) ,
D = fih(a+u 2v2i+dmL+*); so the Hamiltonian H=iid+$iA~,+D,
(2.3) (2.4)
(2.1) has the following form:
(2.5)
where A=w,-2~.
(2.6)
V. Bartzis I Two-photon Jaynes-Cummings
model
351
It is easy to prove that [C,H]=[C,B]=O
(2.7)
[q3, Bl = 20 >
(23)
[Us,, D] = 2B.
(2.9)
and
For the calculation of the time evolution of the operator a, (represents the atom population inversion) we work in the Heisenberg picture. The equation of motion for the operators a3 and B are ih&, = [We, H] = [c~, D] = 2B ,
(2.10)
i?iB = [B, H] = -fiAD
(2.11)
The commutator
+ [B, D] .
of B and D has the form
[B, D] = ?i2A2{q[aZ I&
$&a”]+
-t [a’ @ii,
6%
a”]} ,
(2.12)
where the symbol [ , ] + represents the anti-commutator. So the equation of motion for a, using (2.10) and (2.11) takes the form of
-2h2[(Wx + 1) u2(Lz+u)u+*+ (Wx- 1) Vz
atza2 Vx]
.
(2.13)
By using the following relations of the a, at algebra [20]:
f(u+u)u+ =u+f(u+u+ 1) uf(u’u)= f(u’u + 1) a
)
)
where f(u’u) is any function of the number operator form
(2.14) (2.15) ata, eq. (2.13) has the
q+W2w3=$wiwC),
(2.16)
0 *=4h2(C3+2C2+C)+A2.
(2.17)
352
V. Bartzis I Two-photon Jaynes-Cummings
The operator
model
C is a constant of motion, so the solution of the above equation is
WO)
q(t) = q(O) cos ot + inw
sin wt + 2
(H - fiwC)(l - cos wt) .
(2.18)
We suppose that the atom is initially at the excited state and the field in a squeezed state. Thus,
(uJt))=C{<+(1-$jcosw,,t}IC.~‘, n wll
(2.19)
n
where W”’ = 4h2[(n + 1)’ + 2(n + l)* + (n + l)] + A2
(2.20)
and the coefficients C, are given by the relations (l.lO), (1.11). For the special cases in which A = 0, a = 4, r = 0.2, 0 = 0 (squeezed state) and A = 0, a = 4, r = 0 (coherent state) we plot ( u3(t)) as a function of time in figs. 1 and 2. From the two figures we deduce that the inversion operator mean value for t = 0 is 1, something which is expected because of the assumption that initially the atom is in the excited state. Starting from the value 1 the mean value of the inversion operator a, executes a damped oscillation and soon takes the value 0. After a while a
Fig. 1. Time evolution of (am)
for A = 0, (Y = 4, r = 0.2, 0 = 0 (squeezed state).
V. Bartzis
I Two-photon
Jaynes-Cummings
353
model
Fig. 2. Time evolution of (a,(t)) for A = 0, (Y = 4, r = 0.0 (coherent state).
revival takes place, which however, is not complete, as (a,(t)) does not take the value 1. Comparing the two curves we note that for the case of the coherent state the oscillation is faster. Concerning the rest, the curves are similar.
3. Field statistics
of the intensity
dependent,
two-photon
Jaynes-Cummings
model
The Hamiltonian C = a+a +
u3
of the system is given by eq. (2.1). We define the operators (3.1)
,
N = $4~~ + h(cT+a21G+
(+_ V&a+‘).
(3.2)
It is easy to show that [N,C]=[H,N]=[H,C]=O. So the time evolution
(3.3)
operator
u(t 7 0) = e(-i/fi)Ht =e
-iwCt
can be written as a product, e -iNrz U,(t, 0) U2(t, 0) .
(3.4)
354
V. Bartzis
I Two-photon
Jaynes-Cummings
We will write the matrix representation two-dimensional atomic subspace. So
u,
(t, 0) =
model
of the operators
U, and U, in the
e-iwa+at (eyU’ ,p_J .
The matrix representation
(3.6)
of the operator
N in the same subspace is
;A N=
(3.7)
/i~u+2
and for the Z-integer we have N2[ = ( z=o,
[ $A’ + h2a2(a+a)a’2]’ 0 0 [ aA’ + A2at2(a+a + 2)a2]’ 1,2,.
’
..
(3.8)
From eqs. (3.7) and (3.8) we have
N
+A[dA’
21+1=
+ A2a2(a+a)atZ]’
h[dA* + A2at2(aia + 2)az]‘&%
A[ f A’ + h2az(ata)at*]‘azl/& at*
- tA[+A’
+ A2at2(ata + 2)aZ]’
I= 0, 1,2,
(3.9)
For the operator
U2(t, 0) we have
U2(t, 0) = eeiN’
(3.10)
(3.11) where
K = cos[tj/$A*
+ h2a2(a’a)at2] - i;A
sin[tj/ $A2 + h2u2(u’u)ut2] v$A2 +
L =
_iAsin[t
$A’ + h2u2(u’u)ut2]
d$A2 + A2u2(a+u)u+2
u21mz
)
A2u2(u+u)u+2
(3.12) ’
(3.13)
V. Bartzis
A4 =
sin[t
-iA
Jaynes-Cummings
model
aA’ + h2at2(ata + 2)a2] a+2VXi ,
+ A2a+2(a+u + 2)u2
j/ad”
N = cos[t~~A*
I Two-photon
+ A2ut2(u’u + 2)u*] + i+A
sin[tj/ iA* + h2ut2(utu
355
(3.14)
+ 2)u2]
+ h2ut2(u+u + 2)~’
j/64’
’
(3.15) so, (3.16) where lj? =
e-‘“‘K
,
(3.17)
Z
emimrL ,
(3.18)
=
y = $-‘M W=
)
(3.19)
e’“‘N .
(3.20)
We can easily show that (3.21)
uu+=lJ+u=1.
Assuming the atom to be initially in the excited state we have the density operator of the field as
U(t, 0) (Pfp ;) u+(t,0) pf(t)= Tratom =e
-ioa+a’ cos[tj/ iA* + A2(u+a + l)(u+u + 2)2] _ i’A sin[tj/iA*
+ A2(utu + l)(u+u + 2)2]
2
+ A2(u+u + l)(u+u + 2)2
j/ad’
)
x p,(O) cos[rj/ aA’ + A*(u+u + l)(u+u + 2)2] ( + i’A sin[t
iA’
2
j/$A2
+ A*(u’u + l)(u’u + 2)*]
+ A*(u+u + l)(u+u + 2)*
356
V. Bar&is
+
I Two-photon
Jaynes-Cummings
model
sin[t-\l iA* + h*(~'a)~(a~a - l)]
A2
a+*l4iTiT p,(O)lLiFZ a2 j/ad"+ A2(a'a)'(afa1) sin[tj/ iA2 + h2(ata)2(a'a - l)] X
j/ad'+ A2(ata)'(ata - 1)
1iwa+at e
(3.22)
’
We consider as initial state of the system a squeezed state (eqs. (1.9)(1.11)). Thus it is easy to find the matix elements of p,(t) in the In)-basis, (m]pr(t)]m’)
= C,CL. e-iw(m-m’)r cos[t~~A2+ A2(m + l)(m + 2)2] _ i’A sin[tI/$A2 + A*(m + l)(m + 2)*] 2
j/$4*+ A2(m + l)(m + 2)2
x cos[tj/ iA* + A2(m’ + l)(m’ + 2)2] + i’A sin[tj/ aA" + A2(m'+ l)(m’ + 2)*] 2
>
d$A'+ A2(m'+ l)(m’+2)*
+ C,_,C~._,
e -idm-m’)tA2,/mmr(m
_
l)(m
_
1)
sin[tv $A2 + Azm2(m - l)] X
j/$A2+A2m2(m-1) x~
sin[tj/ aA' + A2mr2(m'- l)] j/~A*+A*m'*(m'- 1)
(3.23) ’
where the coefficients C, are defined in relation (1.10). We can calculate the effect of the field statistics in the mean photon number
i(t) = Tr,i,,,[atadt)l = C n (nl dt)ln >. n We plot the mean photon number as a function of time for A = r = 0.2, 8 = 0 (squeezed state) and for A = 0, cx= 4, r = 0 (coherent figs. 3 and 4. We observe that the two curves are similar but differ on the initial t = 0; for the squeezed state the initial value is 10.76 whereas for the state the corresponding value is 16.00. Also the oscillation for the state is faster.
(3.24)
0, a = 4, state) in moment coherent coherent
V. Barbs
I Two-photon Jaynes-Cummings
357
model
10 2
1
Fig. 3. Time evolution
3
at-b
ii(t) for A = 0, (Y = 4, r = 0.2, 0 = 0 (squeezed
of
state).
Ts. 1
16. 17 16. 15. 2
1
Fig. 4. Time evolution
3
At-
ii(t) for A = 0, (Y = 4, r = 0.0 (coherent
of
We next consider the time development operators,
state).
of the dispersions of the quadrature
x, = &u+a+))
x, = ;
a’)
(a -
(3.25) (3.26)
)
which have the form
((A W’) = ; [ 1 +
c P~(4&)1~) n
+ I& + l)(n + 2) ((fl+ 2ldW
+ blP&)b + WI
+ h+Wlf4 + (4PfWl~+ WI)‘] 7
- ( F P=%(n
(3.27) ((A
X2)')
=
$
[l
-I&
+T
P~(h-d~)l~)
+
l)(n
+ (c w-w+
+
2)
((n
+
2lPfWl4 + (4PrWb + 2))l
+ llPfWb) - blP&>l~+ w1)2] . (3.28)
358
V. Bartzis
We plot the above 8 = 0 (squeezed (coherent
dispersions
state)
state)
I Two-photon
Jaynes-Cummings
as functions
in figs.
5 and
of time
6 whereas
model
for A = 0, a = 4, r = 0.2, for
A = 0, a = 4, r = 0.0
in figs. 7 and 8.
By studying the previous curves, we can conclude that oscillation damps at later times and their values become
the initially
Afterwards,
of the quadrature
in both cases we can observe
almost
in the dispersion
constants.
<(AX,)*> t
Fig. 5. Time evolution
of ((A X,)‘)
for A = 0, a = 4, I = 0.2, 0 = 0 (squeezed
state).
4 <(AX*)*> 11. lo. 9. 6 7. 6. 6’ 4. 32.
0,
1
I
I
2 3
Fig. 6. Time evolution
ht-
of ((A X2)2) for A = 0, a = 4, I = 0.2, 0 = 0 (squeezed
large
state).
V. Bard
<(AX,
I Two-photon Jaynes-Cummings
model
359
I*> t
Fig. 7. Time evolution
of ((A X,)‘)
for A = 0, a = 4, r = 0.0 (coherent
state).
component X, oscillations near the value $ and a weak quantum noise squeezing phenomenon, but at the following moments the quantum noise increases and there is not a squeezing phenomenon in either of the two quadrature components. A <(AX*)*> 18. 17. 16. 15. 14. 13. 12. 11. loQ87. 65. 4. 3. 2. 1’ Ol
Fig. 8. Time evolution
1
of ((A X2)‘)
2
3
ht -
for A = 0, a = 4, r = 0.0 (coherent
state).
360
V. Barks
It is worth noting quadrature
I Two-photon
Jaynes-Cummings
that in the case of the squeezed
components
take lower
values
than
model
state,
the dispersions
that of the coherent
in the
state.
4. Conclusions In this study we studied in depth the intensity dependent, two-photon Jaynes-Cummings model. We have examined the time evolution of the mean value of the atom inversion operator, of the mean photon number and of the dispersions of the two quadrature components of the field as well. As initial state of the radiation we have considered a squeezed state, which, as previously mentioned, is the most general Gaussian pure state. The most important conclusion that is deduced is that there is initially a weak squeezing phenomenon in quadrature component X,, whereas at later moments there is nothing like that in either of the two quadrature components. But for the case of the squeezed state the dispersions components take lower values than that of the coherent case of the squeezed
of the two quadrature state, which is a special
state.
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [IO] [ll] [12] [13] 1141 [15] [16] [17] [18] [19] [20]
E.T. Jaynes and F.W. Cummings, Proc. IEEE 51 (1963) 89. M. Tavis and F.W. Cummings, Phys. Rev. 188 (1969) 692. S. Stenholm, Phys. Rep. 6 (1973) 1. C.V. Sukumar and B. Buck, Phys. Lett. A 83 (1981) 211. B. Buck and C.V. Sukumar. Phys. Lett. A 81 (1981) 132. C.V. Sukumar and B. Buck, J. Phys. A. Math. 17 (1984) 885. S. Singh, Phys. Rev. A 25 (1982) 3206. N. Nayak and B.K. Mohanty, Phys. Rev. A 19 (1979) 1204. B. Nicolaus, D.Z. Zhang and P.E. Toschek, Phys. Rev. Lett. 47 (1981) 171. M. Brune, J.M. Raimond, P. Goy, L. Davidovich and S. Haroche, Phys. Rev. Lett 59 (1987) 1899. C.C. Gerry, Phys. Rev. A 37 (1988) 2683. CC. Gerry and P.J. Moyer, Phys. Rev. A 38 (1988) 5665. D. Stoler, Phys. Rev. D 1 (1970) 3217. H. Yuen, Phys. Rev. A 13 (1976) 2226. D.F. Walls, Nature 306 (1983) 141. R.A. Fisher, M.M. Nieto and V.D. Sandberg, Phys. Rev. D 29 (1984) 1107. B.L. Schumaker, Phys. Rep. 135 (1986) 317. A. Jannussis and V. Bartzis, Nuovo Cimento B 102 (1988) 33. V. Bartzis, E. Vlahos and A. Jannussis, Nuovo Cimento B 103 (1989) 537. W. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).
INTENSITY
347-360
DEPENDENT,
TWO-PHOTON
JAYNES-CUMMINGS
MODEL
V. BARTZIS Department of Mathematics, University of Manchester Institute of Science and Technology, P.O. Box 88, Manchester M60 lQD, UK and Department of Physics, University of Patras, Patras 26110, Greece
Received 14 September 1989 Revised manuscript received 3 January
1990
In this study we consider the intensity dependent two-photon generalized JaynesCummings model. As initial state of the radiation mode we consider a squeezed state, which, as is well known, is the most general Gaussian pure state. We study the time evolution of the mean value of the inversion operator of the atom, of the mean photon number and of the dispersions of the two quadrature components of the electric field of the electromagnetic wave. We also examine the existence of the quantum noise squeezing phenomenon in the quadrature components.
1. Introduction
The Jaynes-Cummings model [l-3] of a two-level atom interacting with a quantized single mode electromagnetic field is very important because it is the simplest solvable model that describes the essential physics of radiation-matter interaction. Its Hamiltonian after the rotating-wave approximation [3] can be expressed in terms of the inversion, raising and lowering operators denoted by 03, (++ , (T_ and the annihilation and creation operators a, u’ of the radiation mode as H =
giw,
CT3 +
hwa+a+ hh(cT+a
+
a-u+> .
(1.1)
Here o,, is the transition frequency of the atom and w is the mode frequency; A is a coupling strength for the radiation-atom interaction. The u’s are 2 x 2 Pauli matrices,
(
1
v3=
0
0378-4371/90/$03.50
0 -1
) 0
>
m+=
( >
1990 - Elsevier
0
1
()
0
Science
9
(+-=
Publishers
(
0
0
1
o
>.
B.V. (North-Holland)
(1.2)
348
V. Bartzis I Two-photon Jaynes-Cummings
The U’S and a’s obey the following
[a,,a,] = +20-, , =
1(1++),
In a series
of articles
u+u_
Jaynes-Cummings
[u+,u-1
relations:
=
a3
)
[a,ai]= 1 ) c; = 1.
Ur+=i(l-q), [4-61 Sukumar
models
with
the
model
and Buck
corresponding
considered interaction
(1.3) (1.4) two generalized Hamiltonians,
respectively, H,,, =
?iA(a+u~ + cl&% a’) ,
Hi,, = KA(u+a” +
u_utm).
(1.5) (1.6)
We note that in the first model (1.5) the coupling strength depends on the number operator II = a’u (or otherwise on the radiation intensity), whereas in the second model (1.6) the transmission of the atom from one level to the other is accompanied by the absorption or emission of m photons. The above authors studied the time evolution of the mean value of the inversion operator of the atom. Singh [7] also studied the photon number distribution and the time evolution of the mean photon number for the above models and for various initial states. The model (1.6) with m = 2 has been studied in ref. [8] to obtain the steady state photon statistics in a two-photon laser in which the decay of the lasing levels was taken into account. The two-photon lasing has been observed by Toschek et al. [9]. However, with the recent interest in Rydberg atoms, Haroche et al. [lo] have recently observed the two-photon laser emission in Rb. The model (1.6) has been widely studied to investigate the dynamics of the field and atomic variables with Rydberg atoms in mind [ll, 121. In this paper by continuing the study of the generalized models we consider one with an interaction Hi”, = hh (u+u ‘a
Hamiltonian
of the form
+ udzz ut2) .
(1.7)
We observe that the transmission of the atom from one level to the other is accompanied by absorption or emission of two photons. The other feature is that the coupling strength is, as in the model (1.5), intensity dependent. In the standard Jaynes-Cummings model it is considered, as is well known, constant. However, observing that the radiation intensity depends on the time (the system is not in a steady state as in ref. [8]) we can consider that the coupling strength depends on the intensity. The exact form of
V. Barks
I Two-photon Jaynes-Cummings
model
349
the dependence is very involved, so our model may be considered to be mathematical, showing the effect of the intensity dependent A. As the initial state of the radiation mode we consider a squeezed state [13-191 which is the most general Gaussian pure state. In theory, a squeezed state is defined as
1%2) = S(z)D(a) lo>>
(1.8)
where D(a) = exp(cya’ - (~*a) is the Weyl displacement z = r emi0 ,
S(z) = exp[ $ (za’ - ~*a+*)] , the squeeze operator. In the n-representation,
operator
and (1.9)
the squeezed state has the form [14]
(1.10) C, =
(~)“‘2Hn[a(2pv)m1’2] exp[ -tlal’ + 5 a*]
&
,
where p = cash r ,
v = eie sinh
r
(1.11)
.
The mean photon number for a squeezed state has the form n= l&l2 + /VI2 )
(1.12)
where ii=p*a-KY*.
The two quadrature
(1.13) components
are defined as
x, = ;(a + a+> )
(1.14)
x2 = i
(1.15)
(a - a’).
So, the electric field of the radiation E(t) =X1 cos wf + X2 sin ot
mode can be expressed as (1.16)
V. Barks
350
/ Two-photon Jaynes-Cummings
model
The dispersions of X, and X, for a squeezed state with 8 = 0 are ((A Xi)‘) = i em2r,
(1.17)
((A X,)‘)
(1.18)
= i e2r ,
which means that the quantum noise in one quadrature component is lower than that of the coherent state (((A Xi)‘) = a, i = 1, 2) and in the other higher. This is called the squeezing phenomenon. In the second section of this paper we study the equation of motion of the inversion operator in the Heisenberg picture and we plot its mean value as a function of time for various values of the field constants. In the third section we deduce the evolution operator for the field, the mean photon number, the quantum noise distribution in two quadrature components and we study the above results for various values of the field constants. Finally, we comment on the above findings and we come to some conclusions .
2. Time evolution
of the atom inversion
The Hamiltonian
operator
of our model is
H = ;nwOC$ + ~wa’a + hh(cT+a2 6%
+ &Zz
at2) .
(2.1)
First of all we define the operators
(2.2)
c = u+a + u3 ) B = hh(a+a* V%z - u_ I&
a+‘) ,
D = fih(a+u 2v2i+dmL+*); so the Hamiltonian H=iid+$iA~,+D,
(2.3) (2.4)
(2.1) has the following form:
(2.5)
where A=w,-2~.
(2.6)
V. Bartzis I Two-photon Jaynes-Cummings
model
351
It is easy to prove that [C,H]=[C,B]=O
(2.7)
[q3, Bl = 20 >
(23)
[Us,, D] = 2B.
(2.9)
and
For the calculation of the time evolution of the operator a, (represents the atom population inversion) we work in the Heisenberg picture. The equation of motion for the operators a3 and B are ih&, = [We, H] = [c~, D] = 2B ,
(2.10)
i?iB = [B, H] = -fiAD
(2.11)
The commutator
+ [B, D] .
of B and D has the form
[B, D] = ?i2A2{q[aZ I&
$&a”]+
-t [a’ @ii,
6%
a”]} ,
(2.12)
where the symbol [ , ] + represents the anti-commutator. So the equation of motion for a, using (2.10) and (2.11) takes the form of
-2h2[(Wx + 1) u2(Lz+u)u+*+ (Wx- 1) Vz
atza2 Vx]
.
(2.13)
By using the following relations of the a, at algebra [20]:
f(u+u)u+ =u+f(u+u+ 1) uf(u’u)= f(u’u + 1) a
)
)
where f(u’u) is any function of the number operator form
(2.14) (2.15) ata, eq. (2.13) has the
q+W2w3=$wiwC),
(2.16)
0 *=4h2(C3+2C2+C)+A2.
(2.17)
352
V. Bartzis I Two-photon Jaynes-Cummings
The operator
model
C is a constant of motion, so the solution of the above equation is
WO)
q(t) = q(O) cos ot + inw
sin wt + 2
(H - fiwC)(l - cos wt) .
(2.18)
We suppose that the atom is initially at the excited state and the field in a squeezed state. Thus,
(uJt))=C{<+(1-$jcosw,,t}IC.~‘, n wll
(2.19)
n
where W”’ = 4h2[(n + 1)’ + 2(n + l)* + (n + l)] + A2
(2.20)
and the coefficients C, are given by the relations (l.lO), (1.11). For the special cases in which A = 0, a = 4, r = 0.2, 0 = 0 (squeezed state) and A = 0, a = 4, r = 0 (coherent state) we plot ( u3(t)) as a function of time in figs. 1 and 2. From the two figures we deduce that the inversion operator mean value for t = 0 is 1, something which is expected because of the assumption that initially the atom is in the excited state. Starting from the value 1 the mean value of the inversion operator a, executes a damped oscillation and soon takes the value 0. After a while a
Fig. 1. Time evolution of (am)
for A = 0, (Y = 4, r = 0.2, 0 = 0 (squeezed state).
V. Bartzis
I Two-photon
Jaynes-Cummings
353
model
Fig. 2. Time evolution of (a,(t)) for A = 0, (Y = 4, r = 0.0 (coherent state).
revival takes place, which however, is not complete, as (a,(t)) does not take the value 1. Comparing the two curves we note that for the case of the coherent state the oscillation is faster. Concerning the rest, the curves are similar.
3. Field statistics
of the intensity
dependent,
two-photon
Jaynes-Cummings
model
The Hamiltonian C = a+a +
u3
of the system is given by eq. (2.1). We define the operators (3.1)
,
N = $4~~ + h(cT+a21G+
(+_ V&a+‘).
(3.2)
It is easy to show that [N,C]=[H,N]=[H,C]=O. So the time evolution
(3.3)
operator
u(t 7 0) = e(-i/fi)Ht =e
-iwCt
can be written as a product, e -iNrz U,(t, 0) U2(t, 0) .
(3.4)
354
V. Bartzis
I Two-photon
Jaynes-Cummings
We will write the matrix representation two-dimensional atomic subspace. So
u,
(t, 0) =
model
of the operators
U, and U, in the
e-iwa+at (eyU’ ,p_J .
The matrix representation
(3.6)
of the operator
N in the same subspace is
;A N=
(3.7)
/i~u+2
and for the Z-integer we have N2[ = ( z=o,
[ $A’ + h2a2(a+a)a’2]’ 0 0 [ aA’ + A2at2(a+a + 2)a2]’ 1,2,.
’
..
(3.8)
From eqs. (3.7) and (3.8) we have
N
+A[dA’
21+1=
+ A2a2(a+a)atZ]’
h[dA* + A2at2(aia + 2)az]‘&%
A[ f A’ + h2az(ata)at*]‘azl/& at*
- tA[+A’
+ A2at2(ata + 2)aZ]’
I= 0, 1,2,
(3.9)
For the operator
U2(t, 0) we have
U2(t, 0) = eeiN’
(3.10)
(3.11) where
K = cos[tj/$A*
+ h2a2(a’a)at2] - i;A
sin[tj/ $A2 + h2u2(u’u)ut2] v$A2 +
L =
_iAsin[t
$A’ + h2u2(u’u)ut2]
d$A2 + A2u2(a+u)u+2
u21mz
)
A2u2(u+u)u+2
(3.12) ’
(3.13)
V. Bartzis
A4 =
sin[t
-iA
Jaynes-Cummings
model
aA’ + h2at2(ata + 2)a2] a+2VXi ,
+ A2a+2(a+u + 2)u2
j/ad”
N = cos[t~~A*
I Two-photon
+ A2ut2(u’u + 2)u*] + i+A
sin[tj/ iA* + h2ut2(utu
355
(3.14)
+ 2)u2]
+ h2ut2(u+u + 2)~’
j/64’
’
(3.15) so, (3.16) where lj? =
e-‘“‘K
,
(3.17)
Z
emimrL ,
(3.18)
=
y = $-‘M W=
)
(3.19)
e’“‘N .
(3.20)
We can easily show that (3.21)
uu+=lJ+u=1.
Assuming the atom to be initially in the excited state we have the density operator of the field as
U(t, 0) (Pfp ;) u+(t,0) pf(t)= Tratom =e
-ioa+a’ cos[tj/ iA* + A2(u+a + l)(u+u + 2)2] _ i’A sin[tj/iA*
+ A2(utu + l)(u+u + 2)2]
2
+ A2(u+u + l)(u+u + 2)2
j/ad’
)
x p,(O) cos[rj/ aA’ + A*(u+u + l)(u+u + 2)2] ( + i’A sin[t
iA’
2
j/$A2
+ A*(u’u + l)(u’u + 2)*]
+ A*(u+u + l)(u+u + 2)*
356
V. Bar&is
+
I Two-photon
Jaynes-Cummings
model
sin[t-\l iA* + h*(~'a)~(a~a - l)]
A2
a+*l4iTiT p,(O)lLiFZ a2 j/ad"+ A2(a'a)'(afa1) sin[tj/ iA2 + h2(ata)2(a'a - l)] X
j/ad'+ A2(ata)'(ata - 1)
1iwa+at e
(3.22)
’
We consider as initial state of the system a squeezed state (eqs. (1.9)(1.11)). Thus it is easy to find the matix elements of p,(t) in the In)-basis, (m]pr(t)]m’)
= C,CL. e-iw(m-m’)r cos[t~~A2+ A2(m + l)(m + 2)2] _ i’A sin[tI/$A2 + A*(m + l)(m + 2)*] 2
j/$4*+ A2(m + l)(m + 2)2
x cos[tj/ iA* + A2(m’ + l)(m’ + 2)2] + i’A sin[tj/ aA" + A2(m'+ l)(m’ + 2)*] 2
>
d$A'+ A2(m'+ l)(m’+2)*
+ C,_,C~._,
e -idm-m’)tA2,/mmr(m
_
l)(m
_
1)
sin[tv $A2 + Azm2(m - l)] X
j/$A2+A2m2(m-1) x~
sin[tj/ aA' + A2mr2(m'- l)] j/~A*+A*m'*(m'- 1)
(3.23) ’
where the coefficients C, are defined in relation (1.10). We can calculate the effect of the field statistics in the mean photon number
i(t) = Tr,i,,,[atadt)l = C n (nl dt)ln >. n We plot the mean photon number as a function of time for A = r = 0.2, 8 = 0 (squeezed state) and for A = 0, cx= 4, r = 0 (coherent figs. 3 and 4. We observe that the two curves are similar but differ on the initial t = 0; for the squeezed state the initial value is 10.76 whereas for the state the corresponding value is 16.00. Also the oscillation for the state is faster.
(3.24)
0, a = 4, state) in moment coherent coherent
V. Barbs
I Two-photon Jaynes-Cummings
357
model
10 2
1
Fig. 3. Time evolution
3
at-b
ii(t) for A = 0, (Y = 4, r = 0.2, 0 = 0 (squeezed
of
state).
Ts. 1
16. 17 16. 15. 2
1
Fig. 4. Time evolution
3
At-
ii(t) for A = 0, (Y = 4, r = 0.0 (coherent
of
We next consider the time development operators,
state).
of the dispersions of the quadrature
x, = &u+a+))
x, = ;
a’)
(a -
(3.25) (3.26)
)
which have the form
((A W’) = ; [ 1 +
c P~(4&)1~) n
+ I& + l)(n + 2) ((fl+ 2ldW
+ blP&)b + WI
+ h+Wlf4 + (4PfWl~+ WI)‘] 7
- ( F P=%(n
(3.27) ((A
X2)')
=
$
[l
-I&
+T
P~(h-d~)l~)
+
l)(n
+ (c w-w+
+
2)
((n
+
2lPfWl4 + (4PrWb + 2))l
+ llPfWb) - blP&>l~+ w1)2] . (3.28)
358
V. Bartzis
We plot the above 8 = 0 (squeezed (coherent
dispersions
state)
state)
I Two-photon
Jaynes-Cummings
as functions
in figs.
5 and
of time
6 whereas
model
for A = 0, a = 4, r = 0.2, for
A = 0, a = 4, r = 0.0
in figs. 7 and 8.
By studying the previous curves, we can conclude that oscillation damps at later times and their values become
the initially
Afterwards,
of the quadrature
in both cases we can observe
almost
in the dispersion
constants.
<(AX,)*> t
Fig. 5. Time evolution
of ((A X,)‘)
for A = 0, a = 4, I = 0.2, 0 = 0 (squeezed
state).
4 <(AX*)*> 11. lo. 9. 6 7. 6. 6’ 4. 32.
0,
1
I
I
2 3
Fig. 6. Time evolution
ht-
of ((A X2)2) for A = 0, a = 4, I = 0.2, 0 = 0 (squeezed
large
state).
V. Bard
<(AX,
I Two-photon Jaynes-Cummings
model
359
I*> t
Fig. 7. Time evolution
of ((A X,)‘)
for A = 0, a = 4, r = 0.0 (coherent
state).
component X, oscillations near the value $ and a weak quantum noise squeezing phenomenon, but at the following moments the quantum noise increases and there is not a squeezing phenomenon in either of the two quadrature components. A <(AX*)*> 18. 17. 16. 15. 14. 13. 12. 11. loQ87. 65. 4. 3. 2. 1’ Ol
Fig. 8. Time evolution
1
of ((A X2)‘)
2
3
ht -
for A = 0, a = 4, r = 0.0 (coherent
state).
360
V. Barks
It is worth noting quadrature
I Two-photon
Jaynes-Cummings
that in the case of the squeezed
components
take lower
values
than
model
state,
the dispersions
that of the coherent
in the
state.
4. Conclusions In this study we studied in depth the intensity dependent, two-photon Jaynes-Cummings model. We have examined the time evolution of the mean value of the atom inversion operator, of the mean photon number and of the dispersions of the two quadrature components of the field as well. As initial state of the radiation we have considered a squeezed state, which, as previously mentioned, is the most general Gaussian pure state. The most important conclusion that is deduced is that there is initially a weak squeezing phenomenon in quadrature component X,, whereas at later moments there is nothing like that in either of the two quadrature components. But for the case of the squeezed state the dispersions components take lower values than that of the coherent case of the squeezed
of the two quadrature state, which is a special
state.
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [IO] [ll] [12] [13] 1141 [15] [16] [17] [18] [19] [20]
E.T. Jaynes and F.W. Cummings, Proc. IEEE 51 (1963) 89. M. Tavis and F.W. Cummings, Phys. Rev. 188 (1969) 692. S. Stenholm, Phys. Rep. 6 (1973) 1. C.V. Sukumar and B. Buck, Phys. Lett. A 83 (1981) 211. B. Buck and C.V. Sukumar. Phys. Lett. A 81 (1981) 132. C.V. Sukumar and B. Buck, J. Phys. A. Math. 17 (1984) 885. S. Singh, Phys. Rev. A 25 (1982) 3206. N. Nayak and B.K. Mohanty, Phys. Rev. A 19 (1979) 1204. B. Nicolaus, D.Z. Zhang and P.E. Toschek, Phys. Rev. Lett. 47 (1981) 171. M. Brune, J.M. Raimond, P. Goy, L. Davidovich and S. Haroche, Phys. Rev. Lett 59 (1987) 1899. C.C. Gerry, Phys. Rev. A 37 (1988) 2683. CC. Gerry and P.J. Moyer, Phys. Rev. A 38 (1988) 5665. D. Stoler, Phys. Rev. D 1 (1970) 3217. H. Yuen, Phys. Rev. A 13 (1976) 2226. D.F. Walls, Nature 306 (1983) 141. R.A. Fisher, M.M. Nieto and V.D. Sandberg, Phys. Rev. D 29 (1984) 1107. B.L. Schumaker, Phys. Rep. 135 (1986) 317. A. Jannussis and V. Bartzis, Nuovo Cimento B 102 (1988) 33. V. Bartzis, E. Vlahos and A. Jannussis, Nuovo Cimento B 103 (1989) 537. W. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).