collapses and revivals in two-level atoms in a superposed state interacting with a single mode...

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International Journal of Modern Physics BVol. 22, No. 17 (2008) 2725–2739c© World Scientific Publishing Company

COLLAPSES AND REVIVALS IN TWO-LEVEL

ATOMS IN A SUPERPOSED STATE INTERACTING

WITH A SINGLE MODE SUPERPOSED

COHERENT RADIATION

HARI PRAKASH∗,† and RAKESH KUMAR∗,‡

∗Department of Physics, University of Allahabad, Allahabad 211 002, India†M. N. Saha Centre of Space Studies, Institute of Interdisciplinary Studies,

University of Allahabad, Allahabad 211 002, India‡Department of Physics, Udai Pratap Autonomous College,

Varanasi 221 002, India†prakash [email protected]

‡r [email protected]

Received 17 May 2007

Collapses and revivals phenomenon in system of a single two-level atom and two two-levelatoms existing in some superposed states and interacting with a single mode superposedcoherent radiation is studied. For superposed Dicke states |1/2, ±1/2〉 of a single two-level atom interacting with even or odd coherent state, only odd revivals occur. Fortwo two-level atoms, it is found that the collapse and revival times for even and oddcoherent states are equal to one half of the corresponding times for coherent state. Forthis system, Rabi oscillations occur with a main mean frequency and also some secondharmonics are present, in general. However, if the two atoms are in the superradiantstate, only the second harmonics with large amplitude are obtained. The appearance ofweak double frequency revivals for superposed atomic and coherent states was studiedand a condition for their disappearance is found.

Keywords: Two-level atoms; coherent radiation; collapses and revivals.

PACS numbers: 42.50.Dv, 42.50.-P, 42.50.ct, 42.50.Md

1. Introduction

The Jaynes–Cummings (J–C) model1–3 of a two-level atom interacting with a single

mode of radiation provides non-classical effects in interaction between electromag-

netic field and matter. One of the important aspects of the J–C model is that it is

exactly solvable for the arbitrary atom–field coupling constant and field strength.

The J–C model describes many interesting effects such as Rabi oscillations,4–9 col-

lapses and revivals of the atomic inversion,4–13 photon antibunching,14–16 quadra-

ture squeezing13,17–19 etc. One interesting phenomenon is the behavior of atomic

inversion under Rabi oscillations. Semi-classical treatment of two two-level atoms

2725

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2726 H. Prakash & R. Kumar

with a classical field or a quantum theory of interaction of two-level atom with a

pure occupation number radiation state involves Rabi oscillations with a fixed fre-

quency. However, if radiation is superposition of several occupation number states,

there is an initial collapse of Rabi oscillations followed by regular revivals that slowly

become broader and less intense and eventually overlap. The revival–collapse phe-

nomenon is a pure quantum mechanical effect having its origin in the granular

structure of the photon number distribution of the field.4–9,20 It has been extended

for multi-atom system21–24 as well as multi-level system.25,26 Some extensive study

of this phenomenon has also been carried out for single photon transition,27 two-

photon coherent state,28,29 binomial state,30 mixed coherent chaotic state31 and

coherent superposition states.29,32 Recently, Faisal A A El-Orany studied the re-

lation between the atomic inversions in particular the occurrence of collapse and

revival phenomenon with, fluctuation of quadrature field,33,34 higher order fluctu-

ation of quadrature fields components35 and wigner function36 in the multiphoton

J–C model. He also studied the same for a single mode field interacting with two

two-level atoms.37 Collapse and revival phenomenon in J–C model has also been

discussed earlier by several authors,38–40 but they consider only excited or de-

excited state atomic assembly interacting with coherent, binomial or superposed

coherent states. Faisal A A El-Orany33,34 considered the superposition of atomic

state interacting with an orthogonal even coherent state and find regular revivals.

Experimental observation of collapses and revivals has been reported by Rempe

et al.41 and Rabi oscillation has been observed by Gentile et al.

42

In this paper, we consider a superposed atomic state having equal weightages

of the lower and upper states, |l〉 and |u〉, interacting with radiation in a super-

posed state of coherent state |α〉 and | − α〉 in equal proportion, i.e., initial state,

1/√

2(|1/2, 1/2〉+eiθ|1/2,−1/2〉)Kγ(|α〉+eiγ |−α〉), where Kγ is the normalization

constant. We obtain only odd revivals in the four special cases, γ = 0 or π and

θ + θα = 0 or π (θα is the phase of the coherent state, i.e., α = |α|eiθα). We also

find that the revival time for γ = 0 or π, is only one half of the revival time for

γ = π/2.

Also, we consider collapses and revival in the interaction of (i) two two-level

atoms with even and odd coherent states and (ii) for initial state 1/√

2(|1, 1〉 +

eiθ|1,−1〉)Kγ(|α〉 + eiγ | − α〉), where Kγ is the normalization constant. We get

oscillations with mean frequencies 2g√

n̄ and 4g√

n̄. We find that the collapse and

revival times for even and odd coherent states are only one half of the corresponding

times for coherent state. For the state |1, 0〉 interacting with even and odd coherent

state of radiation, only mean frequency 4g√

n̄ is obtained and oscillations occur

with large amplitude. For atoms initially in states 1/√

2(|1, 1〉+eiθ|1,−1〉)Kγ(|α〉+eiγ | −α〉), weak double frequency revivals are seen to occur if θ + 2θα 6= π at times

midway between the main revivals. We find that the amplitude of oscillations of

double frequency revival depends on the value of θ+2θα. It is largest for θ+2θα = 0

and disappears for θ + 2θα = π.

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Collapses and Revivals in Two-Level Atoms 2727

In Secs. 2 and 3, we study collapses and revivals for a single two-level atom and

for two two-level atoms in superposed states, respectively, interacting with a single

mode superposed coherent state. In Sec. 4, we discuss our results.

2. Collapses and Revivals in a Single Two-Level Atom in

Superposed State Interacting with a Single-Mode Superposed

Coherent Radiation

For a single two-level atom interacting with a single mode coherent radiation, the

Hamiltonian43,44 is

H = H0 + HI , H0 = HF + HA , (1a, b)

HF = ω0N , HA = ω0Sz , HI = g(aS+ + a+S−) , N = a+a , (2a, b, c)

where subscripts F, A, and I refer to field, atoms, and interaction; N , a and a+

are number, annihilation, and creation operators for radiation, respectively, g is the

coupling constant and S±,z are the Dicke’s collective atom operators44 and ω0 is

the common atomic and radiation frequency. We note that

[H0, HI] = 0 (3)

and hence the complete time evolution operator U = e−iHt can be written as

U = e−iHt = U0UI , U0 = e−iH0t , UI = e−iHt

I (4)

where UI is the time evolution operator in the interaction picture and may be

written as45–47

UI = exp(−iHIt) =

cos gt√

N + 1 −iasin gt

√N√

N

−ia +sin gt

√N + 1√

Ncos gt

√N

. (5)

Let us consider the initial state,

1√2

(∣

∣

∣

∣

1

2,1

2

⟩

+ eiθ

∣

∣

∣

∣

1

2,−1

2

⟩)

Kγ(|α〉 + eiγ | − α) , (6)

where

Kγ =1

√

2(1 + cos γe−2n̄)(7)

is the normalization constant. For γ = 0 or π, the state Kγ(|α〉+eiγ |−α〉) becomes

even or odd coherent state, respectively.

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Using Eqs. (5)–(7) the atomic inversion is given by

〈Sz〉 =e−n̄

4(1 + cos γe−2n̄)

∞∑

n=0

n̄n

n![1 + cos γ(−1)n]

×[

cos 2√

n + 1 gt − cos 2√

n gt + 2 sin(θ + θα)sin 2

√n + 1 gt√

n + 1. (8)

Equation (8) can be broken into two parts and written as

〈Sz〉 = 〈Sz1〉 + sin(θ + θα)〈Sz2〉 , (9)

where

〈Sz1〉 =e−n̄

4(1 + cos γe−2n̄)

∞∑

n=0

n̄n

n![1 + cos γ(−1)n][cos 2

√n + 1 gt − cos 2

√n gt] ,

(10)

〈Sz2〉 =e−n̄

2(1 + cos γe−2n̄)

∞∑

n=0

n̄n

n![1 + cos γ(−1)n]

[

sin 2√

n + 1 gt√n + 1

]

. (11)

We note that while 〈Sz1〉 is a feature in all cases, contribution of 〈Sz2〉 is propor-

tional to sin(θ+θα), and it may be zero for θ+θα = 0 or π. Dependence on γ comes

through the factor [1 + cos γe−2n̄] in the denominator and [1 + cos γ(−1)n] in the

numerator in the summation over n. Since for practical cases of interest, n̄ � 1 and

one may take e−2n̄ ∼= 0 and the factor [1 + cos γe−2n̄] in the denominator becomes

inconsequential.

For γ = 0 or π (i.e., cos γ = 1 or −1 which correspond to the even or odd

coherent states, respectively), summation is over even or odd terms only. It is

seen that in both cases there is no material difference in the envelope, and the

characteristics are rather decided by the fact that the values of n increase in intervals

of two.

Thus, for even and odd coherent states (γ = 0 or π, respectively), 〈Sz1〉 gives

revivals when, for n close to n̄, all terms cos(2√

n + 1 gt) in Eq. (10) contribute in

phase to each other and out of phase to cos(2√

n gt) terms. Thus, we should have all

terms cos(2√

n gt) in Eq. (10) with even n contributing in the same phase and out

of phase to all terms cos(2√

ngt) with n odd. This happens when 2gt√

n̄ + r + 1−2gt

√n̄ + r = (2m+1)π, or gt = gtrevival = (2m+1)π

√n̄, m = 1, 2, . . . which shows

that only odd revivals occur. These odd revivals of 〈Sz1〉 are shown in Fig. 1 the

case, for n̄ = 64, which gives gtrevival = (2m + 1)8π, i.e., odd multiples of 25.12.

For even or odd coherent state (γ = 0 or π, respectively), 〈Sz2〉 gives revivals

when all terms sin(2√

n + 1 gt) in Eq. (11) contribute in phase. Since in the sum-

mation over n, n increases by two, this demands

2gt√

n̄ + r + 2 − 2gt√

n̄ + r = 2mπ

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Fig. 1. Variation of 〈Sz1〉 with coupling time gt for (Kγ/√

2)(|1/2, 1/2〉 + eiθ |1/2,−1/2〉)(|α〉 +eiγ | − α〉) (similar envelope for γ = π).


2)(|1/2, 1/2〉 + eiθ |1/2,−1/2〉)(|α〉 +eiγ | − α〉) (similar envelope for γ = π).

or

gt = gtrevival = mπ√

n̄ ,

which shows all revivals. The revivals of 〈Sz2〉 are shown in Fig. 2 for the typical

case, n̄ = 64, and revivals are seen to occur at gt = 25.12, 50.24, etc. Since 〈Sz〉 =

〈Sz1〉 + sin(θ + θα)〈Sz2〉, for θ + θα = 0 and π, 〈Sz〉 = 〈Sz1〉 and only odd revivals

are seen which have been shown in Fig. 1 and revivals seen at gt = 25.12, 75.36,

etc. Variation 〈Sz〉 = 〈Sz1〉 + 〈Sz2〉 with gt, for θ + θα = π/2 is shown in Fig. 3.

The variation for θ + θα = −π/2 also gives a similar envelope.

For γ = π/2, consider 〈Sz1〉 first. Here, summation in Eq. (10) runs over all

integral values of n. At time gt = (2m+1)π√

n̄, according to what we discussed for

the case γ = 0 or π, terms cos(2√

n gt) contribute in one phase for n even and out

of phase with terms for n odd. For γ = π/2, therefore, the cos(2√

n + 1 gt) term

tends to cancel with cos(2√

n gt) term for each value of n. Hence, we get revivals

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Fig. 3. Variation of 〈Sz〉 with coupling time gt for (Kγ/√

2)(|1/2, 1/2〉 + eiθ |1/2,−1/2〉)(|α〉 +eiγ | − α〉) for (γ, θ + θα) = (0, π/2) (similar envelope are seem for all four cases (γ, θ + θα) =(0,±π/2) and (π,±π/2)).

only at gt = 2mπ√

n̄ at which these two terms have phase difference 2π and hence

support each other. For 〈Sz2〉, similarly, all terms contribute in phase only when,

2gt√

n̄ + r + 1 − 2gt√

n̄ + r = 2mπ or gt = gtrevival = 2mπ√

n̄ .

Thus, for both 〈Sz1〉 and 〈Sz2〉, and therefore, 〈Sz〉 = 〈Sz1〉 + sin(θ + θα)〈Sz2〉,revival time is (2π

√n̄/g) which is double of the earlier obtained revival time π

√n̄/g

for γ = 0 and π. These revivals are shown in Fig. 4 and for θ+θα = π/2 and n̄ = 64.

Revivals are seen to occur at gt = 2mπ√

n̄ = 16mπ = 50.4× m.

The cases for γ = ±π/2 and θ + θα = 0 or π have an additional feature in that

a node occurs at the exact revivals (see Fig. 5). Occurrence of a node in the middle

of revivals, i.e., at gt = 2mπ√

n̄ exactly, for the cases γ = π/2 and θ + θα = 0 or

π (see Fig. 5) can be explained by combining the two cosine terms. We can write


2)(|1/2, 1/2〉 + eiθ |1/2,−1/2〉)(|α〉 +eiγ | − α〉) (similar envelope for θ + θα = −π/2).

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2)(|1/2, 1/2〉 + eiθ |1/2,−1/2〉)(|α〉 +eiγ | − α〉) (similar envelope for θ + θα = π).

〈Sz〉 = 〈Sz1〉 and, for terms with n ∼= n̄, i.e., for the terms that matter, the pairs of

two cosine terms in square brackets in Eq. (10) are of the form

cos 2√

n + r + 1 gt − cos 2√

n + r gt

= cos 2√

n̄

(

1 +r + 1

2n̄

)

gt − cos 2√

n̄(

1 +r

2n̄

)

gt

= sin n̄

(

2 +2r + 1

2√

n̄

)

gt singt

2√

n̄,

which vanishes at exact revival times gt = 2mπ√

n̄. This explains the appearance

of nodes.

3. Collapses and Revivals in Two Two-Level Atoms in Superposed

State Interacting with a Single-Mode Superposed Coherent

Radiation

The time evolution operator for two two-level atoms in the form48 is

UI =exp(−iHIt) =

1 + (N + 1)C(N + 1) −iS(N + 1)a C(N + 1)a2

−ia+S(N + 1) cos(gt√

4N + 2) −iS(N)a

a+2C(N + 1) −ia+S(N) 1 + NC(N − 1)

,

(12)

N = a+a, and functions C(N) and S(N) are defined as

C(N) = [cos(√

4N + 2 gt) − 1]/(2N + 1) ,

S(N) = [sin(√

4N + 2 gt)]/√

(2N + 1) .(13)

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For a system of two two-level atoms in the states |1, 1〉, |1,−1〉, and |1, 0〉 inter-

acting with coherent radiation, it is seen that

〈Sz〉 = e−n̄

∞∑

n=0

n̄n

n![{1 + (n + 1)C(n + 1)}2 − (n + 1)(n + 2){C(n + 1)}2] , (14)

〈Sz〉 = e−n̄

∞∑

n=0

n̄n

n![{n̄C(n + 1)}2 − {1 + nC(n − 1)}2] , (15)

and

〈Sz〉 = e−n̄

∞∑

n=0

n̄n

n![{√

n̄ S(n + 1)}2 − {√

n + 1 S(n)}2] , (16)

respectively.

It has been seen that38,39 the |1,±1〉 give stronger Rabi oscillations with Rabi

frequencies 2g√

n together with weak oscillations with double frequencies 4g√

n.

Revival times for mean frequencies 2g√

n̄ and 4g√

n̄ are 2π√

n̄/g and π√

n̄/g,

respectively. For the state |1, 0〉, however, oscillations with frequency 2g√

n̄ are

suppressed as is clear from Eq. (16) which involves only S2 terms and no S terms.

Behavior of initial states |1, 1〉 and |1, 0〉 are shown in Figs. 6 and 7. State |1,−1〉gives an envelope for collapse and revivals similar to that |1, 1〉.

From Eqs. (14)–(16) and definition Eq. (13), we note the features:

(i) appearance of non-zero average value terms in 〈Sz〉 for the state |1, 0〉, which

is small and becomes noticeable for this state [see Eq. (16)] where amplitude of

oscillations and revivals are small (see, e.g., Fig. 7), and appearance of collapse and

revivals (ii) for mean frequency 2g√

n̄ (for the states |1,±1〉) as well as (iii) for the

mean frequency 4g√

n̄ (for the states |1,±1〉 and |1, 0〉).Revival times of mean Rabi frequencies 2g

√n̄ and 4g

√n̄ for the initial states

|1,±1〉 can be evaluated by considering reinforcement of terms of the form

Fig. 6. Variation of 〈Sz〉 with coupling time gt for |1, 1〉|α〉 (similar envelope for |1,−1〉|α〉).

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Fig. 7. Variation of 〈Sz〉 with coupling time gt for |1, 0〉|α〉.

cos gt√

4n + 2 and cos 2gt√

4n + 2 in 〈Sz〉, respectively. Former (oscillation with

mean frequency 2g√

n̄) gives revivals at gt√

4n̄ + 4r + 6 − gt√

4n̄ + 4r + 2 = 2mπ,

which gives, gtrevival = 2mπ√

n̄, m = 1, 2, . . . .

For latter (oscillations with mean frequency 4g√

n̄) similarly, we get

2gt√

4n̄ + 4r + 6 − 2gt√

4n̄ + 4r + 2 = 2mπ, which leads to gtrevival = mπ√

n̄.

Now, consider initial states |1, 1〉K±(|α〉 ± | − α〉), |1,−1〉K±(|α〉 ± | − α〉), and

|1, 0〉K±(|α〉 ± | − α〉), which leads to

〈Sz〉 = 2K2±e−n̄

∞∑

n=0

n̄n

n![1 ± (−1)n][{1 + (n + 1)C(n + 1)}2

− (n + 1)(n + 2){C(n + 1)}2] , (17)

〈Sz〉 = 2K2±e−n̄

∞∑

n=0

n̄n

n![1 ± (−1)n][{n̄C(n + 1)}2 − {1 + nC(n − 1)}2] , (18)

and

〈Sz〉 = 2K2±e−n̄

∞∑

n=0

n̄n

n![1 ± (−1)n][{

√n̄ S(n + 1)}2 − {

√n + 1S(n)}2] , (19)

respectively, with K± = K0,π and Kγ = 1/√

2(1 + cos γe−2n̄) [see Eq. (7)].

We find that the collapse and revival times for the case of even and odd coherent

states are only one half of the corresponding times for coherent state. It is clear from

a comparison of Figs. 8 and 9 with Figs. 6 and 7. This happens because summation

over n is for values at interval of 2. Revival times for mean frequency 2g√

n̄ (usual

revivals) can be estimated for the state |1, 1〉K±(|α〉 ± | − α〉) using Eq. (17) by

writing gt√

4n̄ + 4r + 14 − gt√

4n̄ + 4r + 6 = 2mπ, which gives, gtrevival = mπ√

n̄,

m = 1, 2, . . . . For revivals of mean frequency 4g√

n̄ (double frequency revivals), we

have 2gt√

4n̄ + 4r + 10 − 2gt√

4n̄ + 4r + 6 = 2mπ, or gtrevival = mπ√

n̄/2. We can

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Fig. 8. Variation of 〈Sz〉 with coupling time gt for K+|1, 1〉(|α〉 + | − α〉) (similar envelope forK−|1, 1〉(|α〉 − | − α〉) and K±|1,−1〉(|α〉 ± | − α〉)).

Fig. 9. Variation of 〈Sz〉 with coupling time gt K+|1, 0〉(|α〉 + | − α〉) (similar envelope forK−|1, 0〉(|α〉 − | − α〉)).

find revival times for the state |1,−1〉K±(|α〉±|−α〉) similarly, and obtain identical

results.

Revival time for the state |1, 0〉K±(|α〉 ± | − α〉), where we have only the mean

frequency 4g√

n̄, can be found using Eq. (19) and demanding that the two cosine

terms are out of phase. This leads to

2gt√

4n̄ + 4r + 6 − 2gt√

4n̄ + 4r + 2 = (2m + 1)π ,

or

gtrevival =1

2(2m + 1)π

√n̄ .

That is, only odd revivals occurs (see Fig. 9). We also note the fact that amplitude

of oscillations at the mean frequency 4g√

n̄ (which is the only mean frequency here)

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is large unlike the previous cases of atomic states |1,±1〉. This can be easily seen

from Eq. (19) by writing the oscillating terms in 〈Sz〉 as

1

2K2

±e−n̄

∞∑

n=0

n̄n

n![1 ± (−1)n]

×[

− n̄

2n + 3cos(2

√4n + 6 gt) +

n + 1

2n + 1cos(2

√4n + 2 gt) . (20)

Let us now consider the collapses and revivals for the initial state, (1/√

2)(|1, 1〉+eiθ|1,−1〉)Kγ(|α〉 + eiγ | − α〉. Straight calculations lead to

〈Sz〉 =e−n̄

2(1 + cos γe−2n̄)

∞∑

n=0

n̄n

n![1 + cos γ(−1)n][2{(n + 1)C(n + 1)}− n{C(n − 1)}

−{n̄2 − n − 1 − 2n̄ cos(θ + 2θα)}{C(n + 1)}2 − {nC(n − 1)}2] . (21)

We can separate the terms involving C(n) and (C(n))2 and write

〈Sz〉 = 〈Sz1〉 + 〈Sz2〉 , (22)

where

〈Sz1〉=e−n̄

(1 + cos γe−2n̄)

∞∑

n=0

n̄n

n![1 + cos γ(−1)n][{(n + 1)C(n + 1)}− n{C(n − 1)}] ,

(23)

〈Sz2〉 =e−n̄

2(1 + cos γe−2n̄)

∞∑

n=0

n̄n

n![1 + cos γ(−1)n]

× [{n̄2 − n − 1 − 2n̄ cos(θ + 2θα)}{C(n + 1)}2 − {nC(n − 1)}2] , (24)

and C(n) is defined by Eq. (13). We see that 〈Sz1〉 gives Rabi oscillations with mean

frequency 2g√

n̄, while 〈Sz2〉 gives Rabi oscillations with double mean frequency

4g√

n̄. These revivals occur at multiples of gt = 25.12 and gt = 12.56 as shown in

Figs. 10 and 11, respectively. We also see the interesting behavior that for 〈Sz1〉,a node occurs at the center of each revival and, for small gt, the amplitude of

oscillations first increases from zero to a maximum and then collapses.

For all values of γ 6= π/2 in summation, values of n change by 2 and hence

revivals for mean frequency 2g√

n̄ occur when gt√

4n̄ + 4r + 14−gt√

4n̄ + 4r + 6 =

2mπ, which gives, gtrevival = mπ√

n̄. For n̄ = 64, revivals are seen to occur at

coupling times which are multiples of π√

n̄ = 25.12 (see Fig. 12).

For γ = π/2, values of n change by one in summation over n and hence revivals

occur when gt√

4n̄ + 4r + 10− gt√

4n̄ + 4r + 6 = 2mπ, giving, gtrevival = 2mπ√

n̄,

and hence for n̄ = 64, revivals occur at coupling times which are multiples of

2π√

n̄ = 50.24 (see Fig. 13).

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Fig. 10. Variation of 〈Sz1〉 with coupling gt for (Kγ/√

2)(|1, 1〉 + eiθ |1,−1〉)(|α〉 + eiγ | − α〉)(similar envelope for γ = π).


2)(|1, 1〉+eiθ |1,−1〉)(|α〉+eiγ |−α〉)(similar envelope for γ = π).


2)(|1, 1〉+ eiθ |1,−1〉)(|α〉+ eiγ | −α〉)(similar envelope for γ = π).

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2)(|1, 1〉 + eiθ |1,−1〉)(α〉 + eiγ | − α〉)(similar envelope for γ = −π/2).

Occurrence of the nodes in the center of revivals of Rabi frequency 2g√

n̄, can

be understood by writing Eq. (21) as

〈Sz〉 = K2±e−n̄

∞∑

n=0

n̄n

n![1 ± (−1)n]

[

n + 1

(2n + 3)

{

cos

(

2√

n̄ +2(n − n̄) + 3

2√

n̄

)

gt

}

− n

(2n − 1)

{

cos

(

2√

n̄ +2(n − n̄) − 1

2√

n̄gt

)}]

.

Since difference of angles in two cosine terms is 2gt/√

n̄ which is a multiple of 2π

at revivals for all cases of γ, the amplitude of oscillations in 〈Sz〉 is nearly

〈Sz〉 = K2±e−n̄

∞∑

n=0

n̄n

n![1 ± (−1)n]

[

n + 1

(2n + 3)− n

(2n − 1)

]

,

≈ K2±e−n̄

∞∑

n=0

n̄n

n![1 ± (−1)n]

[

− 1

n̄

]

≈ − 1

2n̄,

which is very small.

For revivals with double frequency 4g√

n̄, we note that 〈Sz2〉 [see Eq. (24)] can

be further simplified by taking approximation, |α|2 = n̄ � 1 and, we get

〈Sz2〉 ≈e−n̄

2(1 + cos γe−2n̄)

∞∑

n=0

n̄n

n![1 + cos γ(−1)n]

× [(−n)(1 + cos(θ + 2θα))][(C(n − 1))2] . (25)

Equation (25) shows that the amplitude of double frequency revivals depend on

(1+cos(θ +2θα)). Hence, the amplitude of oscillation for double frequency revivals

is largest for θ + 2θα = 0 and disappears for θ + 2θα = π.

For 〈Sz2〉, when γ 6= π/2, values of n increase by 2 in summation and hence

revival occurs when 2gt√

4n̄ + 4r + 14−2gt√

4n̄ + 4r + 6 = 2mπ, or gt = gtrevival =

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(mπ√

n̄/2) For n̄ = 64, the variation shown in Figs. 11 and 12 show first revival at

gt = gtrevival = (π√

n̄/2) = 12.56.

When γ = π/2, values of n increase by one in summation and hence revival occur

when 2gt√

4n̄ + 4r + 10 − 2gt√

4n̄ + 4r + 6 = 2mπ which gives, gt = gtrevival =

mπ√

n̄. For n̄ = 64, Fig. 13 shows the first revival at double frequency at gt = 25.12.

4. Conclusion and Summary

In this paper, we studied collapses and revivals of Rabi oscillations in interaction

of (i) a single two two-level atom having equal weightage of lower and upper states

and (ii) two two-level atom with radiation in superposed state ∼ (|α〉 + eiγ | − α〉).For (i), we find that only odd revivals in special cases γ = 0 or π and θ + θα = 0

or π. We also find that the time of revival for γ = 0 or π, is just one half of the

time for revival for γ = π/2. For γ = π/2 and θ + θα = 0 or π, we also note the

occurrence of nodes in the middle of revivals.

For (ii), in general, Rabi oscillation with mean frequencies 2g√

n̄ and 4g√

n̄

are obtained. For superradiant atomic state, however, the fundamental is absent

and the second harmonic has large amplitude. When radiation is in even or odd

coherent state, the collapse and revival times are exactly one half of those for a

coherent state. Revival time for second harmonic of Rabi frequency is obtained at

one half of that for Rabi frequency. The second harmonic oscillations are absent if

θ + θα = π.

Acknowledgments

We are thankful to Professors N. Chandra and R. Prakash for their interest and

critical comments, and to Dr. R. S. Singh, Dr. D. K. Singh, Dr. Pankaj Kumar,

Mr. D. K. Mishra, Mr. A. Dixit, Miss P. Shukla, Miss Shivani, Miss S. Shukla, Miss

N. Shukla and Mr. Anurag Singh for helpful and stimulating discussions. One of

the authors (HP) is grateful to Indian Space Research Organization, Bangalore for

financial support.

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collapses and revivals in two-level atoms in a superposed state interacting with a single mode...

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