Physics Letters A 376 (2012) 1601–1607

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Physics Letters A

www.elsevier.com/locate/pla

Nonclassicality generated by propagation of atoms through a cavity field

Arpita Chatterjee ∗

School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 26 November 2011Received in revised form 7 March 2012Accepted 29 March 2012Available online 30 March 2012Communicated by P.R. Holland

Keywords:NonclassicalityAtom–cavity interactionCoherent stateThermal state

We successively pass two V -type three-level atoms through a single-mode cavity field. Consideringthe field to be initially in a classical state, we evaluate various statistical properties such as thequasiprobability Q function, Wigner distribution, Mandel’s Q parameter and normal squeezing of theresulted field. We notice that the sequential crossing of atoms induces nonclassicality into the characterof a pure classical state (coherent field). The initial thermal field shows sub-Poissonian as well assqueezing property after interacting with the V atoms.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Generation and manipulation of nonclassical light field hasbeen a major field of interest in quantum optics and quantuminformation processing [1]. The study of these states provides afundamental understanding of quantum fluctuations and opens anew way of quantum communication or imaging beating the stan-dard quantum noise limit. Also nonclassical states have many reallife applications. For example, squeezed states are used to reducethe noise level in one of the phase-space quadratures below thequantum limit [2], entangled states produced in down-conversionprocess are employed to test fundamental quantum features suchas non-locality [3] and to realize quantum information transmis-sion schemes (cryptography [4,5] or teleportation [6]). Quantumsuperpositions of fields with different classical parameters are usedto explore the quantum or classical boundary and the decoherencephenomenon [7].

In this context, theoreticians as well as experimentalists haveproposed various schemes to prepare nonclassical states of opti-cal field. Among them, subtracting photons from and/or addingphotons to traditional quantum states provide an useful way togenerate nonclassical state. Agarwal and Tara [8] first proposeda method for producing the photon-added coherent state. An-other way of creating photon-added or photon-subtracted state isthrough a beam-splitter [9]. Dakna [10] showed that if the ini-tial state and a Fock state are injected at the two input channels,then the photon number counting of the output Fock state reducesthe other output channel into a corresponding photon-added or

* Tel.: +91 11 8527635535.E-mail address: mailtoarpita@rediffmail.com.

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photon-subtracted state. The photon-added coherent states allowone to witness the gradual change from the spontaneous to thestimulated regimes of light emission [11]. Moreover, photon sub-traction can be applied to improve entanglement between Gaus-sian states [12,13], loophole-free tests of Bell’s inequality [14,15]and quantum computing [16].

Single-photon Fock state, a nonclassical state, is an indispens-able resource in an all-optical quantum information processing de-vice [17]. These states can be prepared by controlling the emissionof a single radiator: molecule [18] or quantum dot [19]. In addi-tion, cavity QED experiments in which atoms interact one at a timewith a high Q resonator can be used for Fock state preparation.A one-photon Fock state is created in this way by a π quantumRabi pulse in a microwave cavity [20] or by an adiabatic passagesequence in an optical cavity [21]. We report here how the passageof two V -type [22] three-level atoms transfers the classical cavityfield into a nonclassical one.

We consider here a very basic model to describe the interac-tion of the quantum field with the atom after letting two V -typethree-level atoms successively pass through it. This fundamentalstructure can be framed by adopting a Lindbladian point of view.We can correspond this model to a very simple situation, wherea primary system interacts with a bath of harmonic oscillators atzero temperature, with an interaction Hamiltonian that resemblesthe Jaynes–Cummings format. We can start with the Born–Markovequation and tracing out the bath degrees of freedom, we can ob-tain an equation in the Lindblad form [23]. This interaction causesadditional decoherence which can also be treated by using theLindbladian approach [24].

This Letter is structured as follows: we describe our prob-lem and derive the wave function for the considered atom–cavity system in Section 2. Section 3 concerns with finding the

1602 A. Chatterjee / Physics Letters A 376 (2012) 1601–1607

Fig. 1. Three-level atom in the V-type configuration.

quasiprobability functions of the field left in the cavity. In Sec-tion 4, we study Mandel’s Q parameter and normal squeezing bytaking the quantized initial field in a coherent or thermal state.The last section ends with a summary of the main results of thisLetter.

2. State vector

We begin by considering a V -type three-level atom having itshigher-energy state |e〉 with energy ωe , intermediate-energy state|i〉 with energy ωi and ground-energy state |g〉 with energy ωg .The atom interacts with a single-mode cavity field of frequency γ .�1 (= ωe −ωg −γ ) and �2 (= ωi −ωg −γ ) represent the respec-tive detunings of the transitions |e〉 ↔ |g〉 and |i〉 ↔ |g〉 from thefield mode as shown in Fig. 1. The RWA leads to the interactionHamiltonian (with h = 1) [25]

Hint = g1(aei�1t |e〉〈g| + a†e−i�1t |g〉〈e|)

+ g2(aei�2t |i〉〈g| + a†e−i�2t |g〉〈i|), (1)

where g1, g2 are the atom-field coupling constants, a (a†) are an-nihilation (creation) operators for the field with canonical commu-tation relation [a,a†] = 1. The solution of the Schrödinger equationwith Hamiltonian (1) gives the state vector |ψ(t1)〉 at any timet1 � 0 for the coupled atom–cavity system. We assume that theatom enters the cavity in a coherent superposition of its eigen-kets |e〉 and |i〉 that means |ψa(0)〉 = 1√

2(|e〉 + |i〉) and if initially

the field is in the superposition of photon number states, i.e.|ψ f (0)〉 = ∑

Fn|n〉 with∑

n |F 2n | = 1 then after evolution, the state

vector of the considered atom-field system becomes [26]∣∣ψ(t1)⟩ = ∑

n

[ce,n−1|e,n − 1〉 + ci,n−1|i,n − 1〉 + cg,n|g,n〉], (2)

where

ce,n−1 = −g1√

nB1

[ei(�/2+β1)t1 − 1

(�/2 + β1)− ei(�/2−β1)t1 − 1

(�/2 − β1)

]

+ 1√2

Fn−1,

ci,n−1 = −g2√

nB1

[ei(�/2+β1)t1 − 1

(�/2 + β1)− ei(�/2−β1)t1 − 1

(�/2 − β1)

]

+ 1√2

Fn−1,

cg,n = B1[e−i(�/2−β1)t1 − e−i(�/2+β1)t1

],

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(3)

with �1 = �2 = � and

B1 = −√

n

2

(g1 + g2)

2β1Fn,

β1 =√

�2/4 + n(g21 + g2

2).

⎫⎪⎬⎪⎭ (4)

Later we assume that after interacting with the cavity field fortime t1 the atom exits the cavity in its ground state only [27,28].Then∣∣ψ(t1)

⟩ = ∑cg,n(t1)|g,n〉. (5)

Next we perform the transit of a second identical atom throughthe cavity. Like the previous one, this atom also enters the cavityin the state 1√

2(|e〉+ |i〉) and stays there for time t2. Then for g1 =

g2 = g and for zero detuning, the system evolves to

∣∣ψ(t2)⟩ = ∑

n

[De,n−1(t2)|e,n − 1〉

+ Di,n−1(t2)|i,n − 1〉 + D g,n(t2)|g,n〉], (6)

where

De,n−1(t2) = 1√2

cg,n(t1) cos(√

2ngt2),

Di,n−1(t2) = 1√2

cg,n(t1) cos(√

2ngt2),

D g,n(t2) = −Fn sin(√

2ngt1) sin(√

2ngt2).

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

(7)

This state vector |ψ(t2)〉 describes the time evolution of the wholeatom–field system but we now concentrate on some statisticalproperties of the single-mode field. The field inside the cavity afterdeparting the second atom is obtained by tracing out the atomicpart of ρ(t2) = |ψ(t2)〉〈ψ(t2)| as

ρ f (t2) = Tra[ρ(t2)

], (8)

where we have used the subscript a( f ) to denote the atom (field).This ρ f (t2) will be of consideration throughout the next sec-

tion to determine the statistical properties of the field left into thecavity.

3. Quasiprobability function

The quasiprobability distribution functions are important forthe statistical description of a quantum mechanical state in phasespace. As the position and momentum cannot be defined simul-taneously with infinite precision, the description of a quantummechanical state in phase space is not unique; there is a familyof quasiprobabilities of which the Glauber–Sudarshan P , Husimi Qand Wigner functions are quite well known. But most of thequasiprobabilities involve troublesome integrations over the phasespace variables. The exception is the Q function, a coherent expec-tation of the field density matrix, and is therefore widely used todescribe the field dynamics in situations where the density matrixcan be computed easily.

3.1. Q function

The Q function for the resulted cavity field is defined as thediagonal elements of the density matrix ρ f in the coherent statebasis [29]

Q(α,α∗) = 1

π〈α|ρ f (t2)|α〉. (9)

Given a coherent state |α0〉 or a thermal field ρthm =∑∞n=0

nn

(n+1)n+1 |n〉〈n| as an initial state, the output state after in-

teraction possesses the Q functions respectively

Q coh(α) = 1

πe−(|α0|2+|α|2)

×[∑

n,m

(α0α∗)n−1(α∗

0α)m−1

(n − 1)!(m − 1)!× sin(

√2ngt1) cos(

√2ngt2)

× sin(√

2mgt1) cos(√

2mgt2)

A. Chatterjee / Physics Letters A 376 (2012) 1601–1607 1603

Fig. 2. (Color online.) A graph showing the Q function of the final cavity field for the initial coherent state with α0 = 2 at different times (a) gt1 = gt2 = π/6, (b) gt1 =gt2 = π , (c) gt1 = gt2 = 7π/6 and (d) gt1 = gt2 = 2π .

+∑n,m

(α0α∗)n(α∗

0α)m

n!m! sin(√

2ngt1)

× sin(√

2ngt2) sin(√

2mgt1) sin(√

2mgt2)

], (10)

and

Q thm(α) = 1

π

e−|α|2

(n + 1)

[∑n

1

(n − 1)!(

n|α|2n + 1

)n−1

× sin2(√

2ngt1) cos2(√

2ngt2)

+∑

n

1

n!(

n|α|2n + 1

)n

sin2(√

2ngt1) sin2(√

2ngt2)

].

(11)

In Figs. 2 and 3, we have sketched the mesh plots of the Qfunctions in the complex α-plane if the atom starts from the su-perposition state of |e〉 and |i〉 and the field is either in coherentor in thermal state respectively.

The mean photon number of the initial coherent state is takenas |α0|2 = 4. Fig. 2(a) shows that the Q function initially rep-resents a simple Gaussian distribution centered around (0,0)

whereas Fig. 2(b) presents that in course of time the one-peakedQ distribution is divided into two peaks of similar amplitude butopposite phase. As time goes one of the peak has reduced its

height with respect to the other and then this truncated peak isfragmented into two small parts [see Figs. 2(c) and (d)]. Thus Fig. 2describes the deformation of the Q function for different values ofgt1 and gt2.

Numerical result for Eq. (11) is presented in Fig. 3, where wehave plotted the Q function of the initial thermal field as a func-tion of α. By taking a fixed interaction time 2π/3, a hollowed-peak Gaussian structure is obtained. It is interesting to note thatthe increase of mean photon number of the thermal field doesnot change the shape of the Q function much more. It slightlydecreases the Gaussianity of the state but broadens the peak abit.

3.2. Wigner distribution

In this section, we analyze how the classical behavior of thecoherent or the thermal field inside the cavity is affected by thepassage of two identical atoms by considering the phase-spacemeasure. For any state having density matrix ρ in the Fock statebasis

∑n,m Cn,m|n〉〈m|, the Wigner function is defined by [27]

W(α,α∗) = 2

π2e2|α|2

∫〈−γ |ρ|γ 〉e−2(γ α∗−γ ∗α)d2γ

= 2

πe2|α|2 ∑

n,m

Cn,m(−1)n+m

2n+m

∂n+m

∂αn∂α∗me−4|α|2 . (12)

1604 A. Chatterjee / Physics Letters A 376 (2012) 1601–1607

Fig. 3. (Color online.) The Q -distribution for the initial thermal field is plottedagainst the parameters Re(α) and Im(α) along x and y axes respectively withgt1 = gt2 = 2π/3, (a) n = 2, (b) n = 12.

3.2.1. Coherent stateIf the radiation field is initially in a coherent state |α0〉, then

integral (12) together with Eq. (8) yields the Wigner function as

Wcoh(α) = 2

πe−(|α0|2+2|α|2)

[∑n,m

2n+m−2

(n − 1)!(m − 1)!× (

α0α∗)n−1(

α∗0α

)m−1sin(

√2ngt1) cos(

√2ngt2)

× sin(√

2mgt1) cos(√

2mgt2)

+∑n,m

2n+m

n!m!(α0α

∗)n(α∗

0α)m

sin(√

2ngt1)

× sin(√

2ngt2) sin(√

2mgt1) sin(√

2mgt2)

]. (13)

Fig. 4 elaborates the Wigner function (13) for a fixed α0 = 2and for different values of gt1 and gt2. It is known that the par-tial negativity of the Wigner function is a sufficient condition fortracing out the nonclassicality of quantum states. In Fig. 4(a), thenegative part is slightly noticeable for gt1 = π/2 and gt2 = 3π/2.The Wigner function includes more pronounced negative dips astime parameter changes from Fig. 4(b) to Fig. 4(d) and the posi-tive multi-peak structure of the Wigner function disappears grad-ually. That means the classical nature of the initial coherent fieldis washed out by the crossing of two atoms in sequence.

3.2.2. Thermal stateA thermal state ρthm with average photon number n, acting as

an input state, results the Wigner distribution

Wthm(α) = 2

π

1

(n + 1)e−2|α|2

×[∑

n

(4n|α|2n + 1

)n−1

sin2(√

2ngt1) cos2(√

2ngt2)

+∑

n

(4n|α|2n + 1

)n

sin2(√

2ngt1) sin2(√

2ngt2)

]. (14)

Eq. (14) is just the analytical expression of the Wigner functionwith mean thermal photon number n. Unlike the coherent stateinput, this function has no negative domain.

4. Statistical properties

Next we investigate two observable nonclassical effects, sub-Poissonian photon statistics and quadrature squeezing. First to de-termine the photon statistics of a single-mode radiation field, weconsider Mandel’s Q parameter defined by [30]

Q M = 〈a†2a2〉 − 〈a†a〉2

〈a†a〉 . (15)

For −1 � Q M < 0 (Q M > 0), the statistics is sub-Poissonian(super-Poissonian); Q M = 0 stands for Poissonian photon statistics.To examine the statistical condition of the resulted cavity field, weobtain

⟨a†a

⟩ = 2∑

n

(n − 1)|Fn−1|2 sin2(√

2ngt1) cos2(√

2ngt2)

+∑

n

n|Fn|2 sin2(√

2ngt1) sin2(√

2ngt2),

and

⟨a†2

a2⟩ = 2∑

n

(n − 2)(n − 1)|Fn−1|2 sin2(√

2ngt1) cos2(√

2ngt2)

+∑

n

(n − 1)n|Fn|2 sin2(√

2ngt1) sin2(√

2ngt2),

where |Fn|2 stands for the initial photon distribution.In order to see the variation of the Q M parameter with α0 (co-

herent field) or n (thermal field), we plot the Q M function againstthe scaled time gt in Fig. 5. Q M always exhibits sub-Poissoniancharacter for both the input states and increases its negativity asα0 (n) increases. This implies that the nonclassicality is enhancedby increasing the initial photon number. We should emphasize thatthough the thermal field has no negative Wigner function but itdisplays the sub-Poissonian property.

Secondly, to analyze the squeezing properties of the radiationfield we introduce two hermitian quadrature operators

X = a + a†, Y = −i(a − a†). (16)

These two quadrature operators satisfy the commutation re-lation [X, Y ] = 2i and, as a result, the uncertainty principle(�X)2(�Y )2 � 1. A state is said to be squeezed if either (�X)2 or(�Y )2 is less than 1. To review the principle of quadrature squeez-ing [31], we define an appropriate quadrature operator [32]

Xθ = X cos θ + Y sin θ = ae−iθ + a†eiθ . (17)

The squeezing of Xθ is characterized by the condition 〈: (�Xθ )2 :〉

< 0 where the double dots denote the normal ordering of opera-

A. Chatterjee / Physics Letters A 376 (2012) 1601–1607 1605

Fig. 4. (Color online.) Three-dimensional plot of W (α) when the field is initially in coherent state, using parameters α0 = 2 and (a) gt1 = π/2, gt2 = 3π/2, (b) gt1 = π/2,gt2 = 5π/2, (c) gt1 = 3π/2, gt2 = 7π/2 and (d) gt1 = 5π/2, gt2 = 7π/2.

Fig. 5. (Color online.) Mandel’s Q M as a function of gt1 = gt2 = gt and with (a) α0 = 1,2 and 3 (from upper to lower curves) for a coherent state input and (b) n = 1,2,3(from upper to lower curves) for a thermal state input.

tors. After expanding the terms of 〈: (�Xθ )2 :〉 and minimizing its

value over the whole angle θ , one can get [33]

Sopt = ⟨: (�Xθ )2 :⟩min

= −2∣∣⟨a†2⟩ − ⟨

a†⟩2∣∣ + 2⟨a†a

⟩ − 2∣∣⟨a†⟩∣∣2

. (18)

For the atomic system under consideration, 〈a†a〉 has been derivedearlier and the other parameters of Eq. (18) are given as

⟨a†⟩ = 2

∑n

√n Fn−1 F n+1 sin(

√2n − 2gt1)

× sin(√

2n + 2gt2) cos(√

2n − 2gt1) cos(√

2n + 2gt2)

1606 A. Chatterjee / Physics Letters A 376 (2012) 1601–1607

Fig. 6. (Color online.) Contour plot for Sopt as a function of gt1 = gt2 = gt; (a) α0 (coherent state) and (b) n (thermal state).

+∑

n

√n + 1 Fn F n+2 sin(

√2n − 2gt1) sin(

√2n − 2gt2)

× sin(√

2n + 4gt1) sin(√

2n + 4gt2)

and

⟨a†2⟩ = 2

∑n

√n√

n + 1 Fn−1 F n+2

× sin(√

2n − 2gt1) sin(√

2n + 4gt2) cos(√

2n − 2gt1)

× cos(√

2n + 4gt2) +∑

n

√n + 1

√n + 2Fn F n+3

× sin(√

2n − 2gt1) sin(√

2n − 2gt2)

× sin(√

2n + 6gt1) sin(√

2n + 6gt2).

Substituting the above expectation values in Eq. (18) we obtainlengthy expressions of Sopt for the initial coherent (thermal) state

when Fn = e−|α0|2/2 α0n√

n! ( nn

(n+1)n+1 ).

Fig. 6(a) presents the contour plot of Sopt as a function of α0and gt . One can see clearly that the traveling of two atoms throughthe cavity inject the squeezing property into the coherent statecharacter. In addition, the field transferred from the initial thermalfield also depicts squeezing effect [see Fig. 6(b)].

5. Conclusion

In this Letter, we have proposed to fly two three-level atomsone after one through the cavity field. Incorporating zero detun-ing assumption in the atom-field coupling, an analytical expressionfor the quasi-Q -distribution is derived when the field is initiallyprepared either in a coherent or in a thermal state. It has beenshown that the movement of atoms slightly changes the shape ofthe quasiprobability Q function for both the input states. Further-more, the nonclassicality of the resulted field is discussed in termsof the negativity of the Wigner function, Mandel’s Q parameterand the quadrature squeezing. The Wigner function of the coher-ent state input always exhibits partially negative region which isa clear evidence of its nonclassical behavior. Figs. 4(a)–(d) demon-strate that the negativity of W function gradually increases withtime. As a consequence of the nonclassicality, the coherent fielddepicts sub-Poissonian photon-number distribution and quadraturesqueezing.

In addition, Wigner function does not always indicate a neg-ative value for the nonclassical state, e.g. the squeezed state is

usually considered as a typical nonclassical state since its quadra-ture noise is less than that of the vacuum, but its Wigner functionis regular and positive [34]. While for the thermal description ofthe initial field, we have obtained an almost similar case. Theresulted field owns a positive Wigner function but exhibits sub-Poissonian photon statistics and squeezing. In particular, the out-put state with the input coherent (thermal) state achieves bettersub-Q property for better coherent amplitude α0 (average photonnumber n).

Acknowledgement

A.C. thanks National Board of Higher Mathematics, Departmentof Atomic Energy, India for the financial support.

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