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Optics Communications 234 (2004) 281–294

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Squeezing spectra in a V-type three-level atominteracting with a broadband squeezed field

M.A. Ant�on, F. Carre~no *, Oscar G. Calder�on

Escuela Universitaria de �Optica, Universidad Complutense de Madrid, C/Arcos de Jal�on s/n, 28037 Madrid, Spain

Received 17 December 2003; received in revised form 30 January 2004; accepted 2 February 2004

Abstract

We investigate the effect of quantum interference in the phase-dependent spectra of a V-type three-level atom where

both transitions from the upper states to the lower state are driven by a single coherent field. In addition, the atom is

damped by a broadband squeezed field. We are particularly interested in the role of the vacuum induced coherence on

the spectrum of quantum fluctuations in phase quadratures of resonance fluorescence. We find that in some circum-

stances, the combined effect of the squeezed field and the vacuum induced coherence (VIC) can significantly modify the

nonclassical behavior of the system. In particular, it is possible to enhance the squeezing in marked contrast with

the case where the atom is damped by the standard vacuum. The squeezing spectrum can be modified by changing the

relative phase of the squeezed vacuum to the external coherent field.

� 2004 Elsevier B.V. All rights reserved.

PACS: 42.50.Hz; 42.50.Gy; 42.50.Dv; 32.80.)t

Keywords: Vacuum induced interference; Squeezed vacuum; Resonance fluorescence

1. Introduction

Resonance fluorescence of a driven atom has

been probed as a source of nonclassical light and amanifold of interesting phenomena have been ob-

served, including photon antibunching [1–3], sub-

* Corresponding author. Tel.: +34-91-394-6856; fax: +34-91-

394-6885.

E-mail addresses: [email protected] (M.A. Ant�on),

[email protected] (F. Carre~no), [email protected]

(O.G. Calder�on).

0030-4018/$ - see front matter � 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.optcom.2004.02.004

Poissonian statistics [4], atomic coherences [5],

among others. Squeezing spectra in resonance

fluorescence [6] has also been predicted by Walls

and Zoller [7]. Collet et al. [8] introduced an op-erational definition of the spectral squeezing and

showed that for a weak coherent excitation, the

fluorescence emitted from a two-level atom is

squeezed, reflecting nonclassical atomic polariza-

tion fluctuations. In contrast to fluorescence

spectra that are detected without phase sensitivity,

squeezing spectra are a phase-dependent resonance

fluorescence spectra, which are obtained by hom-odyne detection of scattered radiation from free

ed.

mail to: [email protected]

282 M.A. Ant�on et al. / Optics Communications 234 (2004) 281–294

atoms that are driven by a resonant field. In these

experiments, the scattered radiation field of the

atoms, E, is mixed with a local oscillator (LO)

field, jELOjeih, having a controllable fixed phase h,relative to the driving field. Thus, the signal P ðtÞreaching the detector is proportional to the inter-ference term between the LO and the scattered

field, that is

P ðtÞ / jELOjxh; ð1Þ

where xh is a quadrature of the scattered field given

by xh ¼ 1=2ðe�ihE þ eihEyÞ. Squeezing in the radi-

ation field can be measured by analyzing the

fluctuations in the detected power. In this way, the

normally ordered variances of the quadrature

components either in total phase quadratures or infrequency components (squeezing spectrum) can

be obtained. In the last case, the field has to be

frequency-filtered and then homodyned with the

local oscillator field [9].

In spite of receiving considerable attention,

squeezing in resonance fluorescence has eluded

experimental observation, one of the problems

being that atomic motion produces phase shiftswhich destroy squeezing [10]. However, in a very

recent work [11], precision near-resonant phase-

dependent spectra have been obtained by using a

novel homodyne detection technique that suppress

excess noise by substracting transmitted power

signals from two identically prepared atomic

samples. With this scheme, Lu et al. [11,12] have

found for the first time, some evidence of squeez-ing by measuring the phase-dependent fluores-

cence spectra of coherently driven 174Yb atoms at a

phase near �45� relative to the exciting field. This

important result have renewed the exploration of

phase-dependent spectra of resonance fluorescence

not only in two-level atoms, but in three or mul-

tilevel atoms. The reason relies in the fact that in

this type of atoms electromagnetically inducedtransparency (EIT) and quantum interference

phenomena can lead transparency, eliminating

absorption and dispersion in the atomic system

which are sources of noise. Besides, the nonlinear

optical response in EIT based schemes is highly

enhanced, thus detailed studies of the phase-de-

pendent noise in these systems are of interest in

order to explore the quadrature squeezing previ-

ously found in two-level atoms [13]. Gao et al. [14]

investigated how quantum interference affects

squeezing spectra in a three-level V-type atom.

They found that for weak excitation, quantum

interference can enhance squeezing. In the present

work, we extend the model analyzed by Gao et al.[14] by considering that the V-atom is damped by a

multimode squeezed reservoir.

The interaction of atomic systems with

squeezed light has been subject to intense activity

since Gardiner [15] showed that the two dipole

quadratures of a two-level atom interacting with a

squeezed vacuum decay at different rates. Also

very recently, Zhou and Swain [16] have found intwo-level atoms that squeezing in resonance fluo-

rescence can be greatly enhanced in a frequency-

tunable cavity [16] or in a squeezed vacuum [17].

Ficek and Swain [18] also showed that a 100%

squeezed output can be obtained in the fluorescent

light from a coherently driven two-level atom in-

teracting with a squeezed field. The effects of a

broadband squeezed vacuum on three-level atomsat different configurations (K, V and ladder con-

figurations) have also been investigated [19–25]. In

these systems there are noticeable changes in the

steady-state populations with regard to the case in

which the atom is damped by standard vacuum

[19–22]. Further work has also been done to study

the resonance fluorescence spectra of three-level

atoms interacting with two coherent lasers and twoindependent squeezed vacua [23–25]. Ferguson

et al. [24] have examined the fluorescence spectrum

for a strongly driven three-level system in which

one of the two one-photon transitions is coupled

to a finite-bandwidth squeezed vacuum field. The

work of Dalton et al. [26] presents a comprehen-

sive review of this topic.

All previous works related to the interaction V-type systems with squeezed fields have been done

in the case of V-type atoms with large upper sub-

level splitting. In this case, the coherence between

upper sublevels is reduced by spontaneous emis-

sion to the ground level. However, it is well-known

how quantum coherence can be created in inter-

actions involving a common bath with a set of

closely lying states. This rather contraintuitivephenomenon has been explained by Agarwal [27].

If the two upper levels are very close and damped

∆3

∆2

γ3

γ2

1

3

2

ωL

Fig. 1. A V-type atom driven by a single-mode laser of fre-

quency xL. c3 and c2 are the decay rates from the excited sub-

levels to the ground level and D3 and D2 are the detunings of the

coherent field with the state j3i and j2i, respectively.

M.A. Ant�on et al. / Optics Communications 234 (2004) 281–294 283

by the usual vacuum interactions, spontaneous

emission cancellation can take place [28–31], which

offers the possibility to trap the population in the

excited levels when some particular conditions

hold [32–35]. Recently, we have analyzed the role

of the quantum interference and the squeezedvacuum on the behavior of V-type atoms driven by

coherent fields [36–38].

It is well known that in two-level atoms, the

fluorescence field quadrature components depend

on the difference between the upper-level popula-

tion and the square of the dipole moment, then it

was claimed [10,39–41] that three-level K atoms

would be more suitable than two-level atoms dueto the possibility for reducing the upper-state

population via coherent population trapping ef-

fect. Motivated by this question, we discuss

squeezing in terms of the squeezing spectrum ra-

ther than in terms of the variance of the quadra-

ture components of the total fluorescent field

scattered by a V-type atom driven by a single

monochromatic coherent field. We assume that thebandwidth of the squeezed vacuum is much larger

than both natural linewidths of the transitions and

the frequency separation of the two upper levels,

that is, a single broadband squeezed vacuum is

coupled to both transitions. Specifically, we ana-

lyze how the combination of quantum interference

and the squeezed vacuum can modify the squeez-

ing spectrum.The paper is organized as follows: Section 2

establishes the model, i.e., the Hamiltonian of the

system and the evolution equation of the atomic

operators assuming the rotating wave approxi-

mation. Section 3 is devoted to present the basic

equations for the phase-dependent resonance

fluorescence spectrum. We present the numerical

results in Section 4, and finally, Section 5 sum-marizes the conclusions.

2. The model

We consider a V-type atom consisting of two

upper sublevels j3i, j2i coupled to a single ground

level j1i by a single-mode laser field E. The energy-level scheme is shown in Fig. 1. The external co-

herent field E is taken linearly polarized and

propagates in the z direction. The laser field E is

given by

E ¼ 1

2~EðtÞei/Le�ixLt þ c:c:; ð2Þ

where~EðtÞ and /L being the amplitude and phase of

the slowly varying field envelope, respectively, and

xL the angular frequency of the field. Spontaneous

and stimulated emission between these states are

governed by the interaction of the atom with a res-

ervoir in a multimode squeezed state. In order toanalyze the coherence effects induced by spontane-

ous emission, the two upper levels j3i and j2i arecoupled by the same vacuum modes to the ground

level j1i. The resonant frequencies between the up-

per levels j3i, j2i and the ground level j1i are x31,

x21, respectively. Note that x31 � x21 ¼ x32, x32

being the frequency separation of the excited levels.

The Hamiltonian of the system in the rotatingwave approximation is given by [42,27]

H ¼ �hX3m¼1

xmjmihmj þ �hXkk

xkkaþkkakk

� �hX3m¼2

Xkk

gmkjmih1jakk �H:c:

� �hX3m¼2

Xmei/Le�ixLtjmih1j �H:c:; ð3Þ

where �hxm being the energies of the atomic levels,

akk (aþkk) is the annihilation (creation) operator of

the kth mode of the vacuum field with polarization


~ekk (k ¼ 1; 2) and angular frequency xkk. The pa-

rameter gmk is the coupling constant of the atomic

transition jmi ! j1iwith the electromagnetic mode

gmk ¼ffiffiffiffiffiffiffiffiffiffiffiffixkk

2�h�0V

rð~l1m �~ekkÞ; ð4Þ

where ~l1m is the dipolar moment of the transition

jmi ! j1i, and Xm ¼~l1m �~E=ð2�hÞ is the Rabi fre-

quency of the transition jmi ! j1i. V is the modevolume of the field.

We now assume that the quantized radiation

field is in a broadband squeezed vacuum state with

carrier frequency xv which is tuned closed to the

frequency of the atomic transitions j3i ! j1i and

j2i ! j1i, that is 2xv ’ x31 þ x21. The bandwidth

of the squeezing is assumed to be broad enough so

that the squeezed vacuum appears as d-correlatedsqueezed white noise to the atom. The correlation

function for the field operators aðxkkÞ and aþðxkkÞcan be written as [15,19]

haðxkkÞaþðx0kkÞi ¼ ½NðxkkÞ þ 1�dðxkk � x0

kkÞ;haþðxkkÞaðx0

kkÞi ¼ NðxkkÞdðxkk � x0kkÞ;

haðxkkÞaðx0kkÞi ¼ MðxkkÞdð2xv � xkk � x0

kkÞ;ð5Þ

where NðxkkÞ and MðxkkÞ are slowly varying

functions of the frequency and characterize the

squeezing. The following inequality holds:

jMðxkkÞj2 6NðxkkÞNð2xv � xkkÞþmin½NðxkkÞ;Nð2xv � xkkÞ�: ð6Þ

Note that M is a complex magnitude so that

MðxkkÞ ¼ jMðxkkÞjei/v , where /v is the phase of the

squeezed vacuum. For MðxkkÞ ¼ 0, Eq. (5) de-

scribes a thermal field at finite temperature T ,NðxkkÞ being the mean occupation number of the

mode kk with frequency xkk.

The system is studied using the density-matrix

formalism. By following the traditional approach

of Weisskopf and Wigner [27,42], we obtain the

evolution equations of the density matrix elements

(see Appendix A)

oq33

os¼ �ðN þ 1Þ 2

aq33 � ðN þ 1Þp q23ð þ q32Þ

þ 2Nq11 þ ix3q13 � ix3q31; ð7Þ

a

oq22

os¼ �2ðN þ 1Þaq22 � ðN þ 1Þp q32ð þ q23Þ

þ 2Naq11 þ ix2q12 � ix2q21; ð8Þ

oq31

os¼ � F31q31 � ðN þ 1Þpq21 � 2jM jeiU 1

aq13

�þ pq12

�� ix3 2q33ð þ q22 � 1Þ � ix2q32; ð9Þ

oq21

os¼ � F21q21 � ðN þ 1Þpq31 � 2jM jeiU aq12ð þ pq13Þ

� ix2 q33ð þ 2q22 � 1Þ � ix3q23; ð10Þ

oq23 ¼ �F23q23 � ðN þ 1Þp q22ð þ q33Þ
osþ 2Npq11 þ ix2q13 � ix3q21; ð11Þ
where we have introduced a dimensionless time s,and dimensionless variables

s ¼ffiffiffiffiffiffiffiffic2c3

p

2t; ð12Þ

a ¼ffiffiffiffic2c3

r; ð13Þ

xj ¼2ffiffiffiffiffiffiffiffic2c3

p Xj; ð14Þ

p ¼ ~l13 �~l12

j~l13jj~l12j¼ cos b; ð15Þ

F31 ¼ ðN�

þ 1Þ 1aþ N a

�þ 1

a

�þ i

2ffiffiffiffiffiffiffiffic3c2

p D3

�;

ð16Þ

F21 ¼ ðN�

þ 1Þaþ N a

�þ 1

a

�þ i


p D2

�; ð17Þ

F23 ¼ ðN�

þ 1Þ a

�þ 1

a

�þ i


p ðD2 � D3Þ�;

ð18Þwhere Fij ¼ F �

ji . The coefficients c2 and c3 are the

decay rates for the j2i ! j1i and j3i ! j1i transi-tions, respectively. Dm � xm1 � xL is the laser de-

tuning from the resonance with state jmi(m ¼ 2; 3). The phase U � 2/L � /v, is the phase

difference between the coherent field and thesqueezing field. The terms related to p ¼ cos b in

Eqs. (7)–(11) represent the effect of the quantum

interference arising from the cross coupling be-

tween spontaneous emissions j3i ! j1i and


j2i ! j1i. They depend on the relative direction of

the two transition dipole moments, i.e., the angle bbetween the dipole moments. The case of maxi-

mum quantum interference takes place with par-

allel dipole moments, where p is equal to unity,whereas p ¼ 0 represents the case with orthogonal

dipole moments (no quantum interference). On the

other hand, we can see that coherences (q31 and

q21) depend on the correlations between pairs of

modes in the reservoir so this leads to a phase

sensitivity in the population decay. This is in

contrast with other treatments of V-type atoms

[20]. Moreover, this dependence is reinforced bythe presence of the quantum interference: see the

term �2pjM jeiUq12 in Eq. (9) and the term

�2pjM jeiUq13 in Eq. (10). The set of Eqs. (7)–(11)

can be written in a compact form as

dqds

¼ Bqþ C ð19Þ

with q defined as

q ¼ hr31ðtÞi; hr13ðtÞi; hr33ðtÞi; hr21ðtÞi;½hr12ðtÞi; hr22ðtÞi; hr32ðtÞi; hr23ðtÞi�T; ð20Þ

where B being a 8� 8 time independent time

evolution matrix and C is a constant vector, both

given by the corresponding coefficients of the Eqs.(7)–(11) (note that hrklðtÞi ¼ qlk).

3. Phase-dependent resonance fluorescence spectrum

Based in the Bloch equations (7)–(11), we may

calculate the optical properties of light scattered

by the atom such as the normally ordered fieldvariance or the squeezing spectrum. For this pur-

pose, the electric field operator at the observation

point~r is required. We introduce a slowly varying

electric-field operator with phase h,

~Ehð~r; tÞ ¼1

2~Eþ

h ð~r; tÞeiðxLtþhÞ þ 1

2~E�

h ð~r; tÞe�iðxLtþhÞ

¼ ~E1ð~r; tÞ cos hþ~E2ð~r; tÞ sin h; ð21Þ

where

~E1ð~r; tÞ ¼1

2~Eþ

h ð~r; tÞeixLt þ 1

2~E�

h ð~r; tÞe�ixLt; ð22Þ

~E2ð~r; tÞ ¼i

2~Eþ

h ð~r; tÞeixLt � i

2~E�

h ð~r; tÞe�ixLt ð23Þ

are the in-phase and out-of-phase quadratures of

the fluorescent field relative to the coherent driving

field, respectively. The positive frequency part of

the fluorescent light emitted by the atom in the

radiation zone takes the form [29]

~Eþh ð~r; tÞ ¼ � 1

c2rx2

31~nh

� ~n�

�~l13

�r13ðt0Þ

þ x221~n� ~n

��~l12

�r12ðt0Þ

ie�ixLt0 ; ð24Þ

where t0 ¼ t � r=c is the retarded time, and ~n is a

unit vector in the direction of observation. Sincethe atomic system is nearly degenerate, we assume

that x31 ’ x21, and ~n is perpendicular to the

atomic dipole moments ~l13 and ~l12, thus Eq. (24)

can be rewritten as

~Eþh ð~r; tÞ ¼ f ðrÞ ~l13r13ðt0Þ

hþ~l12r12ðt0Þ

ie�ixLt0 ; ð25Þ

where f ðrÞ ’ x231=c

2r. Squeezing is characterized

by the condition that the normally ordered vari-

ance h: ðDEhÞ2 :i of the electric-field quadraturecomponent Eh is negative. The normally ordered

variance of Eh was defined in [7] as

h: ðDEhð~rÞÞ2 :i ¼1

2p

Z 1

�1dxZ 1

�1dse�ixsT

� h: Ehð~r; tÞ;Ehð~r; t þ sÞ :i; ð26Þ

where hA;Bi ¼ hABi � hAihBi, and T being thetime ordering operator. Following to Collet et al.

[8], we consider the squeezed spectral density de-

fined as

h: Sð~r;x; hÞ :i ¼ 1

2p

Z 1

�1dse�ixsT

� h: Ehð~r; tÞ;Ehð~r; t þ sÞ :i: ð27Þ

Taking into account the action of the time order-

ing operator T , the function under the integral sign

of Eq. (27) reduces to


T h: Ehð~r; tÞ;Ehð~r; t þ sÞ :i

¼ 1

4½h~Eþ

h ð~r; t þ sÞ;~Eþh ð~r; tÞieixLð2tþsÞei2h

þ h~E�h ð~r; t þ sÞ;~E�

h ð~r; tÞie�ixLð2tþsÞe�i2h

þ h~Eþh ð~r; t þ sÞ;~E�

h ð~r; tÞieixLs

þ h~E�h ð~r; t þ sÞ;~Eþ

h ð~r; tÞie�ixLs�: ð28Þ

Inserting the positive and negative parts of the

fluorescent field (25) into Eq. (28), we can express

the spectrum as

h: Sð~r;x; hÞ :i

¼ f ðrÞ2

4pR

Z 1

�1dse�ixs½ðl2

13hr13ðt þ sÞ; r13ðtÞi

þ~l13 �~l12hr12ðt þ sÞ; r13ðtÞiþ~l13 �~l12hr13ðt þ sÞ; r12ðtÞiþ l2

12hr12ðt þ sÞ; r12ðtÞiÞeið2hþxLr=cÞ

þ l213hr31ðt þ sÞ; r13ðtÞi

þ l212hr21ðt þ sÞ; r12ðtÞi

þ~l13 �~l12hr21ðt þ sÞ; r13ðtÞiþ~l13 �~l12hr31ðt þ sÞ; r12ðtÞi�: ð29Þ

In order to obtain the two-time correlation func-

tions in Eq. (29), we introduce the deviation Drij of

the dipole polarization operator from its mean

steady-state value

Drij ¼ rijðt0Þ � hrijð1Þi; i; j ¼ 1; 2; 3; ð30Þwhich, obviously satisfies

dhDrijðsÞids

¼ BhDrijðsÞi; i; j ¼ 1; 2; 3; ð31Þ

B being the matrix defined in Eq. (19). The two-time correlation function of the deviations can be

obtained by invoking the quantum regression

theorem [42] together with the optical Bloch

equations (7)–(11). To do this, it is practical to

define the column vector

U ij ¼ hDr31ðsÞDrijð0Þi; hDr13ðsÞDrijð0Þi;�hDr33ðsÞDrijð0Þi; hDr21ðsÞDrijð0Þi;hDr12ðsÞDrijð0Þi; hDr22ðsÞDrijð0Þi;hDr32ðsÞDrijð0Þi; hDr23ðsÞDrijð0Þi

;

i; j ¼ 1; 2; 3: ð32ÞAccording to the quantum regression theorem and

Eq. (31), the vector U ij satisfies

dU ijðsÞds

¼ BU ijðsÞ; j ¼ 1; 2; 3: ð33Þ

By following the same procedure as described in

[14], and working in Laplace space we obtain the

steady-state fluorescence spectrum

h: Sð~r;x; hÞ :i

¼ f ðrÞ2

4pR

Xk¼8

k¼1

R1kðizÞ c3U13k ð0Þ

�h(þ c21U

23k ð0Þ

�þ R2kðizÞ c2U

23k ð0Þ

�þ c32U

13k ð0Þ

�iei2ðhþxLr=cÞ

þXk¼8

k¼1

R1kðizÞ c3U31k ð0Þ

�hþ c21U

32k ð0Þ

�

þ R2kðizÞ c2U32k ð0Þ

�þ c32U

31k ð0Þ

�i); ð34Þ

where U ijmð0Þ stands for the steady-state compo-

nent of the vector U ijmðsÞ. RjkðizÞ is the ðj; kÞ ele-

ment of the matrix RðizÞ ¼ ½ðizI � BÞ�1þð�izI � BÞ�1�, I being the 8� 8 identity matrix,

z � 2iðx� xLÞ=ffiffiffiffiffiffiffiffic2c3

p, and c23 ¼

ffiffiffiffiffiffiffiffic2c3

pp.

4. Squeezing spectra of the fluorescence field

quadratures. Numerical results

In this section we will analyze numerically theinterplay between the squeezed vacuum field and

the quantum interference on the behavior of the

squeezing spectra. In this analysis we consider, for

simplicity, c2 ¼ c3 � c [43], and j~l31j ¼ j~l21j, thusx2 ¼ x3 � x. We assume a perfect squeezing con-

dition, i.e., jM j ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNðN þ 1Þ

p, and the squeezed

phase U ¼ 0. We also assume e2ixr=c ¼ 1, and scale

the squeezing spectrum by l231f

2ðrÞ=ð2pcÞ. In thefollowing, we consider the external coherent field

on resonance with transition j3i ! j1i, i.e., D3 ¼ 0

(D2 ¼ �x32). We will also consider the cases in

which the quantum interference is absent (p ¼ 0)

and when quantum interference is nearly maximal

with p ¼ 0:99. Note that with p ¼ 1 (maximum

quantum interference) the resonance fluorescence

signal is completely quenched when the atom isdamped by the standard vacuum. In both cases

(p ¼ 0, and p ¼ 0:99) the influence of the input

squeezed field on resonance fluorescence signal will


be analyzed. In the rest of the work, x stands for

2ðx� xLÞ=c.Let us start by considering the squeezing spec-

trum of the out-of-phase quadrature for the de-

generate case (x32 ¼ 0). The squeezing spectrum is

displayed in Fig. 2(a) for a weak driving field. Notethat squeezing occurs in the vicinity of x ¼ 0 when

the atom is damped by standard vacuum (N ¼ 0)

and quantum interference is absent/present (da-

shed-dotted/solid line). Furthermore, the level of

squeezing is larger in the absence of quantum in-

terference in agreement with Gao et al. (see

Fig. 1(a) in [14]). In this regime of Rabi frequen-

cies, the input squeezed vacuum destroys thenonclassical features present in the standard vac-

uum case when considering or not quantum in-

terference (dashed/dotted line, respectively).

Besides, squeezing spectrum in the p ¼ 0 case dis-

plays the typical Lorentzian lineshape (dotted

line), whereas in the p ¼ 0:99 case the squeezing

spectrum is formed by the combination of two

Lorentzians curves with very different linewidths(dashed line). By increasing the Rabi frequency to

moderate values, the maximum of squeezing shifts

to the wings of the spectrum (see Fig. 2(b)). Note

Fig. 2. Squeezing spectra of the out-of-phase quadrature

Sp=2ðxÞ as a function of x for a driving field (a) x ¼ 0:02, (b)

x ¼ 0:3, (c) x ¼ 0:6, and (d) x ¼ 10 in the cases: p ¼ 0, N ¼ 0

(dashed-dotted line), p ¼ 0:99, N ¼ 0 (solid line), p ¼ 0,

N ¼ 0:015 (dotted line), and p ¼ 0:99, N ¼ 0:015 (dashed line).

A perfect squeezing condition is assumed (M ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNðN þ 1Þ

p)

and U ¼ 0. Other parameters are x32 ¼ 0, and D3 ¼ 0.

that the combined effect of quantum interference

and the input squeezing vacuum produces larger

values of squeezing in all frequency range except

for the appearance of a high ultranarrow central

peak (dashed line). This extremely narrow line

results from the combined effect of quantum in-terference and squeezed input vacuum and it is the

remainder of the incoherent part of the intensity

fluorescence spectrum. We have showed in a pre-

vious work [36], that this central line is obtained as

a superposition of two Lorentzians, where one of

them has a subnatural width. With a further in-

crease of Rabi frequency (see Fig. 2(c)) the influ-

ence of quantum interference appears to be crucialin order to obtain squeezing. A closer look at

Fig. 2(c) reveals that in the case with p ¼ 0 the

squeezing disappears in the central range of fre-

quencies when N ¼ 0, and N 6¼ 0 with M 6¼ 0. In

the case with p ¼ 0:99 the effect of the input

squeezed vacuum is to enhance the squeezing in

the wings at the expense of the increase of fluctu-

ations at x ¼ 0 (see dashed lines in Fig. 2(c) and2(b)). Finally at large values of the Rabi frequency

(see Fig. 2(d)) the squeezing disappears for all

cases: this result resembles that obtained for two-

level atoms [44].

One could ask oneself to what extent the re-

duction of fluctuations showed in Figs. 2(b) and

2(c) when both quantum interference and squeezed

vacuum are present, arises from the two-photoncorrelations. To answer this question we present in

Fig. 3 the squeezing spectra when the atom is

damped by a thermal field (N > 0 and M ¼ 0), for

the same Rabi frequencies considered in Fig. 2.

For low and high Rabi frequencies (see Figs. 3(a)

and 3(d), respectively) a thermal field produces

similar results as those obtained for squeezed

vacuum (see Figs. 2(a) and 2(d)). The most strikingresults are obtained at moderate driving intensities

as displayed in Figs. 3(b) and 3(c). In the case with

p ¼ 0, the squeezing spectra (dotted lines) resem-

bles in shape the results obtained for the squeezing

field showed in Figs. 2(b) and 2(c) (dotted lines). In

the case with p ¼ 0:99 the thermal field does not

produces an appreciable reduction of fluctuation

in comparison to the case when the atom isdamped by the squeezed field (see dashed lines in

Figs. 2(b), (c) and 3(b), (c)). We could conclude


Sp=2ðxÞ as a function of x for a driving field (a) x ¼ 0:02,

(b) x ¼ 0:3, (c) x ¼ 0:6, and (d) x ¼ 10 in the cases: p ¼ 0,

N ¼ 0:015 (dotted line), and p ¼ 0:99, N ¼ 0:015 (dashed line).

A thermal field is assumed (M ¼ 0). Other parameters are

x32 ¼ 0, and D3 ¼ 0.


Sp=2ðxÞ as a function of x for a driving field (a) x ¼ 0:02,

(b) x ¼ 0:3, (c) x ¼ 0:4, (d) x ¼ 0:6, and (e) x ¼ 10 in the cases:

p ¼ 0, N ¼ 0 (dashed-dotted line), p ¼ 0:99, N ¼ 0 (solid line),

p ¼ 0, N ¼ 0:015 (dotted line), and p ¼ 0:99, N ¼ 0:015 (dashed

line). A perfect squeezing condition is assumed (M ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNðN þ 1Þ

p) and U ¼ 0. Other parameters are x32 ¼ c, and

D3 ¼ 0.


from Figs. 2 and 3 that the combined effect of

quantum interference and the squeezed vacuum is

responsible for the reduction of fluctuation in the

fluorescent signal.

Let us consider the nondegenerate case, i.e.,x32 6¼ 0. In the case of the standard vacuum, Gao

et al. [14] showed that the out-of-phase quadrature

presents squeezing for weak excitation, whereas

for large Rabi frequencies an ultranarrow central

peak appears, due to the quantum interference.

The effect of the squeezed vacuum field in the

phase-dependent fluorescent spectra is shown in

Fig. 4. This figure shows the spectra of the out-of-phase quadrature versus the frequency for different

values of the coherent driving field. We use the

same parameters as in Fig. 2 except for x32 ¼ c.We observe that for a weak excitation driving field

(see Fig.4(a)), squeezing appears around the

atomic transition frequency when squeezed vac-

uum is absent. For this weak coherent field

(x < N ), the presence of squeezed field destroys thenonclassical feature present when the system in-

teracts with normal vacuum modes. Note that in

this weak-field regime, there is essentially no co-

herent field to provide a phase reference and the

interaction with the squeezed vacuum produces the

same result as a thermal noisy light (not shown in

Fig. 4). The behavior of the fluorescent signal

changes dramatically when the Rabi frequency

increases by an order of magnitude while otherparameters remain unaltered. We can clearly ob-

serve in Figs. 4(b) and (c) that significant squeezing

is obtained around x ¼ 0 when quantum inter-

ference and squeezed vacuum are both present.

This reduction almost disappears when the atom is

damped by a thermal field, thus revealing the rel-

evance of two-photon correlation of the squeezed

vacuum. The presence of the squeezed field greatlyenhances the squeezing over all frequency com-

ponents except for a small region at the wings.

This indicates that a three-level atom with quan-

tum interference and damped by a squeezed vac-

uum can generate larger squeezing than in the case

in which the atom is damped by the standard

vacuum. As the Rabi frequency increases

Fig. 6. Three-dimensional spectra of the out-of-phase quadra-

ture Sp=2ðxÞ as a function of x and N for the dimensionless

Rabi frequency x ¼ 0:3. Other parameters as in Fig. 5.


(Fig. 4(d)), the squeezing disappears at the center

and shifts to the wings of the spectrum. The

spectrum has a double minimum which corre-

sponds to the Rabi splitting of the dressed atomic

levels. Again, the effect of the squeezed vacuum is

to enhance the squeezing of the fluorescent field.Finally, in Fig. 4(e) we plot the squeezing spectrum

for a large Rabi frequency compared to the photon

number of the squeezed field. In this case, the co-

herent part of Bloch equations (7)–(11) dominates

and the Autler–Townes doublet appears (not

shown in Fig. 4(e)), but there is not squeezing in

the fluorescent field.

In order to show an overall view of the varia-tion of the squeezing spectra as a function of the

driving field, we present in Fig. 5 a three-dimen-

sional squeezing spectrum of the out-of-phase

quadrature of the fluorescent field for the same

previous parameters. We can clearly see that the

maximum squeezing occurs at zero frequency

(x ¼ 0) for a weak coherent field x ¼ 0:3. Now let

us analyze the variation of the squeezing spectrumwith the mean photon number N of the squeezed

field. We present in Fig. 6 a three-dimensional plot

of the spectrum of the out-of-phase quadrature as

a function of the mean photon number N for the

Rabi frequency x ¼ 0:3. We see clearly that

squeezing occurs in a narrow region of small

photon numbers (0 < N 6 0:06) around the central

frequency and it is enhanced with regard to thecase in which the atom is damped by normal

–4–2

02

4

0

0.2

0.4

0.6–0.1

0

0.1

ωx

Sπ/

2(ω)

Fig. 5. Three-dimensional spectra of the out-of-phase quadra-

ture Sp=2ðxÞ as a function of x and the dimensionless Rabi

frequency x for x32 ¼ c, D3 ¼ 0, p ¼ 0:99, N ¼ 0:015, and

U ¼ 0.

vacuum (N ¼ 0). In order to emphasize this fact,

we plot in Fig. 7 the spectrum of squeezing (solid

line) versus N at the central line (x ¼ 0). For

comparison, we also present the squeezing of the

input broadband squeezed vacuum (dashed line)which is given by [45]

SINp=2 ¼

1

2ðN � jM jÞ: ð35Þ

It is clear from Fig. 7 that the squeezing of the

fluorescent field is enhanced compared to the

squeezing of the input squeezing vacuum for smallphoton numbers. This result indicates that, as in

the case of a two-level atom [17,18], the three-level

0.00 0.02 0.04 0.06 0.08 0.10

-0.12

-0.08

-0.04

0.00

0.04

0.08

N

INsπ/2

s π/2(0)

Fig. 7. Squeezing spectrum Sp=2ð0Þ at the central line as a

function of N (solid line). Spectrum of squeezing of the input

broadband squeezed vacuum (dashed line). Other parameters as

in Fig. 5.

0 0.02 0.04 0.060

0.2

0.4

0.6

0.8

1

N

p–0.1

–0.08

–0.06

–0.04

–0.02

0 0.02

(a)

0 0.02 0.04 0.060

0.2

0.4

0.6

0.8

1

N

p

0.1

0.07

0.05

0.02 0

–0.02

–0.04

(b)


V-type atom with quantum interference may be

applied as an nonlinear element to amplify the

squeezing.

We would like to point out that the enhance-

ment of the squeezing arises from the combinationof quantum interference and the squeezed vacuum

field. To demonstrate this point, we plot in Fig. 8

the amplitude of the squeezing spectrum

Sp=2ðx ¼ 0Þ at the central line as a function of the

Rabi frequency in the four possible cases when the

quantum interference and the squeezed vacuum

are considered or not. It is clear from Fig. 8 that

the combined effect of quantum interference andsqueezed driving field (dashed line) generates lar-

ger squeezing compared to the other situations.

The interplay between quantum interference and

squeezed vacuum is further analyzed in the central

line of the spectrum Sp=2ðx ¼ 0Þ in Fig. 9(a) which

provides a contour map in the (N ; p) diagram for a

moderate driving field x ¼ 0:3, and x32 ¼ c. Note

that when the mean photon number N is fixedaround 0.06, by increasing p the fluorescent signal

changes from above to below the quantum limit.

On the other hand the role of the squeezed vacuum

can be derived from Fig. 9(a) for a fixed and ar-

bitrary value of p. For example, for p ¼ 0:6, theincrease of N (N < 0:02) is first accompanied by a

reduction of fluctuations, and by further increas-

ing N (N > 0:02) the fluctuations approach to zeroand, eventually, squeezing disappears. This be-

Fig. 8. Squeezing spectrum Sp=2ð0Þ at the central line versus thedimensionless Rabi frequency x for p ¼ 0, N ¼ 0 (dashed-dot-

ted line), p ¼ 0, N ¼ 0:015 (dotted line), p ¼ 0:99, N ¼ 0 (solid

line), and p ¼ 0:99, N ¼ 0:015 (dashed line). Other parameters

as in Fig. 5.

Fig. 9. Contour plot of squeezing spectra of the out-of-phase

quadrature Sp=2ðxÞ at the central line in plane ðN ; pÞ for x ¼ 0:3,

U ¼ 0, x32 ¼ c. We assume a perfect squeezing condition

(M ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNðN þ 1Þ

p).

havior is absent when considering the atom

damped by a thermal field as shown in Fig. 9(b).

We note that for a fixed value of p, the increase ofN only produces a monotonic increase of the

fluctuations (see Fig. 9(b)).

Thus the combined effect of quantum interfer-

ence and squeezed driving field generates larger

squeezing compared to the squeezing obtained in

the absence of the quantum interference (p ¼ 0) or

in absence of squeezed vacuum (N ¼ 0). It is worth

noting that the maximum squeezing obtained isrelated, as in two-level atoms, to the purity of the


Sp=2ðxÞ as a function of x for (a) x ¼ 0:2, (b) x ¼ 0:4,

(c) x ¼ 0:6, and (d) x ¼ 10 for the cases with standard vacuum

(solid line) and with squeezed vacuum field (dashed line):

N ¼ 0:015 and U ¼ 0. The other parameters are x32 ¼ 5c,D3 ¼ 0, and p ¼ 0:99.


three-level atomic system. In two level atoms

damped by a squeezed vacuum, it has been well

established that optimal squeezing in resonance

fluorescence occurs for pure atomic states [46]. In

the V-type three-level atomic system, the maxi-

mum squeezing is also obtained when the puritydefined as

R ¼ Trq2 ¼ q211 þ q2

22 þ q233

þ 2ðjq12j2 þ jq13j

2 þ jq23j2Þ; ð36Þ

reaches a maximum. This can be seen in Fig. 10

where we plot the amplitude of squeezing spectrum

of the out-of-phase quadrature at x ¼ 0, and the

atomic purity, as a function of the Rabi frequency

for two different values of the squeezed photonnumber. The solid and dashed lines represent

Sp=2ð0Þ and R, respectively. Note that in this case,

the atom does not reach a pure state, thus no

optimal squeezing is obtained.

We have also studied the influence of the split-

ting of the excited sublevels on the squeezing

spectrum in Fig. 11. Nonspecial qualitative chan-

ges appear at the central line of the spectrum. Themost significant effect of the splitting is the ap-

pearance of the inner sidebands. Again, in the re-

gime of weak driving field, the presence of the

squeezed vacuum enhances the squeezing. As the

field increases (see Fig. 11(d)) squeezing almost

disappears in both cases, and an ultranarrow

central peak develops due to the combination of

quantum interference and squeezed vacuum [36].Now we address the influence of the relative

phase between the squeezed vacuum and the

Fig. 10. Sp=2ð0Þ and R as functions of the dimensionless Rabi

frequency x, and (a) N ¼ 0:015, (b) N ¼ 0:5. The solid and

dashed lines represent Sp=2ð0Þ and R, respectively.

driving field on the squeezing spectra. Fig. 12

shows Sp=2ðxÞ for three different values of the

squeezed phase U and the same photon number

N ¼ 0:015. We present the behavior for different

values of the Rabi frequency. For comparison, the

case of the standard vacuum (N ¼ 0) is also plot-

ted in this figure. For the case of a weak Rabifrequency (x � N ) (see Fig. 12(a)), the squeezed

vacuum destroys the squeezing of the fluorescent

field in all frequency range and the relative phase

does not change qualitatively the behavior of the

system. As the Rabi frequency increases compared

to the mean photon number of the squeezed field,

the squeezing spectra exhibit dramatic changes

depending on the value of U. This important roleof the phase can be appreciated in Figs. 12(b) and

(c) where the squeezing can be larger or lower than

the case of the standard vacuum (solid line) de-

pending on the value of the phase. This figure

clearly reveals that the squeezing features at the

central peak can be controlled by changing the

phase from zero to p.Finally in Fig. 13, squeezing spectra of different

phase quadrature are presented. It can be seen that

minimum squeezing occurs at h ¼ p=2, althoughsqueezing also occurs for other quadratures.

Fig. 12. Squeezing spectra of the out-of-phase quadrature of

the fluorescent field Sp=2ðxÞ as a function of x for (a) x ¼ 0:2,

(b) x ¼ 0:4, (c) x ¼ 0:6, and (d) x ¼ 10 for a squeezed vacuum:

N ¼ 0:015, and different values of the phase, U ¼ 0 (dashed

line), U ¼ p=2 (dotted line), and U ¼ p (dot-dashed line). Other

parameters are x32 ¼ c, D3 ¼ 0, and p ¼ 0:99. For comparison,

the case N ¼ 0 is plotted (solid line).

–4–2 0

24

0

0.5

1–0.1

0.0

0.1

ω

θ/π

Sθ(ω

)

Fig. 13. Squeezing spectra of phase quadratures ShðxÞ. The

parameters are x32 ¼ c, D3 ¼ 0, p ¼ 0:99, N ¼ 0:015, and

U ¼ 0.


5. Conclusions

In this work we have considered the response of

a V-type three-level atom driven by a coherent

optical field in a broadband squeezed reservoir with

the aim of calculating the squeezing phase-depen-

dent resonance fluorescence spectrum. This setting

has already been addressed in many investigations

in two-level atoms. Here we stress that the addition

of quantum coherence and interference effects ap-

pearing in three-level atoms causes qualitative

changes in the shape of squeezing spectra.

We have shown that the phase-dependent reso-

nance fluorescence spectrum is enhanced by thecombined effect of the quantum interference and

the squeezed vacuum. Furthermore, the fluctua-

tions can be controlled by changing the relative

phase of the squeezed vacuum to the coherent field.

We have found that, as in two-level atoms, this

enhancement of the squeezing occurs for photon

number and Rabi frequency values where the pu-

rity of the atomic system reaches a maximum.

Acknowledgements

This work was supported by the Project No.

BFM2000-0796 (Spain). We are gratefully to

Tom�as Lorca for the correction of the manuscript.

Appendix A

The master equation for the reduced density

matrix of the atomic system, qIs, in the Born and

Markov approximation and in the interaction

picture reads

oqIs

ot¼ � i

�hH I

ex; qIs

� � 1

2

X3i;j¼2

Nðxi1Þ½ þ 1�cij

� ðSþi S

�j q

Is

h� S�

j qIsS

þi Þeixij t

þ ðqIsIS

þj S

�i � S�

i qIsS

þj Þe�ixijt

i� 1

2

X3i;j¼2

Nðxi1Þcij ðS�j S

þi q

Is

h� Sþ

i qIsS

�j Þeixij t

þ ðqIsS

�i S

þj � Sþ

j qIsS

�i Þe�ixijt

i� 1

2

X3i;j¼2

Mðxi1Þgij ðSþj q

IsS

þi

h� Sþ

j Sþi q

IsÞ

þ ðSþi q

IsS

þj � qI

sSþi S

þj Þie�ið2xv�xi1�xj1Þt

� 1

2

X3i;j¼2

M�ðxi1Þg�ij ðS�j q

IsS

�i

h� S�

i S�j q

IsÞ

þ ðS�i q

IsS

�j � qI

sS�i S

�j Þieið2xv�xi1�xj1Þt: ðA:1Þ


In the above equation we have introduced the

following notation for the atomic operators:

Sþ2 ¼ S�

2

�y ¼ j2ih1j;

Sþ3 ¼ S�

3

�y ¼ j3ih1j:ðA:2Þ

The coefficients cij and gij are defined as [19]

cij ¼ pgiðxi1Þg�j ðxi1Þ;

gij ¼ pgiðxi1Þgjð2xv � xi1Þ;ðA:3Þ

where cij ¼ cji and gij ¼ gji. The coefficients cii � ci(i ¼ 2; 3) in Eq. (A.1) are the decay rates for the

j3i ! j1i, and j2i ! j1i transitions. The additionaldamping terms cij (i 6¼ j) are particularly important

when x32 ’ c2; c3, and they arise due to the cou-

pling of the two transitions j3i ! j1i and j2i ! j1iwith the same vacuum mode. They are responsible

for the quantum interference between the two de-cay channels [20,21,24]. These terms oscillate at the

frequency difference D ¼ xi1 � xj1, thus when D is

large enough, they may be dropped. This is the case

treated in reference [20]. The present discussion is

based on the situation where xi1 ’ xj1, so such

nonsecular terms must be retained. Moreover,

the presence of squeezing and the fact that

haðxkkÞaðxk0k0 Þi 6¼ 0, introduces the additionaldamping constants gij which oscillate at

2xv � xi1 � xj1. It must be noted that these terms

disappear (g22 ¼ g33 ¼ 0) in a ladder configuration

[47], and in a V-type atomic configuration when

x31 x21 [20]. In the last case, the atomic opera-

tors do not depend on correlations between pairs of

modes and this fact leads to the absence of phase

sensitivity in population decay. However, in the V-type atomic configuration considered here, the

central frequency of the squeezed vacuum is near

the center of the doublet, thus 2xv ’ xi1 þ xj1, and

all terms in gij must be retained. The main conse-

quence of this fact is that some optical properties of

a V atom with closely lying sublevels become phase

dependent as shall be shown later.

Finally, H Iex represents the interaction between

the atom and the external driving fields in the in-

teraction picture

H Iex ¼ þ�h

X3m¼2

Xmei/Le�iðxL�xm1Þtjmih1j þH:c: ðA:4Þ

The radiative shifts (Lamb and Stark shifts) have

been ignored. In addition, it can be shown [27] that

c23 ¼ g23 ¼ffiffiffiffiffiffiffiffic3c2

p ~l13 �~l12

jl13�!jjl12

�!j

!; ðA:5Þ

where the transition dipole moments ~l13 and ~l12

are assumed to be real valued. The quantum in-

terference is maximum if the transition moment~l13 is parallel to ~l12, and it disappears if they areperpendicular. Now we eliminate the explicit

temporal dependence of the density matrix equa-

tion through an appropriate unitary transforma-

tion q ¼ UþNq

IsUN, where UN ¼ eiðD3j3ih3jþD2j2ih2jÞt,

Dm ¼ ðxm1 � xLÞ (m ¼ 2; 3) being the laser detun-

ing from the resonance with the state jmi.In this frame the evolution equations for the

density matrix elements are given by Eqs. (7)–(11).

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