Interference pattern with a dark center in resonance fluorescence from two atoms driven by a squeezed vacuum field

Ž . Optics Communications 153 1998 245–250

Interference pattern with a dark center in resonance fluorescence from two atoms driven by a squeezed vacuum field

Z. Ficek ^a, R. Tanas´ ^b

aDepartment of Physics and Centre for Laser Science, The UniÕersity of Queensland, Brisbane 4072, Australia

bNonlinear Optics DiÕision, Institute of Physics, A. Mickiewicz UniÕersity, Poznan, Poland´ Received 17 February 1998; revised 4 May 1998; accepted 4 May 1998

Abstract

We study the resonance fluorescence from two interacting atoms driven by a squeezed vacuum field and show that this system produces an interference pattern with a dark center. We discuss the role of the interatomic interactions in this process and find that the interference pattern results from an unequal population of the symmetric and antisymmetric states of the two-atom system. We also identify intrinsically nonclassical effects versus classical squeezed field effects. q 1998 Elsevier Science B.V. All rights reserved.

PACS: 42.50

1. Introduction

It has been demonstrated in recent years that trapping of two atoms or ions at small interatomic distances enables one to observe collective phenomena and interference in the fluorescence field. For example, DeVoe and Brewer 1w x have observed the phenomena of superradiance and subra- diance 2 in the fluorescence field emitted by two ionsw x trapped at a distance comparable to the radiation wave- length and excited by weak laser pulses. At small inter- atomic distances the emission from each ion interferes with the others in a way that depending on the initial phase of the atomic dipoles the resulting radiation is either

Ž . Ž .

reduced subradiance or enhanced superradiance . Eichmann et al. w x3 have observed the Young-type interference in the fluorescence field of two trapped ions.

In this experiment, the ions were continuously driven by a weak coherent laser field and, depending on the polariza- tion, different degrees of interference were observed. The interference pattern in the Young-type experiment depends not only on the polarization but also on the intensity as well as on the direction of propagation of the driving field

w x

with respect to the interatomic axis 4–6 . For a weak driving field an interference pattern is observed due to the predominantly elastic scattering of the incident field. Un- der strong excitation, inelastic scattering dominates result- ing in a reduction of the fringe visibility.

Kochan et al. 7 have shown that the fringe visibility ofw x strongly driven atoms can be partially improved by cou- pling the atoms to a cavity mode. The cavity induces the interatomic correlations which alter the fringe visibility.

Recently, Meyer and Yeoman 8 have reported an evenw x stronger cavity-induced modification of the interference pattern that occurs when the coherent driving field is replaced by an incoherent field. They have shown that in contrast to the coherent excitation, the incoherent field produces an interference pattern with a dark center. They have interpreted this modification as an intrinsically non- classical effect arising from the destructive quantum inter- ference.

In this paper, we report the existence of a similar contrast in behaviour of the interference pattern of two atoms driven in free space by a squeezed vacuum field.

We discuss the dependence of the interference pattern on

0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.

Ž .

PII S 0 0 3 0 - 4 0 1 8 9 8 0 0 2 4 9 - 1

the interatomic separation and on the matching of the squeezed modes to vacuum modes surrounding the atoms.

We point out that the observation of an interference pattern with a dark center does not require the perfect matching of the squeezed modes to the vacuum modes, and therefore does not require the squeezed modes to occupy a major part of the full solid angle of the vacuum modes coupled to the atoms. We also distinguish effects which are attributed to nonclassical correlations contained in the squeezed vac- uum field.

2. The model

We consider a system of two identical atoms interact- ing with a quantized multimode electromagnetic field. The atoms are modelled as two-level systems with the ground

< : Ž . < :

states g_i i s 1,2 and the excited states e_i separated by the transition frequency v₀ and connected by an elec- tric dipole transition with the dipole moment m. We assume that the atoms are driven by a squeezed vacuum field which is propagated in the direction perpendicular to the atomic axis and is focused at the center of the inter- atomic axis. The system can be described by the reduced atomic density operator r, which time evolution obeys the following master equation 9 ,w x

Er 2

q y q y

w x

s yi v⁰Ý S S , r y i^{i i} ÝV^{i j} S S , r^{i j}

E t _{is 1 i/j}

y1whN v_{Ž .}q1x

0 2

=ÝG^{i j}ŽS^qi S^yj r q r S^qi S^yjy2 S^yj r S^qi .

i , j

1 y q y q q y

y²hN vŽ ⁰.ÝG^{i j}ŽS S r q r S S y 2 S r S^{i j i j j i} .

i , j

yhM vŽ 0.e^{2 i vst}

=ÝG^{i j}ŽS^qi S^qj r q r S^qi S^qjy2 S^qj r S^qi .

i , j

yhM⁾Žv0.e^{y2 i vst}

=ÝG^{i j}ŽS^yi S^yj r q r S^yi S^yjy2 S^yj r S^yi ., Ž .1

i , j

where S^q_i and S^y_j are the atomic dipole operators for the ith atom, G s G s G is the spontaneous emission rate,_{ii j j}

Ž .

and G_{i j} and V_{i j} i / j describe collective damping and collective shift of the atomic levels, respectively, and determine the collective properties of the two-atom system.

w x

The collective parameters are defined as 10–13 sin krŽ .

2 i j 2

G s Gi j 32

½

= cos krŽ i j. sin krŽ i j.

y , Ž .2

2 3

5

kr kr

Ž i j. Ž i j.

cos krŽ .

2 i j

V s Gi j 32

½

sin krŽ . cos krŽ .

2 i j i j

q 1 y 3 m P rŽˆ ˆi j. ^{Žkri j.}2 q ^{Žkri j.}3

5

The parameter h, which appears in Eq. 1 , describesŽ . the matching of the squeezed vacuum to the modes sur- rounding the atoms. It depends on the profile of the incident squeezed field and on a solid angle over which the

w x

squeezed field is propagated 14,15 . For perfect matching h s 1, whereas h - 1 for imperfect matching. The perfect

Ž .

matching condition h s 1 is achieved for an ideal squeezed field–atom coupling so that the atoms interact exclusively with squeezed modes of the radiation field. In free space this requires a ‘‘three-dimensional’’ squeezed vacuum, which couples to the atoms through the full 4p solid angle. Such a requirement of large solid angle cou- pling does not suit the present sources of squeezed light, which generate a squeezed beam encompassing only very

w x

small solid angles. It has been suggested 14,15 , that nearly perfect matching conditions could be achieved in an experiment performed with atoms trapped inside an optical cavity. The cavity may be configured so that modes in one spatial dimension dominate and the atoms may interact with the squeezed field through these cavity modes. In other words, the cavity ‘‘tailors’’ the 4p solid angle of the modes coupled to the atoms to a small solid angle which can be easily covered by the squeezed field giving the effective h s 1 coupling.

Ž . Ž . < Ž . <

The parameters N v₀ and M v₀ s M v₀

Ž . < Ž .<²

=exp ic characterize squeezing such that M v F

s 0

Ž .w Ž . x

N v₀ N 2 v y v_{s 0} q1 , where the equality holds for a minimum-uncertainty squeezed state, c is the squeezing_s phase and v_s is the carrier frequency of the squeezed

Ž .

vacuum field. The parameter N v₀ is proportional to the mean number of photons in the squeezed field, and the

Ž .

parameter M v₀ measures the correlations between the squeezed vacuum modes separated in frequency by v₀ either side of the carrier frequency v . The parameters_s depend on the specific mechanism in which the squeezed vacuum is generated. In the following, we consider a degenerate parametric oscillator as a source of the squeezed

w x

vacuum field 16,17 . In this case N vŽ 0.

l²ym² 1 1

s ₂y ₂ ,

2 2

ž /

4 m q v y vŽ s 0. l q v y vŽ s 0. Ž .4

M vŽ 0.

l²ym² 1 1

s ₂q ₂ ,

2 2

ž /

4 m q v y vŽ s 0. l q v y vŽ s 0. Ž .5 where

1 1

l s g q E2 c E , m s g y E2 c E , Ž .6

with g the damping constant of the parametric oscillator_c cavity and EE its amplification constant.

The parameters l and m determine the bandwidth of the squeezed field, and we shall assume that l, m 4 G , in order to satisfy the Markov approximation used in the derivation of Eq. Ž .1 . Moreover, we shall restrict our calculations to the case where the carrier frequency v is_s equal to the atomic transition frequency v . In this case,₀ the squeezing parameters satisfy the relation

'

M s N N q 1 ,Ž . Ž .7

< < < Ž .< Ž . Ž .

where M s M v₀ and N s N v₀ sN 2 v y v ._{s 0} Relation 7 shows that the output of a degenerate paramet-Ž . ric oscillator is in a minimum-uncertainty squeezed state.

Such a state is a quantum state of the electromagnetic field w x

with no classical analogue 18 . The nonclassical character of the squeezed field is manifested by the term q1 under the square root in Eq. 7 showing that the magnitude ofŽ . the correlation between two different photons is greater

than the magnitude of the correlation of a photon with itself. For a field with a classical analogue, the field

< <

correlations can have values M F N. Thus, mode correla-

< <

tions with 0 - M F N may be generated by a classical

< < '

field, whereas correlations with N - M F N N q 1Ž . can only be generated by a quantum field which has no

< <

classical analogue. A field with M F N is called a ‘‘cor- related classical field’’ or ‘‘classically squeezed field’’,

< < '

whereas a field w ith N - M F N N q 1Ž . is called a ‘‘quantum squeezed field’’. The output of a degenerate parametric oscillator, as described by Eqs. 4Ž . and 5 , is an example of such a quantum squeezed field.Ž .

3. The fringe contrast factor

Our objective is to calculate the fringe contrast factor w7,8x

²S S q S S^q₁ ^y₂ ^q₂ ^y₁ :

C s q y q y , Ž .8

²S S q S S_{1 1 2 2} :

for the stationary fluorescence field emitted by two atoms separated by a distance r₁₂ and driven by a squeezed vacuum field. The fluorescence field is detected in the direction perpendicular to the atomic axis and the detector is located in the far field zone of the radiation emitted by the two-atom system. The factor C can be positive as well

< < '

Fig. 1. The fringe contrast factor C as a function of the interatomic separation r12 for N s 0.1, m H r , M sˆ ˆ12 N N q 1Ž . and different

Ž . Ž . Ž .

h: h s 1 solid line , h s 0.2 dashed line , h s 0.04 dash-dotted line .

< < w x as negative, and C is simply the fringe visibility 7,8,19 . For positive values of C, the interference pattern exhibits a

Ž .

maximum at line center bright center , whereas for nega-

Ž .

tive C there is a minimum dark center . The optimum

Ž . Ž .

positive negative value is C s 1 C s y1 , and there is no interference pattern when C s 0.

Ž . < < ' Ž . < <

Fig. 2. The fringe contrast factor C as a function of the interatomic separation r12 for h s 1, a M s N N q 1 , bŽ . M s N, and

Ž . Ž . Ž .

different N: N s 0.5 solid line , N s 5 dashed line , N s 50 dash-dotted line .

The steady-state values of the correlation functions appearing in Eq. 8 can be found from the master equationŽ . Ž .1 . A straightforward calculation leads to the following steady-state value of the fringe contrast factor,

2 2

4 ah M

C s y ₂ , Ž .9

3 2 2 2

n Žn y 1 q 4h. M Ža q n y n .

where

n s 2hN vŽ 0.q1, a s G rG .12 Ž10.

Expression 9 shows that the fluorescence field emittedŽ . from two atoms driven by a squeezed vacuum exhibits an interference pattern. In contrast to the thermal driving field, this effect does not require the presence of an optical

w x

cavity 7,8 . Moreover, depending on G₁₂ the interference pattern can exhibit a bright or dark center. For G ) 0 the₁₂ factor C is negative, whilst C is positive for G - 0. In₁₂ Fig. 1, we plot the factor C as a function of the interatomic separation for N s 0.1 and different h. We see that the factor C oscillates with the interatomic separation and the amplitude of the oscillations decreases as the separation increases. For some separations the factor C is negative indicating that the interference pattern possesses a dark center. The graphs also show that the oscillation frequency of the contrast factor is independent of the matching h.

The matching h alters only the amplitude of the oscilla- tions and the interference pattern, as seen in Fig. 1, can be observed even for relatively small h.

It is interesting to compare the results for the contrast factor of two atoms in a squeezed field produced by a degenerate parametric oscillator with that obtained for a classical squeezed field. In Fig. 2, we plot the factor C as a function of the interatomic separation for the squeezed

< < '

field with M s N N q 1 , Fig. 2a, and for a classicalŽ .

< <

squeezed field with M s N, Fig. 2b. An interference pattern with a dark center is observed in both cases, indicating that the effect is not a signature of a quantum field. The interference pattern with a dark center although not unique to the quantum squeezed field does however contain quantitatively distinctive differences between the cases of excitation with classical versus quantum squeezed field. In particular, the amplitude of C for the case of a classical squeezed field is significantly smaller than for a quantum squeezed field. Moreover, for the classical

< <

squeezed field with M s N the interference pattern van- ishes as N increases, whereas for the case of the quantum squeezed field C approaches y1₂ when N ™ `.

The behaviours of the fringe contrast factor can be easily understood in terms of the collective states of the

w x

two-atom system 2,11 . In this representation, the two- atom system is equivalent to a single four-level system

< : < : < :

with one upper state 2 s e₁ e₂ , two intermediate states:

'

< : Ž < : < : < : < :.

the symmetric state q s e₁ g₂ qg₁ e₂ r 2 ,

< : Ž < : < : < : < :.

and the antisymmetric state y s e g yg e r

1 2 1 2

'2 , and one ground state 1 s g< : < ₁: <g₂:. The energies of

Ž .

these states are E s 2 " v , E s " v " V_{2 0 " 0 12} and E s₁ 0, respectively. In the basis of the collective states the factor C can be written as

r_qqyr_yy

C s , Ž11.

rqqqryyq2 r22

where r_ii are the populations of the collective states. The expression shows that a two-atom system produces an interference pattern when the symmetric and antisymmet- ric states are unequally populated. Moreover, the factor C can be negative when ryy)rqq.

From the master equation 1 , we find the steady-stateŽ . populations of the collective states,

2 2 2 2

n y 1 a h M 2 n y 1

Ž . Ž .

r₂₂s ₂ q ,

D 4 n

2 2 2 2

n y 1 ah M Ž2 n y a.

rqqs 4 n2 y D ,

2 2 2 2

n y 1 ah M Ž2 n q a.

ryys 4 n2 q D , Ž12.

where

2 4 2 2 2 2

D s n n q 4h M Ža y n . . Ž13.

Then

2 2

4 ah M

rqqyryys y , Ž14.

D

indicating that for a ) 0 the antisymmetric state is more populated than the symmetric state, and the population difference is induced by the squeezed vacuum field. The

< <

population difference is proportional to M , which means that can be produced by a classical squeezed field. There- fore, the interference pattern is not a pure quantum effect.

However, there are quantitatively distinctive differences between the cases of excitation with classical versus quan- tum squeezed field. For N 4 1 and a classical squeezed field excitation, the populations of all collective states are equal and approach . Hence, the strong classical squeezed1₄

field does not produce an interference pattern. The situa- tion is different when the atoms are excited by a strong quantum squeezed field. In this case the populations ap- proach the following values

1 1

r22sr11s4, ryys2, rqqs0, Ž15.

< <

showing that for a strong quantum squeezing with M

'

s N N q 1Ž . the population is trapped between the col-

< : < : < :

lective states 2 , 1 , and y , with no population in the symmetric state. This effect leads to a non-zero interfer-

ence pattern with a dark center, which is unique to quan- tum squeezing.

4. Summary

We have calculated the fringe contrast factor of the fluorescence field emitted by two-atoms driven by a squeezed vacuum field. We have shown that the system can produce an interference pattern with a dark center. The effect results from an excess of population in the antisym- metric state over that contained in the symmetric state induced by the squeezed field. We have found that the effect does not require squeezing of all 4p solid angle of the modes surrounding the atoms. The effect can be ob- served even with a classical squeezed field. In this case, however, the interference pattern can appear for small intensities N of the squeezed field and vanishes for large N. When the atoms are driven by a squeezed field with

< < '

nonclassical correlations M s N N q 1Ž . the inter- ference pattern appears for all intensities N and can exhibit the contrast factor of an amplitude up to y0.7. This is in sharp contrast to excitation by a classical squeezed field and results from the population trapping between collec- tive two-atom states.

Acknowledgements

This work has been supported in part by the University of Queensland Travel Awards for International Collabora- tive Research, the Australian Research Council and the

Polish Scientific Research Committee KBN grant 2 P03BŽ 73 13 ..

References

w x1 R.G. DeVoe, R.G. Brewer, Phys. Rev. Lett. 76 1996 2049.Ž . w x2 R.H. Dicke, Phys. Rev. 93 1954 99.Ž .

w x3 U. Eichmann, J.C. Bergquist, J.J. Bollinger, J.M. Gilligan, W.M. Itans, D.J. Wineland, M.G. Raizen, Phys. Rev. Lett. 70 Ž1993 2359..

w x4 Th. Richter, Optics Comm. 80 1991 285.Ž .

w x5 H. Huang, G.S. Agarwal, M.O. Scully, Optics Comm. 127 Ž1996 243..

w x6 T. Wong, S.M. Tan, M.J. Collett, D.F. Walls, Phys. Rev. A

Ž .

55 1997 1288.

w x7 P. Kochan, H.J. Carmichael, P.R. Morrow, M.G. Raizen,

Ž .

Phys. Rev. Lett. 75 1995 45.

w x8 G.M. Meyer, G. Yeoman, Phys. Rev. Lett. 79 1997 2650.Ž . w x9 Z. Ficek, Phys. Rev. A 44 1991 7759.Ž .

w10 M.J. Stephen, J. Chem. Phys. 40 1964 669.x Ž . w11 R.H. Lehmberg, Phys. Rev. A 2 1970 882; 889.x Ž .

w12 G.S. Agarwal, in: G. Hohler Ed. , Quantum Optics, Springerx Ž . Tracts in Modern Physics, vol. 70, Springer, Berlin, 1974.

w13 Z. Ficek, R. Tanas, S. Kielich, Physica 146A 1987 452.x ´ Ž . w14 A.S. Parkins, C.W. Gardiner, Phys. Rev. A 40 1989 3796.x Ž . w15 Z. Ficek, P.D. Drummond, Europhys. Lett. 24 1993 455.x Ž . w16 M.J. Collett, R. Loudon, C.W. Gardiner, J. Mod. Optics 34x

Ž1987 881..

w17 P.D. Drummond, M. Reid, Phys. Rev. A 41 1990 3930.x Ž . w18 For references see the special issues of J. Mod. Optics 34x

Ž6r7. Ž1987 , and J. Opt. Soc. Am. B 4 10. Ž . Ž1987 .. w19 M. Born, E. Wolf, Principles of Optics, Macmillan, Newx

York, 1964.