H. Freedhoff and T. Quang Vol. 12, No. 1 /January 1995 /J. Opt. Soc. Am. B 9

Absorption and dispersion by a driven atom in a cavity

Helen Freedhoff and Tran Quang*

Department of Physics and Astronomy, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada

Received June 9, 1994

We investigate the absorption and dispersion of a probe field by a two-level atom driven by a strong externalfield and coupled to a single cavity mode. For moderate atom–field coupling the absorption spectrumcontains ultrasharp lines, and the real part of the linear susceptibility changes rapidly at the Mollow sidebandfrequencies. For large coupling the absorption and dispersion spectra split into multiplets whose structuresdepend on the photon-number distribution of the cavity field. A large index of refraction accompanied byvanishing absorption is predicted.

PACS numbers: 42.50.Hz, 32.80.-t, 42.50.Dv.

1. INTRODUCTIONThe absorption of a weak probe beam in a medium satu-rated by an intense driving field (saturation spectroscopy)is a well-established technique of laser spectroscopy(Refs. 1 and 2 and references therein). In his pioneerwork3 Mollow showed that the absorption spectrum ofa two-level atom driven by an intense detuned exter-nal field consists of two sidebands, one absorbing andone amplifying. The physical origin of these featuresis explained clearly by the dressed-atom model.4 Thedispersive properties of strongly driven atoms were alsoinvestigated recently,5–8 in the context of a search formaterials that display high dispersion accompanied byvanishing absorption.

In this paper we investigate the influence of thecavity on the absorption spectrum and dispersion of aprobe beam by a strongly driven atom. For moderateatom–field coupling, the absorption spectrum containsultrasharp lines,9 and the real part of the linear sus-ceptibility changes very rapidly at the Mollow sidebandfrequencies. For strong coupling, the absorption anddispersion spectra split into multiplets whose structuresdepend strongly on the photon-number distribution ofthe cavity field. Thus measurement of these spectraprovides a nonintrusive method of measuring the photon-number distribution in the cavity. In both cases a largeindex of refraction accompanied by vanishing absorptionis also predicted.

2. MASTER EQUATIONThe system under investigation consists of a two-levelatom coupled to a single mode of an optical cavity withcoupling constant g and driven by a strong coherent ex-ternal field. The atom has excited state j2l, ground statej1l, and transition frequency va and is damped at the rateg by spontaneous emission into modes other than the priv-ileged cavity mode. The cavity mode has a frequency v

and a decay rate k. For simplicity, in this section wetreat the external driving field classically and work in theinteraction picture. The master equation for the systemof atom 1 driving field 1 cavity field has the form10–14

0740-3224/95/010009-06$06.00

≠

≠tr ­ 2ifH , rg 1

12

kLcr 112

gLar , (2.1)

where

H ­ Dcaya 1 1/2 Dass22 2 s11d

1 igsays12 2 s21ad 1 ess21 1 s12d , (2.2)

Lcr ­ 2aray 2 ayar 2 raya , (2.3)

Lar ­ 2s12rs21 2 s21s12r 2 rs21s12 . (2.4)

Here a and ay are the cavity-mode (boson) annihilationand creation operators, respectively; sij are the atomicoperators sij ­ jil k j jsi, j ­ 1, 2d; Da ­ va 2 vL and Dc ­v 2 vL are the detuning of the atomic resonance fre-quency va and of the cavity mode frequency v, respec-tively, from the driving field frequency vL; and 2e is theRabi frequency at resonance of the atom–driving-field in-teraction. As is well known, fluorescence of the atom oc-curs at the laser frequency vL and at the two sidebandfrequencies vL 6 2V, where 2V ­ 2se2 1 Da

2y4d1/2 is theRabi frequency in the detuned field. In this paper we fo-cus on the case of Dc ­ 2V; i.e., the cavity field is tuned toexact resonance with the high-frequency sideband. How-ever, we note that the tuning Dc ­ 22V will produce thesame results with Da ) 2Da

Equation (2.1) is written in the basis of the atomicHilbert space jil, i ­ 1, 2. We introduce instead thedressed states4,13

j1l ­ cos fj1l 2 sin fj2l , (2.5)

j2l ­ sin fj1l 2 cos fj2l , (2.6)

where

cos2 f ­ 1/2 1d

2p

1 1 d2, (2.7)

d ­ Day2e is the scaled detuning, and the rotation anglef belongs to the interval f0, py2g. We make the unitarytransformation

r ­ expsiH0tdr exps2iH0td , (2.8)

1995 Optical Society of America

10 J. Opt. Soc. Am. B/Vol. 12, No. 1 /January 1995 H. Freedhoff and T. Quang

where

H0 ­ VsR22 2 R11d 1 Dcaya , (2.9)

Rij ­ ji l k j j , (2.10)

and ignore rapidly oscillating terms at frequencies 2V and4V. The master equation (2.1) in the dressed-state basisreduces to

≠

≠tr ­ g1sayR12 2 R21a, rd 1 1/2 Lcr

1 sgy8dsin2s2fd s2R3rR3 2 rR32 2 R3

2rd

1 sgy2dcos4 fs2R12rR21 2 rR21R12 2 R21R12rd

1 sgy2dsin4 fs2R21rR12 2 rR12R21 2 R12R21rd

­ Lr , (2.11)

where

R3 ­ R22 2 R11 (2.12)

and g1 ­ g cos2 f is the effective coupling constant.

3. ABSORPTION AND DISPERSIONOF A PROBE FIELD: NUMERICALCALCULATIONWe now suppose that the system is weakly perturbed bya monochromatic probe field at frequency n, in whoseabsorption and dispersion we are interested. The linearsusceptibility xsnd of the probe field is given in terms ofthe Fourier transform of the average value of the two-timecommutator of the atomic operator s12 as 3,4,9

xsnd ­ iAZ `

0kfs12std, s21gls expsintddt , (3.1)

where A is a normalization constant and the index s in-dicates an average over the steady-state solution of themaster equation (2.11). For high Rabi frequency V, thecentral component at vL vanishes ( just as it does in freespace3,4), and xsnd consists of two well-separated side-bands, centered at the frequencies vL 6 2V:

xsnd ­ iA sin4 fZ `

0kfR21std, R12gls

3 expfisn 2 vL 1 2Vdtgdt

1 iA cos4 fZ `

0kfR12std, R21gls

3 expfisn 2 vL 1 2Vdtgdt . (3.2)

The equations of motion for kR21stdl and kR12stdl are foundfrom the master equation (2.11) in the form

≠

≠tkR21stdl ­ g1kaystdR3stdl 2 GkR21stdl , (3.3)

≠

≠tkR12stdl ­ g1kR3stdastdl 2 GkR12stdl , (3.4)

where

G ­ sgy2d s1 1 2 sin2 f cos2 fd (3.5)

is the linewidth of the sidebands in free space. Using the

quantum regression theorem,15 we write the equations forkfR21std, R12gls as

≠

≠tkfR21std, R12gls ­ g1kfaystdR3std, R12gls

2 GkfR21std, R12gls , (3.6)

≠

≠tkfR12std, R21gls ­ g1kfR3stdastd, R21gls

2 GkfR12std, R21gls . (3.7)

Equations (3.6) and (3.7) have the formal solutions

kfR21std, R12gls ­ kR3ls exps2Gtd

1 g1

Z t

0exps2Gst 2 td

3 kfaystdR3std, R12glsdt , (3.8)

kfR12std, R21gls ­ kR3ls exps2Gtd

1 g1

Z t

0exps2Gst 2 td

3 kfR3stdastd, R21glsdt . (3.9)

The correlation function kfaystdR3std, R12gls is calculatedfrom the following relation, valid for Markovian systems:

kfaystdR3std, R12gls ­ TrhayR3 expsLtd fR12, rssdgj ,

­ TrfayR3rstdg , (3.10)

where rstd is the solution of Eq. (2.11) with the initialcondition that

rs0d ­ fR12, rssdg (3.11)

and rssd is the steady-state solution of Eq. (2.11), obtainedin an earlier publication.16 The correlation functionkfR3stdastd, R21gls can be calculated analogously. Weintegrate Eq. (2.11) numerically, using a truncated ba-sis of photon-number states; one can also evaluate it byusing a vector recurrence-relation approach.17 A nu-merical continued fraction technique, described in detailin Ref. 13, is used to evaluate the correlation functionkfaystdR3std, R12gls and the linear susceptibility xsnd. Itwas shown in Ref. 16 that the steady-state density matrixof the cavity has nonvanishing elements only along thediagonal. Using this property, one can easily show fromEqs. (3.8) and (3.9) that

kfR21std, R12gls ­ 2kfR12std, R21gls , (3.12)

so that the left and right sidebands of xsnd differ only bya factor of 2tan4 f, as in free space. In what follows, wewill therefore focus only on the left sideband, at frequencyvL 2 2V.

The linear susceptibility [Eq. (3.2)] can be written as

xsnd ­ x 0snd 1 ix 00snd , (3.13)

where the real s x 0d and the imaginary s x 00d parts of x

determine the index of refraction and the absorption co-efficient, respectively, of the probe field.

H. Freedhoff and T. Quang Vol. 12, No. 1 /January 1995 /J. Opt. Soc. Am. B 11

Fig. 1. Real x 0 (dashed curve) and imaginary x 00 (solid curve)parts of the linear susceptibility (in units of g ­ 1) for d ­ 20.8,k ­ 0.05, and g ­ 1.25.

In Fig. 1 we plot xsnd for moderate coupling, g ­ 1.25g.The absorption spectrum, x 00snd, consists of a single ul-trasharp (amplification) line at n ­ vL 2 2V, with widthof order kyknl, together with broad shoulders. The in-dex of refraction, x 0snd, changes very rapidly at n ­vL 2 2V —behavior that may make this system useful asa quantum frequency filter. Furthermore, x 0snd remainslarge at frequencies at which the absorption vanishes.

In Figs. 2 and 3 below we plot xsnd for strong coupling,g ­ 10g and g ­ 50g. Instead of the single sharp featureof Fig. 1, the absorption and dispersion spectra displaya complex multiplet structure, including both absorptionand amplification lines, and a number of frequencies atwhich the material is transparent, while the index ofrefraction remains large. We explain the physical originof the system’s behavior, both for strong and for moderatecoupling, in Section 4, using a doubly dressed atom model.

4. ABSORPTION SPECTRUM:ANALYTICAL CALCULATIONIn this section we calculate analytically the absorptionspectrum of the system, using the doubly dressed atommodel of Ref. 13.

The Hamiltonian Hda of the system atom 1 drivingfield has the eigenstates ji, Nl si ­ 1, 2d that satisfy theeigenvalue equation

Hdaji, Nl ­ fNvL 1 s21diVgji, N l , (4.1)

where

j1, Nl ­ cos fj1, N l 2 sin fj2, N 2 1l , (4.2)

j2, Nl ­ sin fj1, Nl 1 cos fj2, N 2 1l , (4.3)

ji, N l is the state in which the atom is in state jil and Nphotons are present in the driving mode, and f is givenby Eq. (2.7). The Hamiltonian Hc of the cavity mode hasthe eigenvalue equation

Hcjnl ­ nsvL 1 2Vdjnl . (4.4)

The noninteracting dressed-atom 1 cavity-mode Hamil-tonian H0 ­ Hda 1 Hc has the eigenvalue equation

H0ji, N , nl ­ fNvL 1 s21diV 1 nsvL 1 2Vdgji, N , nl ,

(4.5)

where ji, N , nl ­ ji, N l ≠ jnl. The state j1, N , 0l is non-degenerate, whereas the states j2, N , nl and j1, N 2

1, n 1 1l form doubly degenerate pairs. When we includethe interaction W ­ gsays12 1 s21ad between the dressedatom and the cavity mode, the degeneracy is lifted, re-sulting in doublets jN , 6nl that satisfy the eigenvalueequation

sH0 1 W djN , 6nl ­ EN , 6njN , 6nl , (4.6)

where

EN , 6n ­ NvL 1 s2n 2 1dV 6 g1p

n , (4.7)

jN , 6nl ­8<: 1p

2 sj2, N 2 n 1 1, n 2 1l 6 j1, N 2 n, nld if n fi 0j1, N , 0l if n ­ 0

,

(4.8)

and g1 ­ g cos2 f.Interaction with the vacuum field allows spontaneous

emission (fluorescence) to occur from the states jN , 6nlto states jN 2 1, 6n0l, where n0 ­ n, n 6 1.13 Similarly,interaction with a weak probe beam gives rise to absorp-tion and stimulated emission between the same pairs oflevels, whereas cavity damping causes transitions to oc-cur from states jN , 6nl to jN 2 1, 6sn 2 1dl. The linearsusceptibility of the system is obtained by use of Eq. (3.1),as described in Section 3. The central component (at vL)vanishes, while the left and right sidebands differ only bya multiplicative factor of 2tan4 f, as in free space.4 Inwhat follows, therefore, we focus only on the sidebandat frequency vL 2 2V, produced by transitions betweenstates jN , 6nl and jN 2 1, 6sn 1 1dl and resulting inspectral lines at frequencies vL 2 2V 6 ns6d

n , where

ns6dn ­ g1s

pn 1 1 6

pn d . (4.9)

We discuss separately the cases of strong and of moderatecoupling.

A. Strong Coupling: Nonoverlapping LinesThe case of strong coupling is illustrated in Figs. 2 s g ­10gd and 3 s g ­ 50gd. We obtain the spectrum by con-sidering the evolution of the rising part of the atomicdipole moment operator:

my ­P

n,n0,Nmn,n0 rn,n0 ,N , (4.10)

where mn,n0 ­ kN , njs21jN 2 1, n0l and rn,n0,N is the off-diagonal element of the density operator,

rn,n0,N ­ jN , nl kN 2 1, n0j . (4.11)

We study the sideband at vL 2 2V by projecting the mas-ter equation (2.1) in turn onto jN 2 1, 6sn 1 1dl on theright and kN , 6nj on the left, in all possible combinations,and then averaging over N. For strong coupling, most ofthe spectral lines do not overlap, and we can neglect inter-ference between them. As in the case of the fluorescencespectrum, it is then straightforward to show that the linesat vL 2 2V 6 ns6d

n correspond to the reduced coherences

12 J. Opt. Soc. Am. B/Vol. 12, No. 1 /January 1995 H. Freedhoff and T. Quang

Fig. 2. Real x 0 (dashed curve) and imaginary x 00 (solid curve)parts of the linear susceptibility (in units of g ­ 1d for d ­ 20.4,k ­ 0.5, and g ­ 10.

Fig. 3. Absorption spectrum x 00snd (solid curve) for d ­ 20.8,g ­ 50, and k ­ 1y8. The dashed curve corresponds to theanalytical spectrum, Eq. (4.15), for the same parameters. Theinset shows the distribution Pn based on Eq. (4.17) for the sameparameters. Some (sample) transition lines are labeled.

r6n,6sn11d ; k r6n,6sn11d,N lN whose equations of motion aregiven by

Ùr6n, sn11d ­ 2 fisvL 2 2V 2 ns6dn d

1 Gn, n11gr6n, sn11d , (4.12)

Ùr6n, sn11d ­ 2 fisvL 2 2V 1 ns6dn d

1 Gn, n11gr6n, 2sn11d , (4.13)

where

Gn, n11

­

8>>><>>>:12

g

√12

1 sin2 f

!1

14

k if n ­ 0

12

g 1 nk if n ­ 1, 2 , . . .

.

(4.14)

Using Eq. (3.1) and the quantum-regression theorem,15

we find that the net intensity of absorption at vL 2 2V 2

ns1dn , for example, is equal to the difference between the

absorption of the probe beam that occurs between thelevels jN , 2nl and jN 2 1, n 1 1l and the stimulated

emission that occurs between the same pair of levels.Thus it is straightforward to show that the absorptionspectrum of the left sideband is given by

x 00snd ­ A sin4 fX

n­0,6

gn,n11Gn,n11

sv 2 vL 1 2V 6 nns6dd2 1 G

2n,n11

3 sPn 2 Pn11d , (4.15)

where

gn,n11 ­ gjkN, njs21jN 2 1, n 1 1lj2

­ 1/4 g sin4 fs1 1 dn,0d , (4.16)

Pn ­ P0

nYm­1

g sin4 f

g cos4 f 1 s2m 2 1dk(4.17)

is the steady-state population of level jN , nl and P0 isdetermined by normalization.13

Expression (4.15) is plotted in Fig. 3 (dashed curve) forg ­ 50g, together with the numerical spectrum as cal-culated in Section 3 (solid curve) and an inset showingthe steady-state populations Pn. The maximum popu-lation Pn occurs for states with n ø 2 and n ø 3, withP2 ø P3. As a result, there is no net absorption at fre-quencies n

s6d2 (absorption upward is balanced by stimu-

lated emission downward), and only small dispersionlikefeatures appear there. Spectral lines at frequencies ns6d

nare absorptive for n , 2 and amplifying for n . 3. Thenumerical and analytical spectra are in very good agree-ment, except in the vicinity of n 2 vL 1 2V ­ 0, whereoverlap occurs between the spectral lines. The overlapinvolves two groups of transitions, those at frequenciesvL 2 2V 1 ns2d

n and those at vL 2 2V 2 ns2dn , n $ 3, and

gives rise to the two sharp features near zero in the (nu-merical) spectrum. The narrowing of these features be-low their width as predicted by the analytical approxima-tion (which neglects overlap) is due to interference amongthe transitions within each group. The mechanism ofthis interference is described in Subsection 4.B.

Apart from these overlap features, it is clear that theintensities of the remaining absorption lines are directlydetermined by the populations Pn, which are related tothe photon-number distribution of the cavity field by

Pn ­ Pn 1 Pn11 . (4.18)

This suggests that a measurement of the absorption spec-trum can provide a nonintrusive measurement of thephoton-number distribution of the cavity field. Such ameasurement is the subject of intense interest in cavityquantum electrodynamics.18,19

B. Moderate Coupling: Line OverlapFor moderate coupling g , g, but with values of d

and k chosen so that knl is relatively large, spacingswithin groups of frequencies become ,, g, and interfer-ence between the corresponding transition coherencesmust be taken into account even within the secularapproximation.4 The frequencies ns2d

n and the spacingsbetween them Dns2d

n have the asymptotic leading terms

H. Freedhoff and T. Quang Vol. 12, No. 1 /January 1995 /J. Opt. Soc. Am. B 13

ns2dn )

g1

2p

n, (4.19)

Dns2dn ­ n

s2dn11 2 ns2d

n )g1

4n3/2. (4.20)

Thus, for the parameters of Fig. 1 s g1 ­ 0.235g, knl ,4.3d, there is coupling in the equations of evolution ofall coherences r6n,6sn11d that correspond to transitionsat the minus frequencies h6ns2d

n j. Similarly, even in thestrong-coupling case of Fig. 3, interference exists betweenthe minus transitions for n $ 3 within the two separategroups at 1ns2d

n and 2ns2dn .

We discuss first the coupling within the group of minustransitions h6ns2d

n j of Fig. 1. These are governed by thecoherences r6n,6sn11d with which are associated the dipoletransition moments p6n:

p6n ; m6n,6sn11dr6n,6sn11d , (4.21)

where

mn,n11 ­ kN , njs21jN 2 1, n 1 1l ­ 2m2n,2sn11d

­

8>>><>>>:12

sin2 f if n . 0

1p

2sin2 f if n ­ 0

. (4.22)

Equations of motion for the associated quantities Pn ­pn 1 p2n and Mn ­ pn 2 p2n are found from the masterequation (1.1) in the form

ÙPn ­ 2ins2dn Mn 2 CnPn 2 AnPn 1 An11Pn11 , (4.23)

ÙMn ­ 2ins2dn Pn 2 DnMn 2 BnMn 1 Bn11Mn11 (4.24)

for n . 0 and by

ÙP0 ­ 2ins2d0 M0 2 sgy2d s1 1 sin2 f cos2 fdP0 1 kP1 ,

(4.25)

ÙM0 ­ 2 ins2d0 P0 2 1/2 fg sin2 fs1 1 sin2 fd 1 kgM0

1 sg cos4 f 1p

2 kdgM1 , (4.26)

where

An ­ sky2d

"n 1

qsn 1 1d sn 2 1d

#, (4.27)

Bn ­ sgy2d ssin4 f 1 cos4 fd

1 sky2d

"qnsn 2 1d 1

qnsn 1 1d

#, (4.28)

Cn ­ G 1 sky2d

"n 2

qsn 1 1d sn 2 1d

#) G 1 ky4n ,

(4.29)

Dn ­ sky2d

"2n 2

qnsn 2 1d 2

qnsn 1 1d

#) ky8n .

(4.30)

Here G and ns2dn are given by Eqs. (3.5) and (4.9),

respectively.Equations (4.23) and (4.24) contain terms ns2d

n , Cn, andDn, which vary slowly with n and which we replace by

ns2dknl , Cknl, and Dknl. When we perform the sum over n

in Eqs. (4.23) and (4.24) and neglect and effects at n ­ 0(valid for large knld, the terms involving An and Bn vanish;we then obtain equations for P ­

Pn Pn and M ­

Pn Mn

in the form

ÙP ­ 2

"G 1

k4knl

#P 2 i

g1

2p

knlM , (4.31)

ÙM ­ 2k

8knlM 2 i

g1

2p

knlP . (4.32)

The eigenvalues of Eqs. (4.31) and (4.32) are

L1 >k

8knl1

g2 cos4 f

4Gknl, (4.33)

L2 > G 2g2 cos4 f

4Gknl1

k4knl

. (4.34)

The two-time averages kp6nstds12l satisfy the same equa-tions of motion as p6nstd (quantum-regression theorem15).The total contribution of the minus transitions to the spec-trum is then proportional to the Fourier transform ofPn

fk mn,n11pnstds12l 1 k m2n,2sn11dp2nstds12lg

> 1/2 sin2 fkMstds12l . (4.35)

Thus the dominant contribution to the spectrum that cor-responds to these transitions arises from the eigenvalueL1, resulting in the ultrasharp line in Fig. 1. Physically,this line results from coherence among the many transi-tions of the entangled dressed-atom 1 cavity system, all ofwhich occur at approximately the same frequency. Thisbehavior provides a plausible microscopic explanation toofor the narrowness of the linewidth of the conventionallaser.

The shoulders on the ultrasharp line are due in smallpart to the second eigenvalue L2 that corresponds to theminus transitions and in large part to the plus transitions.The 1ns1d

n frequencies are separated from the 2ns1dn fre-

quencies by distances $ g and hence evolve separatelyin the secular approximation. Their individual lines arenot resolved, however, and give rise to the (symmetric)broad shoulders on the sharp central line.

A second (if less dramatic) example of the effects of co-herence among several transitions can be seen in Fig. 3,involving the two (separate) groups of transitions at vL 2

2V 1 ns2dn and at vL 2 2V 2 ns2d

n , n $ 3. A similar calcu-lation to that described above predicts that these groupswill give rise to two sharp features, centered at vL 2 2V 6

0.25g1, each with a width of Gy2 1 3ky16knl ø 0.04g1, ver-sus the neglecting-overlap approximate width of ,0.1g1.This result is in agreement with the ,50% narrowing forthe numerical spectral features at vL 2 2V 6 0.24g1 ob-served in Fig. 3.

5. CONCLUSIONSWe have investigated, numerically and analytically, ab-sorption and dispersion of a probe field by a two-levelatom driven by a strong external field and coupled to asingle cavity mode. For moderate atom–field coupling,

14 J. Opt. Soc. Am. B/Vol. 12, No. 1 /January 1995 H. Freedhoff and T. Quang

the absorption spectrum contains ultrasharp lines, andthe real part of the linear susceptibility changes veryrapidly at the Mollow sideband frequencies. For strongcoupling, the absorption and dispersion spectra split intomultiplets whose structures are strongly dependent onthe populations Pn, so that a measurement of these spec-tra provides a nonintrusive method of measuring thephoton-number distribution in the cavity. A large indexof refraction accompanied by vanishing absorption is alsopredicted.

ACKNOWLEDGMENTThis research was supported in part by the Natural Sci-ences and Engineering Research Council of Canada, towhich the authors extend their thanks.

*Present address, Department of Physics, University ofToronto, 60 St. George Street, Toronto, Ontario, M5S 1A7Canada.

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