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OpticsOptikOptikOptik 120 (2009) 689–695

0030-4026/$ - se

doi:10.1016/j.ijl

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Probe absorption spectra of a V-type atom embedded in PBG reservoir

Han Zhanga, Xin Bo Lib, Yang Liua, Han Zhuang Zhanga,�

aCollege of Physics, Jilin University, Changchun 130023, PR ChinabCollege of Communication Engineering, Jilin University, Changchun 130025, PR China

Received 13 September 2007; accepted 15 February 2008

Abstract

In this paper, the probe absorption spectra of a V-type atom embedded in photonic band gap (PBG) reservoir havebeen investigated under conditions that quantum interference among decay channels is important. The effect of theprobe polarization on the absorption amplitude and spectral structure is investigated in detail. Comparing with similarmodels located in vacuum reservoir studied earlier, the study here shows that the probe polarization has some differenteffects on the absorption spectra in PBG reservoir.r 2008 Elsevier GmbH. All rights reserved.

PACS: 42.50.Gy; 42.50.Ct; 42.70.Qs

Keywords: Photonic band gap; Quantum interference; Probe absorption spectrum

1. Introduction

Quantum interference [1–9], which plays an essentialrole in quantum mechanics, has been manifested invarious branches of physics. In atomic, molecular andoptical physics, absorption [1–3], stimulated emission[4–6] and spontaneous emission processes [7–9] subjectto quantum interference resulting from different transi-tion processes have been of considerable interest formany years. The weak probe absorption spectra ofvarious atomic models in vacuum reservoir have beeninvestigated by several authors [1,2,4,5,10]. The verydifferent effects have attracted much attention.

In recent years, photonic band gap (PBG) structureshave been shown to have different DOS compared withthe free-space field [11,12]. The study of quantum andnonlinear optical phenomena for atoms embedded in

e front matter r 2008 Elsevier GmbH. All rights reserved.

eo.2008.02.018

ing author. Tel.: +86431 88498034.

ess: zhanghanjlu@yahoo.com.cn (H.Z. Zhang).

such PBG reservoirs leads to the prediction of manyinteresting effects, for example, localization of light[13–15], photon-atom bound states [11–13,15,16] andother phenomena [17–25]. Even though many studieshave been carried out on the PBG structures, there arefew papers referring to the absorption spectra of anatom located in PBG reservoir as far as we know[19,26–31].

It is well known that in vacuum reservoir, thespontaneous emission process is incorporated in thereduced master equation by introducing the decay rates[32]. The susceptibility due to absorption has beencalculated by using the master equation method in thelinear response theory. In this paper, we present adetailed deduction of the ‘‘decay rate’’ terms of an atomlocated in PBG reservoir. The numerical emulation ofthe probe absorption spectrum of a V-type atomembedded in PBG reservoir is also performed. Theprobe polarization direction has been taken into accountfor a V-type atom with a closely spaced doublet under

ARTICLE IN PRESSH. Zhang et al. / Optik 120 (2009) 689–695690

conditions that quantum interference among decaychannels is important. Comparing with similar modelslocated in vacuum studied earlier, the study here showsthat the probe polarization has some different effects onthe absorption spectra in PBG reservoir. Most interest-ingly, the frequency shifts of atomic levels resulting frominteraction with the PBG reservoir lead to quite differentquantum interference effects.

The paper is organized as follows. In Section 2, theequations of the ‘‘decay rate’’ terms of an atom locatedin PBG reservoir are deduced in detail. In Section 3, thetheoretical deduction of the probe absorption spectra ofa V-type atom with the transitions coupled to PBGmodes are investigated. The main results and discussionsare given in Section 4. The major conclusions aresummarized in Section 5.

2. The deduction of ‘‘decay rate’’ terms in PBG

reservoir

We consider a two-level atom damped by a reservoirof simple harmonic oscillators described by annihilation(and creation) operators bk (and bþk ), and the index k

labels the momentum and polarization indices of the kthfield mode whose frequency is ok. The levels of the atomare denoted by j0i (the ground state) and j1i (the excitedstate). In the interaction picture and the rotating-waveapproximation, the Hamiltonian is simply

Hi ¼ _X

k

gkðbþk s�e

iðmk�DpÞt þ sþbke�iðmk�DpÞtÞ, (1)

where s� ¼ j0ih1j and sþ ¼ j1ih0j. In the interactionpicture, within the Born approximation, the evolution ofthe atomic system is described by the following reducedmaster equation:

_ratom ¼ � i=_TrR½HðtÞ; ratomðtiÞ � rRðtiÞ�

� 1=_2TrR

Z t

ti

½HðtÞ; ½Hðt0Þ; ratomðt0Þ � rRðtiÞ��dt0

¼ � iX

k

gkhbþk i½s�; ratomðtiÞ�e

iðmk�DpÞt

�

Z t

0

dt0Xk;k0

gkgk0 f½s�s�ratomðt0Þ � 2s�ratomðt

0Þs�

þ ratomðt0Þs�s��eiðmk�DpÞtþiðmk�DpÞt

0

hbþk bþk0 i

þ ½s�sþratomðt0Þ � sþratomðt

0Þs��eiðmk�DpÞt�iðmk�DpÞt0

hbþk bk0 i þ ½sþs�ratomðt0Þ � s�ratomðt

0Þsþ�

�e�iðmk�DpÞtþiðmk�DpÞt0

hbkbþk0 i þH:c:; (2)

where the expectation values refer to the initial state ofthe reservoir.

hbki ¼ hbþk i ¼ 0; hbþk bk0 i ¼ n̄kdkk0 ,

hbkbþk0 i ¼ ðn̄k þ 1Þdkk0 ; hbkbk0 i ¼ hbþk bþk0 i ¼ 0. (3)

Here n̄k is the mean quantum numbers of the reservoirmodes under thermal equilibrium, we consider thephotonic reservoir to be initially in its vacuum staten̄k ¼ 0. The Born approximation assumes a weakcoupling between the atomic system and the radiationreservoir of the photonic crystal, and also that changesin the photonic reservoir as a result of atom-reservoirinteraction are negligible. In anisotropic PBG reservoir,it has been shown that the non-Markovian effectsassociated with the fast variation of the density of statesat the band-edge frequency can be analyzed within theframework of the Born approximation [30,31]. Usingthe statistical characteristics of the reservoir fieldoperators, it can be easily shown that the reducedmaster equation is given below:

_ratom ¼ �Z t

0

dt0X

k

g2k½sþs�ratomðt

0Þ

� s�ratomðt0Þsþ�e�iðmk�DpÞðt�t0Þ

�

Z t

0

dt0X

k

g2k½ratomðt

0Þsþs�

� s�ratomðt0Þsþ�eiðmk�DpÞðt�t0Þ

¼ �

Z t

0

dt0G10ðt� t0Þ½j1ih1jratomðt0Þ

� j0ih1jratomðt0Þj1ih0j�

�

Z t

0

dt0G01ðt� t0Þ½ratomðt0Þj1ih1j

� j0ih1jratomðt0Þj1ih0j�, (4)

where G10ðt� t0Þ and G01ðt� t0Þ are the delay Greenfunction which can be expressed as follows, for theanisotropic PBG reservoir [33],

G10ðt� t0Þ ¼X

k

g2ke�iðmk�DpÞðt�t0Þ

¼1

2a10 �

ei½d10c2ðt�t0Þþp=4�þiDpðt�t0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4pðt� t0Þ3

q8><>:

þei½d10c1ðt�t0Þ�p=4�þiDpðt�t0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4pðt� t0Þ3q

9>=>;.

G01ðt� t0Þ ¼X

k

g2ke

iðmk�DpÞðt�t0Þ

¼1

2a10 �

e�i½d10c2ðt�t0Þþp=4��iDpðt�t0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4pðt� t0Þ3

q8><>:

þe�i½d10c1ðt�t0Þ�p=4��iDpðt�t0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4pðt� t0Þ3q

9>=>;. (5)

ARTICLE IN PRESSH. Zhang et al. / Optik 120 (2009) 689–695 691

Eq. (4) can be expressed in the form of the densitymatrix equations as follows:

_r11 ¼ �Z t

0

dt0G10ðt� t0Þr11ðt0Þ

�

Z t

0

dt0G01ðt� t0Þr11ðt0Þ,

_r10 ¼ �Z t

0

dt0G10ðt� t0Þr10ðt0Þ,

_r01 ¼ �Z t

0

dt0G01ðt� t0Þr01ðt0Þ,

_r00 ¼Z t

0

dt0G10ðt� t0Þr11ðt0Þ

þ

Z t

0

dt0G01ðt� t0Þr11ðt0Þ. (6)

For the vacuum reservoir, these equations simplify to

_r11 ¼ �g10r11; _r10 ¼ �g102

r10,

_r01 ¼ �g102

r01; _r00 ¼ g10r11, (7)

where the Green function G10ðt� t0ÞðG01ðt� t0ÞÞ ¼12g10dðt� t0Þ. Now we incorporate the relaxation process

by introducing the ‘‘decay rate’’ terms into the densitymatrix equations in PBG reservoir successfully.

3. The absorption spectrum of a V-type atom

located in PBG reservoir

Consider a V-type three-level atom as shown inQJ;Fig. 1(a), where two close excited states j1i and j2iseparated in frequency by o21 decay to the ground statej0i. d10 and d20 are the transition dipole moments. Thedegree of quantum interference can be denoted byp ¼ cos y, where y denotes the angle between thespontaneous emission dipole matrix elements. Thiseffect of quantum interference results from the cross-coupling between two decay channels j1i—j0i and j2i—j0i. When ya90�, interference between the two sponta-neous emission paths can arise. Maximal quantuminterference is seen when the dipole moments areparallel or antiparallel, i.e. p ¼ �1. On the other hand,the system exhibits no interference effect when p ¼ 0.A weak, frequency-tunable probe beam is used to probethe transitions j1i—j0i and j2i—j0i. Op1 and Op2 arethe Rabi frequencies of the probe laser field corres-ponding to the transitions j1i—j0i and j2i—j0i,respectively.

In the interaction picture, the resultant masterequations for the reduced density operator r takethe form (including the ‘‘decay rate’’ terms intro-duced in Section 3), where we also introduce the

interference terms:

_r11 ¼ Op1r01 þ O�p1r10

�

Z t

0

dt0G10ðt� t0Þr11ðt0Þ �

Z t

0

dt0G01ðt� t0Þr11ðt0Þ

�p

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRefG10gRefG20g

pðr12 þ r21Þ,

_r22 ¼ Op2r02 þ O�p2r20

�

Z t

0

dt0G20ðt� t0Þr22ðt0Þ �

Z t

0

dt0G02ðt� t0Þr22ðt0Þ

�p

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRefG10gRefG20g

pðr12 þ r21Þ,

_r21 ¼ Op2r01 þ O�p1r20 � io21r21

�

Z t

0

dt0G10ðt� t0Þr21ðt0Þ �

Z t

0

dt0G20ðt� t0Þr21ðt0Þ

�p

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRefG10gRefG20g

pðr11 þ r22Þ,

_r10 ¼ Op1ðr00 � r11Þ þ Op2r12 þ iD1r10

�

Z t

0

dt0G10ðt� t0Þr10ðt0Þ �

p

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRefG10gRefG20g

pr20,

_r20 ¼ Op2ðr00 � r22Þ þ Op1r21 þ iD2r20

�

Z t

0

dt0G20ðt� t0Þr20ðt0Þ �

p

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRefG10gRefG20g

pr10,

r00 þ r11 þ r22 ¼ 1,

rij ¼ r�ji. (8)

By taking the Laplace transforms of Eq. (8), the steady-state solutions for r10ðtÞ, r20ðtÞ can be obtained:

r10ð1Þ ¼ ser10ðsÞjs!0

¼ð�iD2 þ eG20ðsÞÞOp1 �

p2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRefG10gRefG20gp

Op2

ð�iD2 þ eG20ðsÞÞð�iD1 þ eG10ðsÞÞ �p2

4RefG10gRefG20g

,

r20ð1Þ

¼ ser20ðsÞjs!0

¼ð�iD1 þ eG10ðsÞÞOp2 �

p2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRefG10gRefG20gp

Op1

ð�iD2 þ eG20ðsÞÞð�iD1 þ eG10ðsÞÞ �p2

4RefG10gRefG20g

,

(9)

where we have chosen r11ð0Þ ¼ 0, r22ð0Þ ¼ 0, andr00ð0Þ ¼ 1.

The eG20ðs! 0Þ, eG02ðs! 0Þ, eG10ðs! 0Þ and eG01ðs! 0Þare the Laplace transforms of the delay Green functions,which have the forms as follows, for the PBG case andthe vacuum case, respectively, PBG:

eG20ðs! 0Þ

¼1

2a20 � f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid20c2 þ D2

p� i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid20c1 þ D2

pg,eG02ðs! 0Þ

¼1

2a20 � f�i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�d20c2 � D2

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�d20c1 � D2

pg,

ARTICLE IN PRESS

|1>

|0>

Δp

|2>

d20

d10

ep

�21

�2 �1

��

Fig. 1. (a) The energy scheme used in this paper. (b) The

arrangement of field polarization and dipole moments. y is the

angle between the two dipole moments and c the angle

between the field polarization and d10.

-4

0.0

0.5

1.0

1.5

2.0

A (δ

p)

δp

1.5

2.0

-2 0 2

H. Zhang et al. / Optik 120 (2009) 689–695692

eG10ðs! 0Þ

¼1

2a10 � f

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid20c2 þ D2

p� i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid20c1 þ D2

pg,eG01ðs! 0Þ

¼1

2a10 � f�i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�d20c2 � D2

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�d20c1 � D2

pg.

(10)

Vacuum:

eG20ðs! 0Þ ¼ eG02ðs! 0Þ ¼1

2g20,

eG10ðs! 0Þ ¼ eG01ðs! 0Þ ¼1

2g10. (11)

The definitions of aj0 and gj0ðj ¼ 1; 2Þ are the same as[33]. Then the absorption AðdpÞ of the system can becalculated from the imaginary part of the susceptibility,by the following equation:

PBG:

w ¼ffiffiffiffiffiffia10p

coscr10ð1Þ þffiffiffiffiffiffia20p

cosðy� cÞr20ð1Þ,

Vacuum:

w ¼ffiffiffiffiffiffig10p

coscr10ð1Þ þffiffiffiffiffiffig20p

cosðy� cÞr20ð1Þ. (12)

Numerical calculations for the AðdpÞ is plotted as afunction of dp. In our calculations, we will consider Op1,Op2, d20c1 and d20c2 as parameters.

-4

0.0

0.5

1.0

A (δ

p)

-2 0 2δp

Fig. 2. The absorption spectra in case 4.1. The parameters

employed are y ¼ p=2, c ¼ 3p=8: (a) PBG reservoir

d20c2 ¼ 2:0, dc2c1 ¼ 1:0, o21 ¼ 0, a10 ¼ a20(solid); o21 ¼ 0,

a20 ¼ 0:1a10(thin); (b) vacuum reservoir o21 ¼ 0, g20 ¼ g10(so-lid); o21 ¼ 0, g20 ¼ 0:1g10 (thin); o21 ¼ 0:5, g20 ¼ 0:1g10 (dash

line).

4. Results and discussions

In order to describe the absorption properties underdifferent parameters, we divide this section into threesubsections according to the choice of the y.

4.1. h ¼ p/2

Since we choose the condition y ¼ p=2, quantuminterference does not occur under this situation. There-fore, the spectrum is the sum of two independent

Lorentzian lines. In a special case o21 ¼ 0, the absorp-tion spectrum can be reduced to

AðDpÞ ¼ R2Reð eG10Þcos

2cEeG10 � 2iDp

þ2Reð eG20Þsin

2cEeG20 � 2iDp

" #. (13)

For the vacuum case, we neglect the frequency shiftsof atomic levels resulting from interaction with thevacuum field in our calculation. When o21 ¼ 0, the twoLorentzian lines are both centred at Dp ¼ 0. And it is tobe noted that the peak value of the absorption Að0Þ isequal to 2, independent on the probe polarizationdirection [10].

Comparing with the vacuum case, in PBG reservoir,there is an absorption gap in the absorption spectrum(Fig. 2(a)). This new feature results from no allowedpropagating electromagnetic modes for the range offrequencies of PBG. In our deduction, the influence ofthe photonic band gap on the atom–radiation interac-tion is embedded in the memory function Gðt� t0Þ,which, in turn, is determined by the DOS of theradiation field. Also, it is to be noted that the locationof the absorption peak is correlated with a for the PBGcase. When a10 ¼ a20, the two absorption peaks are bothcentred at the same point. However, for a10aa20, thereis a distance between the two peaks corresponding to thetwo absorption transitions. And, it is interesting that the

ARTICLE IN PRESS

-4

0.0

0.5

1.0

1.5

2.0

A (δ

p)

δp

-4

0.0

0.5

1.0

1.5

2.0

-2 0 2

A (δ

p)

δp

-2 0 2

Fig. 4. The absorption spectra in case 4.2 for o21 ¼ 0: (a) PBG

reservoir a10 ¼ a20(bold line), a20 ¼ 0:1a10(thin line); (b)

vacuum reservoir g20 ¼ g10(bold line), and g20 ¼ 0:1g10(thinline). The other parameters are the same as Fig. 3.

1.5

2.0

H. Zhang et al. / Optik 120 (2009) 689–695 693

spectrum of the atom located in PBG reservoir foro21 ¼ 0 (the thin line in Fig. 2(a)) is much similar to thespectral line of the vacuum case for o21a0 (the dash linein Fig. 2(b)).

4.2. h ¼ 0

This transparency reported here is certainly due to theeffect of quantum interference. It is to be noted mostimportantly that there is a super narrow hole bored intothe broad spectrum. The super narrow hole andtransparency originates from quantum interference.When g10 ¼ g20 or a10 ¼ a20, the linewidth of the holecan be greatly narrowed as o21! 0, as shown in Fig. 3.However, for the vacuum case, when o21 becomes zero,the super narrow hole disappears, it is surprising to seethat the absorption spectrum is a single Lorentzian line(Fig. 4). This conclusion is also attributed to quantuminterference: the two absorption transitions j1i—j0i andj2i—j0i are totally correlated and thus indistinguishablein this situation. As a result, we can regard thetransitions j1i—j0i and j2i—j0i as a single transition[10]. This is the same situation for the case of PBG,when a10 ¼ a20. However, for a10aa20, there is adistance between the two peaks corresponding to thetwo absorption transitions, and the absorption transi-tions j1i—j0i and j2i—j0i in PBG reservoir can’t beregarded as a single transition. Then the absorptionspectrum is not a single Lorentzian line any more

20-2-4

0.0

0.5

1.0

1.5

2.0

A (δ

p)

δp

20-2-4

0.0

0.5

1.0

1.5

2.0

A (δ

p)

δp

Fig. 3. The absorption spectra in case 4.2. The parameters

employed are y ¼ 0, c ¼ 0, o21 ¼ 0:5(thin), o21 ¼ 1:0(bold):(a) PBG reservoir d20c2 ¼ 2:0, dc2c1 ¼ 1:0, a10 ¼ a20; (b)

vacuum reservoir g20 ¼ g10.

-4

0.0

0.5

1.0

-4

0.0

0.5

1.0

1.5

2.0

A (δ

p)

δp

-2 0 2

A (δ

p)

-2 0 2δp

Fig. 5. The absorption spectra in case 4.3. The parameters

employed are c ¼ y=2, o21 ¼ 0:3, y ¼ p=2(dash dot), y ¼ 0:5(solid), y ¼ 0(dash): (a) PBG reservoir a10 ¼ a20; (b) vacuumreservoir g20 ¼ g10.

(shown in Fig. 4(a)). Most interestingly, it is to be notedthat there is a transparency point in the broad spectrumdue to the effects of quantum interference. That is to say

ARTICLE IN PRESSH. Zhang et al. / Optik 120 (2009) 689–695694

the two transitions are still distinguishable due to thefrequency shifts of atomic levels resulting from interac-tion with the PBG reservoir.

4.3. ha 0, p/2

For simplicity, we investigate here for o21a0, a10 ¼a20 case only. The discussion about o21 ¼ 0 and a10aa20case is almost the same with the above section. It is clearthat the quantum interference results in a super narrowhole and absorption reduction. When y ¼ 0, i.e. thequantum interference is maximal, the transparencyphenomenon takes place. As y increases, the absorptionreduction becomes smaller. Finally, when y ¼ p=2, thenarrow hole disappears and the absorption spectrumbecomes the sum of two independent Lorentzian lines(shown in Fig. 5).

5. Conclusions

In this paper, we present a detailed deduction of the‘‘decay rate’’ items in the master equations of an atomlocated in PBG reservoir. The absorption spectra of aV-type atom embedded in PBG reservoir in the presenceof quantum interference have also been investigatedwith the help of the density matrix. Also, the presentstudy distinguishes itself from previous work by takinginto account the probe polarization direction in PBGreservoir. Comparing with similar models located invacuum studied earlier, the frequency shifts of atomiclevels resulting from interaction with the PBG reservoirlead to quite different quantum interference effects.Particularly, when o21 ¼ 0 and a10aa20, there is atransparency point due to the effects of quantuminterference in the broad spectrum in PBG reservoir.That is to say the two transitions are still distinguishabledue to the frequency shifts.

The methodology, we put forward here, can beapplied to other types of atomic models by smallmodification. It can be used to investigate the variousquantum interference effects in PBG materials, includ-ing the electromagnetically induced transparency (EIT),amplification and lasing without population inversion(AWI), and others. All these investigations will providethe theoretical foundation for the experimental applica-tion of the quantum interference effects in PBGmaterials, such as the designation of optical switching,low-threshold lasing and other phenomena.

Acknowledgements

This work was supported by the National NaturalScience Foundation of China (Nos. 10774060.

J0730311), and the Jilin Province Natural ScienceFoundation of China (No. 20070512).

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