Role of temperature effects in the phenomenon of ultraslow electromagnetic pulsesin Bose-Einstein condensates of alkali-metal atoms

Yurii Slyusarenko* and Andrii SotnikovAkhiezer Institute for Theoretical Physics, NSC KIPT, 1 Akademichna Street, Kharkiv 61108, Ukraine

�Received 2 April 2009; published 6 November 2009�

We study the temperature dependence of optical properties of dilute gases of alkali-metal atoms in the statewith Bose-Einstein condensates. The description is constructed in the framework of the microscopic approachthat is based on the Green’s-functions formalism. We find the expressions for the scalar Green’s functionsdescribing a linear response of a condensed gas to a weak external electromagnetic field �laser�. It is shown thatthese functions depend on the temperature, other physical properties of a system, and on the frequencydetuning of a laser. We compare the relative contributions of the condensate and noncondensate particles in thesystem response. The influence of the temperature effects is studied by the example of two- and three-levelsystems. We show that in these cases, which are most commonly realized in the present experiments, the groupvelocity and the absorption rate of pulses practically do not depend on the gas temperature in the region fromthe absolute zero to the critical temperature. We discuss also the cases when the temperature effects can playa significant role in the phenomenon of slowing of electromagnetic pulses in a gas of alkali-metal atoms withBose-Einstein condensates.

DOI: 10.1103/PhysRevA.80.053604 PACS number�s�: 03.75.Hh, 05.30.�d, 42.25.Bs

I. INTRODUCTION

Bose-Einstein condensate �BEC� is one of the most im-pressive examples when the matter demonstrates its quantumnature on the macroscopic level. Now this system is interest-ing also due the possibility of observing the electromagneticpulses propagating with extremely slow group velocities in it�1�.

Up to now, in the theoretical investigations pretending todescribe the mentioned phenomenon in a BEC �see Refs.�2,3��, the authors assumed that the temperature of a gas issmall in comparison to the critical temperature. In otherwords, this assumption corresponds to the consideration ofthe zero-temperature limit. But in real systems, the tempera-tures can be of the same order as the critical temperature.Therefore, one needs to study the account of temperatureeffects in the ultraslow light phenomenon. Naturally, it isalso important to compare the theoretical results to the ex-perimental data �1� describing the dependence of the groupvelocity of a signal on the temperature of a system.

In the present paper, we generalize the approach devel-oped earlier in Ref. �3� for the uniform �nontrap� systems onthe case of finite �nonzero� temperatures. This approach isbased on the Green’s-functions formalism �4� and an ap-proximate formulation of the second-quantization method�5�. An object of the mentioned generalization is a study ofthe influence of temperature effects on the dispersion char-acteristics of the system and, as a result, on the propagationproperties of a signal in it.

II. LINEAR RESPONSE OF A GAS IN A BEC STATE ATFINITE TEMPERATURES: GREEN’S FUNCTIONS

To describe the optical properties of gases consisting ofalkali-metal atoms that are used in the BEC-related experi-

ments, it is most convenient to use the model of an ideal gasof hydrogenlike atoms in the stationary state �the limits ofthis approach are discussed in Ref. �3��. By the term “sta-tionary state” we mean that the atoms can be found only inthe states whose lifetimes are much greater than the relax-ation time of the system �e.g., hyperfine levels of the groundstate� and in the states whose occupations are stimulated byan external electromagnetic field �e.g., a laser radiation�. Inthe case of a BEC presence in this gas, the density distribu-tion of atoms in the quantum state � by the momentum p atnonzero temperatures �0�T�Tc, T is the temperature in en-ergy units� can be set equilibrium, therefore, it can be writtenas follows �see also Ref. �4��:

���p� = ���1 − �T/Tc��3/2���p� + g��2���−3

��exp����p� − ��/T� − 1�−1, �1�

where �� is the total density of atoms in the � state

�� =� dp���p� ,

where Tc� is the temperature of the transition of a gas ofatoms in the � state to the BEC phase, ��p� is the Dirac deltafunction, and g�= �2F�+1� is the degeneracy order of thestate by the total momentum F� of an atom in this state. Inthe next calculations, we assume that an external field ispresent in the system, i.e., we set g�=1. Here, also � is thePlanck constant, ��p�=�+ p2 /2m, where � is the energyof an atom in the � state, m is the atomic mass �m=mp+me, mp, and me are masses of the atomic core and electron,respectively�, and � is the chemical potential an atom inthis state.

Now we must note the following. It is known that a Bose-Einstein condensation is a collective effect. The atoms occu-pying the same quantum state � are identical. Because of thisfact, one cannot definitely state what particles participate in*slusarenko@kipt.kharkov.ua

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the condensation process. Therefore, the terms “condensate”and “noncondensate” particles do not correspond directly tothe selected atoms in the system. Strictly speaking, the termcondensate particles corresponds only to the fraction of allatoms, which participate in the formation of the Bose-condensed component. Hence, the number of noncondensateparticles can be found from the difference between the totalnumber of atoms and number of condensate particles in thesystem. By the use of these terms, now we can say that thefirst summand in Eq. �1� corresponds to the contribution ofcondensate particles in the distribution function and the sec-ond summand corresponds to the normal component. Thus inthe next calculations, we differentiate contributions from thedifferent types of particles in the response of the system tothe external perturbation by an electromagnetic field.

It is shown in Ref. �3� that the linear response of an idealgas of hydrogenlike atoms with BECs to a weak electromag-netic field can be studied from the first principles in theframework of the Green’s-function formalism. There it wasshown that the Fourier transform for the scalar Green’s func-tion can be defined by the relation �see also Ref. �6� fordetails�

G�k,�� =1

V�p

��,�

���k�2f��p − k� − f��p�

��p� − ��p − k� + � + i���

.

�2�

Here, by k and � we denote the wave vector and the fre-quency of the external perturbing field, respectively, and ���

denotes the linewidth related to the probability of a sponta-neous transition between the � and � states.

The quantity ���k� defines the matrix elements of thecharge density of hydrogenlike atoms. This quantity can beexpressed in terms of the wave functions �� and �� of theatoms in � and � states �see also Refs. �3,5��, respectively,

���k� = e� dy����y����y�exp�i

mp

mky�

− exp�− ime

mky� ,

where e is the elementary charge. In particular, in the casewhen the dipole transition is allowed between the � and �states, in the linear order over small term ky�1, one gets

���k� � ikd��, d�� = e� dyy����y����y� ,

where d�� is the matrix element of the atomic dipole mo-ment. It should be mentioned that in next calculations, weuse only the quantities that are proportional to ���k�2.Therefore, for the allowed dipole transitions, these quantitiescan be expressed in terms of the average dipole moment d��,

���k�2 � k2d��2 /3. �3�

The quantity f��p� in Eq. �2� corresponds to the Bosedistribution function of atoms by the momentum p,

f��p� = �exp����p� − ��/T� − 1�−1. �4�

Note that one can use the Bose distribution in the form �4�only in the case of the thermal equilibrium in the system.Thus, in the next calculations, we consider only the stateswhose lifetimes are much greater than the system relaxationtime and the states whose existence in the system is stimu-lated by an external field. Hence, for a studied gas �a con-densed phase can be formed by atoms with the energy ��,one gets the condition for the chemical potential � �see Ref.�7� for details�

��T � Tc� = �.

At temperatures 0�T�Tc in Eq. �2�, one can substitutethe distribution function �4� by the density distribution �1�.To this end, one needs also use the rule

1

V�p

f��p� ¯ =� dp���p� ¯ .

Next let us note that the relation �2� is linear over the distri-bution function f��p�. Therefore, in accordance with Eq. �1�,it can be divided into two summands that define the contri-butions of the condensate and noncondensate particles in thesystem response

G�k,�� = G�c��k,�� + G�n��k,�� , �5�

where the summands G�c��k ,�� and G�n��k ,�� are defined by

G�c��k,�� = ��,�

���k�2���1 − t�

3/2����� + i���

,

G�n��k,�� = �2���−3��,�

���k�2

��0

� 2�p2dp

exp�p/T� − 1�

−1

1 dy

���� + pky/m + i���

.

�6�

Here, t�=T /Tc� denotes the relative temperature of a gas �inthis paper, we consider the case t��1�, ����=�+��� isthe frequency detuning taken in the energy units, ���=�

−�, p= p2 /2m denotes the kinetic energy of an atom, andthe integration variable y denotes the cosine of the polarangle �, y�cos �. Here and below, we neglect of the recoilenergy r=�2k2 /2m that is small enough to do a significantcontribution into the effect �r����; strictly speaking, it canbe accounted by redefining the quantity ����, i.e., by shift-ing the resonant frequency by r�. For the definiteness, in thenext description, the summand G�c��k ,�� in Eq. �5� is calledas the condensate Green’s function and the summandG�n��k ,�� is called as the noncondensate Green’s function.

A. Condensate Green’s function at finite temperatures

Firstly, we consider the scalar Green’s function corre-sponding to the contribution of condensate particles. For aconvenience, let us study in detail the real and imaginaryparts of it. It is shown below that in the region of transpar-

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ency, the real part makes a main contribution to the refractiveindex of a gas and imaginary part makes a main contributionto the absorption rate of light pulses. In accordance with Eq.�6�, the real and imaginary parts of this function can be writ-ten as

Re G�c� = ��,�

���k�2���1 − t�

3/2�����

������2 + �����2 ,

Im G�c� = − ��,�

���k�2���1 − t�

3/2����

������2 + �����2 . �7�

To simplify the description, in the next calculations, weuse a model of a two-level system ��=1, �=2�. Below, bythe index “1” we denote the set of quantum numbers corre-sponding to the ground state and by the index “2” we denotethe set of quantum numbers corresponding to the excitedstate. It is also assumed that the occupation of the exitedstate is stimulated by a low-intensity laser pulse. Due to alow intensity of the pumping field, we can consider that thedensity of the atoms in the excited state is small in compari-son to the density of atoms in the ground state �see Ref. �3�for details�

�2 � �1 � � .

According to this relation in Eq. �7� we neglect of the sum-mands that are proportional to the density �2. Therefore, onecan get

Re G�c� = 12�k�2��1 − t3/2��−1F���� ,

Im G�c� = 12�k�2��1 − t3/2��−1F���� , �8�

where we introduce the quantity ���12 and the dimension-less functions F��x�=x / �x2+1� and F��x�=−1 / �x2+1�. Thedependencies of these functions on the relative frequencydetuning � ��=�� /�� are shown in Fig. 1.

Let us note that the functions F� and F� do not depend onthe temperature. Hence, it is easy to see from Eq. �8� that thereal and imaginary parts decrease with temperature by apower law as t3/2. Note also that the condensate Green’sfunction is proportional to the density of condensate particlesin a gas, ��c�=��1− t3/2�, and it equals to zero at the criticaltemperature, t=1.

B. Noncondensate Green’s function at finite temperatures

Now we study the scalar Green’s function G�n� corre-sponding to the noncondensate particles �see Eq. �6��. Takingthe real and imaginary parts, for the two-level system �set-ting the indexes �=1 and �=2, see above�, we come to therelations

Re G�n� = 12�k�2m2T

4�2�4kI���,t� ,

Im G�n� = 12�k�2m2T

4�2�4kI���,t� , �9�

where we introduce the dimensionless functions I��� , t� andI��� , t� that depend on the frequency of an external field andon the system parameters �including the temperature of agas�

I���,t� = �0

� xdx

ex2− 1

ln� 1 + �� + x���t��2

1 + �� − x���t��2� ,

I���,t� = 2�0

� xdx

ex2− 1

�arctan�� − x���t��

− arctan�� + x���t��� . �10�

Here, analogously to the formulas for the condensate par-ticles �see Eq. �8��, �=�� /� is the relative frequency detun-ing, t=T /Tc is the relative temperature,

Tc = 2����3/2��−2/3�2�2/3m−1 �11�

is the critical temperature of the transition to the BEC phase,and ��x� is the Riemann zeta function. In Eq. �10�, we alsointroduce the dimensionless parameter �,

� =�k

��2Tc

m�

�k

�m��1/3 �12�

that depends on the physical properties of a gas and the wavenumber k of the external electromagnetic field. It is shownbelow that the value of this parameter defines the character-istics of the normal component response to an externalperturbation.

Let us estimate the value of the parameter � for the reso-nant radiation that corresponds to the sodium D2 line. Thelaser pulses tuned to the components of this line were used inRef. �1�. Taking m=3.82�10−23 g, �=8.1�10−21 erg, k=1.07�105 cm−1, and Tc=435 nK, we get ��0.025. Usingthis parameter value, it is easy to get the dependencies for theintegrals I��� , t� and I��� , t� that are shown in Fig. 2. There,one can study also a behavior of the integral I� with a de-

-10 -8 -6 -4 -2 0 2 4 6 8 10

-1.0

-0.5

0.0

0.5

F’(Δ)

Fun

ctio

nva

lue

Detuning, Δ=δω/γ

F’’(Δ)

FIG. 1. �Color online� Dependencies of the functions F� and F�corresponding to the response of the condensate particles on thefrequency detuning �.

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crease of the temperature to the values 0.5Tc and 0.1Tc, re-spectively �note that the integral I� has an analogous tem-perature dependence�.

From Fig. 2, one can see that the dependencies of thefunctions I� and I� are similar to the dependencies of thecondensate functions F� and F� �cf. Fig. 1�. The reasons forthis similarity become evident below, but now it is easy tosee that the values of the functions I� and I� strongly dependon the gas temperature. In particular, from Eq. �9� and Fig. 2,we can conclude that the contribution of the noncondensateGreen’s functions in the total Green’s function �5� increaseswith temperature. Let us study this effect in detail in the nextsection.

C. Dependence of the scalar Green’s functionon the temperature

Let us note that the functions I� and I� characterizing thecontribution of the noncondensate particles to the total re-sponse of a gas in the case ��1 can be simplified at arbi-trary values of the detuning �. Really, due to the stronglydecreasing function exp�−x2� in the integrand, the main con-tribution to the integral give small values of the integrationvariable x, x�1. This fact allows us to expand the integrandsinto series over �x��t��1. As a result, using Eqs. �8� and�10�, accurate within quadratic summands, we get

I���,t� � 4��tF������0

� x2dx

ex2− 1

+ ���t�2

�4

3F�2��� + F���� �

0

� x4dx

ex2− 1� ,

I���,t� � 4��tF������0

� x2dx

ex2− 1

+ ���t�2

�F�2��� −1

3F�2��� �

0

� x4dx

ex2− 1� . �13�

According to the relations

�0

� x2dx

ex2− 1

=��

4��3/2�, �

0

� x4dx

ex2− 1

=3��

8��5/2� ,

the Green’s functions �9� with account of Eqs. �11�–�13� canbe written as

Re G�n� � 122�t3/2�−1F�����1 + �2tS����� ,

Im G�n� � 122�t3/2�−1F�����1 + �2tS����� , �14�

where we introduced the functions

S���� = 2��5/2���3/2�F�2��� +

3

4F���� ,

S���� =3

2

��5/2���3/2�F�2��� −

1

3F�2��� .

Note that Eq. �14� explains also the similarity in the behaviorof the functions F�, F� and I�, I� at ���t��1 that is men-tioned above �see Figs. 1 and 2�.

It is easy to see that the “total” Green’s function in theregion from the absolute zero to Tc weakly depends on thetemperature. This dependence appears only in the quadraticterms over ���t��1. Really, by the use of Eqs. �5�, �8�, and�14�, one can get

Re G � 122��−1F�����1 + �2tS����� ,

Im G � 122��−1F�����1 + �2tS����� . �15�

But in the case ���t��1, the expansions �13� becomeincorrect and analytical formulas for the functions I� and I�cannot be found. Therefore, in this case, we use the numeri-cal calculations at the definite values of the detuning � andparameter � �see Figs. 3 and 4�.

In Fig. 3, one can see the dependencies of the condensateand noncondensate Green’s functions on the temperature.The real parts of these functions are normalized to a value ofthe real part of the condensate Green’s function at t=0 andcorresponding detuning �. Analogously, the imaginary partsof these functions are normalized to a value of the imaginarypart of the condensate Green’s function at t=0 and corre-sponding detuning �. In other words, in Fig. 3, the followingdependencies are shown:

Re G�c,n��t,��/Re G�c��t = 0,�� ,

Im G�c,n��t,��/Im G�c��t = 0,�� .

One can see that due to this normalization, the inequalityIm G�t ,�� / Im G�c��0,���0 takes place. But note that theimaginary part of all Green’s functions is negative due to thefunction F���� �see Eqs. �8� and �14��.

-10 -8 -6 -4 -2 0 2 4 6 8 10

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

t=0.1

t=0.5

t=1.0

Inte

gral

valu

e

Detuning, Δ=δω/γ

I’ (Δ,t)

I’’ (Δ,t)

FIG. 2. �Color online� Dependencies of the functions I��� , t�and I��� , t� on the frequency detuning at T=Tc �bold lines�. By thinlines, the dependencies of the function I� at temperatures T /Tc

=0.5 and 0.1 are shown ��=0.025�.

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Analyzing the dependencies shown in Fig. 3, one can con-clude that the real and imaginary parts of the condensateGreen’s function demonstrate a similar behavior �drop-downbold curve�. This dependence agrees with the obtained rela-tions. But at the same time, the relative values of the real andimaginary parts of the noncondensate Green’s functionsstrongly depend on the parameter � and detuning �. One canconclude that at ��1, by the use of the mentioned normal-ization, the real and imaginary parts practically coincide. Inthe opposite case, ��1, as one can see in Fig. 3, the real andimaginary parts demonstrate different dependencies. Thisfact must have a strong impact on the dependence of the totalGreen’s function �the sum of the condensate and nonconden-sate Green’s functions� on the temperature.

In Fig. 4, the dependencies of the total Green’s functionon the temperature are shown at different values of the pa-rameters � and �. Here, it is important to pay attention to thecentral curve that is practically horizontal �the case ��1�.This curve demonstrates a weak dependence of the totalGreen’s function �including its real and imaginary parts� onthe temperature. Note that one can come to the same conclu-sion by analyzing Eq. �15�. Thus at ��1, the influence ofthe temperature effects in the response of a BEC to an exter-nal field is insignificant. The problem on obtaining the re-sponse of this system can be solved with a good accuracy bysetting the temperature of a gas equal to zero �exactly thisapproximation was used in Refs. �3,6,8,9��. In other cases �at��1�, as one can see from Fig. 4, the influence of the tem-perature effects can be rather significant.

Evidently, the effects studied in this section in some way�significantly or not� can influence on the dispersion proper-ties of a condensed gas. Let us recall that the resonant pecu-liarities of the dispersion characteristics play an essential rolein the ultraslow-light phenomenon in a BEC of alkali-metalatoms.

III. DISPERSION CHARACTERISTICS OF A GAS IN ABEC STATE AT FINITE TEMPERATURES

It is known that in the framework of the linear approach,the permittivity ��k ,�� of a gas can be expressed in terms ofthe scalar Green’s function �see in this case Refs. �4,6�� as

�−1�k,�� = 1 +4�

k2 G�k,�� .

Therefore, the real and imaginary parts of the permittivity�k ,��,

��k,�� = ���k,�� + i���k,�� ,

can be written as

���k,�� =1 + 4�k−2 Re G

�1 + 4�k−2 Re G�2 + �4�k−2 Im G�2 ,

���k,�� =− 4�k−2 Im G

�1 + 4�k−2 Re G�2 + �4�k−2 Im G�2 . �16�

At the same time, the refractive index and damping factorcan be written in terms of the real and imaginary parts of thepermittivity

n��k,�� =1

�2����2 + ��2 + ��,

n��k,�� =1

�2����2 + ��2 − ��. �17�

Hence, basing on the derived equations for the scalar Green’sfunctions, one can study the propagation properties of weakelectromagnetic pulses through a BEC at finite �nonzero�temperatures. As the derived analytical expressions for therefractive index and damping factor in the general case may

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0 ReG(n)

ImG(n)

Δ=0

Δ=0

κ=10

κ=1

κ<<1

Rel

ativ

eV

alue

ofG

reen

Fun

ctio

ns

Temperature, T/TC

κ=4Δ=4.0

G(c)

FIG. 3. �Color online� Relative dependencies of the condensateand noncondensate Green’s functions on the temperature. All func-tions are normalized to a value of the condensate Green’s functionat t=0 and corresponding detuning �.

0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Tot

alV

alue

ofG

reen

Fun

ctio

ns

Temperature, T/TC

ReGImG Δ=0

Δ=0

κ=10

κ=1

κ<<1

κ=4Δ=4.0

κ=2Δ=4.0

FIG. 4. �Color online� Relative dependencies of the totalGreen’s functions on the temperature. All functions are normalizedto a value of the condensate Green’s function at t=0 and corre-sponding detuning �.

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have rather lengthy form �see Eq. �6��, let us consider someparticular cases.

A. Temperature effects in two-level systems

Firstly, let us consider a case when the frequency of anexternal field is close to the energy spacing between twodefinite quantum states of atoms. Thus, using the most com-mon method �see, e.g., Ref. �10��, in calculations we canconsider only the resonant terms corresponding to this tran-sition. In other words, we can consider atoms as a two-levelsystem. Therefore, by the use of Eqs. �5�, �8�, �9�, and �12�,the real and imaginary parts of the permittivity �see Eq. �16��take the form

���k,�;t� =1 + aG1

�1 + aG1�2 + �aG2�2 ,

���k,�;t� =− aG2

�1 + aG1�2 + �aG2�2 , �18�

where

G1 = �1 − t3/2�F���� + btI���,t;b� ,

G2 = �1 − t3/2�F���� + btI���,t;b� , �19�

a = 4� 12�k�2�

k2�, b = ������3/2��−1. �20�

In particular, for the condensed sodium vapor with thedensity �=1.44�1012 cm−3 that interacts with the resonantradiation corresponding to the D2 line, in accordance withEqs. �3� and �20�, we get a�4.19d2� /�. Next, consideringd2�SFF��3.52er0�2 �r0 is the Bohr radius, e is the elementarycharge, and SFF� is the relative strength of the dipole-allowedtransition F→F� �11��, F=2, F�=2, and S22=1 /4, we cometo a�0.015 and b�20.38. For the two-level system withthese parameters, the characteristic dependencies are shownin Fig. 5 �straight bold lines on the upper and lower graphs;there also the dependencies for the case �=10 are shown�.Let us note that these dependencies �bold lines� also corre-spond to the case t=0 independently of the parameter �value �see also Eq. �15��.

Now we use the expressions that define the values of thegroup velocity and intensity of the transmitted light

vg�t� = c�n��t� + ���n��t�/����−1,

I�t� = I0 exp�− n��t�kL� , �21�

where L is the characteristic size of the atomic cloud �incalculations we set L=0.004 cm�. For the mentioned physi-cal characteristics, one can obtain the dependencies corre-sponding to the signals tuned up exactly to the resonant fre-quency ��=0�. The corresponding curves are shown in Fig.6. As one can conclude from Fig. 6, the propagation velocityof the light pulse at the temperatures lower than the criticaltemperature in the case ��1 �as it is expected� practicallydoes not depend on the temperature.

Note that inequality ��1 usually takes place for dilutegases of alkali-metal atoms that interact with a laser field���0.025 for the parameters of the experiment �1��. Prob-ably, this inequality takes place for the most BEC-relatedexperiments at the present moment. Really, as it comes fromthe definition �12� of the parameter �, to increase its value inone order of magnitude, the density of atoms in a condensatemust be increased by 3 orders of magnitude. This require-ment results in the instability of the BEC phase due to anincrease of the number of three-body collisions �see Ref.�12��. Also it is easy to see that the use of the high-frequencyradiation �with the large wave number k� cannot result to theconsiderable increase of the parameter �. It comes from thefact that the high-frequency transitions correspond to thehigh levels of the atomic spectrum. But the linewidth ofthese levels is much greater than the linewidth correspondingto the lower states. Hence, probably, the only way to a sig-nificant increase of the � value is to use the states with the

-10 -8 -6 -4 -2 0 2 4 6 8 100.000

0.002

0.004

0.006

0.008t=0.0t=0.5t=1.0

Dam

ping

Fac

tor

Detuning, Δ=δω/γ

0.996

0.998

1.000

1.002

1.004

κ=10

t=0.0t=0.5t=1.0

Ref

ract

ive

Inde

x κ=10

FIG. 5. �Color online� Dispersion dependencies of the two-levelsystem in the BEC state. The used value for the parameter a=0.015.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

Tra

nsm

itte

dL

ight

Temperature, T/TC

0

100

200

300

400

500

600

κ=1.0

κ=10.0

κ=10.0

κ=1.0

κ=0.011

κ=0.011

Gro

upV

eloc

ity

(m/s

)

FIG. 6. �Color online� Dependencies of the group velocity andintensity of the transmitted light on the temperature for the two-level system with different values of the parameter �.

YURII SLYUSARENKO AND ANDRII SOTNIKOV PHYSICAL REVIEW A 80, 053604 �2009�

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smaller linewidth � �see Eq. �12��. It means that to increasethe mentioned value, one needs to select the levels whoseprobability of a spontaneous transition is much less than wasused in the above calculations.

Therefore, the cases ��1 that are shown in Fig. 6 corre-spond to the “long-living” levels. One may relate these statesto the levels with the forbidden dipole transitions �e.g., hy-perfine levels of the ground state, see in this case Ref. �8�� orthe sublevels, whose relative transition intensity is ratherlow. One can conclude that in this case, the temperature ef-fects must have a strong impact on the ultraslow light phe-nomenon in a BEC.

B. Temperature effects in three-level systems

Note that two-level systems from the standpoint of theexperiments dealing with the ultraslow-light phenomenon ina BEC can be inconvenient. As it is known, this phenomenonis realized mostly in three-level systems. These systems havesome characteristic advantages: the absorption rate of thesignal can be decreased by a special tuning of the laser fre-quency, one can get positive time delays of the pulses �posi-tive sign of the group velocity�, and also one can use themagnetic field to control the group velocity of the ultraslowpulses �9� or use an additional coupling laser to provide theelectromagnetically induced transparency �13�.

Therefore, in the framework of the developed approach,let us study the temperature dependencies of the group ve-locity and intensity of the transmitted light in a three-levelsystem. To this end, let us consider the system that is sche-matically illustrated in Fig. 7 �upper part�.

Note that in the framework of the developed approach, wedo not account quantum interferences in the system. There-fore, in this case, the derived above expressions change in-significantly. In particular, due to the additive contribution ofall quantum states to the Green’s functions �see Eq. �6��, thefunctions G1 and G2 characterizing the response of all par-ticles in the system can be written as follows �cf. Eq. �19��:

G1 = �1 − t3/2��F��� + �/2� + F��� − �/2��

+ bt�I��� + �/2,t;b� + I��� − �/2,t;b�� ,

G2 = �1 − t3/2��F��� + �/2� + F��� − �/2��

+ bt�I��� + �/2,t;b� + I��� − �/2,t;b�� , �22�

where �=�mag /� and �mag is the energy spacing betweenthe excited states. Here and below, we consider that the split-ted levels belong to the same multiplet, i.e., 122 /�12= 132 /�13. Hence, in accordance with Eq. �20�, we can seta12=a13=a. Thus the formulas �16�, �17�, and �21� that de-fine the dependence of the dispersion characteristics on thetemperature have the same form.

To get the numerical estimates, let us consider the con-densed gas of sodium atoms with the density �=5�1012 cm−3. For this gas, we get a=0.052, �=0.016, andb=13.46. Also we consider that there are two nearby excitedlevels in the system �states 2� and 3� in Fig. 7�. We also setthe spacing between them several times larger than the line-width �, �mag=8�. For this system, one can get the graphscharacterizing the dispersion characteristics of the systemthat are shown in Fig. 7 �bold curves�. Note that in this case���1�, the functions n���� and n���� practically do not de-pend on the temperature. Let us emphasize that we regard thesystem as a superposition of two two-level systems �Autler-Townes treatment�. This is the reason that the absorption ratedoes not go to zero at �=0 �Fig. 7, bottom� as in theelectromagnetically-induced transparency �EIT� regime�1,13�.

Within the developed approach, one can also study thedependencies in the case when the temperature effects have asignificant influence on the system response �as it is men-tioned above, this corresponds to the case ��1�. In particu-lar, taking �=10 and the same values of the other param-eters, we get the dispersion curves at different temperaturesthat are shown in Fig. 7. It should be noted that in this case,the steepness of the slope of the refractive index decreaseswith the temperature. At the same time, the absorption in thecentral region increases with the temperature. This behaviorcan be explained by the fact that the density distribution ofthe noncondensate particles by the momentum p in the case��1 is not so “sharp” as for the condensate particles �seeEq. �1��. As a result, the absorption rate of the pulses tunedup to the resonance ���1� decreases and the absorption rateof the pulses detuned from the resonance ���1� increases�see Figs. 4, 5, and 7� with the temperature.

The mentioned behavior of the main macroscopic param-eters with an account of Eq. �21� results in the dependenciesthat are shown in Fig. 8. As one can see from the graphs, thevelocity of a light pulse at the temperature lower than thecritical temperature in the case ��1 �analogously to the

Δεmag

|1

|2

|3

3S1/2

3P3/2

εres

-20 -16 -12 -8 -4 0 4 8 12 16 200.00

0.01

0.02

Δεmag

=8γ

t=0.0t=0.5t=1.0

Dam

ping

Fac

tor

Detuning, Δ=δω/γ

Δεmag

=8γ

κ=10

κ=10

0.99

1.00

1.01t=0.0t=0.5t=1.0

Ref

ract

ive

Inde

x

(b)

(a)

FIG. 7. �Color online� Scheme and dispersion dependencies ofthe three-level system in a BEC state. The used value for the pa-rameter a=0.052.

ROLE OF TEMPERATURE EFFECTS IN THE PHENOMENON… PHYSICAL REVIEW A 80, 053604 �2009�

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two-level case� is practically constant. It corresponds to thefact that the refractive index profile and absorption rate inthis case change insignificantly. Let us emphasize that in thiscase, for the calculations, we used the same parameters of agas with a BEC as were realized in Ref. �1�.

At ��1, the situation differs from the two-level case. Letus note that here one can see that both the group velocity andabsorption rate of the pulse increase with the temperature. Itcan be explained by the fact that for the detuned pulses, thesteepness of the slope of the refractive index decreases whilethe absorption increases with the temperature �see Fig. 7�.Thus the system becomes not so convenient for the realiza-tion of the ultraslow light phenomenon as in the zero-temperature limit.

IV. CONCLUSION

In this paper, we studied some optical properties of dilutegases in the BEC state at finite temperatures. The analysis isbased on the Green’s-function formalism. We found analyti-cal expressions for the scalar Green’s functions that charac-terize a linear response of a gas to an external electromag-netic field �laser�. We studied the characteristic dependenciesof these functions on the temperature and frequency detun-ing. We made a comparison of the contribution of the con-densate and noncondensate particles into the effect. Theultraslow-light phenomenon in a BEC is studied both on theexamples of two-level and three-level systems.

In particular, it is shown that for a light pulse, which istuned up close to the dipole-allowed transitions, the disper-sion characteristics of a gas of alkali-metal atoms weaklydepend on the temperature in the region from the absolutezero to the critical temperature. Therefore, the group velocityof the pulses in this system weakly depends on the tempera-ture. This fact allowed us to conclude that the results of theprevious works, where the authors studied the response of agas in the limit of zero temperatures, T→0, are correct andcan be used also in the region 0�T�Tc.

It is significant to note that in the experiment �1�, datarelating to the dependence of the group velocity of lightpulses on the temperature of an ultracold gas of sodium at-oms were obtained. From these results, one can conclude thatthe group velocity weakly depends on the temperature in theregion 0�T�Tc. In the present paper, in the case of a three-level system, we use the same parameters of a gas as in Ref.�1� and we come to the same conclusion in our research. Butthe statement that the results of our theory are verified by thementioned experiment is not quite correct. It should be notedthat in Ref. �1�, the effect of the electromagnetically inducedtransparency was used. But in the framework of the intro-duced approach that is based on the Green’s-functions for-malism, we cannot account for quantum interference effectsin the system �see in this case Ref. �3��. Probably, the simi-larity of the results may be a sign that the quantum interfer-ence effects in the mentioned experiment have a weak influ-ence on the dependence of the group velocity on thetemperature of a gas with a BEC. But, obviously, this state-ment requires an additional experimental verification.

We also studied the cases when the temperature effectscan have a strong impact on the dispersion characteristics ofgases with a BEC. In our opinion, this situation may berealized when the frequency of an external field is tunedclose to the transitions between “long-living” states. Thesestates correspond to the excited levels with the forbiddendipole transitions �upper hyperfine levels of the ground stateof alkali-metal atoms� or the levels with a low relative inten-sity of the transition. In this case, as one can conclude fromthe results of the paper, the parameters of slowing impairwith the temperature increase. Thus, the achievement of thelower temperatures of a gas is more necessary in this case.

ACKNOWLEDGMENT

This work is partly supported by the National Fund ofFundamental Research of Ukraine Grant No. 25.2/102 andthe National Academy of Sciences of the Ukraine Grant No.55/51–2009.

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0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

κ=4.0

κ=0.016

Tra

nsm

itte

dL

ight

Temperature, T/TC

κ=0.016

κ=4.0

κ=10.0

200

400

600

800

1000

κ=10.0

Gro

upV

eloc

ity

(m/s

)

FIG. 8. �Color online� Dependencies of the group velocity andintensity of the transmitted light on the temperature for the three-level system with different values of the parameter �.

YURII SLYUSARENKO AND ANDRII SOTNIKOV PHYSICAL REVIEW A 80, 053604 �2009�

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