PHYSICAL REVIEW A 83, 023601 (2011)

Propagation of relativistic charged particles in ultracold atomic gases withBose-Einstein condensates

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

Department of Physics and Technology, Karazin National University, 4 Svoboda Square, Kharkiv 61077, Ukraine(Received 23 November 2010; published 2 February 2011)

We study theoretically some effects produced by a propagation of the charged particles in dilute gases ofalkali-metal atoms in the state with Bose-Einstein condensates. The energy change of the high-speed (relativistic)particle that corresponds to the Cherenkov effect in the condensate is investigated. We show that in the studiedcases the particle can both lose and receive the energy from a gas. We find the necessary conditions for theparticle acceleration in the multicomponent condensate. It is shown that the Cherenkov effect in Bose-Einsteincondensates can be used also for defining the spectral characteristics of atoms.

DOI: 10.1103/PhysRevA.83.023601 PACS number(s): 03.75.Hh, 41.60.Bq, 32.70.−n

I. INTRODUCTION

Physical studies devoted to the charged particles propagat-ing in the refractive medium have a rather long history. TheCherenkov radiation [1], the ionization losses of energy [2],and the nonlinear Doppler effect in the refractive medium [3]are some of the most vivid examples. At the end of thepast century, experimentalists achieved the Bose-Einsteincondensation (BEC) phase, which is often called a new stateof matter. This state most commonly is observed in dilutegases consisting of alkali-metal atoms. In these systems, thecritical transition is achieved at ultralow temperatures (ontheorder of nanokelvin) and relatively low densities, ν �1014 cm−3. Besides that, a Bose-Einstein condensate is knownas a coherent state of matter, that is, the state with practicallysimilar behavior of atoms. Because of their unusual require-ments and properties, gases in the BEC state manifest uniqueeffects that are inherent only to these systems (see Ref. [4]for details). The extreme conditions for the temperatures canresult in more vivid manifestation of the physical phenomenathat are well studied for other systems. For example, one ofthese effects is the ultraslow light phenomenon in BECs [5,6].Therefore, now Bose-Einstein condensates are considered bymany physicists as unique systems from the standpoint ofexperimental observations and theoretical studies.

Taking into account all the mentioned facts, we have studiedthe propagation of the charged particles in a gas with Bose-Einstein condensates. Within this paper, we focus on someaspects of this problem.

II. FORMALISM

It is known that alkali-metal atoms are most effectively usedfor the studies of the phase with a BEC in atomic gases. Fora description of the electromagnetic properties of dilute gasesin some cases, one may use the ideal gas approximation (seeRef. [6] for details). As it is easy to see, this model is alsoconvenient for studying the effects that take place during theinteraction of the charged particle with a BEC in vapors of

*slusarenko@kipt.kharkov.ua

alkali-metal atoms. For gases with comparably low densi-ties of atoms, ν � 1014 cm−3, the microscopic approach isdeveloped [6]. In the framework of this model, it is shown thatthe dispersion characteristics of a gas can be described withgood accuracy by analyzing the relation for its permittivity. Inthe case of the system considered in the BEC state, the Fouriertransform of the permittivity can be written as follows:

ε−1(k,ω) = 1 + 4π

k2

∑a,b

(νa − νb)|σab(k)|2ω − �εab + i�ab/2

, (1)

where ω and k are the wave frequency and wave vector,respectively (here and in the following we set h̄ = 1), andindices a and b denote the sets of quantum numbers thatcorrespond to the certain states of alkali-metal atoms. Thequantities νa and νb are the atomic densities in these states,�εab is the transition energy between defined states, and �ab

is the natural linewidth that corresponds to the probability ofthe spontaneous transition between the states. The quantityσab(k) in Eq. (1) is the matrix element of the charge densityof atoms that is defined in the framework of the approximateformulation of the second quantization method in the case ofthe presence of the bound states (atoms) in the system (seealso Ref. [7] for details)

σab(k) = e

∫dyϕ∗

a (y)ϕb(y)

[exp

(imp

mk · y

)

− exp

(− i

me

mk · y

)]. (2)

Here ϕa is the wave function of an atom in the quantum state a,and mp and me are the masses of the atomic core and electron,respectively (m = mp + me).

A quantity of the energy that is absorbed by the gas in theinterval of wave frequencies dω and wave numbers dk duringa propagation of the charged particle can be described by theformula (see, e.g., Ref [8]):

Qωk = − 2

(2π )4Im

(4π

ωε|j‖|2 + 4πω|j⊥|2

ω2ε − c2k2

), (3)

where j‖ and j⊥ are the longitudinal and transversal com-ponents of the density of the external current [j = j‖ + j⊥,j‖ = (k · j)k/k2]. Note that here and in the following for the

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YURII SLYUSARENKO AND ANDRII SOTNIKOV PHYSICAL REVIEW A 83, 023601 (2011)

sake of simplicity in the derived relations we set the magneticpermeability of the gas equal to unity.

In the case of the smallness of the energy losses (we makeestimates that confirm this approximation), the particle motioncan be considered as uniform. Thus the current density j thatis introduced in Eq. (3) can be defined as

j(x,t) = evδ(x − vt),

where e and v are the particle charge and velocity, respectively,and δ(x) is the Dirac δ function. Therefore, the Fouriertransform of the current density produced by the propagatingparticle has the form

j(ω,k) = 2πevδ(ω − k · v). (4)

Then, we can use Eq. (4) to find the quantity (3). Aftersimple mathematical transformations, one gets the relationthat defines the energy change of the charged particle per unitof time, Eωk = −Qωk/T ,

Eωk = e2ω

π2δ(ω − k · v)Im

v2/c2 − 1/ε

ω2ε/c2 − k2. (5)

To get this result, it is necessary to use also the formula

δ2(ω − k · v) = T

2πδ(ω − k · v),

where T is the flight time of the particle through the systemunder consideration.

Hence, the total change of the energy per unit of length isgiven by

dEdx

= 1

v

∫dωd3kEωk. (6)

It should be noted that, in the general case, the charged particlecan both lose the energy (the case of slowing, dE/dx < 0)and receive the energy (the case of acceleration, dE/dx > 0).As one can see from Eq. (5), the realization of these casesdepends on the initial velocity of the particle and dispersioncharacteristics of the system. In the next section, we studyin detail some possibilities of the realization of these casesin ultracold gases of alkali-metal atoms with Bose-Einsteincondensates.

III. ENERGY CHANGE OF THE CHARGED PARTICLETHAT PROPAGATES IN A BEC

Note that dilute gases of alkali-metal atoms have an indexof refraction close to unity even in the resonance regions. Inparticular, the change of the refractive index of a gas of sodiumatoms with the density ν ∼ 1013 cm−3 in the frequency regioncorresponding to the D2 line is relatively small, �n ∼ 0.01.Therefore, the effects resulting from the pole 1/ε [see Eq. (5)]do not appear in this system. Evidently, in some particularcases, another pole, ω2ε − c2k2 = 0, can play a key role. Itshould be noted that this pole corresponds to a contributionof the Cherenkov effect in the energy change of the chargedparticle.

We must emphasize that the Cherenkov radiation in thedispersive medium is observed only in the region of thefrequencies with the following condition:

n(ω)β � 1, (7)

where β = v/c,

n2(ω) = (√

ε′2 + ε′′2 + ε′)/2, (8)

and where ε′ and ε′′ are the real and imaginary parts of thepermittivity, respectively, ε = ε′ + iε′′.

Hence, we conclude that in a medium with n(ω) ≈ 1 theenergy (5) changes only at the relativistic values of the velocityof the propagating particles. As it is easy to see from directcalculations, for example, for the maximal value nmax = 1.01it is necessary to accelerate particles up to velocities v ≈ 0.99c

that correspond to the Lorentz factor γ ≈ 7. In other words,the Cherenkov effect in the system under consideration can beobserved only for the particles with the kinetic energy that, atleast, is greater than the rest energy by several times.

To analyze the peculiarities of the energy change of thecharged particle in an atomic BEC, we use the model of atwo-level system. To this end, we fix in Eq. (1) the indicescorresponding to the chosen quantum states in atoms, a = 1and b = 2 (ε1 < ε2 < 0). In other words, we consider the casewhen the frequency of the potential radiation is close to theenergy spacing between the defined states. As a result, fromEq. (1) one gets

ε(k,ω) =[

1 + 4π

k2|σ (k)|2 (ν1 − ν2)

δω + i�/2

]−1

, (9)

where δω is the frequency detuning, δω = ω − ε1 + ε2. Notethat in the quantities σ (k), δω, and � we omit the pair index“12” that corresponds to the chosen resonant transition in theatom. Then, the permittivity (9) can be divided into the realand imaginary parts. Within the approximation (ε′ − 1) � 1,the energy change (5) is written in the form

Eωk ≈ −ε′′ e2ω3β2

π2c2

[(ω2ε′

c2− k2

)2

+(

ω2ε′′

c2

)2]−1

, (10)

where

ε′ = δω(δω + α) + �2

(δω + α)2 + �2, ε′′ = α�

(δω + α)2 + �2, (11)

α(k) = 4π

k2|σ (k)|2(ν1 − ν2). (12)

The relation (10) contains an important information aboutthe energy change of the charged particle in the two-levelsystem. It should be mentioned that the sign of the quantity Eωkdepends on the sign of the imaginary part ε′′ of the permittivity.This quantity, in turn, is proportional to the occupationdifference (ν1 − ν2) of the quantum states in the system[see Eqs. (11) and (12)]. Therefore, we come to a result thatthere are two particular cases for the energy change of thecharged particle. In the system with the normal population,ν1 > ν2, the Cherenkov losses can be observed, Eωk < 0.Otherwise, in the system with the inverse population, ν1 <

ν2, the particle can be accelerated by a condensate. Thedependencies of the imaginary part of the permittivity on thefrequency detuning for the both cases are shown in Fig. 1.

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PROPAGATION OF RELATIVISTIC CHARGED PARTICLES . . . PHYSICAL REVIEW A 83, 023601 (2011)

0.00

0.01

0.02

0.03

0.04

Imε

(a )

|1

ν1>ν2

|2

|1

ν1<ν2

|2

-6 -4 -2 0 2 4 6-0.04

-0.03

-0.02

-0.01

0.00

Imε

Frequency Detuning

(b)

FIG. 1. (Color online) Dependencies of theimaginary part of the permittivity on the frequencydetuning (� = 2δω/�) in the case of the normal (a)and inverse (b) population of the two-level system.

Then, we use Eqs. (6) and (10) for defining the totalenergy change in the resonance region. To this end, thecondition (7) must be taken into account. This relationcharacterizes an appearance of the Cherenkov effect in thesystem. After the integration over the angles and absolute valueof the wave number, we come to the result

dEdx

= − 2e2

πv2

∫ ωr

ωl

dωωε′′(ω)∫ θ0(ω)

0

sin θdθ

cos3 θ

×[(

ε′ − c2

v2 cos2 θ

)2

+ ε′′2]−1

, (13)

where the limiting angle θ0(ω) can be found from the conditionn(ω)β cos θ0 = 1, and the quantities ωl,r define the limits of thefrequency interval where the effect is observed, n(ωl,r )β = 1(see also Fig. 2).

After integrating the internal integral over polar angle, onegets

dEdx

= − e2

πc2

ε′′

|ε′′|∫ ωr

ωl

dωω

{arctan

[β2(ε′2 + ε′′2) − ε′

|ε′′|]

− arctan

[β2(ε′2 + ε′′2) − n2ε′

|ε′′|n2

]}. (14)

FIG. 2. Dependence of the refractive index on the frequency de-tuning. The hatched area corresponds to the region of the observationof the Cherenkov effect.

Therefore, Eq. (14) describes the energy change of the chargedparticle on the frequency interval that corresponds to theresonant transition in the gas of atoms forming BEC. Naturally,to calculate the total energy change, it is necessary to sum theseparate contributions of the resonant transitions where thecondition (7) is satisfied.

For the sake of simplicity and visibility, we do not considerthe particular cases with a large number of resonant transitionsin alkali-metal atoms. To get the qualitative results, one canestimate the energy change by an order of magnitude in theregion of a single resonance. Considering that the differenceof the arctangents in the integrand of Eq. (14) does not exceedthe π -number value and using the approximation (ωr − ωl) ≈�, � � ω0, where ω0 is the frequency corresponding to theenergy of the resonant transition, � ≡ �12, we find

dEdx

∼ e2

c2ω0�. (15)

By using this approximate formula, it is easy to make estimatesfor the electron propagating in a gas of alkali-metal atoms. Forexample, for a gas of sodium atoms we can use the data thatcorrespond to the resonant D2 line, � ∼ 108 s−1 and ω0 ∼1015 s−1. Thus, from Eq. (15) we find dE/dx ≈ 10−5 eV/cm.Next, based on the fact that typical sizes of atomic sampleswith condensates are of the order of submillimeters, we finallyget the change of energy corresponding to a single resonance(the transition a ↔ b) in an ultracold gas with a BEC,�Eab ≈ 10−7 eV.

It should be noted that in more general cases the atomsmust be considered as multilevel systems. Therefore, asmentioned, the effect takes place for an ensemble of theresonant transitions where the relation (7) is satisfied. Evena relatively large (but, naturally, finite) number of quantumstates do not change significantly the kinetic energy E0 of aparticle. Taking the Lorentz-factor value γ = 7, one gets theratio for the energies, �Eab/E0 ∼ 10−11.

We also note that the effect can be enhanced by usingmore dense gases (this condition increases the hatched area inFig. 2) and also by using multiply ionized atoms or nu-clei with the charge value Ze that enhances the effect inZ2 times.

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YURII SLYUSARENKO AND ANDRII SOTNIKOV PHYSICAL REVIEW A 83, 023601 (2011)

A. Possibility of particle acceleration in a BEC

It is shown previously that the energy change of thepropagating particle in the system under consideration is small.Probably, the changes in the velocity value resulting from theCherenkov effect cannot be directly measured for the caseof dilute gases. However, in our opinion, additional attentionshould be paid to the possibility of particle acceleration byan ultracold atomic gas. It is mentioned previously that theparticle acceleration in the two-level model can be reachedfor the inverse-populated atomic states. As for the physicalinterpretation of this phenomenon, the analogy from the laserphysics can be used. One can say that the propagating particlestimulates a transition from the upper to the lower state, and apart of the released energy is transferred to the charged particle,resulting in its acceleration.

Going beyond the two-level approximation, it is easy tosee that on certain resonant transitions the particle can receivethe energy from a medium and on other transitions it canemit the energy. Therefore, the particle interacts with a set ofquantum transitions and loses the energy in the general case.In some particular cases, by a special tuning of the particlevelocity and occupations of the atomic states, it is possible toaccelerate the charged particle by the gas. This particular caseis shown in Fig. 3. As one can see, in the region correspondingto the transition with the inverse population, the condition (7)is satisfied. At the same time, the velocity of the particle is notenough for the Cherenkov effect on the transitions with thenormal population.

Apparently, the most appropriate quantum states for theexperimental realization of this effect are the levels of thehyperfine structure of the ground state of alkali-metal atoms.This statement is based on the fact that these states have largelifetimes in comparison with the levels with the nonzero orbitalmoment. From the standpoint of the present experimentalconditions, it is not difficult to populate the hyperfine statesin such a way as to provide the particle acceleration (see the

FIG. 3. (Color online) Dispersion characteristics of a gas thataccelerates the particle on the resonant transition with the inversepopulation. The density difference for this transition is taken to betwice as large as the transitions with normal population (for sake ofsimplicity, the linewidths �ab are set the same).

scheme in Fig. 3). If for the the upper (optical) transitions inthis system the condition (7) is not satisfied, the particle will beaccelerated by a multicomponent BEC. Otherwise, it receivesthe energy on the microwave transitions and emits the energyon the optical transitions.

Also, we should note the case when the inverse populationis provided by means of the optical pumping. This effectis usually produced by the additional lasers that are tunedup close to the resonant transitions. The effect in this casecould be even larger, according to the estimates in Eq. (15),but the dispersion characteristics of a gas in the case ofoptical pumping cannot be studied in an appropriate wayin the framework of the present paper. Really, the influenceof additional strong fields result in the repopulation of thequantum states and thus it is necessary to go beyond the limitsof the linear response theory [6], which are used in this paper.

Therefore, it is shown that the particle that has a large kineticenergy (it can be larger than the rest energy in several times)in principal can additionally receive the energy from ultracoldgases, consisting of atoms that have extremely small (or evenzero) values of the kinetic energy.

IV. DEFINING THE SPECTRAL CHARACTERISTICSOF THE ATOMS

By registering the Cherenkov radiation produced by thepropagation of the charged particle in a BEC, it is possibleto define the spectral characteristics of atoms that formthis condensate. In particular, in this section, we use theCherenkov effect for finding the natural linewidth � value. Thisquantity can be defined both for the optical (dipole-allowed)and microwave (dipole-forbidden) transitions in alkali-metalatoms. The last problem may be actual also from the standpointof the atomic-clock experiments (see, e.g., Ref. [9]).

The main idea of this method consists in defining thefrequency that corresponds to the maximum of the refractiveindex in the region of the certain resonant transition. Thisproblem can be solved by varying the initial velocity of thepropagating particle. As one can see from Fig. 2, by reducingthe particle velocity, the height of the hatched area becomesmaller, and in the limit 1/β → nmax we get ωl,r → ωc,ωc = ω(nmax).

Here and in the following discussion we consider the case ofthe transmittance region, that is, |ε′′| � ε′ ≈ 1. As one can see,in the zero order of the perturbation theory over (ε′′/ε′) � 1,from Eq. (8) it comes that n(ω) ≈ √

ε′(ω). Thus we get therelation ω(0)

c = ω0 − �/2 from the condition for the maximum,∂n/∂ω = 0, and Eq. (9). The mentioned expression, in turn,transforms to the following formula:

�(0) = 2(ω0 − ωc). (16)

Therefore, the linewidth �(0) can be found in the case whenthe frequency ωc is defined from the spectral peak of theCherenkov radiation in a BEC. Note that to get tangible resultsfrom Eq. (16), it is necessary also to use the ω0 value that mustbe determined with relatively good accuracy.

In the case when the transition frequency ω0 is not knowna priori, it is possible to define the linewidth by the use of

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PROPAGATION OF RELATIVISTIC CHARGED PARTICLES . . . PHYSICAL REVIEW A 83, 023601 (2011)

an additional condition nmax = 1/β. Then, in accordance withEq. (11), one gets

�(0) = α(k)

2(1/β − 1). (17)

Hence, it is possible to find the natural linewidth �(0) fromthe microscopic characteristics of the system. As is easy tosee from Eqs. (2) and (17), in this case one need to know theexplicit form for the wave function in two certain quantumstates.

It should be noted that it is not difficult to find more explicitrelations for the linewidth � by the use of the terms appearingfrom the further expansion into series in the perturbation theoryover (ε′′/ε′) � 1. Note that this strict inequality corresponds tothe approximation |α(k)| � �. Thus, after some mathematicaltransformations, we come to the following expression:

�(1) = 2(ω0 − ωc) + 3α(k)/2. (18)

Using similar expansions for the undefined ω0 value, one gets

�(1) = α(k)

2(1/β − 1)

[1 − (1/β − 1)

2

]. (19)

We must emphasize that basing on the Cherenkov effect ina BEC it is possible to define also other characteristics of thesystem under consideration. For example, using the describedmethod with the defined value of the natural linewidth one canfind the transition frequency ω0, the density νa of atoms in thecertain quantum states, or the velocity v of the propagatingparticle.

V. CONCLUSION

In this paper, we investigate some effects that occur duringthe propagation of the charged particle in Bose-Einsteincondensates of alkali-metal atoms. It is shown that it isnecessary to use the charged particles with high (relativistic)velocities for the Cherenkov effect in the system. This condi-tion corresponds to the fact that the refractive index for diluteatomic gases is close to unity even in the regions of resonantfrequencies.

Basing on the relations for the permittivity of a gas in theBEC state, we find the energy change of the charged particle.It is shown that the effect of deceleration (acceleration) of theparticle is very small for this system. The principal scheme forthe preparation of the ultracold atomic gas that accelerates therelativistic charged particle is studied.

Based on the Cherenkov effect, we also consider thepossibility of defining the spectral characteristics of atomsthat form the condensate. It is shown that the region ofthe local maximum of the refractive index can be found byvarying the initial velocity of the charged particle. We proposea method for finding the natural linewidth value from theCherenkov radiation in a BEC based on the knowledge ofthe additional spectral or microscopic characteristics of thesystem.

ACKNOWLEDGMENT

This work is partly supported by the National Academy ofSciences of Ukraine, Grant No. 55/51-2010.

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