Exact density oscillations in the Tonks-Girardeau gas and their optical detection

Exact density oscillations in theTonks-Girardeau gas and their optical

detection

Mihaly G. Benedict,* Csaba Benedek, and Attila CzirjakDepartment of Theoretical Physics, University of Szeged,

H-6720 Szeged, Tisza Lajos krt. 84-86, Hungary

*[email protected]

Abstract: We construct the exact time dependent density profile for asuperposition of the ground and singly excited states of a harmonicallytrapped one dimensional Bose gas in the limit of strongly interactingparticles, the Tonks-Girardeau gas. We obtain analytic results that al-lows one to determine the number of particles in the gas, as well as thequantum amplitudes in the superposition, from measurement results in anoff-resonant light scattering experiment.

© 2010 Optical Society of America

OCIS codes: (020.1475) Bose-Einstein condensates; (260.2710) Inhomogeneous optical me-dia; (120.4290) Nondestructive testing; (140.7010) Laser trapping.

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#130517 - $15.00 USD Received 22 Jun 2010; accepted 21 Jul 2010; published 30 Jul 2010(C) 2010 OSA 2 August 2010 / Vol. 18, No. 16 / OPTICS EXPRESS 17569

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1. Introduction

Density fluctuations in dilute Bose-Einstein condensates were traditionally described by usinga mean-field theory [1–3], and a similar approach has also been proposed specifically for theexplanation of the properties of elongated pencil shaped samples [4], observed also in exper-iments [5, 6]. Theoretical descriptions of such quasi one-dimensional systems have used thehydrodynamic approximation [7,8] and have described the properties of the fluctuations as cor-rections to an approximate static density. Problems, however with time dependent mean fieldtheories have been pointed out in [9]. More recent experiments reported the confinement of Rbatoms in a quantum wire geometry [10, 11], where the ratio of the interaction to kinetic energysignificantly exceeds unity thus approaching the Tonks-Girardeau (TG) limit. These works haveturned the purely mathematical model considered in classic papers [12–14] into a real physicalsystem with potential applications. For recent reviews see [15, 16].

In the case of strongly interacting bosons, when the TG model can be applied [17–20] onecan start from the many body wave function of the system and consider the exact time depen-dence of the density determined by the trapping frequency. These space and time dependentoscillations give rise to a modulation of a weak probing field which can provide information onthe properties of the Bose gas without destroying it.

2. Density oscillations in the Tonks-Girardeau gas

We give first a microscopic explanation of the observed oscillations based on the theory of astrongly interacting, one dimensional, harmonically trapped Bose gas. In this framework theground state wave function of the system is equivalent to a noninteracting one dimensionalFermi system [13]. Its density oscillations are the consequence of the quantum mechanicalsuperposition of the ground state with energy E0, and one or a few excited states. Consideringthe simplest case of having only a single excitation with energy hω , where ω is the angular


frequency determined by the harmonic trapping potential, the wave function of the system is

ΨS(x1, . . . ,xN , t) = c0Ψ0 exp

[−i

E0

ht

]+ c1Ψ1 exp

[−i

(E0

h+ω

)t

](1)

where Ψ0 is the ground state and Ψ1 is the first excited stationary state, both being symmetricfunctions of the particle coordinates: x1, . . . ,xN . This wave function yields the time dependentparticle density

ρ(x, t) = N∫

|ΨS(x,x2 . . . ,xN , t)|2 dx2 . . .dxN (2)

which will obviously exhibit oscillations with frequency ω. The explicit form of these are

ρ(x, t)= |c0|2ρ0(x)+ |c1|2ρ1(x)+2Rec∗1c0ρ01(x)exp(−iωt) (3)

where ρ0 and ρ1 are the particle densities of the ground and excited states respectively, while

ρ01(x) = N∫

Ψ∗0(x,x2 . . . ,xN)Ψ1(x,x2 . . . ,xN)dx2 . . .dxN (4)

determines the spatial pattern of the density oscillations.In order to calculate explicitly the terms in Eq. (3) we recall the construction given for Ψ0

[13, 21]. The ground state N particle wave function can be written as

Ψ0 =1√N!

(N−1,N)det

(n, j)=0,1ϕn(x j) ∏

1≤ j<k≤Nsign(xk − x j) (5)

where ϕn(x j) = (2n n!�√

π)1/2 exp(−x2j/2�2)Hn (x j/�) is the nth normalized harmonic oscilla-

tor eigenfunction, � = (h/mω)1/2, m is the mass of a particle, while the product of the signfunctions ensures the symmetric nature of the total wave function. In the case of real one-particle eigenfunctions – as is the case now – instead of multiplying the determinant with theproduct of the sign function we can take the absolute value of the determinant. Similarly we ob-tain the first excited many body state Ψ1 by replacing the last row in the determinant in Eq. (5)by functions of the N-th excited state of the oscillator. Then, expanding according to the lastrow we get

Ψ1 =

∣∣∣∣∣1√N!

N∑j=1

(−1)N+ j−1ϕN(x j)Dj

∣∣∣∣∣ (6)

where Dj denotes the (N, j)-th minor of the determinant in Ψ0. Due to the orthogonality of thesingle particle functions we obtain

ρ0(x) =N−1∑

n=0|ϕn(x)|2 , ρ1(x) =

N−2∑

n=0|ϕn(x)|2 + |ϕN(x)|2 . (7)

The time dependent cross term contains the product of two determinants both of which can beexpanded by their last rows and we obtain

Ψ0Ψ1 =1

N!

N∑j=1

(−1) j−1ϕN−1(x j)Dj ×N∑j=1

(−1) j−1ϕN(x j)Dj (8)

Because of the orthogonality of the single particle functions we obtain

ρ01(x) = ϕN−1(x)ϕN(x) (9)


which is exact in the one dimensional Tonks model with harmonic trapping.Figure 1 shows the time dependent density for 10 atoms in comparison with the ground state

average density [18] ρ(x) = ρ0(1−(x/xT )2)1/2, where ρ0 =√

2N/(π�), and xT =√

2N� is theground state 1D radius of the system. ρ(x, t) shown here is the fundamental swinging mode ofthe system, and as the number of atoms increases, it approaches the average density, while thenumber of spatial oscillations increases. The attached video file (Media 1) shows an animationof the density oscillations vs. time.

�1.5 �1.0 �0.5 0.0 0.5 1.0 1.5

0

1

2

3

4

5

6

7

x�xT

�

Fig. 1. Plot of the time dependent particle density ρ(x, t) for 10 atoms as function of xmeasured in units of the radius xT . The thin solid line (red in color) is at t = 0, the thicksolid line (blue in color) is at t = π/2ω and the thin dotted line (orange in color) is att = π/ω . We plot the average density ρ(x) for comparison as a black dashed line. Theattached video file (Media 1) shows an animation of the density oscillations vs. time.

3. Optical detection of the density oscillations

We now consider the signatures of these oscillations in the time-dependent transmission ofa weak CW field E (x, t), which is sent through the TG gas along its axis. Theories of lightscattering and proposals of such experiments have been put forward previously [23–28]. Inour treatment the probing field is assumed to be far-detuned from a resonant atomic transition,and will create a polarization density along the sample [29]. The latter can be given as P =N d2El/h(Δ− iγ), where N is the volume density of the atoms, d is the dipole matrix elementand γ is the width of the transition in question, while Δ = ω0 −ωL is the detuning between theresonant transition frequency ω0 and the carrier of the probing field ωL. El is the local fieldwhich is a superposition of the external field and the secondary field of the atomic dipoles. Wewrite N = N aρ(x, t), where Na is the average number of atoms in unit cross section. In realexperiments [5, 10, 11] one has a lattice of pencil shaped samples, therefore the actual valueof Na is the inverse of the cross section of one such “pencil”. (We do not consider here theeffect of light propagating between these pencils.) The polarization, P leads to a space andtime dependent susceptibility and index of refraction with the simple local field correction:

n(x, t) =(

1+βρ(x, t)

1−βρ(x, t)/3

)1/2

(10)

with β = Nad2/(ε0h(Δ− iγ)). This model assumes identical detunings and level widths of theatoms, while the nonsymmetric positions of the atoms in the chain leads to a nonuniform levelshift. The latter is below 1 MHz in relevant experiments with Rb atoms [5]. We shall assume adetuning Δ which is at least three orders of magnitude larger than this nonuniform level shift,which means that we can safely neglect it.


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As the time dependence of the harmonic trap is very slow with respect to the frequency ofthe optical fields, instead of the wave equation we shall solve the one dimensional amplitudeequation for the electric field:

∂ 2

∂x2 E(x, t)+n2(x, t)k2E(x, t) = 0 (11)

where k = 2π/λ0 is the corresponding wave number in vacuum. Here E(x, t) is the temporallyslowly varying amplitude of the full electric field: E (x, t) = E(x, t)exp(−iωLt). The solutionof this equation with a given incoming plane wave from the negative x direction shall yield thetransmitted wave Etr(t)exp(ikx) for x � xT , as well as a reflected wave at x �−xT . The resultsof a numerical solution will be discussed below.

In order to get a better insight into the nature of the problem we also present an approximateanalytic solution to Eq. (11) by a 2nd order WKB approximation obtained for the forwardpropagating wave as

Ef(x, t) =1√

n(x, t)E0 exp

(ik

∫ x

x0

n(x′, t)dx′ − in′(x)

4kn2(x)− i

∫ x

x0

(n′(x′))2

8kn3(x′)dx′

), (12)

where E0 is the incident field amplitude at x0, far before the trapped boson gas. The transmissioncoefficient of the system depends only on time for x � xT , i.e. far beyond the system:

T (t) =∣∣∣∣Ef(x, t)

E0

∣∣∣∣2

= exp

(−2k

∫ x

x0

Imn(x′, t)dx′ +14k

∫ x

x0

Im(n′(x′))2

n3(x′)dx′

)(13)

since n(x, t) = 1 for x � xT . In the following we consider the light source at x0 = −∞, anda photodetector at x = ∞. In case of an off-resonant external field we can expand the indexof refraction given by Eq. (10) under the integrals up to second order in βρ as n(x, t) = 1 +βρ(x, t)/2+β 2ρ2(x, t)/24. Substituting this expansion into Eq. (13) we have:

T (t) = exp

(−NkImβ +

kImβ 2

12

∫ ∞

−∞ρ2(x, t)dx+

Imβ 2

16k

∫ ∞

−∞(ρ ′(x, t))2dx

)(14)

We write ρ(x, t) = r0(x)+ r1(x)cos(ωt +α), where r0(x) = |c0|2ρ0(x)+ |c1|2ρ1(x) is an evenfunction and r1(x) = 2|c0||c1|ρ01(x) is an odd function of x, and α = argc0−argc1. Substitutingthis into Eq. (14) and using the parity of r0 and r1, we obtain the following formula:

T (t) = TN exp [ζ cos(2(ωt +α))] , (15)

where

TN = exp

(−NkImβ − k

12Im(β 2)(R0 +R1/2)+

Imβ 2

16k(R01 +R11/2)

)(16)

does not depend on time, and

ζ = Im(β 2)(−kR1/24+R11/32k), (17)

with the integrals

R0 =∫ ∞

−∞r20(x)dx, R1 =

∫ ∞

−∞r21(x)dx, R01 =

∫ ∞

−∞(r′0(x))

2dx, R11 =∫ ∞

−∞(r′1(x))

2dx (18)

depending only on the number of particles in the sample and on the coefficients c0 and c1.


We can expand the time dependent factor in Eq. (15) into a Jacobi form [30], which is directlyrelated to the discrete Fourier transform of the time dependent transmission:

T (t) = TN

[I0(ζ )+2

∞

∑s=1

Is(ζ )cos(2s(ωt +α))

](19)

where Is(ζ ) are the modified Bessel functions. This means that the complex Fourier coefficientsT of the time-dependent transmisson are proportional to modified Bessel functions with thesame argument:

T (2sω) = TNIs(ζ )exp(−2siα), s = 0,1,2, ..., (20)

and all the other Fourier coefficients vanish.In particular, the time averaged transmission is T (0) = TNI0(ζ ). According to Eq. (22), ζ

is of the order of Imβ 2 << 1, therefore I0(ζ ) = 1 and thus T (0) = TN . The formula for TN

involves a term with Imβ , which does not depend on the sign of the detuning, and a term withImβ 2, which has the sign of Δ. If the transmisson is measured for detunings with opposite signs,then the product of the time averaged transmissions gives the number of atoms:

N = −(2k Imβ )−1 ln[TΔ(0)T−Δ(0)

]. (21)

Once we know N, we can calculate the integrals contained in R0 and R01, and the only unknownquantities are |c0| and |c1| in Eq. (16). Using |c0|2 + |c1|2 = 1, we can calculate both of them.

The relative phase α can be obtained the following way: based on |c0| and |c1| we cancalculate ζ from Eq. (17), the I1(ζ ) is real and odd, therefore the phase of T (2ω) yields therelative phase of the states Ψ0 and Ψ1, according to Eq. (20) as: α = − 1

2 arg[T (2ω) signζ

](the sign of ζ is the sign of I1(ζ )).

A well known relation for the Bessel functions [30] enables us to calculate ζ also directlyfrom the transmission spectrum:

ζ =2I1(ζ )

I0(ζ )− I2(ζ )=

2T (2ω)exp(2iα)T (0)−|T (4ω)| (22)

Since the expression of ζ involves |c0||c1| (via the integrals R1 and R11), the measurement ofthe time dependent transmission gives an alternative way for the calculation of |c0| and |c1|.

We illustrate the use of the modulated transmission by processing a simulated transmis-son signal which we obtain from the numerical solution of the second order amplitude equa-tion, Eq. (11), using the software Mathematica 7. We consider a system where 87Rb atomsare trapped in an array of pencil shaped samples [5, 10, 11], containing 30 atoms pro pen-cil, in a superposition state defined by Eq. (1) with c0 = c1 = 1/

√2. The laser light for the

transmission measurement is assumed to be detuned with an angular frequency Δ = 2π × 250MHz from the center of the D1 line (λ0 =794.978 nm), and we use γ = 1.80647×107 1/s andd = 1.4651×10−29 Cm [31]. We set the longitudinal trap angular frequency to ω = 2π ×100Hz. The relevant Fourier amplitudes of the time-dependent transmission result from the simu-lation as TΔ(0) = 0.96039700 and T−Δ(0) = 0.96076446. A calculation based on these data andEqs. (21) and (16), reproduces correctly that the sample contains 30.0054 atoms, and the su-perposition coefficients are |c0|= |c1|= 0.707. Both ζ and T (2ω) are real and negative, whichleads to α = 0. Since the time averaged transmisson is around 96 %, heating of the sample canbe avoided.

4. Conclusions

We have constructed an exact many body superposition state for the Tonks gas, exhibiting atime-dependent particle density in a swinging mode. Generalizations for superpositions involv-ing higher excited modes are straightforward. The model for the interaction of the Tonks gas


with a weak laser beam opens the possibility of measuring the effect of these density oscilla-tions as a weak but measurable oscillation in the transmission signal. The approximate analyticformula obtained for this time-dependent transmission allows for the calculation of the quanti-ties which characterize the TG gas and its interaction with the CW field. The number of atomsin the Tonks gas and the coefficients of the many body superposition state could be measuredwithout destroying the sample.

Acknowledgements

This work was supported by the Hungarian Scientific Research Fund OTKA under ContractsNo. K81364, T48888, M045596. We thank P. Domotor and P. Foldi for useful discussions.


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Exact density oscillations in the Tonks-Girardeau gas and their optical detection