1. Introduction

Density fluctuations in dilute Bose-Einstein condensates were traditionally described by using a mean-field theory [1–3], and a similar approach has also been proposed specifically for the explanation of the properties of elongated pencil shaped samples [4], observed also in experiments [5, 6]. Theoretical descriptions of such quasi one-dimensional systems have used the hydrodynamic approximation [7,8] and have described the properties of the fluctuations as corrections to an approximate static density. Problems, however with time dependent mean field theories have been pointed out in [9]. More recent experiments reported the confinement of Rb atoms in a quantum wire geometry [10, 11], where the ratio of the interaction to kinetic energy significantly exceeds unity thus approaching the Tonks-Girardeau (TG) limit. These works have turned the purely mathematical model considered in classic papers [12–14] into a real physical system 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] one can start from the many body wave function of the system and consider the exact time dependence of the density determined by the trapping frequency. These space and time dependent oscillations give rise to a modulation of a weak probing field which can provide information on the 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 a strongly interacting, one dimensional, harmonically trapped Bose gas. In this framework the ground state wave function of the system is equivalent to a noninteracting one dimensional Fermi system [13]. Its density oscillations are the consequence of the quantum mechanical superposition of the ground state with energy E ₀, and one or a few excited states. Considering the simplest case of having only a single excitation with energy h̄ω, where ω is the angular frequency determined by the harmonic trapping potential, the wave function of the system is

where Ψ₀ is the ground state and Ψ₁ is the first excited stationary state, both being symmetric functions of the particle coordinates: x ₁, …, x_N. This wave function yields the time dependent particle density

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

where ρ ₀ and ρ ₁ are the particle densities of the ground and excited states respectively, while

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

where φ_n(x_j) = (2ⁿ n!ℓ√π)^1/2 exp(−x ² _j/2ℓ ²)H_n (x_j/ℓ) is the nth normalized harmonic oscillator eigenfunction, ℓ = (h̄/mω)^1/2, m is the mass of a particle, while the product of the sign functions 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 the product of the sign function we can take the absolute value of the determinant. Similarly we obtain 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 last row we get

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

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

Because of the orthogonality of the single particle functions we obtain

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] $\overline{\rho }$(x) = $\overline{\rho }$ ₀(1 − (x/x_T)²)^1/2, where ${\overline{\rho }}_0=\frac{\sqrt{2N}}{\left(\pi \ell \right)}$ , and $x_T=\sqrt{2N}\ell$ is the ground state 1D radius of the system. ρ(x, t) shown here is the fundamental swinging mode of the system, and as the number of atoms increases, it approaches the average density, while the number of spatial oscillations increases. The attached video file (Media 1) shows an animation of the density oscillations vs. time.

 

Fig. 1. Plot of the time dependent particle density ρ(x, t) for 10 atoms as function of x measured in units of the radius x_T. The thin solid line (red in color) is at t = 0, the thick solid line (blue in color) is at t = π/2ω and the thin dotted line (orange in color) is at t = π/ω. We plot the average density $\overline{\rho }$(x) for comparison as a black dashed line. The attached video file (Media 1) shows an animation of the density oscillations vs. time.

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3. Optical detection of the density oscillations

We now consider the signatures of these oscillations in the time-dependent transmission of a weak CW field ℰ(x, t), which is sent through the TG gas along its axis. Theories of light scattering and proposals of such experiments have been put forward previously [23–28]. In our 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 d ² ℰ_l/h̄(∆ − iγ), where 𝒩 is the volume density of the atoms, d is the dipole matrix element and γ is the width of the transition in question, while ∆ = ω₀ − ω_L is the detuning between theresonant transition frequency ω ₀ and the carrier of the probing field ω_L. ℰ_l is the local field which is a superposition of the external field and the secondary field of the atomic dipoles. We write 𝒩 = 𝒩 _aρ(x, t), where 𝒩_a is the average number of atoms in unit cross section. In real experiments [5, 10, 11] one has a lattice of pencil shaped samples, therefore the actual value of 𝒩_a is the inverse of the cross section of one such “pencil”. (We do not consider here the effect of light propagating between these pencils.) The polarization, 𝒫 leads to a space and time dependent susceptibility and index of refraction with the simple local field correction:

with β = 𝒩_ad ²/(ε ₀ h̄(∆ − iγ)). This model assumes identical detunings and level widths of the atoms, while the nonsymmetric positions of the atoms in the chain leads to a nonuniform level shift. The latter is below 1 MHz in relevant experiments with Rb atoms [5]. We shall assume a detuning ∆ which is at least three orders of magnitude larger than this nonuniform level shift, which means that we can safely neglect it.

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

where k = 2π/λ ₀ is the corresponding wave number in vacuum. Here E(x, t) is the temporally slowly varying amplitude of the full electric field: ℰ(x, t) = E(x, t)exp(−iω_Lt). The solution of this equation with a given incoming plane wave from the negative x direction shall yield the transmitted wave E _tr(t)exp(ikx) for x ≫ x_T, as well as a reflected wave at x ≪ −x_T. The results of a numerical solution will be discussed below.

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

where E ₀ is the incident field amplitude at x ₀, far before the trapped boson gas. The transmission coefficient of the system depends only on time for x ≫ x_T, i.e. far beyond the system:

since n(x, t) = 1 for x ≫ x_T. In the following we consider the light source at x ₀ = −∞, and a photodetector at x = ∞. In case of an off-resonant external field we can expand the index of refraction given by Eq. (10) under the integrals up to second order in βρ as n(x, t) = 1 + βρ(x, t)/2 + β ² ρ ²(x, t)/24. Substituting this expansion into Eq. (13) we have:

We write ρ(x, t) = r ₀(x) + r ₁(x) cos (ωt + α), where r ₀(x) = |c ₀|² ρ ₀(x) + |c ₁|² ρ ₁(x) is an even function and r ₁(x) = 2|c ₀| |c ₁|ρ ₀₁(x) is an odd function of x, and α = argc ₀ − argc ₁. Substituting this into Eq. (14) and using the parity of r ₀ and r ₁, we obtain the following formula:

where

does not depend on time, and

with the integrals

depending only on the number of particles in the sample and on the coefficients c ₀ and c ₁.

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

where I_s (ζ) are the modified Bessel functions. This means that the complex Fourier coefficients T̃ of the time-dependent transmisson are proportional to modified Bessel functions with the same argument:

and all the other Fourier coefficients vanish.

In particular, the time averaged transmission is T̃(0) = T_NI ₀(ζ). According to Eq. (22), ζ is of the order of Imβ ² ≪ 1, therefore I ₀(ζ) = 1 and thus T̃(0) = T_N. The formula for T_N involves a term with Imβ, which does not depend on the sign of the detuning, and a term with Imβ ², which has the sign of ∆. If the transmisson is measured for detunings with opposite signs, then the product of the time averaged transmissions gives the number of atoms:

Once we know N, we can calculate the integrals contained in R ₀ and R ₀₁, and the only unknown quantities are |c ₀| and |c ₁| in Eq. (16). Using |c ₀|² + |c ₁|² = 1, we can calculate both of them.

The relative phase α can be obtained the following way: based on |c ₀| and |c ₁| we can calculate ζ from Eq. (17), the I ₁(ζ) is real and odd, therefore the phase of T̃(2ω) yields the relative phase of the states Ψ₀ and Ψ₁, according to Eq. (20) as: $\alpha =-\frac{1}{2}\arg \left[\tilde{T}\left(2\omega \right)\mathrm{sign}\zeta \right]$ (the sign of ζ is the sign of I ₁(ζ)).

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

Since the expression of ζ involves |c ₀| |c ₁| (via the integrals R ₁ and R ₁₁), the measurement of the time dependent transmission gives an alternative way for the calculation of |c ₀| and |c ₁|.

We illustrate the use of the modulated transmission by processing a simulated transmisson signal which we obtain from the numerical solution of the second order amplitude equation, Eq. (11), using the software Mathematica 7. We consider a system where ⁸⁷Rb atoms are trapped in an array of pencil shaped samples [5, 10, 11], containing 30 atoms pro pencil, in a superposition state defined by Eq. (1) with c ₀ = c ₁ = 1/√2. The laser light for the transmission measurement is assumed to be detuned with an angular frequency ∆ = 2π × 250 MHz from the center of the D ₁ line (λ ₀ = 794.978 nm), and we use γ = 1.80647 × 10⁷ 1/s and d = 1.4651 × 10⁻²⁹ Cm [31]. We set the longitudinal trap angular frequency to ω = 2π × 100 Hz. The relevant Fourier amplitudes of the time-dependent transmission result from the simulation as T̃ _∆(0) = 0.96039700 and T̃ _−∆(0) = 0.96076446. A calculation based on these data and Eqs. (21) and (16), reproduces correctly that the sample contains 30.0054 atoms, and the superposition coefficients are |c ₀| = |c ₁| = 0.707. Both ζ and T̃ (2ω) are real and negative, which leads to α = 0. Since the time averaged transmisson is around 96%, heating of the sample can be avoided.

4. Conclusions

We have constructed an exact many body superposition state for the Tonks gas, exhibiting a time-dependent particle density in a swinging mode. Generalizations for superpositions involving 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 oscillations as a weak but measurable oscillation in the transmission signal. The approximate analytic formula obtained for this time-dependent transmission allows for the calculation of the quantities which characterize the TG gas and its interaction with the CW field. The number of atoms in the Tonks gas and the coefficients of the many body superposition state could be measured without destroying the sample.