The general astigmatic matter wave

Jun Chen; Qiang Lin; Yunxian Liu

doi:10.1364/OE.16.003368

1. Introduction

With the rapid progress on the cooling and trapping of atoms [1], the investigation on the characteristics of coherent matter waves attracts more and more attentions [2, 3, 4]. Depending on the various experimental setups, the BEC or cold atom cloud may be cigar-shaped [5, 6] or pancake-shaped [7]. That means there is astigmatism in the matter wave. Experimentally, the astigmatism of matter waves originates from the anisotropic magnetic trap and the anisotropic light fields. The astigmatism leads to the spatial anisotropy of the density distribution and the spatial anisotropy of the phase distribution, just as that of Gaussian light beams [8, 9]. For the gaussian-distributed atom cloud, its density profile and phase front are ellipses in the cross section. If the orientation of the ellipse of the density profile is parallel to that of the phase front, the astigmatism is simple. In this case, the evolutions in each dimension are independent, and can be treated separately in individual dimensions [10, 11, 12, 13]. For a general astigmatic matter wave, however, the evolutions in each dimension are coupled, therefore can not be separated.

In this paper, the characteristics and the evolution of the general astigmatic matter wave are treated by using a tensor-formed matrix formalism. A tensor ABCD law for describing Gaussian-shaped astigmatic matter wave is developed. As an example, the evolution of a matter wave consisted of noninteracting atoms and without spontaneous emissions is calculated by using this tensor ABCD law. We show that, the lens pulses with oblique orientations can bring the general astigmatism to an isotropic atom cloud. During the following evolution, the orientations of the density profile and the phase front of the matter wave rotate continuously.

2. Tensor ABCD law for general astigmatic matter waves

The ABCD formalism for atom optics, which was derived firstly by Bordé [4, 14], is a convenient way to treat the evolution problem of the single atom wave function in the gravitational field. For a coherent matter wave without interaction among atoms, the many-atom system can also be described by the single atom wave function. To calculate this wave function, the motion of the center mass and the action of the atom are needed.

The motion of the atom from one space-time point (r ₀,t ₀) to another (r ₁,t ₁) is described by [14]:

(\begin{matrix} r_{1} \\ v_{1} \end{matrix}) = (\begin{matrix} ξ \\ \dot{ξ} \end{matrix}) + M (\begin{matrix} r_{0} \\ v_{0} \end{matrix}),

where r _j=(x _j,y _j, z _j)^T and v _j=(v _xj,v _yj,v _zj)^T (j=0,1) denote the position and the velocity, respectively. The superscript T means the transpose of the matrix. ξ and $\dot{ξ}$ are the displacement and the additional velocity caused by the gravitational field. M is called the evolution matrix, which is constituted of four 3×3 sub-matrices:

M = (\begin{matrix} A & B \\ C & D \end{matrix}) .

The sub-matrices A, B, C, and D can be determined by solving the Hamilton-Jacobi equation.

The action of the atom is given by [14]

S = m \tilde{\dot{ξ}} (r_{1} - ξ) + \int_{t_{0}}^{t_{1}} L (t) d t - \int_{t_{0}}^{t_{1}} V (t) d t

where m is the mass of the atom, L(t) is a partial Lagrangian given by L(t)=m(| $\dot{ξ}$ |²+ $\tilde{ξ}$ γξ+2g̃ξ)/2, g and γ representing the gravitational acceleration and the gravitational gradient respectively, and V(t) is an external potential. The tilde ^~ indicates the transpose of the vector.

The action S can be expressed in the tensor form:

S_{(r_{1}, t_{1}, r_{0}, t_{0})} = m \tilde{\dot{ξ}} (r_{1} - ξ) + S_{0} + \frac{m}{2} {(\begin{matrix} r_{0} \\ r_{1} - ξ \end{matrix})}^{T} U (\begin{matrix} r_{0} \\ r_{1} - ξ \end{matrix}),

where

S_{0} = \int_{t_{0}}^{t_{1}} L (t) d t - \int_{t_{0}}^{t_{1}} V (t) d t

depends only on the evolution time, and U is a 6×6 matrix:

U = [\begin{matrix} B^{- 1} A & - B^{- 1} \\ (C - D B^{- 1} A) & D B^{- 1} \end{matrix}] .

Since the action S is a scalar, the matrix U must have transpose symmetry, which leads to:

{\begin{matrix} {(B^{- 1} A)}^{T} = B^{- 1} A, \\ - {(B^{- 1})}^{T} = C - D B^{- 1} A, \\ {(D B^{- 1})}^{T} = D B^{- 1} . \end{matrix}

According to the path integral method, the wave function after evolution is:

ψ (r_{1}, t_{1}) = \int K (q_{1}, t_{1}, r_{0}, t_{0}) ψ (r_{0}, t_{0}) d r_{0} .

The parameter K is the quantum-mechanical propagator and can be obtained from Van Vleck’s formula[15]:

K (r_{1}, t_{1}, r_{0}, t_{0}) = {(\frac{m}{2 π i \bar{h}})}^{\frac{3}{2}} {∣ \det B ∣}^{- \frac{1}{2}} \exp [\frac{iS}{\bar{h}}],

where S is the action in Eq. (4), hˉ is the Plank constant. The function ψ(r ₀,t ₀) in Eq. (8) is the initial wave function e.g. the lowest-order Gaussian mode[14]:

ψ (r_{0}, t_{0}) = \exp [\frac{im}{2 \bar{h}} {(r_{0} - r_{0 c})}^{T} Q_{0}^{- 1} (r_{0} - r_{0 c})] \exp [\frac{im {\tilde{v}}_{0} (r_{0} - r_{0 c})}{\bar{h}}],

where Q ^-1 ₀ is the complex curvature tensor of the matter wave, r _0c is the initial central position and v ₀ is the initial velocity. The complex curvature tensor is a 3×3 matrix carrying the information about the density distribution and the phase distribution, defined as:

Q_{0}^{- 1} = R_{0}^{- 1} + 2 i \frac{\bar{h}}{m} W_{0}^{- 2},

where R ^-1 ₀ is a matrix of the phase front curvature and W ₀ is a matrix of the cloud density width. For the sake of simplicity, the origin of the coordinate system can be set at the initial central position of the matter wave: r _0c=0 (as we do in this papter). As a result, the analytic form of the wave function can be obtained after the vector integration:

ψ (r_{1}, t_{1})

where the final complex curvature is related to the initial one as follows:

Q_{1}^{- 1} = (C + D Q_{0}^{- 1}) {(A + B Q_{0}^{- 1})}^{- 1},

and S _c is called the classical action, given by:

S_{c} = \frac{m}{2} {\tilde{v}}_{0} {DBv}_{0} + m \tilde{\dot{ξ}} {Bv}_{0} + \frac{m}{2} \int_{t_{0}}^{t_{1}} ({∣ \dot{ξ} ∣}^{2} + \tilde{ξ} γ ξ + 2 \tilde{g} ξ) dt - \int_{t_{0}}^{t_{1}} V (t) dt .

The parameters r _c and v _c represent the central position of the matter wave and the velocity in the classical limit, respectively:

r_{c} = {Bv}_{0} + ξ, v_{c} = {Dv}_{0} + \dot{ξ} .

Equation (13) is the tensor ABCD law for coherent matter waves with general astigmatism. The tensor ABCD law Eq. (13) together with the wave function evolution formula Eq. (12) are very useful in treating the general astigmatic matter waves, which can be seen in the following section.

3. Evolution of the general astigmatic matter wave

In atom optics, the light pulses play the role of cylindrical lenses [13, 16]. The electro-magnetic wave (EMW) with a smooth intensity profile

E (x) = E_{0} (1 - \frac{x^{2}}{w_{x}^{2}}) u,

where E ₀ is the amplitude of the electric field, w _x is the spot width along the x-direction and u is the unit vector, causes a position-dependent AC Stark phase shift:

Ω_{ac} (x) = Ω_{ac}^{0} (1 - 2 \frac{x^{2}}{w_{x}^{2}}) .

If the EMW is impinging on a matter wave, it will imprints a quadratic phase to the latter[13]:

ψ (t_{0} + τ) = ρ_{1} \exp [- i \frac{2 Ω_{ac}^{0} τ}{w_{x}^{2}} x^{2}] ψ (t_{0}; v_{0} + \frac{\bar{h} k_{eff}}{m}, Q_{0}^{- 1}) .

The coefficient ρ ₁ contains the position-independent phase shift of the matter wave. ψ(t ₀;v ₀+hˉ k _eff/m,Q ^-1 ₀) is the initial wave function of the matter wave with the complex curvature Q ^-1 ₀ and the velocity v ₀+hˉ k _eff/m, where k _eff is the effective wave vector of the EMW pulses. ψ(t ₀+τ) is the wave function after the interaction with the EMW pulses. τ is the duration of the EMW pulse.

Fig. 1. One EMWpulse with a “focus length” f ₁, focuses the matter wave in the x-direction. After an interval time t ₁, another EM pulse whose “focus length” is f ₂, focuses the matter wave in a direction with an angle α to the x-direction in the x-y plane. The gravitational acceleration g goes in the z-direction.

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Using the tensor form to express the wave function, we recast Eq. (18) into:

ψ (t_{0} + τ) = ρ_{1} ψ (t_{0}; v_{0} + \frac{\bar{h} k_{eff}}{m}, Q_{1}^{- 1}),

where Q ^-1 ₁ is given by Eq. (13), and the evolution matrix M is

M = (\begin{matrix} A & B \\ C & D \end{matrix}) = (\begin{matrix} I & 0 \\ F & I \end{matrix}),

where I is a 3×3 unit matrix, 0 is a 3×3 zero matrix and F is called the focus matrix. In the case of the x-direction focusing, the matrix F is:

F = - \frac{1}{f} (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}),

where f is the “focus length” of the “cylindrical lens” in the temporal domain, given by:

f = \frac{{mw}_{x}^{2}}{4 \bar{h} Ω_{ac}^{0} τ} .

The situation we consider here is that two “cylindrical lens” pulses, whose “focus lengths” are f ₁ and f ₂ respectively, impinge separately on an atom cloud with the interval time t ₁, and the angle of the two focus-directions is α, as shown in Fig.1. The evolution matrix of this process is a product of the f ₁-lens matrix, the t ₁-free-evolution matrix and the f ₂-lens matrix, according to the temporal sequence multiplied from the right to the left:

M = M_{2} M_{t_{1}} M_{1} .

M _1,2 is the “cylindrical lens” matrix:

M_{1, 2} = (\begin{matrix} I & 0 \\ F_{1, 2} & I \end{matrix}),

with

F_{1} = - \frac{1}{f_{1}} (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}),

and

F_{2} = - \frac{1}{f_{2}} (\begin{matrix} \cos^{2} α & \sin α \cos α & 0 \\ \sin α \cos α & \sin^{2} α & 0 \\ 0 & 0 & 0 \end{matrix}) .

$M_{t_{1}}$ is the free evolution matrix:

M_{t_{1}} = (\begin{matrix} I & t_{1} I \\ 0 & I \end{matrix}) .

If the initial atom cloud is an isotropic matter wave, its complex curvature can be expressed by:

Q_{0}^{- 1} = (\begin{matrix} q^{- 1} & 0 & 0 \\ 0 & q^{- 1} & 0 \\ 0 & 0 & q^{- 1} \end{matrix}),

where q ^-1=R ^-1+2i(hˉ/m)W ^-2 is a complex quantity representing the complex curvature in each coordinate direction, R and W are the phase front radius and the cloud density width. By using the tensor ABCD law in Eq. (13) and the evolution matrix in Eq. (23), the complex curvature after the interactions with the two EMW pulses can be found as follows:

Q_{2}^{- 1} = (\begin{matrix} \frac{q^{- 1} - \frac{1}{f_{1}}}{1 - \frac{t_{1}}{f_{1}} + t_{1} q^{- 1}} - \frac{\cos^{2} α}{f_{2}} & - \frac{\sin α \cos α}{f_{2}} & 0 \\ - \frac{\sin α \cos α}{f_{2}} & \frac{q^{- 1}}{1 + t_{1} q^{- 1}} - \frac{\sin^{2} α}{f_{2}} & 0 \\ 0 & 0 & \frac{q^{- 1}}{1 + t_{1} q^{- 1}} \end{matrix}) .

According to the definition in Eq. (11), the matrix of Q ^-1 ₂ can be separated into real and imaginary parts. Its real part is a matrix which represents the curvature of the phase front R ^-1 ₂, and its imaginary part is a matrix describing the curvature of the density profile W ^-2 ₂. If α=0 or π/2, the matrix of Q ^-1 ₂ is diagonalized. The real part and the imaginary part are both oriented in one coordinate direction, and the matter wave is simple astigmatic. If α≠0 or π/2, the matrix of Q ^-1 ₂ can’t be diagonalized. The real part is rotated with respect to the coordinate direction while the orientation of the imaginary part is unaffected. In this case, the atom cloud is a general astigmatic matter wave.

To make the matter wave configuration more visual, we introduce real-valued parameters. In the following example, all lengths are in units of [hˉ/(2mω)]^1/2 and the time is in units of ω ^-1, where ω is the trapping frequency of the initial atom cloud.

Considering the initial isotropic matter wave described in Eq. (28), we assume that it has a flat phase front R ^-1=0 and a Gaussian density distribution with W=√2, then the complex curvature in each coordinate direction is q ^-1=2i. The angle α of the two focus-directions is assumed to be π/4, and the interval time t ₁=0.1. The focusing effect we consider here belongs to the strong focus [16, 17], thus the “focus lengths” of the pulses can be assumed as f ₁=f ₂=0.2. The total evolution matrix of the matter wave is

M_{t} = M_{t_{2}} M,

where

M_{t_{2}} = (\begin{matrix} I & t_{2} I \\ 0 & I \end{matrix})

is a free evolution matrix, t ₂ is the evolution time of the matter wave after the interaction with the last EMW pulse. According to the ABCD law of Eq. (13) we can obtain the final complex curvature Q ^-1 ₃ by the knowledge of Q ^-1 ₀ in Eq. (28) and M _t in Eq. (30). The exact expression of Q ^-1 ₃ is quite complicated and will not be given here. Instead, we will present some figs. to show the evolution procedure of the general astigmatic matter wave. By substituting the finial complex curvature into Eq. (12), the phase front and the density distribution of the matter wave can be obtained.

Fig. 2. The phase front (a)-(d) and the density distribution (e)-(f) of the atom cloud after the interaction with the second pulse: (a) and (e) t ₂=0; (b) and (f) t ₂=0.06; (c) and (g) t ₂=0.2; (d) and (h) t ₂=0.3, where t ₂ is the evolution time. The horizontal axis of figs. (e)-(f) is the x-coordinate and the vertical axis is the y-coordinate. The origin of the coordinate system is the center of mass of the atom cloud. The initial trapping frequency ω is 100kHz, and the atoms are Rubidium 87.

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Figure 2 is the calculation result of the phase front and the density profile on the x-y plane at different evolution times after the interaction with the last lens pulse. We can see clearly that, during the evolution, the phase front changes from the concave surface to the convex ones. In a certain plane, the phase front is flat, and the matter wave is focused. After the focusing, the convex phase front tends to be an ellipsoidal surface. As t ₂ increases, the ellipsoidal surface would change to a spherical surface with an increasing radius. Meanwhile, the density distribution expands more and more widely. During the procedure, the orientations of the phase front and the density profile rotate continuously. If the evolution time is long enough, the phase front tends to be a flat surface, and the orientation of the density profile would reach an limit angle. This rotation feature originates from the anisotropy of the evolution system of the matter wave.

4. Conclusion

In this paper, a new kind of matter wave with the general astigmatism is proposed. Due to the coupling of the evolutions in each dimension, the general astigmatic matter wave can not be separated into individual dimensions. A 3×3 complex curvature tensor M ^-1 is introduced to describe the Gaussian-shaped general astigmatic matter wave. A generalized tensor ABCD law is derived, which is very convenient to treat the evolution of the general astigmatic matter wave. As an example, we demonstrate the origin of the general astigmatism. It is revealed that the two oblique “cylindrical lens” pulses can bring the general astigmatism to the matter wave. During the evolution, the orientations of the phase front and the density profile rotate continuously. The density distribution and the phase distribution are closely related. The formalism presented in this paper is a powerful tool in treating the evolution problems of the Gaussian-shaped matter waves with general astigmatism.

Acknowledgments

The authors wish to acknowledge the supports from the Ministry of Science and Technology of China (grant nos. 2006CB921403 & 2006AA06A204) and the National Natural Science Foundation of China (grant no. 60478042) and the Zhejiang Provincial Qian-Jiang-Ren-Cai Project of China (grant no. 2006R10025).

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The general astigmatic matter wave

Abstract

1. Introduction

2. Tensor ABCD law for general astigmatic matter waves

3. Evolution of the general astigmatic matter wave

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (2)

Equations (39)

Optics Express