Nonlinear evolution of the interacting Gaussian-shaped Matter Waves

Jun Chen; Zhang Zhang; Yunxian Liu; Qiang Lin

doi:10.1364/OE.16.010918

1. Introduction

The rapid progress on the cooling and trapping of atoms makes the matter wave optics a new and exciting field of physics[1, 2]. A lot of matter wave controlling elements emerge such as the matter wave mirrors, lenses and diffraction gratings[3] and so on. The evolution of the coherent matter wave becomes a hot topic both in the experimental and theoretical investigations. The analytic formula of the matter wave based on the ABCD formalism[4] is very useful in dealing with the evolution problems of the noninteracting coherent matter waves, such as the angular divergence of a noninteracting atom laser[5], the phase shift caused by the gravitational field in an atom interferometry[6, 7, 8], the improvement of the measuring precision of the atom gravimeter[9], the design of matter wave cavity[10], and so on. The matrix formalism has been recently extended to include the non-Gaussian atom laser[11, 12], the general astigmatic matter wave[13] and the partially coherent matter wave[14].

For the evolution of the interacting matter waves, however, the usual analytic solution is not applicable, because the evolution satisfies the nonlinear Schrödinger equation which has to be solved numerically in general case[15, 16, 17]. Nevertheless we recall that an approximate analytic method based on the effective complex curvature (ECC) has been developed to treat the paraxial evolution of Gaussian light beams in nonlinear media[18, 19].

In this paper, the ECC method of nonlinear optics is extended to treat the evolution of the Gaussian-shaped matter wave with the weak interaction among atoms. The effective complex curvature of the Gaussian-shaped matter wave, which contains the interaction coefficient of the matter wave is given, and the evolution equation of the interacting matter wave is linearized. The conditions under which this extension is appropriate are presented. As an example, focusing of a one-dimensional interacting coherent matter wave is calculated and compared with the noninteracting matter waves.

2. The effective complex curvature (ECC) method of the matter wave

The evolution and propagation of a coherent matter wave can be described by the Schrödinger equation

i \overset{}{\bar{h}} \frac{\partial}{\partial t} ϕ (r, t) = (- \frac{{\overset{}{\bar{h}}}^{2}}{2 m} \nabla^{2} + V (r, t)) ϕ (r, t),

where m is the mass of atom, V(r,t) is the total potential. For quasi-continuous matter waves, the wave function ϕ(r,t) is a product of two parts, one is related to the transverse coordinates, the other is related to the longitudinal coordinate[5, 12]:

ϕ (r, t) = ψ (x, y, t) \exp [\frac{i}{\bar{h}} σ (z, t)] .

Here ψ(x,y,t) is the transverse wave function, and depends on the transverse coordinates x, y and the evolution time t. σ (z,t) is a complex quantity related to the longitudinal properties of the matter wave. We discuss the case where the potential can be separated as

V (r, t) = V_{t} (x, y, t) + V_{z} (z, t) .

V _t(x,y,t) is the potential on the transverse plane such as the transverse confining potential or the transverse interaction potential, and V_z(z,t) is the potential along the longitudinal axis. In this case it is straightforward to show that the Schrödinger equation Eq.(1) can be separated into the following two equations:

- \frac{\partial}{\partial t} σ (z, t) = - \frac{i \bar{h}}{2 m} \frac{\partial^{2}}{{\partial z}^{2}} σ (z, t) + \frac{1}{2 m} {(\frac{\partial}{\partial z} σ (z, t))}^{2} + V_{z} (z, t) σ (z, t),

i \bar{h} \frac{\partial}{\partial t} ψ (x, y, t) = (- \frac{{\bar{h}}^{2}}{2 m} \nabla_{⊥}^{2} + V_{t} (x, y, t)) ψ (x, y, t),

where ∇² _⊥=∂ ²/∂x ² +∂ ²/∂y ². Equation(5) describes the evolution of the matter wave on the transverse plane.

If the low-energy two-body collision among atoms is taken into account, in the free evolution case, the transverse potential is given by[3]

V_{t} (x, y, t) = U {∣ ψ (x, y, t) ∣}^{2},

where U is the interaction coefficient. Then Eq.(5) turns out to be a two-dimensional nonlinear Schrädinger equation:

i \bar{h} \frac{\partial}{\partial t} ψ (x, y, t) = (- \frac{{\bar{h}}^{2}}{2 m} \nabla_{⊥}^{2} + U {∣ ψ (x, y, t) ∣}^{2}) ψ (x, y, t) .

Equation(7) is similar in form to the evolution equation of a light beam in nonlinear media, therefore the method of solving this kind of equation can be borrowed from optics[18, 19].

We assume that the interaction among atoms is weak, and the profile of ψ(x,y,t) in Eq.(7) is in Gaussian shape:

ψ (x, y, t) = A (t) \exp [\frac{im}{2 \bar{h}} \frac{x^{2} + y^{2}}{q (t)}],

where A(t) is the probability amplitude at the evolution time t, and

\frac{1}{q (t)} = \frac{1}{ρ (t)} + 2 i \frac{\bar{h}}{m} \frac{1}{{w (t)}^{2}}

is the complex curvature of the matter wave, ρ(t) and w(t) are the phase front radius and the matter wave density width on the transverse plane, respectively. Under the paraxial approximation, the nonlinear term in Eq.(7) can be expanded by the Taylor series. If we preserve the first two terms, we get

{∣ ψ (x, y, t) ∣}^{2} \approx {∣ A (t) ∣}^{2} - 2 {∣ A (t) ∣}^{2} \frac{(x^{2} + y^{2})}{{w (t)}^{2}} .

Therefore the complex curvature 1/q(t) satisfies the following equation:

{(\frac{1}{q (t)})}^{′} + \frac{1}{q {(t)}^{2}} - \frac{{4 U ∣ A (t) ∣}^{2}}{mw {(t)}^{2}} = 0,

where the prime ′ stands for ∂/∂t, and the probability amplitude A(t) satisfies:

\frac{{A (t)}^{′}}{A (t)} + \frac{1}{q (t)} + i \frac{U {∣ A (t) ∣}^{2}}{\bar{h}} = 0 .

In most cases, U is real, and Eq.(12) can be rewritten as

\frac{{({∣ A (t) ∣}^{2})}^{'}}{{∣ A (t) ∣}^{2}} + \frac{2}{ρ (t)} = 0 .

The imaginary part of Eq.(11) produces the following relation between ρ(t) and w(t):

\frac{1}{ρ (t)} = \frac{{w (t)}^{'}}{w (t)} .

Equation (13) is the equation for the probability density, which can be solved with the help of Eq.(14). The result is

{∣ A (t) ∣}^{2} = {∣ A (0) ∣}^{2} \frac{{w (0)}^{2}}{w {(t)}^{2}},

where A(0) is the initial probability amplitude, and w(0) is the initial matter wave density width. Substituting Eq.(15) into Eq.(11) we can obtain the following equation:

{(\frac{1}{ρ (t)})}^{'} + 2 i \frac{\bar{h}}{m} {(\frac{1}{{w (t)}^{2}})}^{'} + \frac{1}{{ρ (t)}^{2}} + 4 i \frac{\bar{h}}{mρ (t) {w (t)}^{2}} - 4 \frac{{\bar{h}}^{2}}{m^{2}} \frac{M^{2}}{{w (t)}^{4}} = 0,

where M ² is a factor related to the interaction coefficient U, the initial probability amplitude A(0) and the initial matter wave density width w(0), given by

M^{2} = 1 + \frac{mU}{{\bar{h}}^{2}} {∣ A (0) ∣}^{2} {w (0)}^{2} .

In optics, M ² is the beam propagation factor which describes the deviation of a real light beam from an ideal Gaussian[19]. Similarly, in the case of matter wave, M ² plays the same role, describing the spread speed of the matter wave.

Equation (16) can be separated into its real and imaginary parts:

{(\frac{1}{ρ (t)})}^{'} + \frac{1}{{ρ (t)}^{2}} - 4 \frac{{\bar{h}}^{2}}{m^{2}} \frac{M^{2}}{{w (t)}^{4}} = 0,

2 \frac{\bar{h}}{m} {(\frac{1}{{w (t)}^{2}})}^{'} + 4 \frac{\bar{h}}{mρ (t) w {(t)}^{2}} = 0 .

For repulsive interacting matter waves, the interaction coefficient U > 0, and the factor M ² > 1. Multiplying Eq.(19) with iM and adding up with Eq.(18), we get:

{(\frac{1}{ρ (t)})}^{'} + \frac{1}{ρ {(t)}^{2}} - 4 \frac{{\bar{h}}^{2}}{m^{2}} \frac{M^{2}}{{w (t)}^{4}} + 2 i \frac{\bar{h}}{m} {(\frac{1}{{w (t)}^{2}})}^{'} M + 4 i \frac{\bar{h} M}{mρ (t) {w (t)}^{2}} = 0 .

By defining an effective complex curvature

\frac{1}{q_{eff} (t)} = \frac{1}{ρ (t)} + 2 i \frac{\bar{h}}{m} \frac{M}{{w (t)}^{2}},

the differential equation Eq.(20) becomes

{(q_{eff} (t))}^{'} - 1 = 0 .

Equation (22) shows that, if we define an effective complex curvature containing the interaction term (Eq.(21)), the nonlinear equation for the complex curvature 1/q(t) (Eq.(11)) can be linearized. Equation (22) has the same form as the equation satisfied by a noninteracting Gaussian-shaped matter wave, whose solution reads:

q_{eff} (t) = q_{eff} (0) + t .

The main advantage of the effective complex curvature (ECC) expression is that it uses the well-known analytic formalism for the Gaussian-shaped matter wave to evaluate the evolution of the interacting matter wave. In this case, the evolution of the equivalent complex curvature can also be described by the ABCD formalism, where all the A, B, C, D elements remain the same for both linear and nonlinear evolution:

q_{eff} (t) = \frac{{Aq}_{eff} (0) + B}{{Cq}_{eff} (0) + D},

with A, B, C, D being the elements for the free evolution

A = 1, B = t, C = 0, D = 1 .

There are three preconditions for the ECC method of the matter wave: (i) the interaction among atoms is weak; (ii) the total potential can be separated into the transverse potential and the longitudinal one; and (iii) the matter wave evolves in the paraxial region.

3. The nonlinear evolution of the matter wave

In order to get an intuitionistic impression, we draw the relative phase distribution and the amplitude profile of the matter wave by means of the ECC method and the exact numerical calculation method. For the sake of simplicity, we take the 1-D problem. All lengths are set in units of [h̄/(2mω)]^1/2, the energy is in units of h̄ω and the time is in units of ω ^-1, with ω being the trapping frequency of the initial atom cloud. We assume the initial wave function to be

ψ (x, t = 0) = \exp [- x^{2} (\frac{1}{4} + \frac{i}{100})] .

Then the factor M ² is

M^{2} = 1 + \frac{U}{\frac{{\bar{h}}^{2}}{(2 m)}} = 1 + Ũ,

where Û is the ratio of the interaction coefficient to the kinetic term coefficient. The relative magnitude of the interaction coefficient determines the application of the ECC method in approximating the amplitude profile of the interacting matter wave. As illustrated in Fig.1(a), the ECC amplitude profile is a good approximation when the interaction effect is the same as the kinetic one. When the interaction coefficient is much higher than the kinetic one, as shown in Fig.1(b) and (c), the exact amplitude profiles tend to be flatter than the ECC ones.

In this case, some other approximation method could be used, such as the Thomas-Fermi approximation[20, 21, 22]. The relative phase distributions of different interacting matter waves are shown in Figs.1(d), (e) and (f). The central phase is set at -π, and it would be the reference for all the other spacial points. The discontinuous jumps in the phase are introduced because of the mapping of the continuous phase of the wave function ψ(x,t) onto (-π, π]. It is shown that, for various M ²-factors, the ECC method can well approximate the phase distribution in the central region where most atoms are located. Totally speaking, if only the interaction energy among atoms is not much larger than the kinetic energy, say 5 times, the ECC method is a good approximation in treating the evolution of matter waves.

The relative phase distributions of the matter wave with different M ²-factors are depicted by means of the ECC method in Fig. 2. It is shown that the phase of the interacting matter wave (M ² > 1) changes with the x-coordinate more rapidly than that of the ideal matter wave (M ² = 1). As illustrated by the M ²=5 curve (green, dot) and the M ²=10 curve (red, dash) in Fig. 2, when the magnitude of the M ²-factor becomes much larger than 1, say 5, the difference between the relative phase distributions caused by the different M ²-factors trends to disappear. In another word, the relative phase distribution of the strong interacting matter waves is more insensitive to the change of the factor M ² than that of the weak interacting ones.

As an application example of the ECC method, we investigate the matter wave focusing problem in the following. The initial state in our calculation is

ψ (x, t = 0) = \exp [- \frac{x^{2}}{4} (1 + i \frac{1}{f})] .

It describes a ground state whose phase has been altered by a lens pulse of a “focus length” f. As illustrated in Fig. 3(a), when the matter wave width w(t) becomes a minimum, focusing occurs. According to Eq. (23) and Eq. (28), the general formula of the focusing time for the matter wave is given by:

t_{f} = \frac{f}{1 + M^{2} f^{2}} .

It is the time duration in which the matter wave shrinks into the minimum size due to focusing. Figure 3(a) illustrates the process that a matter wave focuses on the axis and then spreads out. The temporal point at which the density of the matter wave comes to its maximum is t_f =0.4. It is the focusing time of this example. Under the weak interacting circumstances, as illustrated in Figs. 3(b), (c) and (d), the effective density profile is a good approximation to the exact density profile (obtained by means of the numerical calculation method).

Figure 4 shows the relation between the focusing time t_f and the “focus length” f in two different situations: in one situation, there is no interaction among atoms (M ²=1); in the other one, there is weak repulsive interaction among atoms (M ²=1.5). It is shown that in the strong focusing case (for small values of f), the focusing properties of the interacting matter wave and the noninteracting matter wave are very similar: the focusing time t_f of the matter wave and the “focus length” f of the lens pulse are in a linear relation. In the weak focusing case (for large values of f), the focusing time of the interacting matter wave is shorter than that of the noninteracting one. This is understandable because the repulsive collisions make the atoms against to assemble. The matter wave with repulsive interactions therefore begins to spread earlier than the noninteracting one. It is a self-defocusing phenomena in matter wave optics. The result of Fig. 4 obtained by the ECC method agrees with that by the numerical calculations in Ref.[21, 22].

The interaction among atoms is determined by the interaction coefficient U. It can be repulsive(U > 0), or attractive(U < 0). The sign and the magnitude of the coefficient U are determined by the atomic species. Most of the experiment works with the coherent matter wave utilize the atomic gas with repulsive interactions. However, some experiments are done with the attractive interactions[23], or with the interactions that can be adjusted to be attractive by the Feshbach resonances[24]. The method we used in this paper is also applicable to the matter wave with attractive interactions. This corresponds to the self-focusing phenomena in optics when the light beam propagates in a nonlinear medium. For the attractive case (U < 0), the matter wave trends to shrink in order to lower the interaction energy. Meanwhile, the kinetic pressure from the kinetic energy of atoms brings the matter wave a tendency to spread. The competition between the attractive interaction and the kinetic pressure is reflected on the sign of the M ²-factor. When the tendency of the matter wave to spread due to the kinetic pressure counteracts the tendency of the matter wave to shrink due to the attractive interaction, the factor M ² > 0. In this case, the matter wave beam retains Gaussian-shape. When the attractive interaction overwhelms the kinetic pressure, the factor M ² < 0. In this case, the matter wave will collapse onto itself [25]. This is corresponding to the light beam self-destructing [26] in the strong self-focusing medium in optics.

Fig. 1. Comparison between the exact amplitude profile (solid line) and the ECC amplitude profile (dashed line). The evolution time is t = 1.4.

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Fig. 2. Relative phase distribution of matter waves with different M ²-factors. The evolution time is t = 1.4.

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Fig. 3. (a) Evolution of the density profile of the effective Gaussian-shaped matter wave after the interaction with a focusing pulse f =1. The M ² factor of the interacting matter wave is 1.5; (b) comparison between the exact density profile (the solid line) and the effective density profile (the dashed line) at t = 0.8; (c) at t = 1.2; (d) at t = 1.6.

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Fig. 4. Focusing time t_f as a function of f , both in dimensionless units. The solid line is the graph of the noninteracting matter wave; The dashed line is the graph for an interacting matter wave with M ² = 1.5.

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4. Conclusions

In this paper, the ECC methods are extended to the field of matter wave. The nonlinear evolution of the interacting matter wave is investigated analytically. The effective complex curvature is presented to describe the weak interacting matter wave, and its analytic formula is derived. As an example, we analyzed the focusing problem of a Gaussian-shaped matter wave with the repulsive interaction among atoms. Self-defocusing of the interacting matter wave is clearly illustrated. These results show that the ECC solution is a simple and convenient method in treating the nonlinear evolution and transformation of interacting matter waves, as long as the interaction among atoms is not very strong compared with the kinetic energy of the atoms. This is the condition that many cold atom systems satisfy.

Acknowledgment

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

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Nonlinear evolution of the interacting Gaussian-shaped Matter Waves

Abstract

1. Introduction

2. The effective complex curvature (ECC) method of the matter wave

3. The nonlinear evolution of the matter wave

4. Conclusions

Acknowledgment

References and links

Cited By

Figures (4)

Equations (29)

Optics Express