Chaotic scattering of atoms at a standing laser wave

A beam of two-level atoms in the z-direction crosses a SW laser field with optical axis in the x-direction (fig. 1). The laser field amplitude has the Gaussian profile $\exp [-(z-z_0)^2/r^2]$ with r being the e⁻² radius at the laser beam waist. The characteristic length of the atom-field interaction is supposed to be 3r because the light intensity at z = z₀ ± 1.5r is two orders of magnitude smaller than the peak value. The longitudinal velocity of atoms, v_z, is much larger than their transversal velocity v_x and is supposed to be constant. Thus, the spatial laser profile may be replaced by the temporal one. The Hamiltonian of an atom in the 1D SW field can be written in the frame rotating with the laser frequency ω_f as follows:

$$\begin{eqnarray} \hat H &=& \displaystyle\frac{P^2}{2m_a}+\displaystyle\frac{\hbar}{2}(\omega_a-\omega_f)\hat\sigma_z-\hbar \Omega_0 \exp\!\left[\!-\left(\!t-\displaystyle\frac{3}{2} \sigma_t\!\right)^2\!\!\!\Bigg/\sigma^2_t\!\right]\nonumber \\ &&\times \left(\hat\sigma_-+\hat\sigma_+\right)\cos{k_f X} -\displaystyle\frac{i\hbar\Gamma}{2}\hat\sigma_+\hat\sigma_-, \end{eqnarray} \tag{ 1 }$$

where $\hat \sigma _{\pm , z}$ are the Pauli operators for the internal atomic degrees of freedom, X and P are the classical atomic position and momentum, Γ, ω_a, and Ω₀ are the decay rate, the atomic transition and maximal Rabi frequencies, respectively, and σ_t ≡ r/v_z, i.e., 3σ_t is the transit time. The wave function for the electronic degree of freedom is |Ψ(t)〉 = a(t)|2〉 + b(t)|1〉, where a ≡ A + iα and b ≡ B + iβ are the probability amplitudes to find the atom in the excited, |2〉, and ground, |1〉, states, respectively.

In the semiclassical approximation, an atom with quantized internal dynamics is treated as a point-like particle (the transversal atomic momentum p is supposed to be, in average, much larger than the photon one, ℏk_f) with the equations of motion written for the real and imaginary parts of the probability amplitudes

$$\begin{eqnarray} &&\hspace{-8pt}\dot x =\omega_r p, \, \dot p =- 2\exp\!\left[-\left(\tau-\frac{3}{2} \sigma_{\tau}\right)^2\!\!\!\!\Bigg/\!\!\sigma^2_{\tau}\right](AB+\alpha \beta)\sin x, \nonumber\\ &&\hspace{-8pt}\dot A = \frac{1}{2} (\omega_rp^2\! - \Delta)\alpha -\! \frac{1}{2} \gamma A - \exp\!\left[\!-\left(\!\tau-\!\frac{3}{2} \sigma_{\tau}\!\right)^2\!\!\!\!\Bigg/\!\!\sigma^2_{\tau}\!\right]\!\beta\cos x, \nonumber\\ &&\hspace{-8pt}\dot \alpha = -\frac{1}{2} (\omega_rp^2\! -\! \Delta)A\! -\! \frac{1}{2} \gamma \alpha\! + \exp\!\left[\!-\!\left(\!\tau-\!\frac{3}{2} \sigma_{\tau}\!\right)^2\!\!\!\!\Bigg/\!\!\sigma^2_{\tau}\!\right]\!B\cos x, \nonumber\\ &&\hspace{-8pt}\dot B = \frac{1}{2} (\omega_rp^2 + \Delta)\beta - \exp\!\left[-\left(\tau-\frac{3}{2} \sigma_{\tau}\right)^2\!\!\!\!\Bigg/\!\!\sigma^2_{\tau}\right]\alpha \cos x, \nonumber\\ &&\hspace{-8pt}\dot \beta = -\frac{1}{2} (\omega_rp^2 + \Delta)B+ \exp\!\left[-\left(\tau-\frac{3}{2} \sigma_{\tau}\right)^2\!\!\!\!\Bigg/\!\!\sigma^2_{\tau}\right]A\cos x,\nonumber\\ \end{eqnarray} \tag{ 2 }$$

where x ≡ k_fX and p ≡ P/ℏk_f are the atomic center-of-mass position and transversal momentum, respectively, and the dot denotes differentiation with respect to the dimensionless time τ ≡ Ω₀t. The recoil frequency, ω_r ≡ ℏk²_f/m_aΩ₀ ≪ 1, the atom-laser detuning, Δ ≡ (ω_f − ω_a)/Ω₀, the decay rate γ = Γ/Ω₀, and the characteristic interaction time, σ_τ ≡ rΩ₀/v_z, are the control parameters.

Without the decay, γ = 0, eqs. (2) constitute the nonlinear Hamiltonian dynamical system with three degrees of freedom describing an atom moving in the six-dimensional phase space. The simple way to know how complicated this motion may be is to compute the quantitative measure of chaos, maximal Lyapunov exponent characterizing the mean rate of the exponential divergence of initially close trajectories. Because of a transient character of chaos [23] we compute the finite-time Lyapunov exponent λ, i.e. the value of the exponent at the moment when atoms leave the interaction region. The result of computation with eqs. (2) at the given value of the recoil frequency, ω_r = 10⁻³, and zero decay rate in dependence on the detuning Δ and the initial atomic transversal momentum p₀ is shown in fig. 2. The gray scale codes the magnitude of the finite-time Lyapunov exponent. In the white regions the values of λ are almost zero, and the internal and translational motion is regular in the corresponding ranges of Δ and p₀. In the shadowed regions positive values of λ imply unstable motion.

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The scheme of the proposed experiment in fig. 1 resembles the scattering problem with particles entering an interaction region along completely regular trajectories and leaving it along asymptotically regular trajectories [24–29]. However, differently from the standard chaotic scattering with a nonattractive fractal invariant set existing over an infinite time, this process may be interpreted as a finite-time chaotic scattering.

There are three possible chaotic regimes of the center-of-mass motion along the SW optical axis [17, 18]. In dependence on the initial conditions and the values of the control parameters, atoms may oscillate chaotically in wells of the optical potential or move ballistically over its hills with chaotic variations of their velocity. Chaotic motion becomes possible in a narrow range of the detuning values, 0 < |Δ| < 1. At Δ = 0, the synchronized electric-dipole component, u = 2(AB + αβ) becomes a constant. That implies the additional integral of motion in the Hamiltonian version of eqs. (2) and the regular motion with λ = 0. Far from the resonance, at |Δ| > 1, the motion is again regular both in the trapping and flight modes. That speculation is confirmed by the Lyapunov map in fig. 2.

Moreover, there is a specific type of motion, chaotic walking in a deterministic optical potential, when atoms can change the direction of motion alternating between flying through the SW and being trapped in its potential wells. We would like to stress that the local instability produces chaotic center-of-mass motion in a rigid SW without any modulation of its parameters differently from the case with periodically kicked and far-detuned optical lattices [7, 9–11]. The trivial time dependence in the Hamiltonian (2) cannot produce chaotic motion, it simply accounts for a modulation of the interaction of atoms with a Gaussian laser beam. Even if the atoms crossed an absolutely homogeneous (in the z-direction) laser beam there would be under appropriate conditions chaotic atomic center-of-mass motion in the transversal x-direction.

Chaotic walking occurs due to the specific behavior of the Bloch-vector component, u, of a moving atom whose shallow oscillations between the SW nodes are interrupted by sudden jumps with different amplitudes while the atom crosses each node [18]. We illustrate in fig. 3 the behavior of the u component with different values of the detuning Δ = 0.2 and Δ = 1 at which the atomic motion in accordance with the λ-map in fig. 2 is chaotic and regular, respectively. The time of the atomic interaction with the SW field is estimated to be 3σ_τ = 1200. So, the jumps of the u variable (if any) disappear after that time in fig. 3. It follows from the second equation in the set (2) that jumps in the variable u = 2(AB + αβ) result in jumps of the atomic momentum while crossing a node of the SW. If the value of the atomic energy is close to a separatrix one, the atom after the corresponding jump-like change in p can overcome the potential barrier and leave a potential well or be trapped by the well, or move as before. The jump-like behavior of u is the ultimate reason of chaotic atomic walking along a deterministic SW.

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It is easy to estimate the range of initial momenta at which atoms are expected to change their direction of motion or move ballistically. At small detunings, |Δ| ≪ 1, the total energy consists of the kinetic one, K = ω_rp²/2, and the potential one, U = u cos x. If K(τ = 0) > |U_max| = 1, then the atom will move ballistically. This occurs if the initial atomic momentum, p₀, satisfies the condition $p_0> \sqrt {2/\omega _r}>44$. If the initial conditions are chosen to give 0 ⩽ K(τ = 0) + U(τ = 0) ⩽ 1, the atoms with $0 \lesssim p_0 \lesssim 44$ are expected to perform a chaotic walking at positive Δ.

Let us consider a spatially uniform and previously focused beam of atoms crossing a Gaussiam laser beam. The position and momentum distributions of atoms are measured after interaction with the SW field. We predict that those distributions would be much broader at those values of Δ at which one expects chaotic walking to occur. Firstly, we perform simulation with a negligible probability of spontaneous emission. To be concrete let us take calcium atoms with the working intercombination transition 4¹S₀-4³P₁ at λ_a = 657.5 nm, the recoil frequency ν_rec ≃ 10 kHz, and the lifetime of the excited state T_sp = 0.4 ms. Taking the maximal Rabi frequency to be Ω₀/2π = 2·10⁷ Hz, the radius of the laser beam r = 0.3 cm, and the mean longitudinal velocity v_z = 10³ m/s, the interaction time is estimated to be 0.9 ms. The normalized recoil frequency is ω_r = 4πν_rec/Ω₀ = 10⁻³ and σ_τ = 400.

We numerically solve the equations of motion (2) with γ = 0 at two values of Δ corresponding to the chaotic and regular regimes of the center-of-mass motion. In accordance with the Lyapunov map in fig. 2, the behavior of the Bloch component u in fig. 3, and the simple estimates given above, we expect the chaotic scattering of atoms at Δ = 0.2 and their regular motion at Δ = 1. Trajectories in the real and momentum spaces for 50 atoms with the same initial momentum, p₀ = 10, and initial positions in the range −π/10 ⩽ x ⩽ π/10 are shown in figs. 4 and 5, respectively, at the fixed value of the recoil frequency ω_r = 10⁻³. The upper panels in figs. 4 and 5 illustrate the broad distributions of the atoms in the x and p spaces in the regime of chaotic scattering that contrasts strictly with those obtained in the case of the regular scattering at Δ = 1 (figs. 4(b) and 5(b)). In order to create narrow atomic beams, one may use a pair of light masks. The first SW with a red large detuning (Δ < 0) splits the initial atomic beam into a number of narrow beams with widths much smaller than the optical wavelength which then cross the second SW. A method for creating narrow wave packets in the nonadiabatic regime of scattering has been proposed in ref. [30].

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We have illustrated in figs. 4 and 5 the Hamiltonian chaotic scattering that may occur in the absence of any losses. To simulate trajectories of spontaneously emitting atoms we use the standard stochastic wave function technique (see, for example, [31–33]) for solving eqs. (2). The integration time is divided into a large number of small time intervals δτ. At the end of the first one τ = τ₁ the probability of spontaneous emission, s₁ = γδτ|a_τ1|²/(|a_τ1|² + |b_τ1|²), is computed and compared with a random number ε from the interval [0,1]. If s₁ < ε₁, then one prolongs the integration but renormalizes the state vector in the end of the first interval at τ = τ⁺₁: $a_{\tau _1^+}=a_{\tau _1}/\sqrt {|a_{\tau _1}|^2 + |b_{\tau _1}|^2}$ and $b_{\tau _1^+}=b_{\tau _1}/\sqrt {|a_{\tau _1}|^2 + |b_{\tau _1}|^2}$. If s₁ ⩾ ε₁, then the atom emits a spontaneous photon and performs the jump to the ground state at τ = τ₁ with A_τ1 = α_τ1 = β_τ1 = 0, B_τ1 = 1. Its momentum in the x-direction changes for a random number from the interval [0,1] due to the photon recoil effect, and the next time step commences.

We simulate lithium atoms with the relevant transition 2S_1/2 − 2P_3/2, the corresponding wavelength λ_a = 670.7 nm, recoil frequency ν_rec = 63 kHz, and the decay time T_sp = 2.73·10⁻⁸ s. With the maximal Rabi frequency Ω₀/2π ≃ 126 MHz and the radius of the laser beam r = 0.05 cm one gets ω_r = 10⁻³, σ_τ = 400, and γ = 0.05. Simulated trajectories in the real space for 50 spontaneously emitting atoms under the same conditions as in fig. 4 are shown in fig. 6. Even though deterministic Hamiltonian chaos is masked by random events of spontaneous emission, the spatial and momentum (not shown) distributions are much broader at those values of Δ at which the Hamiltonian center-of-mass motion is chaotic. Namely the chaotic Hamiltonian walking is eventually responsible for divergency of atomic beams in the real and momentum spaces.

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To simulate a real experiment we consider a beam of 10⁴ lithium atoms with initial Gaussian distribution (the rms σ_x = σ_p = 2 and the average values, x₀ = 0, and momentum, p₀ = 10) and compute their distribution at a fixed moment of time. In fig. 7(a) we compare the atomic position distributions at τ = 1000 for the chaotic scattering at Δ = 0.2 (bold curve) and the regular scattering at Δ = 1 (dashed curve) when neglecting spontaneous emission. The difference is evident. In the regime of the chaotic scattering at Δ = 0.2 atoms are distributed more or less homogeneously over a large distance of 8 wavelengths along the x-axis whereas they form a few peaks in a much narrower interval under the conditions of regular scattering at Δ = 1. Figure 7(b) demonstrates the distributions of spontaneously emitting atoms at the normalized decay rate γ = 0.05 under the same conditions as in fig. 7(a). The regularly scattered atoms at Δ = 1 (dashed curve) form the contrast atomic relief with the bifurcated peaks around the first few SW nodes at x = ± 1/4, x = ± 3/4 and x = 5/4. The distribution of chaotically scattered atoms at Δ = 0.2 (bold curve) has the peaks without any bifurcation at x = ± 1/4 and x = 3/4 with a smaller number of atoms in each one. Moreover, this distribution is less contrast as compared to the previous one. Thus, we predict that under the conditions of chaotic scattering there should appear less contrast and more broadened atomic reliefs as compared to the case of regular scattering because a large number of atoms are expected to be deposited between the nodes as a result of chaotic walking along the SW axis. The effect is expected to be more prominent under coherent evolution but it seems to be observable with spontaneously emitting atoms as well.

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We predict that experiments on the scattering of atomic beams at a SW laser field can directly image chaotic walking of atoms along the SW. In a real experiment the final spatial distribution can be recorded via fluorescence or absorption imaging on a CCD, commonly used methods in atom optics experiments yielding information on the number of atoms and the cloud's spatial size. The other possibility is a nanofabrication where the atoms after the interaction with the SW are deposited on a silicon substrate in a high-vacuum chamber. In this case the spatial distribution can be analyzed with an atomic force microscope. As to the momentum distribution, it can be measured, for example, by a time-of-flight technique. The modern tools of atom optics enable to create narrow initial atomic distributions in position and momentum, reduce coupling to the environment and technical noise, to create one-dimensional optical potentials, and to measure spatial and momentum distributions with high sensitivity and accuracy [9].