Fast manipulation of Bose–Einstein condensates with an atom chip

In this section we introduce the atom chip model and the trapping potential used in the present study. Atom chips designed for the manipulation of neutral atoms are insulating substrates with conducting micro-structures such as metal wires [12, 41, 42]. The wire geometry design can easily be adapted for a particular application [43]. DC wire currents generate inhomogeneous magnetic fields which can be used to trap atoms near the chip surface where high magnetic field gradients produce high trap frequencies and enable fast evaporation. This allows high-flux BEC creation of typically 10⁵ atoms s⁻¹ [8].

We consider here the case of a Z-shaped chip configuration [44], as shown in figure 1, in the presence of a time-dependent homogeneous magnetic bias field B_bias(t). If the bias field varies slowly, the spins of the atoms remain adiabatically aligned with the total magnetic field. In the weak field approximation and in the absence of gravity, the trapping potential can be expressed as

$$\begin{eqnarray}&&V({\bf{R}},t)={m}_{F}\,{g}_{F}\,{\mu }_{{\rm{B}}}\,{\rm{B}}({\bf{R}},t),\end{eqnarray} \tag{ 1 }$$

where μ_B is the Bohr magneton, g_F is the Landé factor, m_F is the azimuthal quantum number, and B(R, t) is the total magnetic field. The three-dimensional spatial position is denoted by ${\bf{R}}\equiv (X,Y,Z)$. As shown in [9], a temporal variation of the magnetic field can be used to transport the atoms. Our goal here is to design and test a fast transportation scheme for a realistic setup. We show how the implementation of such a scheme is feasible by specializing our discussion to the hyperfine state $| F=2,{m}_{F}=2\rangle$ of the ground 5S_1/2 state-of ⁸⁷Rb as a study case. This hyperfine state is a low-field seeking state with g_F = +1/2.⁵

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The Z-shaped wire is represented schematically on the left side of figure 1. In a first approximation the wires are considered as infinitely thin. The two wires aligned along the Y-axis are 16 mm long. The wire along X measures 4 mm. They carry a DC current I_w = 5 A. The magnetic bias field B_bias(t) points along Y and its magnitude varies between B_bias(0) = 21.5 G (initially) and B_bias(t_f) = 4.5 G (at the end of the displacement). These parameters are close to those used in the Quantus experiment [8]. Slices of the initial trapping potential V(R,0) at time t = 0 in the (YZ) and (XY) planes are shown in panels (a) and (b) of figure 1, respectively. As shown in the left panel of figure 1, Z denotes the distance to the chip surface. The atoms are initially trapped at a distance Z_i ≈ 0.45 mm from the chip surface directly under the origin of the axes. The shape of the trap seen in panel (a) shows a strong confinement in the Y and Z-directions with similar trap frequencies ν_Y(0) and ν_Z(0). On the contrary, the cigar shape seen in panel (b) reveals that ${\nu }_{X}(0)\ll {\nu }_{Y}(0)\approx {\nu }_{Z}(0)$. The initial trap is thus characterized by a strong two-dimensional confinement in the Y and Z-directions. The initial potential shows a small tilt angle θ(0) = θ_i ≈ 1.53° in the (XY) plane. In figure 1, the positions of the initial and final potential minima are marked by a white + sign. The trapping potential V(R, t_f) obtained at the end of the transport (t = t_f) is shown in panels (c) and (d) of figure 1. At this time the minimum of the potential is located at a distance ${Z}_{f}\approx 1.65\,\mathrm{mm}$ from the chip surface and is again centered in the (XY) plane. The BEC transport takes place over a total distance Z_f − Z_i ≈ 1.2 mm. This distance is much larger than the typical size of the BEC, of a few μm. The comparison of panels figures 1(a) and (c), and of panels figures 1(b) and (d), shows that during the transport the size of the trap along Y and Z decreases a lot while remaining of the same order of magnitude along X. Thus, at t_f the aspect ratio is not as large as initially, and ν_X(t_f) < ν_Y(t_f) ≈ ν_Z(t_f). The tilt of the potential has increased to θ(t_f) = θ_f ≈ 12.5°. In order to calculate the three eigenfrequencies of the rotated trap, one can diagonalize the Hessian matrix associated to the potential. This allows to rotate the coordinate system by the tilt angle θ(t), and to define the new coordinates ${\bf{r}}\equiv (x,y,z)$, with z = Z, associated with the three eigen-axes of the trap at any time t. The rotated axes x and y are shown as black dotted lines in figures 1(b) and (d).

3.1.1. Mean-field approach

In the mean-field regime, the evolution of a BEC in a time-dependent potential V(r, t) can be described by the time-dependent macroscopic condensate wave function Ψ(R, t) solution of the Gross–Pitaevskii equation [35, 36]

$$\begin{eqnarray}&&{\rm{i}}{\rm{\hslash }}\,{\partial }_{t}{\rm{\Psi }}({\bf{R}},t)=\left[-\displaystyle \frac{{{\rm{\hslash }}}^{2}}{2m}{{\rm{\Delta }}}_{{\bf{R}}}+{V}_{a}({\bf{R}},t)+{gN}{| {\rm{\Psi }}({\bf{R}},t)| }^{2}\right]{\rm{\Psi }}({\bf{R}},t),\end{eqnarray} \tag{ 4 }$$

where m denotes the atomic mass and $g=4\pi {{\rm{\hslash }}}^{2}{a}_{s}/m$ is the scattering amplitude. a_s is the s-wave scattering length [46] and N denotes the number of condensed atoms. The nonlinear term ${gN}{| {\rm{\Psi }}({\bf{R}},t)| }^{2}$ describes the mean-field two-body interaction energy [47]. The Gross–Pitaevskii equation is written here in the fixed coordinate system ${\bf{R}}\equiv (X,Y,Z)$, with

$$\begin{eqnarray}{V}_{a}({\bf{R}},t) & = & \displaystyle \frac{1}{2}m\phantom{\rule{}{3.4ex}}[\,{\omega }_{X}^{2}(t)\,{X}^{2}+{\omega }_{Y}^{2}(t)\,{Y}^{2}+2\,{\omega }_{{XY}}(t)\,{XY}\\ & & \left.+{\omega }_{Z}^{2}(t)\,{(Z-{z}_{t})}^{2}\left(1+\displaystyle \frac{2(Z-{z}_{t})}{3{L}_{3}(t)}\right)\,\right],\end{eqnarray} \tag{ 5 }$$

and

$$\begin{eqnarray}&&{\omega }_{X}^{2}(t)={\omega }_{x}^{2}(t){\cos }^{2}[\theta (t)]+{\omega }_{y}^{2}(t){\sin }^{2}[\theta (t)],\end{eqnarray} \tag{ 6a }$$

$$\begin{eqnarray}&&{\omega }_{Y}^{2}(t)={\omega }_{x}^{2}(t){\sin }^{2}[\theta (t)]+{\omega }_{y}^{2}(t){\cos }^{2}[\theta (t)],\end{eqnarray} \tag{ 6b }$$

$$\begin{eqnarray}&&{\omega }_{Z}^{2}(t)={\omega }_{z}^{2}(t),\end{eqnarray} \tag{ 6c }$$

$$\begin{eqnarray}&&{\omega }_{{XY}}(t)=[{\omega }_{x}^{2}(t)-{\omega }_{y}^{2}(t)]\cos [\theta (t)]\sin [\theta (t)].\end{eqnarray} \tag{ 6d }$$

In the present work, the time-dependent Gross–Pitaevskii equation (4) describing the BEC dynamics is solved numerically using the second order split-operator method [48] and the initial ground state is calculated with the imaginary time propagation technique [49]. To efficiently describe the transport over 1.2 mm while resolving the μm scale BEC shape on a numerical grid, we use a co-moving frame [50–52]. This transformation eliminates the center of mass motion and defines a new time-dependent BEC wave function.

$$\begin{eqnarray}&&{\rm{\Phi }}({\bf{R}}^{\prime} ,t)={{\rm{e}}}^{-{\rm{i}}[{{\bf{K}}}_{a}(t)\cdot {\bf{R}}^{\prime} +{\varphi }_{a}(t)]}\,{\rm{\Psi }}({\bf{R}}^{\prime} +{{\bf{R}}}_{a}(t),t),\end{eqnarray} \tag{ 7 }$$

where ${\bf{R}}^{\prime} ={\bf{R}}-{{\bf{R}}}_{a}(t)$ defines the new time-dependent variable grid, translated by the vector R_a(t) compared to a fixed laboratory frame, and where

$$\begin{eqnarray}&&{\rm{\hslash }}{{\bf{K}}}_{a}(t)=m{\dot{{\bf{R}}}}_{a}(t),\end{eqnarray} \tag{ 8a }$$

$$\begin{eqnarray}&&{\rm{\hslash }}{\varphi }_{a}(t)=\displaystyle \frac{m}{2}\,{\int }_{0}^{t}| {\dot{{\bf{R}}}}_{a}(t^{\prime} ){| }^{2}\,{\rm{d}}{t}^{\prime} .\end{eqnarray} \tag{ 8b }$$

Using this transformation, the new time-dependent BEC wave function Φ(R', t) verifies the transformed Gross–Pitaevskii equation

$$\begin{eqnarray}&&{\rm{i}}{\rm{\hslash }}\,{\partial }_{t}{\rm{\Phi }}({\bf{R}}^{\prime} ,t)=\left[-\displaystyle \frac{{{\rm{\hslash }}}^{2}}{2m}{{\rm{\Delta }}}_{{\bf{R}}^{\prime} }+{V}_{a}({\bf{R}}^{\prime} +{{\bf{R}}}_{a},t)+m\,{\ddot{{\bf{R}}}}_{a}\cdot {\bf{R}}^{\prime} +{gN}{| {\rm{\Phi }}({\bf{R}}^{\prime} ,t)| }^{2}\right]\,{\rm{\Phi }}({\bf{R}}^{\prime} ,t),\end{eqnarray} \tag{ 9 }$$

that we solve numerically in a splitting approach which separates the kinetic energy operator from the potential and interaction energy [48]. To limit the size of the numerical grid describing the new coordinate R' = R − R_a(t), a good choice for R_a(t) is to take it as the instantaneous position of the center of mass of the BEC [50–52]. Moreover, it is well known that, if the potential remains (to a good approximation) harmonic at all time, the average position of the condensate

$$\begin{eqnarray}&&{{\bf{R}}}_{a}(t)\equiv ({X}_{a}(t),{Y}_{a}(t),{Z}_{a}(t))=\langle {\rm{\Psi }}({\bf{R}},t)| {\bf{R}}| {\rm{\Psi }}({\bf{R}},t)\rangle \end{eqnarray} \tag{ 10 }$$

follows Newton's classical equation of motion [53]. Since the potential minimum in the X- and Y-directions remains at the origin, ${X}_{a}(t)={Y}_{a}(t)=0={x}_{a}(t)={y}_{a}(t)$ applies during the entire propagation. As a consequence the following relations X' = X, $Y^{\prime} =Y$ and $Z^{\prime} =Z-{Z}_{a}(t)$ also hold. Note finally that, along z, in the harmonic approximation and in the absence of gravity, the center of mass position Z_a(t) = z_a(t) simply follows

$$\begin{eqnarray}&&{\ddot{z}}_{a}(t)+{\omega }_{z}^{2}(t)({z}_{a}(t)-{z}_{t})=0.\end{eqnarray} \tag{ 11 }$$

In practice, to take into account the eventual influence of anharmonicities we consider that the classical transport trajectory z_a(t) is a solution of the anharmonic equation

$$\begin{eqnarray}&&{\ddot{z}}_{a}(t)+{\omega }_{z}^{2}(t)({z}_{a}(t)-{z}_{t})\left(1+\displaystyle \frac{{z}_{a}(t)-{z}_{t}}{{L}_{3}(t)}\right)=0,\end{eqnarray} \tag{ 12 }$$

in agreement with equations (3) and (5).

As a conclusion, the numerical procedure described in the present section allows to solve the time-dependent Gross–Pitaevskii equation (9). It is possible, if necessary, to obtain the solution ${\rm{\Psi }}({\bf{R}},t)$ of the Gross–Pitaevskii equation (4) in the fixed frame of reference at any time t by simply inverting the relation (7).

3.1.2. Scaling laws

The introduction of the frame transformation (7) in our numerical approach dramatically improves the computational efficiency of the procedure. In three-dimensions, the procedure can remain computationally expensive, depending on the exact propagation time t_f and on the evolution of the trap parameters (position and frequencies). We therefore introduce an alternative, approximate approach, that we compare, in section 4, with the solution of the Gross–Pitaevskii equation.

In the Thomas–Fermi regime of large atom numbers [47] and within the harmonic approximation, one can calculate the evolution of the size of the BEC in the rotating frame by solving three coupled differential equations [54, 55]

$$\begin{eqnarray}&&{\ddot{\lambda }}_{\alpha }(t)+{\omega }_{\alpha }^{2}(t){\lambda }_{\alpha }(t)=\displaystyle \frac{{\omega }_{\alpha }^{2}(0)}{{\lambda }_{\alpha }(t)\,{\lambda }_{x}(t){\lambda }_{y}(t){\lambda }_{z}(t)},\end{eqnarray} \tag{ 13 }$$

for α = x, y and z. The scaling factors λ_α(t) describe the evolution of the size of the BEC in the three directions, providing that the initial conditions verify λ_α(0) = 1 and ${\dot{\lambda }}_{\alpha }(0)=0$. These 'scaling laws' assume that the BEC keeps its Thomas–Fermi parabolic shape at all time and that the condensate follows adiabatically the rotation of the trap in the (XY) plane.

The initial size of the BEC in the rotating frame is then defined by [47]

$$\begin{eqnarray}&&{R}_{\alpha }^{\mathrm{TF}}(0)={a}_{{\rm{osc}}}\,{\left(\displaystyle \frac{15{{Na}}_{s}}{{a}_{{\rm{osc}}}}\right)}^{1/5}\,\displaystyle \frac{\bar{\omega }(0)}{{\omega }_{\alpha }(0)},\end{eqnarray} \tag{ 14 }$$

where $\bar{\omega }(0)={[{\omega }_{x}(0){\omega }_{y}(0){\omega }_{z}(0)]}^{1/3}$ is the geometric mean of the three oscillator frequencies and ${a}_{{\rm{osc}}}={[{\rm{\hslash }}/m\bar{\omega }(0)]}^{1/2}$ is the characteristic quantum mechanical length scale of the 3D harmonic oscillator. The characteristic size of the BEC along the three principal axes of the trap α = x, y and z is then given at any time t by the relation

$$\begin{eqnarray}&&{R}_{\alpha }^{\mathrm{TF}}(t)={\lambda }_{\alpha }(t)\,{R}_{\alpha }^{\mathrm{TF}}(0).\end{eqnarray} \tag{ 15 }$$

Knowing the parabolic shape of the wave function, these three typical sizes ${R}_{\alpha }^{\mathrm{TF}}(t)$ can be related to the three widths Δα(t) (i.e. the standard deviations in the directions α = x, y and z) of the BEC wave function, using

$$\begin{eqnarray}&&{\rm{\Delta }}\alpha (t)={R}_{\alpha }^{\mathrm{TF}}(t)/\sqrt{7}.\end{eqnarray} \tag{ 16 }$$

Numerically, we also evaluate the three widths ${\rm{\Delta }}x(t)$, Δ y(t) and Δ z(t) in the rotating frame from the solution Ψ(R, t) of the time-dependent Gross–Pitaevskii equation in the laboratory frame using

$$\begin{eqnarray}&&{\rm{\Delta }}x(t)={[{({\rm{\Delta }}X)}^{2}{\cos }^{2}\theta +2({\rm{\Delta }}{XY})\cos \theta \sin \theta +{({\rm{\Delta }}Y)}^{2}{\sin }^{2}\theta ]}^{1/2},\end{eqnarray} \tag{ 17a }$$

$$\begin{eqnarray}&&{\rm{\Delta }}y(t)={[{({\rm{\Delta }}X)}^{2}{\sin }^{2}\theta -2({\rm{\Delta }}{XY})\cos \theta \sin \theta +{({\rm{\Delta }}Y)}^{2}{\cos }^{2}\theta ]}^{1/2},\end{eqnarray} \tag{ 17b }$$

$$\begin{eqnarray}&&{\rm{\Delta }}z(t)={\rm{\Delta }}Z,\end{eqnarray} \tag{ 17c }$$

where

$$\begin{eqnarray}&&{\rm{\Delta }}X(t)={[\langle {\rm{\Psi }}| {X}^{2}| {\rm{\Psi }}\rangle -{\langle {\rm{\Psi }}| X| {\rm{\Psi }}\rangle }^{2}]}^{1/2},\end{eqnarray} \tag{ 18a }$$

$$\begin{eqnarray}&&{\rm{\Delta }}Y(t)={[\langle {\rm{\Psi }}| {Y}^{2}| {\rm{\Psi }}\rangle -{\langle {\rm{\Psi }}| Y| {\rm{\Psi }}\rangle }^{2}]}^{1/2},\end{eqnarray} \tag{ 18b }$$

$$\begin{eqnarray}&&{\rm{\Delta }}Z(t)={[\langle {\rm{\Psi }}| {Z}^{2}| {\rm{\Psi }}\rangle -{\langle {\rm{\Psi }}| Z| {\rm{\Psi }}\rangle }^{2}]}^{1/2},\end{eqnarray} \tag{ 18c }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{XY}(t)=\langle {\rm{\Psi }}| {XY}| {\rm{\Psi }}\rangle -\langle {\rm{\Psi }}| X| {\rm{\Psi }}\rangle \langle {\rm{\Psi }}| Y| {\rm{\Psi }}\rangle .\end{eqnarray} \tag{ 18d }$$

3.1.3. Collective excitation modes

We use the mean-field equation (4) together with the scaling approach (13) to describe the characteristic size excitations of the BEC which arise in the final trap at the end of the transport protocol, due to the fast anisotropic trap decompression over the transport. These excitations can be described as a sum of different collective modes with different amplitudes [54–61].

The first low-lying collective excitation modes of a BEC in a cigar shape potential are well known [56]. They can be easily described if we approximate the atom chip trapping potential at time t_f by

$$\begin{eqnarray}&&V({\bf{r}},{t}_{f})\approx \displaystyle \frac{1}{2}m{\omega }_{\perp }^{2}({\eta }^{2}{x}^{2}+{r}_{\perp }^{2}),\end{eqnarray} \tag{ 19 }$$

where ${r}_{\perp }=\sqrt{{y}^{2}+{z}^{2}}$ and ${\omega }_{\perp }={\omega }_{y}({t}_{f})\approx {\omega }_{z}({t}_{f})$. The trap aspect ratio is denoted here by $\eta ={\omega }_{x}({t}_{f})/{\omega }_{\perp }$. For a low degree of excitation, these modes form a basis of six possible excitations, as depicted schematically in figure 3.

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These modes are associated with specific, characteristic frequencies

$$\begin{eqnarray}&&{\omega }_{{D}_{\perp }}={\omega }_{\perp },\end{eqnarray} \tag{ 20a }$$

$$\begin{eqnarray}&&{\omega }_{{D}_{x}}=\eta \,{\omega }_{\perp }={\omega }_{x}({t}_{f}),\end{eqnarray} \tag{ 20b }$$

$$\begin{eqnarray}&&{\omega }_{Q1}={[2+3{\eta }^{2}/2+\delta /2]}^{1/2}\,{\omega }_{\perp },\end{eqnarray} \tag{ 20c }$$

$$\begin{eqnarray}&&{\omega }_{Q2}=\sqrt{2}\,{\omega }_{\perp },\end{eqnarray} \tag{ 20d }$$

$$\begin{eqnarray}&&{\omega }_{{{Sc}}_{{xy}}}={[1+{\eta }^{2}]}^{1/2}\,{\omega }_{\perp },\end{eqnarray} \tag{ 20e }$$

$$\begin{eqnarray}&&{\omega }_{M}={[2+3{\eta }^{2}/2-\delta /2]}^{1/2}\,{\omega }_{\perp },\end{eqnarray} \tag{ 20f }$$

where $\delta ={[9{\eta }^{4}-16{\eta }^{2}+16]}^{1/2}$. The dipole modes ${D}_{\perp }$ and D_x show a classical oscillation of the center of mass of the condensate at the trap frequencies ${\omega }_{{D}_{\perp }}={\omega }_{\perp }$ and ${\omega }_{{D}_{x}}=\eta \,{\omega }_{\perp }={\omega }_{x}({t}_{f})$, respectively. The first quadrupole mode Q₁ shows a simultaneous expansion of the two strong axes, while the weak axis is compressed. In the second quadrupole mode, the weak axis does not oscillate and the size oscillations are only present along the two strong axes. The scissors mode Sc_xy shows the effect of the trap rotation about the direction of transport, and the monopole mode M, also called breathing mode, shows an alternating compression and expansion of the condensate in the three directions in phase.

We present here the method of reverse engineering, used to find a perturbation-free transport of the center of mass of the BEC [62, 63] within a STA approach. This reverse engineering protocol works as follows: we set the classical trajectory of the atoms, z_a(t), according to fixed boundary conditions, which have to be fulfilled experimentally to ensure an optimal transport, i.e. initially and finally the center of mass has to be at rest, at the position of the minimum of the trap. This leads to the following boundary conditions

$$\begin{eqnarray}&&{z}_{a}(0)={z}_{i};\quad {\dot{z}}_{a}(0)=0;\quad {\ddot{z}}_{a}(0)=0\end{eqnarray} \tag{ 21a }$$

and

$$\begin{eqnarray}&&{z}_{a}({t}_{f})={z}_{f};\quad {\dot{z}}_{a}({t}_{f})=0;\quad {\ddot{z}}_{a}({t}_{f})=0,\end{eqnarray} \tag{ 21b }$$

where z_i and z_f denote the initial and final positions, respectively. To account for experimental constraints, we also wish the trap-to be at rest initially and finally. We therefore impose

$$\begin{eqnarray}&&{z}_{t}(0)={z}_{i};\quad {\dot{z}}_{t}(0)=0;\quad {\ddot{z}}_{t}(0)=0,\end{eqnarray} \tag{ 22a }$$

and

$$\begin{eqnarray}&&{z}_{t}({t}_{f})={z}_{f};\quad {\dot{z}}_{t}({t}_{f})=0;\quad {\ddot{z}}_{t}({t}_{f})=0.\end{eqnarray} \tag{ 22b }$$

The conditions on the second derivatives of the positions are imposed to enforce smooth magnetic field changes. Inserting these last six constraints in Newton's equations (11) or (12) shows that they are equivalent to the additional four constraints

$$\begin{eqnarray}&&{z}_{a}^{(3)}(0)=0;\quad {z}_{a}^{(4)}(0)=0\end{eqnarray} \tag{ 23a }$$

and

$$\begin{eqnarray}&&{z}_{a}^{(3)}({t}_{f})=0;\quad {z}_{a}^{(4)}({t}_{f})=0,\end{eqnarray} \tag{ 23b }$$

where the exponent (n) denotes the nth time derivative. These four extra boundary conditions (23a), (23b) can be seen as additional robustness constraints against oscillations of the center of mass of the BEC in the final trap. The simplest polynomial solution to the ten boundary conditions can be fulfilled with the polynomial function of order nine

$$\begin{eqnarray}&&{z}_{a}(t)={z}_{i}+({z}_{f}-{z}_{i})\,[126\,{u}^{5}-420\,{u}^{6}+540\,{u}^{7}-315\,{u}^{8}+70\,{u}^{9}],\end{eqnarray} \tag{ 24 }$$

where u = u(t) denotes the rescaled time t/t_f. The second derivative of the polynomial function (24) presents a sine-like variation due to the presence of an acceleration stage followed by a deceleration step. This suggests a non-trivial Ansatz for z_a(t) in the form

$$\begin{eqnarray}&&{z}_{a}(t)={z}_{i}+\left(\displaystyle \frac{{z}_{f}-{z}_{i}}{12\pi }\right)[6v-8\sin (v)+\sin (2v)],\end{eqnarray} \tag{ 25 }$$

whose second derivative presents a similar sine-like shape, and where

$$\begin{eqnarray}&&v=v(t)=2\pi \left(\displaystyle \frac{1+a\,u+b\,{u}^{2}}{1+a+b}\right)\displaystyle \frac{t}{{t}_{f}},\end{eqnarray} \tag{ 26 }$$

is a 'chirped' function of time. The constants a and b act here as two additional control parameters, making this solution more powerful than the simple polynomial one. These parameters can be optimized to limit the impact of the anharmonic term in equation (3) in order to recover a BEC at rest after the transport. Note that, according to equation (12), to limit the anharmonic effects, one has to fulfill the following criterium

$$\begin{eqnarray}&&\chi (t)=\left|\displaystyle \frac{{z}_{a}(t)-{z}_{t}}{{L}_{3}(t)}\right|\ll 1,\quad \forall t.\end{eqnarray} \tag{ 27 }$$

The elaborate form (25) of z_a(t) is used in section 4 with a = −1.37 and b = 0.780. With such parameters the maximum value reached by χ(t) during the transport is 0.03 while it reaches 0.09 without any chirp (i.e. for a = b = 0). Once z_a(t) is defined, one can extract the magnetic bias field B_bias(t) from equation (11) since the trap parameters ω_z(t) and z_t are related to B_bias(t) unambiguously. The technical procedure used to extract these parameters, and thus B_bias(t), is described in appendix A.

In figure 4(a), the atomic cloud position is shown during and after the implementation of the STA protocol in the cases of a chip trap assumed to be harmonic (thin solid blue line) and more realistically including the cubic term of equation (3) (dashed red line). In both cases, the classical solution of Newton's equation is indistinguishable from the average position of the wave packet solution of the Gross–Pitaevskii equation. The position of the atomic cloud during the transport is plotted in the left part of the figure. The dashed vertical line signals the end of the displacement and the beginning of a holding period in the final trap. The upper panel (a) of the graph shows the appropriateness of the transport ramp to guide the atoms over more than 1.2 mm with no noticeable residual center of mass oscillations. To be more convinced that the STA ramp works out thanks to the careful optimization described in the previous section and not because the transport time is long enough to approach the adiabatic limit, we also plot the classical solution for the same displacement time but with a linear ramp (green solid line). The contrast with the optimized solutions is clear, with large residual oscillations with an amplitude of the order of 100 μm after t_f = 75 ms. This clearly shows that the chosen ramp duration is far from the adiabatic time scale, which would trivially bring the atoms at rest in the final trap.

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The STA ramp devised in this case allows for the position of the atomic cloud to deviate from the trap position during the transport. This becomes visible in figure 4(b), which shows the offset [z_a(t) − z_t] between the positions of the BEC and the time-dependent trap. In this graph, the Gross–Pitaevskii solutions are indicated at chosen times by empty black squares (harmonic potential) and by plain red circles (cubic term included). For the chosen ramp time, the maximum offset is about 14 μm. This relatively large offset is responsible for limiting the quality of the transport, as quantified by the amplitude of residual oscillations in the anharmonic case (dashed red line and circles). Indeed, the harmonic solution found for the BEC trajectory by solving equation (11) becomes less appropriate the more the atoms explore trap anharmonicities which show up when leaving the trap center. This effect is clearly noticed when comparing the holding trap oscillations in figure 4(b) between the harmonic case (no visible residual oscillations) and the one with a cubic term, which shows an oscillation amplitude of about 0.7 μm around the trap center. It is interesting to note that the chirp introduced in equation (26) drastically reduces the residual oscillations of the BEC. Indeed, the oscillation amplitude would reach approximately 6 μm with a = b = 0. Similarly, with no chirp, the same oscillation amplitude of 0.7 μm would require a ramp time t_f > 300 ms. The quantum mechanical solution found by computing the average position of the BEC wave function rigorously lies on the Newtonian trajectory in both cases. This was predictable in the harmonic case since it is a consequence of Ehrenfest's theorem applied to our problem. One can also notice that the quantum mechanical solutions are following here the Newtonian trajectory in the anharmonic case. This is because the anharmonic effect is small for such a transport time of 75 ms.

In order to quantitatively assess the magnitude of the anharmonic term during the transport, we plot in figure 5(a) the maximum offset to trap center reached by the BEC as a function of the ramp time t_f. The solid curve is again corresponding to the Newton's equation solution and the red dots are depicting the Gross–Pitaevskii solution. As expected, short ramps lead to atomic positions departing further from the trap center in both the classical and quantum case since the adiabaticity criterion is less respected. The larger this spatial offset is, the higher the magnitude of the cubic term in equation (3) is, the worse the harmonic trap-based reverse engineering for the chip trap trajectory is, and the larger the final residual oscillations are. This is perfectly visible when analyzing figure 5(b) giving the magnitude of the cubic term (in percent) relative to the one of the harmonic term. As a consequence, the residual oscillation amplitudes shown in figure 5(c) are larger for shorter ramp times, as expected. In all cases, the quantum solution is in a good agreement with the classical one, leading to the conclusion that regarding the position of the wave packets, BECs can here be safely treated as classical point-like particles. As a result, knowing the maximum oscillation amplitude tolerated in an experiment, one can implement our treatment to find the fastest transport ramp.

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To assess the practical feasibility of the proposed fast BEC transport, it is necessary to estimate the impact of small experimental imperfections. The present robustness study, therefore, characterizes the residual oscillation amplitude induced by ramp timing errors, denoted here by δ t_f, and offsets δ B_bias in the time-dependent magnetic bias field applied to drive the chip trap.

Considering the more complete case where cubic potential terms are present, we use Newton's equations (12), where ω_z, z_t and L₃ are implicit functions of B_bias(t). The average position of the condensate can be written as

$$\begin{eqnarray}&&{z}_{a}(t)={z}_{a}^{0}(t)+{\epsilon }_{z}(t),\end{eqnarray} \tag{ 28 }$$

where ${z}_{a}^{0}$(t) denotes the unperturbed trajectory. A lowest order perturbative treatment applied to Newton's equation (12) yields

$$\begin{eqnarray}&&\ddot{{\epsilon }_{z}}+{\omega }_{z}^{2}(t)({\epsilon }_{z}-\delta {z}_{t})+\delta {\omega }_{z}^{2}({z}_{a}^{0}-{z}_{t})+\displaystyle \frac{{\omega }_{z}^{2}(t)}{{L}_{3}(t)}{({z}_{a}^{0}-{z}_{t})}^{2}=0,\end{eqnarray} \tag{ 29 }$$

where $\delta {z}_{t}$ and δω_z denote first order perturbations to the trap position and to the trap frequency, respectively. In the following, we solve equation (29) for the harmonic (i.e. ${L}_{3}\to \infty$) and anharmonic trapping cases.

Figure 6 shows the residual oscillation amplitude as a function of the perturbations δ B_bias in panel (a) and δ t_f in panel (b). This figure confirms the robustness of our transport method. Indeed, δ B_bias = 1 mG of control error in the bias field only leads to an offset of about 0.5 μm in the final position of the BEC. Moreover, the same order of infidelity in the final position of the BEC requires ramp timing errors better than 1 ms, a limit which is easily matched experimentally. Both limits are therefore considered to be safely within state-of-the-art capabilities of standard cold-atom laboratories.

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In this section, the time-dependent spatial density distribution of the transported BEC is considered. By applying a similar treatment as reported in [62, 63], it is possible to suppress residual holding trap oscillations of the wave packet center as well as size excitations after the transport. This is achieved, however, at the cost of a long transport time. In this article, we would like to highlight the potential of atom chip-generated STA protocols in the metrological context, i.e. with fast enough transport to allow for short duty cycles.

The gallery in figure 7 shows typical BEC wave packet size oscillations occurring during and after the transport ramp considered in the last section. An adiabatic or long enough transport would bring the BEC to its ground state in the final holding trap, reflected in trivial flat lines starting at 75 ms for the three sizes of the left panel. We observe instead a breathing of the wave packet in the three space directions with the largest amplitude occurring in the weak frequency axis x. Although the transport is performed in our simulations solely in the z-direction, we clearly witness a size oscillation of the atomic wave packet in the two other directions due to the mean-field interactions connecting all spatial directions.

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The left panel of figure 7 illustrates the results of simulations based on the scaling approach (harmonic approximation, solid blue curve) on one hand and on a numerical solution of the Gross–Pitaevskii equation in the harmonic case (black empty squares) on the other. An introduction of the anharmonicities in the Gross–Pitaevskii equation yields the solid red circles and the dashed green line is the most complete case including both, anharmonicities and trap rotation during the transport.

Qualitatively, the four configurations show a similar behavior. The numerical results being similar with and without the cubic term suggests that our trade-off ramp time versus anharmonicities magnitude, previously made for the atomic cloud center, is conclusive regarding BEC size dynamics as well. This is one of the main results of this study since it demonstrates the benign effect of anharmonicities in typical atom chip traps even with fast STA non-adiabatic transports.

The right panel of figure 7 is a density plot complementing the left part with the density probability distribution during the transport and for 150 ms of holding time. The quasi-cylindrical symmetry of the trap is reflected in the collective excitation modes observed. Indeed, the strongly trapped directions y and z are subject to in phase size oscillations. The size along the weak axis x is subject to larger-amplitude size oscillations since the trapping frequency is weak along this axis. The excited modes responsible for these oscillations will be identified by the quantitative study of next section.

4.4.1. Collective excitations in the holding trap

To gain insight into the impact of the transport speed on the collective excitation of the BEC in the final trap, we plot in figures 8(a) and (c) the extracted BEC size oscillations resulting from the ramp of equation (25) for a total transport time of 75 ms and 750 ms, respectively. In order to compare to analytical results, we consider a cylindrical symmetry suggested by figure 2(b) where ν_y is chosen to be strictly equal to ν_z. We plot the sizes normalized to the ones at the end of the transport in the directions x (solid blue line) and y or z (dashed red line) as a function of the holding time t_hold = (t − t_f) in the final trap.

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In both cases, the final holding time is chosen to be 500 ms and one easily notices the complex shape of the residual size oscillations for the fast ramp compared to the slower one, where a simple periodic evolution of the size of the BEC is obtained. This difference occurs due to the rapid variation of the trap aspect ratio in the fast ramp. Indeed, in the transport of figure 8(a), the aspect ratio (ω_x/ω_⊥) varies by one order of magnitude in 10 ms only, when a similar variation happens in 100 ms for the slow transport of figure 8(c).

In figures 8(b) and (d) we plot the Fourier transforms of the Thomas–Fermi radius in Log scale, as a function of the oscillation mode frequency ν for the two cases mentioned previously of t_f = 75 ms and 750 ms. These graphs reveal the main collective modes and their harmonics present in the holding trap after the end of the transport. The vertical dashed lines in these plots denote the analytically found collective excitation frequencies according to the treatment described in section 3 and reported in [56]. This treatment is an approximation in the case of small perturbations. It is clearly not valid for the faster transport reported here. It is, nevertheless, a useful indicator to identify the excitation modes presenting the largest magnitude.

The slow ramp is characterized by the presence of a single quadrupole mode Q₁ explaining the simple periodicity of the size oscillation behavior, with the two strong axes in phase and the weak axis out of phase. Note that the oscillation magnitude is, in this case, quite negligible, the size departing only by about 1% from the one at the end of the transport. The fast transport ramp is, however, exciting several collective modes explaining the more complex size oscillation periodicity and the larger-amplitude variation to about ±70% change relative to the final transport size in the weak axis x.

This analysis is useful on many levels. The predominance of the quadrupole Q₁ mode suggests, for example, the optimization we discuss in the next section. By taking advantage of the symmetry of certain modes, one can also imagine, in further studies, designing a transport protocol forbidding or enhancing them.

4.4.2. Optimization of the expansion dynamics

The designed quantum states studied in this article would serve as an input of a precision atom interferometry experiment [64]. In such measurements, it is beneficial to work with the slowest cloud expansion possible since this increases the maximum interferometry time available, with an impact on the density threshold for detection and hence, on the sensitivity of the atomic sensor [65]. Moreover, long free evolution times of several seconds are beneficial for micro-gravity [39, 66–70] and atomic fountain experiments [4, 6, 71]. To largely reduce the expansion rate of the atomic samples, the DKC technique [37–40] is commonly applied. It consists in re-trapping an expanding cloud of atoms for a brief duration in order to align its phase-space density distribution along the position coordinate axis, therefore minimizing its momentum distribution width in preparation for a further expansion. This is in analogy with the collimating effect of a lens in optics and DKC is often referred to as an atomic lens. It is worth noticing that the phase-space density is conserved in such a process which does therefore not achieve a cooling in the sense of reducing the phase-space density. This method was successfully implemented and led to record long observation times of several seconds [40, 72].

If the trap is anisotropic, as the quasi-cylindrical case considered in this paper, the lensing effect is different in every direction and would typically be negligible in the weak frequency axis when the two others are well collimated. To overcome this problem, we take advantage of the collective excitations described in the preceding section to release the BEC at a well-defined time, soon after a maximum size amplitude of the weak trapping direction such that the subsequent expansion velocity is naturally reduced. This timing is chosen such that the kinetic energy associated with the natural re-compression of the cloud is quickly balanced by the repulsive mean-field interaction energy which naturally leads to an expansion of the cloud in this direction.

To illustrate this optimization, we consider in this section the case of a transport from ${z}_{i}\simeq 0.45\,\mathrm{mm}$ to ${z}_{f}\simeq 1.35\,$ mm in 75 ms. This transport is realized with the chip DC current I_w = 5 A and a bias magnetic field which varies between B_bias(0) = 21.6 G (initially) and B_bias(t_f) = 5.9 G (at the end of the displacement). The final trap is characterized by the frequencies ν_x = 12.5 Hz, ν_y = 50 Hz and ν_z = 49.5 Hz. This is realized following the reverse engineering technique described in section 3. The final trap is used to hold the atoms after the end of the transport. The result of this optimization is shown in figure 9(a) where the blue curves show the variation of the size of the released BEC in the x-direction for three different holding times of 29.4, 31.4 and 33.4 ms. A natural choice is to consider the switch-off time of 31.4 ms leading to a collimated subsequent free expansion. Indeed, a holding time slightly below leads to an immediate fast increase of the condensate size (see the dashed–dotted blue line in figure 9(a)) while a holding time slightly above leads to a transient compression of the BEC (see the dashed blue line in figure 9(a)) soon followed by a very fast expansion.

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Following the intermediate and optimal choice ${t}_{{\rm{hold}}}=31.4$ ms, after 100 ms of free expansion the mean-field interaction energy is almost entirely in the form of kinetic energy and an atomic lens (DKC pulse) can be applied. It is realized by switching-on a cylindrical trap of frequencies ν_x = 1.7 Hz and ν_y = ν_z = 7.2 Hz for Δ t_lens = 4.84 ms, created with a DC current of intensity I_w = 0.1 A and a magnetic bias field of B_bias = 0.12 G, leaving the trap minimum at z_f = 1.35 mm. The collimation effect is dramatic in the y and z-directions (red dashed line and green dashed–dotted line in figure 9(b)). The expansion observed after the application of the DKC pulse corresponds to an average speed in the three spatial directions of about 25.3 μm s⁻¹, equivalent to a temperature of 2.2 pK (see appendix B for details). Figure 9(c) finally shows the robustness of the procedure in case of timing errors for the holding time t_hold and for the lens duration Δ t_lens. With timing errors as large as 0.5 ms the expansion temperature remains below 20 pK. This demonstrates the marginal influence of relatively large timing errors for the 3D collimation effect proposed here.

To illustrate the importance of taking advantage of the collective oscillations, we plot in the same figure the virtual case of an adiabatic transport in the x-direction (thin dotted blue line). If one applies a mere adiabatic decompression as suggested by this latter curve, the expansion temperature would be much larger, higher than 550 pK, even if we consider very well collimated y and z-directions, the x-direction being hardly affected by the magnetic lens. It is therefore crucial to control the release timing of the BEC in order to implement low-velocity expansions.