An ultra-bright atom laser

Atom laser beams can be generated in a number of different ways: pulsed atom lasers have been produced from magnetically trapped BECs by short blasts of rf-radiation [3] ⁷ . Atom lasers based on spilling atoms from a purely optical [7] and from magnetically compensated optical potentials [24] have been demonstrated for small laser fluxes. Bright quasi-continuous atom lasers are usually outcoupled from magnetically trapped BECs using weak electromagnetic fields, which couple the Zeeman levels of the atoms and thus the trapped states to propagating ones. The transition can either be direct (rf output coupler) [6] or via a virtual state (Raman output coupler) [5]. Even though this process is irreversible [25], it is coherent in the sense that the atoms in the beam retain a well-determined phase relationship the atoms in the condensate and with the rf-field [26]. After the atoms have been transferred to the propagating states they accelerate out of the condensate and form the atom laser beam. This acceleration is due to a combination of gravity, magnetic forces, and interaction with the atoms remaining in the condensate.

For very weak fields, the flux of the atom laser increases proportionally with the intensity of the rf-field [27]. For stronger coupling fields, however, a bound state appears, which eventually shuts off the lasers output and therefore severely limits the maximal atom flux that can be achieved from a given condensate [28, 29].

The beam quality of the weakly coupled atom laser is affected by the geometry of the outcoupling region. Since the atoms are outcoupled only where the rf-frequency is resonant with the Zeeman splitting due to the local magnetic field, this will usually occur somewhere inside the condensate. In this case the chemical potential of the remaining condensate will act as as a matter-wave lens, which can lead to sever distortions of the atom laser beam [30].

Here we present a novel type of continuous atom laser, which is based on time-dependent adiabatic potentials (TDAP). Rather than using a weak rf-field to slowly outcouple a small fraction of the condensate, we use strong rf-fields to deform the trapping potential such that the condensate spills slowly out of a small opening at the bottom of the trap. As opposed to the atom laser based on the weak-field output coupler, the TDAP can emit atoms from the condensate at an almost arbitrary rate. This allowed us to generate atom lasers with much larger flux than can be achieved in the weak coupling regime. The TDAP also produces the coldest thermal atom beams to date. Examples can be seen in figure 1: the central panel shows an extremely bright atom laser, the left panel a highly collimated atom laser, and the right image for the first time a combined thermal and atom laser beam.

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Due to the dynamic nature of the TDAP together with the non-trivial shape of the outcoupling region, the exact form of the outcoupled atom beam is difficult to predict analytically. However, we can distinguish different regimes of output coupling by comparing the trap frequencies $({{\omega }_{\text{z}}},{{\omega }_{\rho }})$ to the output coupling rate of the condensate $({{\Omega }_{\text{oc}}})$, which we define as the inverse of the time that it takes to outcouple the entire condensate. In the slow outcoupling regime, where the outcoupling is slow compared to any other time scale $({{\Omega }_{\text{oc}}}{{\omega }_{\text{z}}}{{\omega }_{\rho }})$, the shape of the condensate adapts adiabatically to the reducing atom number [34]. The output coupling will then only occur in a very small region below the centre of the condensate. At somewhat larger outcoupling rates $({{\Omega }_{\text{oc}}}\simeq {{\omega }_{\text{z}}}{{\omega }_{\rho }})$ the outcoupling region grows larger and shape oscillations are excited in the condensate, which affects the intensity and the shape of the atom laser beam. In the intermediate regime, where the output coupling rate is large compared to the axial frequency but is still small compared to the radial trapping frequency $({{\omega }_{\text{z}}}{{\Omega }_{\text{oc}}}{{\omega }_{\rho }})$, the shape of the condensate remains essentially frozen in the axial direction [21, 35] and the output coupling will occur below the whole length of the cigar-shaped condensate. Finally, in the fast outcoupling regime, when the sweep of the rf-frequency is fast compared to the radial trapping frequency $({{\omega }_{\text{z}}}{{\omega }_{\rho }}{{\Omega }_{\text{oc}}})$, the atoms will form a single accelerating shell-shaped matter-wave pulse like the one seen in [3].

A number of factors lead to the atom laser being very well-collimated both in the slow $({{\Omega }_{\text{oc}}}{{\omega }_{\text{z}}}{{\omega }_{\rho }})$ and intermediate $({{\omega }_{\text{z}}}{{\Omega }_{\text{oc}}}{{\omega }_{\rho }})$ outcoupling regimes: the absence of matter-wave lensing, the smooth shape of the potential, the absence of shape oscillations, and the vertical acceleration of the atoms after they leave the trap: matter-wave lensing occurs, when a weak rf outcouples the atom laser beam somewhere within the condensate. The chemical potential of the remaining condensate then acts as a matter-wave lens, which distorts the atom laser beam. In the case of the TDAP atom laser, matter-wave lensing is completely absent because the atoms are outcoupled from the very edge of the condensate where the chemical potential is negligible. Furthermore, the shape of the confining potential at the outcoupling point tends to be very smooth in the direction transverse to the atom beam and the transverse gradient and curvature of the adiabatic potential become vanishingly small for very low output coupling rates $({{\Omega }_{\text{oc}}}{{\omega }_{\text{z}}}{{\omega }_{\rho }})$. Shape oscillations of the condensates are absent because in the slow and the intermediate outcoupling regimes the internal dynamics of the condensate are negligible. Finally, since the outcoupled atoms are in the anti-trapped magnetic hyperfine state they are strongly accelerated downwards, thus minimizing the influence of the transverse momentum component.

Our numerical simulations of the time-dependent Schrödinger equation confirm that the TDAP atom laser can indeed produce well-collimated quasi-continuous atom laser beams (see appendix A.5).

For many applications it is desirable for the atom laser beam to have a low transverse and a large longitudinal velocity. Examples include collision physics [36, 37], surface science [38, 39] and atom nano-lithography [40]. The acceleration of the TDAP atom laser due to the atoms being in the anti-trapped state $({{m}_{\text{F}}}=-2)$ provides these high velocities, reaching $2\;\text{m}\;{{\text{s}}^{-1}}$ after only 1 cm of travel.

For other applications a slower beam might be more useful. In this case it is sufficient to introduce a second rf-field at a frequency larger than the first one. This transfers the atoms from the ${{m}_{\text{F}}}=-2$ back to the ${{m}_{\text{F}}}=+2$ state. The atom laser beam then decelerates and eventually comes to a standstill thus providing access to a very large range of longitudinal velocities [41].

Absorption images of the atom laser in the y–z plane are taken by shining resonant laser light along the x-axes, image it onto a CCD camera, and calculate the atom column density for each pixel. We then cut the images into horizontal slices, integrate each slice in the vertical direction, and fit them with the following binomial profile:

$$\begin{eqnarray}&&f=a+b\;{{e}^{-{{\left( \frac{x-{{x}_{0}}}{\Delta {{x}_{\text{t}}}} \right)}^{2}}}}+c\text{Re}{{\left[ 1-{{\left( \frac{\text{x}-{{\text{x}}_{0}}}{\Delta {{\text{x}}_{\text{c}}}} \right)}^{2}} \right]}^{3/2}}.\end{eqnarray} \tag{ 2 }$$

The first term accounts for a technical, global offset. The second term represents the Gaussian momentum distribution of a thermal contribution. The third term has been chosen ad hoc for the apparent absence of wings in the experimental images of pure atom lasers. It implies an inverted-parabola shape of the transverse density profile, which is integrated in the direction of the imaging beam. Examples of such fits can be found in the online supplementary material (available at stacks.iop.org/njp/16/033036/mmedia).

From the fits to the individual integrated atom slices we can then determine the atom number and sizes of both the coherent and thermal components. From the position of a slice we can calculate the velocity and output coupling time of the atoms in it and subsequently the divergence, temperature, and local flux of the atom beam. If no reliable fit can be obtained for the bimodal fit, we force either $b=0$ or $c=0$ in order to obtain a fit for only an atom laser or only a thermal beam respectively.

The creation of the TDAP atom laser is rather straight forward: a single linear sweep of the rf-frequency cools a thermal cloud of magnetically trapped atoms into a BEC and then outcouples the atom laser from the trap.

Our experimental apparatus has been described elsewhere [42, 43]. We load ${{10}^{9}}$ atoms into a magnetic Ioffe–Pritchard trap. Within one second we compress this trap to its final state by ramping up the currents in the Ioffe and pinch coils whilst simultaneously reducing ${{B}_{0}}$ to 0.5–1 G. The final trapping frequencies are then 17 Hz in the axial and 561–793 Hz in the radial direction. Then we lower the rf-frequency down to its final value in a linear ramp of 10 s duration. The sample cools by forced evaporation until the critical temperature is reached and a BEC of about $1.5\times {{10}^{5}}$ atoms forms. A thermal atom beam emerges from the bottom of the trap. As the rf-frequency drops even lower, the trap depth becomes smaller than the chemical potential of the condensate and an atom laser is outcoupled from the lower edge of the condensate (figure 3(b)) until finally the trap vanishes (figure 3(c)). We then switch off the magnetic trap, let the atoms evolve in free flight for 1–10 ms, and take a resonant absorption picture of y–z plane.

Note that the requirements on the reproducibility of ${{B}_{0}}$ are relatively low: whereas the traditional atom laser needs to aim the weak rf exactly to the bottom of the BEC, the TDAP atom laser only needs to sweep across it. In our case the magnetic field reproducibility is of the order of a few milli-Gauss with a shot-to-shot noise of less than one milli-Guass and a 50 Hz modulation of 40 mG.

In this section we describe in detail three different atom lasers: a well-collimated pure atom laser (figure 1(a)), an atom laser with a very high atom-flux (figure 1(b)), and finally an atom beam containing at the same time an atom laser and a thermal atom beam (figure 1(c)). The trap had a gradient $\alpha =440\;\text{G}\;{\text{cm}^{-1}}$ and a curvature $\beta =170\;\text{G}/{\text{cm}^{-2}}$. The rf-frequency was ramped down from 50 MHz at a speed of ${{\dot{\omega }}_{\text{rf}}}/2\pi =5$ MHz ${{\text{s}}^{-1}}$. The output coupling was therefore at an intermediate rate $({{\omega }_{\text{z}}}{{\Omega }_{\text{oc}}}<{{\omega }_{\rho }})$. We generated the rf using a direct digital synthesis card (Analog Devices AD9854/PCBZ) and an rf amplifier (Amplifier Research 25A250A), which is coupled to two antennas in parallel (40 mm diameter, 70 mm distance, three windings each).

We determine the absolute coupling strength ${{\Omega }_{\text{rf}}}$ by measuring the rf-frequencies ${{\omega }_{\text{rf}}}_{1,2}$ at which the trap vanishes for two different coupling strengths ${{\Omega }_{\text{rf}}}_{1,2}$. Knowing the relative coupling strength ${{\Omega }_{\text{rf}}}_{2}/{{\Omega }_{\text{rf}}}_{1}=10$ and difference between the frequencies at which the traps vanish $({{\omega }_{\text{rf}}}_{2}-{{\omega }_{\text{rf}}}_{1})/2\pi =16$ kHz, we can read the coupling strength off figure A.1 ${{\Omega }_{\text{rf}}}/2\pi =78\pm 20$ kHz without having to determine ${{B}_{0}}$.

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The three experiments described below differ in the initial temperature, the atom number, the offset field $({{B}_{0}})$, and the duration of the time-of-flight free expansion (1–10 ms).

4.2.1. A well-collimated pure atom laser

A well-collimated pure atom laser of $\sim 4.5$ mm length and 2 ms duration can be seen in figure 1(a). It originated from a trap with (non-dressed) axial and radial trapping frequencies of 16.6 Hz and 561 Hz, respectively, and is therefore in the intermediate regime ${{\omega }_{\text{z}}}{{\Omega }_{\text{oc}}}{{\omega }_{\rho }}$. Since there are no observable thermal wings to the atom laser, we fit only the atom laser part of equation (2) and find a peak flux of $2.5\times {{10}^{7}}$ atom ${{\text{s}}^{-1}}$ with a total of $3.1\times {{10}^{4}}$ atoms.

As discussed in section A.4, we expect the TDAP to be very well-collimated. Fitting a straight line to the width as a function of distance (figure 4), we find a divergence of only 10 mrad, which compares well with the lowest divergences reported for any atom laser [44, 45]. The width of the atom laser at its origin is $50\pm 5\;\mu \text{m}$. A condensate containing all $3.1\times {{10}^{4}}$ atoms detected in the atom laser beam would have an axial Thomas–Fermi radius of 44 μm. The agreement between the width of the atom laser beam and the Thomas–Fermi radius in the trap confirms that in the regime ${{\omega }_{\text{z}}}{{\Omega }_{\text{oc}}}{{\omega }_{\rho }}$ the condensate dynamics remain frozen in the axial direction and the outcoupling occurs over the full width of the condensate.

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4.2.2. A very high flux atom laser

Much effort has been focused on trying to maximize the flux available from a given BEC [25, 46, 47]. The main obstacle to achieving very large fluxes from magnetically trapped BECs using weak rf-fields is the appearance at larger coupling strengths of a bound state, which shuts off the atom laser [28, 29]. The reason for this is that it is not possible to adiabatically transfer the atomic population of a bare state to a single dressed state simply by ramping up the rf-intensity. In contrast to this, the TDAP atom laser relies on ramping the rf-frequency of a strong rf. This transfers the entire atomic population adiabatically from the trapped to the anti-trapped state. We verified this in a Stern–Gerlach experiment, where we found all atoms in the ${{m}_{\text{F}}}=-2$ state after the rf-frequency had swept through the bottom of the trap. We also confirmed by rf-spectroscopy that the chemical potential of the original condensate matches the atom number detected in the atom laser beam.

The flux and transverse size of the atom laser of figure 1(b) can be seen in figure 5(a) and (b). The trapping frequencies were 16.6 Hz and 793 Hz and the ramping rate was ${{\dot{\omega }}_{\text{rf}}}/2\pi =5$ MHz s⁻¹. The atom laser was therefore well in the intermediate regime of outcoupling. The horizontal axis is the position below the condensate after a free expansion of 1 ms. Atoms imaged at higher values of the position have been outcoupled earlier and at higher dressing frequencies $({{\omega }_{\text{rf}}})$. Since there are no discernible thermal wings, we fit the atom laser part of equation (2) only. For smaller values of ${{\omega }_{\text{rf}}}$, i.e. smaller distances in figure 5(b), we find a well-collimated beam. At about 1.6 mm, however, we find a sudden increase in the divergence of the beam. The signal to noise ratio below this point did not permit us to analyse the exact shape of the atom beam. The absence of thermal wings for lower rf-frequencies together with the sudden change in divergence of the atom beam indicate a laser threshold, i.e. the instant at which the trap depth becomes smaller than the chemical potential of the BEC.

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The flux in figure 5(a) reaches $7.4\times {{10}^{7}}$ atoms ${{\text{s}}^{-1}}$ originating from only $9\times {{10}^{4}}$ atoms in the BEC. This is more than seven times larger than the previous maximum flux even though our initial BEC had only less than half the number of atoms [25].

4.2.3. An ultra-cold thermal beam

Figure 1(c) shows for the first time an atom beam that contains an atom laser and a thermal atom beam at the same time. The upper part of figure 1(c) is an atom laser beam whereas the lower part is an ultra-cold thermal beam. In the central part of the figure the thermal and the atom laser overlap. We analyse the image by fitting the full equation (2) to the integrated image slices. A plot of the integrated slices and fits can be found in the supplementary material. We find $7.2\times {{10}^{4}}$ atoms in thermal beam and $2.1\times {{10}^{4}}$ atoms in the atom laser. Figure 6(a) plots the thermal and atom laser fluxes against the position of the slice (bottom axis) and the value of the rf-frequency at the time of output coupling (top axis). As the rf-frequency ramps down (right to left on the plot), initially only thermal atoms are outcoupled with a peak rate of $1.5\times {{10}^{8}}$ atoms ${{\text{s}}^{-1}}$. At an rf-frequency of about 5 kHz above the trap bottom atom laser emission sets in. It reaches a flux of up to $9\times {{10}^{7}}$ atoms ${{\text{s}}^{-1}}$ until the trap opens up completely. The divergence of the atom laser is 22 mrad and its duration 0.35 ms.

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Figure 6(b) shows the temperature of the thermal beam as calculated from its width and total expansion time. The solid line is the critical temperature of Bose–Einstein condensation (450 nK) in the non-dressed trap for the total number of atoms in both beams. The temperature of the thermal beam stays constant because at this point the collision rate is no longer sufficient to thermalize the atoms remaining in the trap. At only 200 nK this is—to our best knowledge—the coldest thermal atom beam reported to date by more than two orders of magnitude [48–51]. Even though the absolute flux of the thermal atoms is very low compared to other atom beam sources, the extremely high energy resolution possible with this ultra-cold thermal beam will be useful e.g. in experiments comparing scattering of coherent and thermal atoms.

Near the trap bottom $(y{{B}_{0}})$ we can write the trapping potential in the direction of gravity $(x=0,z=0)$ as

$$\begin{eqnarray}&&V(y)={{m}_{\text{F}}}\hbar \sqrt{{{\left( \frac{\alpha _{\omega }^{2}{{y}^{2}}}{2{{\Omega }_{\text{L}}}}-\Delta {{\omega }_{\text{rf}}} \right)}^{2}}+\Omega _{\text{rf}}^{2}}\;+M{{g}_{\text{e}\,}}y,\end{eqnarray} \tag{ A.1 }$$

where $\Delta {{\omega }_{\text{rf}}}={{\omega }_{\text{rf}}}-\left| {{g}_{\text{F}}} \right|{{\mu }_{\text{B}}}{{B}_{0}}/\hbar$ is the detuning of the rf-frequency from the trap bottom in the absence of the dressing field, and ${{\alpha }_{\omega }}$ is the gradient of the radial quadrupole field written in angular frequency units $({{\alpha }_{\omega }}={{g}_{\text{F}}}{{\mu }_{\text{B}}}\alpha /\hbar )$. If the rf-field is linearly polarized and orthogonal to the z-axis, then close to the centre of an elongated Ioffe–Pritchard trap the coupling strength can be simplified to ${{\Omega }_{\text{rf}}}=\left| {{g}_{\text{F}}} \right|{{\mu }_{\text{B}}}{{B}_{\text{rf}}}/(2\hbar )$. We calculate the trap depth $(\Delta {{V}_{\text{trap}}})$ by taking the difference $(\Delta {{V}_{\text{trap}}}={{V}_{\text{out}}}-{{V}_{0}})$ of the trap minimum $({{V}_{0}})$ and output coupling point $({{V}_{\text{out}}})$.

Figure A.1 shows the trap depth for our typical experimental parameters as a function of the coupling strength ${{\Omega }_{\text{rf}}}$ and of the rf-frequency relative to the trap bottom. Note that because of gravity and the weakening of the trap by the dressing field, the depth of shallow traps is much smaller than the detuning of the rf-frequency from ${{B}_{0}}$ $(\Delta {{V}_{\text{trap}}}\hbar \;\Delta {{\omega }_{\text{rf}}})$.

During the lasing phase the atoms are accelerated both by gravity $({{g}_{\text{e}}})$ and by the gradient of the magnetic field $({{a}_{\alpha }}={{m}_{\text{F}}}{{g}_{\text{F}}}{{\mu }_{\text{B}}}\alpha /M)$. During the time-of-flight expansion they feel only earth's acceleration. The atoms in the slice located at position x of the image have been accelerated by the magnetic gradient for a time:

$$\begin{eqnarray}&&{{t}_{\text{L}}}=\sqrt{t_{\text{E}}^{2}+\frac{2x}{{{g}_{e}}+{{a}_{\alpha }}}}-{{t}_{\text{E}}}.\end{eqnarray} \tag{ A.2 }$$

For convenience, we set x=0 at the position of the BEC after time-of-flight expansion of duration $({{t}_{\text{E}}})$. The velocity of the atoms just after the time-of-flight expansion is given by ${{v}_{\text{E}}}=({{g}_{\text{e}}}+{{a}_{\alpha }})\;{{t}_{\text{L}}}+{{g}_{\text{e}}}\;{{t}_{\text{E}}}$. The flux just after the lasing phase but before the time-of-flight expansion is then

$$\begin{eqnarray}&&j(x)={{n}_{\text{1D}}}\left( {{a}_{\alpha }}+{{g}_{e}} \right)\left( {{t}_{\text{E}}}+{{t}_{\text{L}}} \right),\end{eqnarray} \tag{ A.3 }$$

where ${{n}_{\text{1D}}}$ is the one dimensional density of the atoms, i.e. the number of atoms at point x in a slice divided by its thickness.

The temperature of a slice of atoms is calculated from the fit of equation (2) as

$$\begin{eqnarray}&&T=\frac{1}{2{{k}_{\text{B}}}}M{{\left( \frac{\Delta {{x}_{\text{t}}}}{{{t}_{\text{L}}}+{{t}_{\text{E}}}} \right)}^{2}},\end{eqnarray} \tag{ A.4 }$$

where ${{k}_{\text{B}}}$ is the Boltzmann constant. This slightly overestimates the temperature, because it does not take into account that the atoms are 'anti-trapped', i.e. it neglects the contribution to the $\Delta {{x}_{\text{t}}}$ that originates from the negative curvature of the potential of equation (1) during the lasing phase before the free expansion phase.

One can determine the divergence $\Theta$ of an atom laser beam either in real space or in momentum space. In real space one takes the arctangent of the slope of a straight-line fit to the size of the atom laser beam $(\Delta {{x}_{\text{c}}})$ versus the position x (figure A.2(a)). In momentum space we take the arctangent of the calculated velocities for each slice separtely (figure A.2(b)). The latter method requires knowledge of the initial size of the beams, which we estimate as the Thomas–Fermi radius of a BEC in the non-dressed trap containing all the atoms detected in the atom laser. The velocities are calculated from the position in the image on the basis of the gravitational and magnetic acceleration.

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The results are not limited by the imaging resolution is 5 μm nor by the effective pixel size of 43 μm. The atoms in image slices from different positions have been outcoupled at different lasing times and might therefore have had different initial sizes. A more thorough analysis of the divergence would therefore have to follow the expansion of each individual image slice.

For the 'pure' laser (figure A.2) we found a divergence of $10\pm 3$ mrad in real and $13\pm 6$ mrad in momentum space. For the 'large flux' laser we found a divergence of $22\pm 1$ mrad in real space (figure 5(b)) and $13\pm 2$ mrad in momentum space (for the region of sufficient signal to noise, 0.7–13 mm).

We consider the dynamics of the Gross–Pitaevskii equation

$$\begin{eqnarray}&&i\hbar \frac{\partial \psi }{\partial t}=-\frac{{{\hbar }^{2}}}{2m}{{\nabla }^{2}}\psi +{{V}_{d}}(y,z)\psi +\frac{4\pi {{\hbar }^{2}}}{m}{{a}_{s}}{{\left| {{\psi }_{1}} \right|}^{2}}\psi \end{eqnarray} \tag{ A.5 }$$

with a dressed potential

$$\begin{eqnarray}&&{{V}_{d}}=\sqrt{{{\left( \hbar {{\Omega }_{\text{rf}}} \right)}^{2}}+{{\left[ \mu \sqrt{{{(\alpha y)}^{2}}+{{\left( {{B}_{0}}+\frac{\beta }{2}{{z}^{2}} \right)}^{2}}}-\hbar {{\omega }_{\text{rf}}} \right]}^{2}}}+{{g}_{\text{e}}}\;M\;y.\end{eqnarray} \tag{ A.6 }$$

In order to reduce the computational complexity of the problem, we consider only the two dimensional problem. We introduce the dimensionless quantities $\tau =t/{{t}_{0}}$, $\xi =y/{{y}_{0}}$, $\zeta =z/{{y}_{0}}$, and $\psi =A\phi$ and then set the kinetic term coefficient to $1/2$ and the nonlinear coefficient to unity by requiring that ${{y}_{0}}=\sqrt{\hbar \;{{t}_{0}}/M}$ and $A=\sqrt{M/(4\pi \hbar {{t}_{0}}{{a}_{s}})}$. The resulting normalized Gross–Pitaevskii equation has the form

$$\begin{eqnarray}&&i\frac{\partial \phi }{\partial \tau }=-\frac{1}{2}{{\nabla }^{2}}\phi +V(\xi ,\zeta )\psi +\gamma {{\left| {{\psi }_{1}} \right|}^{2}}\psi \end{eqnarray} \tag{ A.7 }$$

with a potential

$$\begin{eqnarray}&&V=\sqrt{{{\kappa }^{2}}+{{\left[ \sqrt{{{({{\alpha }_{y}}y)}^{2}}+{{\left[ F+{{({{\alpha }_{z}}z)}^{2}} \right]}^{2}}}-\Delta (\tau ) \right]}^{2}}}+Gy\end{eqnarray} \tag{ A.8 }$$

and we have defined $\kappa ={{\Omega }_{\text{rf}}}{{t}_{0}}$, ${{\alpha }_{y}}={{t}_{0}}\;{{\mu }_{\text{B}}}\alpha {{y}_{0}}/\hbar$, $F={{B}_{0}}{{t}_{0}}\;{{\mu }_{\text{B}}}/\hbar$, ${{\alpha }_{z}}={{t}_{0}}\;{{\mu }_{\text{B}}}\beta y_{0}^{2}/(2\hbar )$, $G=gM{{y}_{0}}{{t}_{0}}/\hbar$, $\Delta (\tau )=2\pi {{\omega }_{\text{rf}}}(t)(\tau {{t}_{0}}){{t}_{0}}$.

Using the numerical values $M=1.44\times {{10}^{-25}}\;\text{kg}$, ${{g}_{\text{e}}}=9.81\;\text{m}\;{{\text{s}}^{-2}}$, ${{m}_{\text{F}}}=2$, ${{g}_{\text{F}}}=1/2$, $\mu ={{g}_{\text{F}}}{{m}_{\text{F}}}\mu B=0.73\;\text{MHz}$,and selecting $\alpha =8\;\text{T}\;{{\text{m}}^{-1}}$, $\beta =3\times {{10}^{5}}\;\text{T}\;{{\text{m}}^{-2}}$, ${{B}_{0}}=0.5\times {{10}^{-4}}\;\text{T}$, ${{\Omega }_{\text{rf}}}=2\pi \;20\;\text{kHz}$,and ${{t}_{0}}=1/(2\pi \;5000)\;\text{s}$,we derive the parameters for the normalized potential. In particular,the space and density scalings are given by ${{y}_{0}}=1.5\times {{10}^{-7}}\;\text{m}$, $A=2.44\times {{10}^{10}}\;{{\text{m}}^{-3/2}}$,while the potential parameters are $G=0.065$, ${{\alpha }_{y}}=3.41$, $F=140$, ${{\alpha }_{z}}=0.099$, $\kappa =4$. To release the condensate from the trapping potential we set the initial value of the frequency to ${{\omega }_{\text{rf}}}=4.6\times {{10}^{6}}\;\text{Hz}$ and gradually decrease its value. The initial state of the condensate is given by the Thomas–Fermi approximation. In the initial states of the dynamics (up to $\tau =120$) we have $\Delta \prime (\tau )=-1.75$ while for $\tau >120$ this value is increased to $\Delta \prime (\tau )=-0.035$. Thus for $\tau >120$, after a 'relaxation' of the BEC dynamics of the condensate, it becomes adiabatic and quasi-stationary in the observed time scale.

In figure A.3, we see the dynamics of an atom laser BEC for different evolution times. We used a forth order split step Fourier method to computationally solve equations (A.7)–(A.8). The scheme involves periodic boundary conditions, and thus, in order to 'absorb' the wave close to the boundaries we introduced a boundary layer with linear loss. In figure A.3 we note that there are only very small differences in the density profiles for $\tau =500,600,700$ and thus the dynamics are adiabatic and quasi-stationary. The small changes in the dynamics observed in the figure are attributed to small changes of the potential with time.

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