Fourier transforming a trapped Bose–Einstein condensate by waiting a quarter of the trap period: simulation and applications

We investigate the property of isotropic harmonic traps to Fourier transform a weakly interacting Bose–Einstein condensate (BEC) every quarter of a trap period. We solve the Gross–Pitaevskii equation numerically to investigate the time evolution of interacting BECs in the context of the Fourier transform, and we suggest potential applications.


Introduction
The Fourier transform is important in optics, particularly in linear optics. For example, it is useful in understanding the behaviour of linear optical systems: every light beam can be described as a superposition of plane waves-the beam's Fourier components-and once a system is understood for the Fourier components the behaviour of general beams in the system can be constructed. Often it is important to physically Fourier transform a light field experimentally; this is used in areas such as optical information processing and holography [1].
Within optical holography, Fourier holography gives the greatest freedom of shaping; for example, the intensity distribution of a light field's Fourier transform can be completely different from that of the original field. For this reason Fourier holography, which shapes a light beam in a plane where it is the Fourier transform of the original beam, can alter the beam most profoundly. The same argument also holds for Fourier holography of non-interacting, i.e. linear, Bose-Einstein condensates (BECs), and an analogous argument-based on the evolution time it takes for an interacting BEC to lose all similarity to the original state-can be made for interacting BECs.
BECs are not usually linear, but they can be: Feshbach resonances [2,3] can be used to tune the nonlinearity of a BEC to almost any arbitrary value, and this technique is becoming increasingly commonplace in BEC experiments. This means that Fourier transform-based experimental techniques for manipulating BECs could potentially be very important.
BECs are already routinely Fourier transformed by switching off the trap and allowing the BEC to expand for a relatively long time (e.g. [4]). While this leads to the BEC to sort itself into its Fourier components, the BEC ends up outside the trapping volume, so this technique is usually destructive.
A BEC at zero temperature is accurately described by the Gross-Pitaevskii equation. Starting with the Gross-Pitaevskii equation for a BEC containing N atoms, (where the wavefunction is normalized as ρ(r) dr = N |ψ(r)| 2 dr = N, N is the number of atoms, V ext = 1 2 mω 2 r 2 is the harmonic external trap potential and g = N4πh 2 a/m describes the two-body collisions between the atoms with the s-wave scattering length a), we show in this paper how, by exploring the dynamics and waiting for a quarter of a trap period, a BEC in an isotropic harmonic trap can be described as a Fourier transform.
The above-mentioned Fourier transform mechanism is strictly speaking valid only for noninteracting BECs. As this does not apply to most BEC experiments, we investigate the evolution of interacting (g = 0) BECs in terms of the Fourier transform, specifically how far into the nonlinear regime our Fourier transform mechanism can be applied.

Fourier transforming by waiting half a trap period
Most BEC traps are harmonic, at least locally. The corresponding density and shape of the ground state depends on the interactions. For a weakly interacting case, it takes the form of a Gaussian, whereas if the chemical potential gρ is larger than the ground-state energy of the harmonic trap, hω, the density takes a parabolic shape, often referred to as the Thomas-Fermi solution. If the ground state is perturbed, interesting non-stationary modes can form [5]- [7]. We study here modes of a harmonic trap which behave like Fourier components. If a trapped BEC is initially in the shape of the ground state but displaced sideways from the trap centre, it will keep its shape and size but its centre of mass will oscillate about the trap centre with the trap period, T (the inverse of the trap frequency). Such a mode is called a dipole mode. If the width of the probability density is initially narrower than that of the ground state, the BEC expands and then contracts again-a so-called breathing mode. This oscillation happens over half a trap period for a weakly interacting gas, but the oscillation time changes with increasing interaction strength.
We consider here the dynamics of states that start off much wider than the ground state and which are positioned off-centre in the trap. To illustrate the main mechanisms, we first show how a non-interacting BEC can be Fourier transformed in a harmonic trap. Figure 1 shows the simulated dynamics of such states in a one-dimensional (1D) BEC. The dynamics of the states contain the characteristics of both the dipole mode and breathing mode, so we call them breathing dipole modes.
Breathing dipole modes can be seen as approximations to Fourier components. An analytical solution of the 1D Gross-Pitaevskii equation with starting conditions (at t = 0) which is a centred Gaussian of width σ with a linear phase gradient proportional to the transverse wavenumber k x , gives for time t 4 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT where length and time are respectively in units of √h /mω and 1/ω of the harmonic oscillator. In these units, the trap period is T = 2π. For time t = T/4 this simplifies to (where A = σ(1−i) √ 2 and τ = 1/σ), which is a flat-phased Gaussian whose x-position is proportional to k x , the phase gradient at time t = 0. If the width of the Gaussian at t = 0, σ, is large, i.e. σ 1, the width of the Gaussian at t = T/4, τ, becomes small, i.e. τ 1. This time evolution is approximately a Fourier transform, in which a uniform, inclined (with transverse wavenumber k x ), plane wave transforms into a δ function at a displacement proportional to k x . In fact, in the limit σ → ∞ the breathing dipole mode behaves exactly like a Fourier component. This is the key observation of this paper.
Another quarter of a trap period later, at t = T/2, ψ is back to a Gaussian peak of the original width σ and with a uniform phase gradient across it that is equal and opposite to that of the wavefunction at t = 0: The time evolution between t = T/4 and t = T/2 corresponds to another Fourier transform. As a Fourier transform is equivalent to an inverse Fourier transform combined with a flipping of the original distribution, two successive Fourier transforms, and therefore time evolution through T/2, correspond to a flipping of the original distribution. So the time evolution through T/4 of breathing dipole modes corresponds to the Fourier transform of individual Fourier components. For the time evolution through T/4 of arbitrary superpositions of breathing dipole modes to correspond to a Fourier transform, the individual Fourier components have to keep their relative phase at the trap centre (x = 0). According to equation (4), this is the case (with a constant phase offset of −π/4).
To demonstrate the Fourier transform property of a BEC's time evolution, we have numerically simulated [10] the time evolution of 2D and 3D non-interacting (g = 0) BECs in isotropic harmonic traps by integration of the Gross-Pitaevskii equation. (The dynamics for anisotropic traps are considerably more complex and outwith the scope of this paper.) We started with an initial wavefunction that is the (discrete) inverse Fourier transform of distinctive shapes: in the 2D case a photo frequently used in image processing, in the 3D case a shape that looks like a circle from one direction and like a square from another. Time evolution through T/4 should then return the BEC into its original shape. Figure 2 clearly shows that this is the case in our simulation.
Generally, through analogy with the propagation of light in quadratic graded-index (GRIN) wave guides [11]- [13], the time evolution of a non-interacting BEC in a harmonic trap can be described as a fractional Fourier transform.

Fourier filtering
As an example for an application of the Fourier transform, we consider here a Fourier-filtering technique borrowed from optics which allows arbitrary shaping of both a wavefunction's that is T/4. In both cases, we set the initial wavefunction (centre; log plot over four orders of magnitude) to the (discrete) inverse Fourier transform of distinctive shapes (left panel)-a well-known photo in the 2D case and a shape that looks circular from one direction and square from another in the 3D case-and simulated their time evolution over T/4, using a split-step technique. In the 2D case, the simulations were run on a grid of 96 × 96, representing a square of side length 34.7 dimensionless units. Density is represented as brightness (black represents zero density, white maximum density). In the 3D case, simulations were run on a 64 × 64 × 64 grid, representing a cube of side length 35 dimensionless units. Density is represented as white opacity in front of a black background; the graphs were created using the Volpack rendering software [8,9]. amplitude and phase when only the amplitude can be shaped directly. (There are also optical techniques for shaping phase and amplitude when only the phase can be shaped directly [14], which, by analogy, would also work for BECs and make direct use of the various techniques for 'phase imprinting' [15,16].) If a wavefunction with a uniform phase and arbitrary density distribution can be created, then part of it can be turned into an arbitrary state ψ a as follows. The state is a superposition of the stateψ a , the inverse Fourier transform of ψ a , travelling in the positive x direction (with additional wavenumber k x ), its complex conjugate travelling in the negative x direction, and a real number r. As the sum of a complex number and its complex conjugate is a real number, this wavefunction is real everywhere, and if r is chosen to be sufficiently large it is also positive everywhere. Strictly speaking, equation (6) is difficult to justify in general because it is only true in the non-interacting case. If, however, the expansion is sufficiently fast so that the cloud separates before collisions have time to destroy any coherence, then equation (6)   In this example, the phase and density of part of a pancake BEC was shaped into a vortex at t = T/4 after only the density of the BEC was manipulated a quarter of a trap period earlier (t = 0). The state at t = 0 was calculated according to equation (6). At t = T/4, about 85% of the remaining BEC is in the central peak, the rest is distributed equally between the doughnut-shaped vortices with circulation +1 and −1 on either side (about 7.5% each). A colour representation of the phase in the areas around both vortices, marked by red squares, is shown above the squares. The simulations for this figure were performed on a 256 × 256 grid representing an area of 28 × 28 dimensionless units. starting point. The state ψ 1 can therefore be created from a BEC ground state, which has uniform phase, by manipulating the density of the BEC to be |ψ 1 | 2 , provided this does not influence the phase distribution. This can be achieved by adding to the harmonic trap a light-induced dipole trap, whose shape can be engineered using holograms. A quarter of a trap period later, after the light-induced dipole trap has been removed but with the harmonic trap still present, the three terms in equation (6) have-as a whole and individually-Fourier transformed and separated spatially (provided k x was chosen to be sufficiently large). The parts of the BEC that correspond to the last two terms can be filtered out using the same technique for removing parts of a BEC, by using a shaped resonant laser beam to selectively transfer atoms into an untrapped state. The remaining part is the desired state ψ a , positioned off-centre in the trap. Figure 3 shows an example in a 2D BEC in which vortices are created from a trap ground state by manipulating the density of the ground state and waiting for a quarter of a trap period. This is an example of the more general idea of shaping only part of a BEC into a fully specified state and accepting that there is a remainder which might have to be removed from the trap in order not to interfere with the desired part at a later stage.

The role of interactions
For the case of strongly interacting BECs, the situation inevitably becomes more complicated. Firstly, the dynamics of each individual Fourier component (breathing dipole mode) are altered as the period of the breathing mode is dependent on g, which means that at t = T/4 it no longer reaches its minimum size. Secondly, the Fourier components interact with each other. Here, we investigate numerically the time evolution of 2D interacting BECs in the context of the Fourier transform. Throughout the paper and in all the figures, we use the dimensionless parameter g = 8πaN/d to describe the role of the interactions. Here, the scattering length a is in units of the harmonic trap size √h /mω and d is the effective thickness of the cloud in the 2D simulations. As an illustration, a g ≈ 100 would correspond to a strongly interaction gas well within the Thomas-Fermi regime.
Again, we select distinctive states P(x, y), set the initial condensate wavefunction to the inverse Fourier transform of P(x, y) and simulate the time evolution in a harmonic trap. At each time step, we calculate the fraction of the current state ψ(t) in the state P(x, y) by projecting ψ(t) on to P, i.e.
This assumes that both P and ψ are normalized. Figure 4 shows f(t) over half a trap cycle for a distinctive pattern P for different values of g. In the case g = 0, the fraction of the condensate in the desired mode reaches almost 1 at t = T/4, confirming that in this case time evolution over a quarter of a trap period corresponds to a Fourier transform. For g = 0, there is still a sharp peak around t = T/4 (shifting to the right, in the direction of half the breathing mode period), but its height decreases with increasing value of g. Only the data points for which more than 20% of the BEC are in the Fourier transformed state are shown. Insets show a greyscale representation of the central 20 × 20 dimensionless units of the density of the state P for each curve, and a colour representation (the colour range red → green → blue represents phases between 0 to 2π) of the phase, unless the phase is uniform. Figure 5 plots the time of the peak, t peak , as a function of g for a variety of states P. The states P include various doughnut-shaped vortex states and states with density cross-sections in the shape of a picture and flat phase. We arbitrarily chose the density distribution of our vortex states to be that of 2D Laguerre-Gaussian (LG) functions, given by where (r, φ) are circular coordinates around the trap centre, r 0 determines the size of the density doughnut, and m is the topological charge of the vortex. Figure 6 shows the peak fraction in state P for various patterns as a function of g.
These results show that waiting for T/4 does not always Fourier transform the wavefunction of an interacting BEC perfectly, and that the detailed behaviour depends on the wavefunction. General trends that can be seen in our limited-size sample are that f(t) has a maximum near T/4, that t peak (g) is longer than T/4 for g > 0, and that f peak generally falls off as |g| increases. However, our sample clearly does not represent all possible behaviours. Had we, for example, chosen the initial state to be stationary, then the graph of f(t) would be a flat line and therefore not have a maximum near T/4. The detailed behaviour also depends on the size of the states: for example, if the soliton states are stretched they are no longer solitons and their behaviour will be different. Again, we chose the size of our states arbitrarily.

Conclusions
We have demonstrated that Fourier transforming a BEC non destructively can be both easy and useful. However, for a reliable Fourier transform the BEC has to be only very weakly interacting. For reasons of computational complexity, most of our simulations were for 2D ('pancake') BECs. Corresponding results for 3D BECs will be an important extension of our study.