Numerical Method

The aim of this section is to point to the relevant elements of the theory in chapter 2, and to clarify the basis of the conclusions drawn from our stochastic simulations.

We present solutions of the Wigner SDE (2.88) in section 2.5, modelling the Bosenova experiment without any significant free parameters. Recall from sec- tion 2.5 that in the stochastic treatment, the equation of motion is very similar to the GPE, with additional noise η added to the initial state and, in the presence of three-body losses, a dynamical noise term dξ in the equation of motion.

For reasons given in section 2.5.6, it is advantageous to solve the stochas- tic equation of motion for Bosenova collapse scenarios in the harmonic oscillator basis. A brief description of the method and definitions for condensed and un- condensed component in this case can also be found in that section.

Eqns. (2.95)-(2.99) show the form of the stochastic GPE in the energy basis, including the required overlap integrals. It has been outlined in Refs. [103, 104] how these integrals can be exactly computed on an appropriately chosen non- equidistant spatial grid. Different spatial grids would however be necessary for the exact solution of integrals involving different powers of the wave function. To remain computationally efficient we chose a grid which allows the exact calcula- tion of Eqns. (2.96) and (2.98). We checked that our results are invariant under

a variation of the grid used for evaluation of the integrals

Finally, a comment is in order regarding our definition of the condensed com- ponent of the stochastic field in section 2.5.6, using the expectation value of the field operator. A more rigorous definition is given by the Penrose-Onsager cri- terion [131]. Exemplary applications of this method can be found in [104, 132]. However to employ the criterion, we would have to average and subsequently diagonalize the one-body density matrix of size Nmodes × Nmodes, which is not

feasible in our case as will be explained very shortly in this section.

Throughout our truncated Wigner studies of collapsing condensates, we use three different levels of approximation of Eq. (2.95):

• GP evolution: Gross-Pitaevskii evolution only. In this case both the noise on the initial state and dynamical noise are omitted (η = 0, dξ= 0).

• TWA with initial noise: Truncated Wigner evolution without dynamical noise (dξ= 0).

• TWA with dynamical noise: Complete truncated Wigner evolution (η= 0,

dξ = 0).

The reasons for studying the GP evolution are two-fold and similar to those motivating section 3.3. Firstly it aids the determination of the required number of oscillator modes by comparison with the established position space results [30]. Secondly it allows us to quantify differences between the classical and quantum field simulations.

For the determination of the required mode numbers, the GP equation is solved in the harmonic oscillator basis to reproduce the atom number curve of Fig. 3.2 (b). In doing so, we have encountered a limitation of the oscillator-basis: Due to the extremely narrow peak of the condensate wave function at the collapse time, described in section 3.3, numerically accurate simulations beyond this point require a very large number of modes 106_{. The condensate evolution beyond}

the moment of collapse is therefore not feasible8. Fortunately it is possible to recognise a potential acceleration of the collapse from the total atom numbers and the maximum of |φ|2 (peak density) until a time shortly before the collapse. This approach was already taken when we interpreted the results of our HFB simulations.

8_{Note that in the oscillator basis we cannot make use of the FFT algorithm, and hence the} tractable mode numbers are lower than for position grid methods.

0 5 10 0 1 2 3 4 0 5 10 3000 4000 5000 6000 0 1 2 3 4 5 6 7 8 9 0 5 a) c) t[ms] t[ms] b) t[ms] Ncond npeak 20 3 _{10 /m} peak n / 10 /m 20 3 E Ncut modes 30 50 100 10 4 5 10 4 4 10 5 38 60 80 2 10 4 1 10 5 2 10 5

Figure 3.10: GP evolution only. (a) Atom numberNcond in the condensate during

a collapse withacollapse =−10a0, wherea0is the Bohr radius. (•) Results obtained on a spatial grid [30]. Thick lines correspond to solutions of Eq. (2.88) in the energy basis. The inset is a table with the number of modes for different Ecut

and a legend. As Ecut increases, the atom number curves approach the correct position basis solution. The position space data presented has a low number of temporal samples, but suffices for the point we make here. (b) Time evolution of the peak density npeak of the condensate. Labels as in (a). (c) Close up of peak

density for the initial stage only.

Fig. 3.10 (a) shows the number of atoms remaining in the condensate for different numbers of oscillator modes employed. In the case of 4×105 modes, the result appears close to convergence against the solution of the GP obtained on a spatial grid [30]. However, we can conclude from the evolution of the peak densities in panel (b) that a cutoff as big as Ecut∼ 150, corresponding to about

1.5×106 modes, would be required to evolve through the collapse.9 We find that the evolution until ∼8 ms can be accurately represented in a basis which is tractable also in the stochastic multi-trajectory treatment (∼ 5×104 modes,

Ecut = 50). This can be seen in Fig. 3.10 (c).

We outlined in section 2.5.3 under which conditions the truncation in the Wigner approach is justified. The requirement was that the condensate density

ncondbe larger than the mode density in the volume where the condensate density

is significant. This criterion, Eq. (2.83), depends on the type of basis and on

9_{The height of the narrowest possible peak in a restricted mode basis is proportional to the} cutoffEcut.

-10 0 -5 0 5 10 200 400 600 800 y n , δ (y,y) cond c

Figure 3.11: Comparison of BEC density ncond (black) and restricted modespace

commutator δc(x,x) (red) along the y-axis at t = 0. Plots are forEcut = 50. As t increases, so does the condensate density, resulting in an even better justified truncation approximation. The values forEcutare given in units of the oscillatory

energy ωx throughout the rest of this chapter.

the employed mode number. For the Bosenova problem it can be fulfilled in the oscillator basis but not in the position basis. We explicitly verified that our choice

Ecut = 50 safely fulfills Eq. (2.83). Fig. 3.11 shows the comparison between the

condensate density of the initial state of our simulation and the mode density

δc. Everywhere but in a negligible region on the cloud surface the criterion is

fulfilled.