Cómo surgen los grupos humanos

The flow towards the sink in the parabolic parent BEC cloud of an atom laser does not establish a sonic horizon outside of the outcoupling region, because of unsuitable conditions for a quasi-stationary sonic horizon. We can change these conditions by using an additional potential hump and turn the BEC flow supersonic. Fig. 5.9 illustrates a cigar shaped BEC cloud. The cloud is sliced by a Gaussian blue detuned (repulsive) laser sheet at the centre16_{. Our localised}

outcoupling mechanism is now placed near the long end of the cigar trap. Due to the outcoupling, the bulk BEC will flow towards that end of the trap to replenish the loss. This flowv and the outcoupled beam of atoms are symbolised by arrows in the sketch 5.9.

v

Figure 5.9: Setup for sonic horizon creation in BEC flow over a hump potential. We imagine the repulsive potential and outcoupling laser of Eq. (5.34) to be realised as laser sheets so that their potentials are (approximately) independent of r.

If the outcoupling is initiated slowly, we expect a quasi-stationary flow after an initial transient period. We can then use the formalism of section 5.3.1 to

predict the behaviour of the flow. The outcoupling strength γ fixes the flux J of the outflow according to J =γn0/2.

According to Ref. [203], the parameterq(x), Eq. (5.10), takes its minimal value at the horizon. If this exceeds the critical value 3/2 the flow will remain subsonic. The critical outcoupling strength γcrit is found fromq(0) = µ−W

m₍JU_m)2/3 = 3 2, as γcrit = ₂ 3(µ−W) 3/2 √ mU n0 . (5.37)

If this is exceeded, the flow turns supersonic at the crest of the potential [203]. As established in section 5.3.1, to return the flow to subsonic velocites, a second hump would be required. We soon explain why a double hump structure in the considered setup cannot be advantageous for the dynamical creation of a transsonic flow. For now we ignore our gloomy prediction and just investigate the behaviour of a flow that breaches the speed of sound at the obstacle by means of numerical solutions of Eq. (5.33). We shall find that our impudence is rewarded with an interesting discovery.

Narrow Obstacle Potential

For our results here we will employ realistic parameters unlike in section 5.5.2. We continue our use of the 1D GPE, as the dynamics that we will encounter would render higher dimensional simulations extremely challenging. However the parameter regime is one where the one dimensional equation should be a good approximation.

We compare two cases. First, we aim to realise the maximal Hawking tem- perature presented in section 5.3.4. The atom of choice is hence sodium, the peak density chosen according to table 5.2 (˜n1D = 100) and the width of the obstacle

is σ = 12 = 35ξ, for the appropriate healing length. Further we assume tight transverse confinement with ω_⊥ = 6800×2π Hz, which results in aosc = 0.25 µm, ˜U1D = 0.022. We use the Thomas-Fermi approximation [34] to estimate

˜

µ1D = ˜n1DU˜1D = 2.2. Finally we employγ = 0.8 (γcrit = 0.3).

The result of a numerical solution of Eq. (5.33) for these parameters is shown in Fig. 5.10. Initially, the region of nonvanishing outflow velocity spreads with the speed of sound. Once this perturbation of the wave function reaches the hump potential at the centre, the flow breaches the speed of sound at the potential crest as expected. No quasi-stationary flow is established, in agreement with our expectations. Instead there is a white hole, which emits a train of gray solitons as

-2000 0 200 100 200 300 400 500 600 700 -2000 0 200 2 4 6 8 10 12 14 0 500 1000 1500 0 50 100

|M|, V [a.u.]

~

Τ 1000

t

x

x

a)

_b)

c)

n

, V/U

~

~~

Figure 5.10: Dynamical evolution of BEC flow over an obstacle potential with

σ = 12. (a) density (solid), ˜V /U˜ (dashed), outcoupling regionO(x) [a.u.] (dotted, at the bottom to the right). Successive samples are moved upwards by 100. Snapshots are at ˜t = 56, 240, 256, 320, 496, 616, 2376. ˜t= 621 corresponds to 15 ms. (b) Mach number (solid, black). Successive samples are moved upwards by 2. Red line indicates M = 1. Scaled potential (dashed). (c) Analogue Hawking temperature ˜TH. ˜TH = 0.04 corresponds to TH = 13 nK.

we have seen in section 5.4. Each time a soliton is created the black hole horizon is strongly perturbed, the third sample in Fig. 5.10 (b) was chosen to show this. As a result, even though the instability originates fom the white hole, the numerically determined analogue Hawking temperature exhibits large variations in time. The evolution for this case is also captured in the movie humpflow q1d bad gpe.mov

on the CD.

Note that in contrast to our studies of the optical tunnel, here we are not interested in the sonic horizons associated with the solitons themselves. We only study the horizons of the background flow.

Wide Obstacle Potential

We get a more interesting result when the width of the hump potential is increased to σ = 60. The evolution is shown in Fig. 5.11. We see soliton train emission as before, however in contrast to the previous case, the black hole horizon stays unperturbed by the dynamical instability of the white hole. As already stated in section 5.4, the soliton width is determined by the condensate healing length ξ. For a sufficiently wide obstacle potential, the black and white holes are separated further than ξ. Hence solitons can form at the white hole and propagate down- stream without affecting the black hole sonic horizon. As a consequence, also the analogue Hawking temperature remains constant over long times, Fig. 5.10 (c).

The whole left hand side of the condensate remains stable for the time span shown in the figure, while the right hand side becomes increasingly turbulent. The left side is shielded by the “analogue event horizon” of the black hole. However as we know, this is only true for phononic excitations. Wavepackets from the particle like, high momentum part of the Bogoliubov spectrum can reach the left side and after times longer than ˜t ∼ 1200 the black hole is perturbed and the BEC at x <0 also suffers turbulence.

A careful inspection of the numerical data shows that the onset of perturba- tions in the previously stable left hand region coincides with a wave reflection from the right cloud edge reaching the black hole. In the moviehumpflow q1d gpe.mov

on the CD this can also be seen. We conjecture that the reflection effect could be avoided in practice by further extending the BEC cloud in the x-direction. Our

x domain was limited by numerical constraints: The simulation presented used 8192 points and has been verified on a doubled grid. As the grid spacing is fixed by the requirement to be smaller than the healing length, a sufficiently large x

-2000 0 200 100 200 300 400 500 600 700 -2000 0 200 2 4 6 8 10 12 14 0 500 1000 1500 0 5

|M|, V [a.u.]

~

_{Τ 1000}

t

x

x

a)

b)

c)

n

, V/U

~

~~

Figure 5.11: Graph as Fig. 5.10 but for σ = 60. (c) In contrast to Fig. 5.10, the analogue Hawking temperature remains almost constant until ˜t ∼1200. The value ˜TH = 6×10−3 corresponds to TH = 1.9 nK.

experiment. The condensate simulated contains only 55000 atoms, an experiment could easily accommodate an order of magnitude more. We added another movie to the CD, humpflow gpe.mov, which shows an even less perturbed black hole horizon, albeit for less realistic parameters.

Comparing the results forσ= 12 andσ= 60, we see that with the requirement of shielding the black hole from the white hole instability, a Hawking temperature as given in table 5.10 cannot be achieved. The value σ = 12 is not even as

narrow asσ = 20ξ, but yet the width is already too small. Besides not being able to reach the temperature limit, this scenario would have another drawback in practice: In the hydrodynamic regime the condensate only flows over the hump as long as ˜U1Dn˜1D,bg > W, see Eq. (5.6). In this expression we refer to the peak

background density away from the hump as ˜n1D,bg. Due to three-body losses at

high densities, ˜n1D,bg will decrease, inescapably violating the condition at some

time. It is possible that the problem could be overcome by dynamically reducing

W by the same rate as ˜n1D,bg decreases. Alternatively the detection of analogue

Hawking phonons has to be fast.

Finally, there should also be a special advantage to this scenario. We can vary it between transsonic and supersonic flows by minimal variation of the parameter

γ, as long as we are near γcrit. This might be helpful for the removal of the

measurement background as discussed in section 5.3.5. However, we did not explicitly verify this feature. Since we strictly do not have a stationary flow, Eq. (5.37) might only be approximately valid.

We still owe an account of complications that occur in attempting to dynam- ically establish flow over a double hump structure, realising a double de-Laval nozzle. Consider two Gaussian humps in the flow, with the outcoupling region to the right of them. The region of non-vanishing flow velocity spreads from the outcoupling region and reaches the rightmost hump first. The flow will first be- come supersonic there, and encounters no further hump downstream to act as a diffuser. Thus the scheme is doomed to fail.

5.5.4

Summary

We have now considered sonic horizons in the parent BEC cloud of an atom laser, a much simpler system than the supersonic optical tunnel. We can create sonic horizons either by strong outcoupling or by the addition of a repulsive hump potential in the way of the flow. In the first case, we establish effectively black holes only, however these necessarily appear inside the outcoupling region. This is an undesired feature. In the second case, we establish a BH-WH pair. We see dynamical instabilities at the white hole similar to those in section 5.4. How- ever, using realistic parameters for a quasi-1D sodium condensate, we numerically showed that for wide humps the soliton emission does not perturb the black hole horizon for times up to t = 30 ms. The realised analogue Hawking temperature corresponds to TH = 1.95 nK. We conjecture that the unperturbed time span

feasibly simulate.