Simulation of classical trajectories

In this section we numerically solve the differential equation for a collection of point particles in a harmonic potential subject to a force arising from a surface potential of the formUs =−C/(z−zc)4. The differential equation is given by

X i ¨ zi+ 2γz˙i+ω02zi− 4C m(zi−zc)5 = 0, (5.19)

wherezi is the position coordinate of the i-th atom. The minimum of the magnetic potential is chosen to be at the origin zt,0 = 0, the cantilever position is zc =

5.8 Simulation of cloud excitation 115

zc,0+acosωt, and γ is the atomic amplitude damping rate which we neglect in the

following. The temperature of the cloud enters via the choice of initial parameters zi(0),z˙i(0). For the calculation we use the ode45solver of MATLAB.

As long as anharmonicities are negligible, we expect a linear rise of the oscilla- tion amplitude on resonance as derived in chapter 5.1. We can use Eq. 5.3 for a crude estimate of the expected cloud amplitude after a coupling interval for typical experimental parameters. In our experiment, the minimum resolvable amplitude is a = 13 nm rms, measured in a trap with ωz = ωm = 2π×10 kHz at d = 1.5 µm. The analysis of the surface potential yields Us= 200C4/(z−zc)4 (which is assumed from now on), and for the chosen distance results in= 0.13 and thusδzt= 1.7 nm for the minimum resolvable amplitude. After a coupling interval of th = 20 ms we expect an amplitude b(20ms) = 200πδzt = 1130 nm. The trap has a radius (i.e. the distance between the trap minimum and the barrier) of ∼ 700 nm, while the cloud has a 1/eradius rth= 150 nm at T =Tc/2. A naive guess would be that loss becomes visible for an amplitude b ∼ 550 nm. This indicates that deviations from Eq. 5.3 will be substantial and that the anharmonicity will dominate the dynamics. Figure 5.14 shows the time evolution of a cloud at T =Tc/2 with a radius rth = 150 nm in a resonant trap for typical coupling parameters as used in the experiments. For clarity we show a representative set of three classes of atomic initial conditions in the trap. The first class covers atoms with energy E = p2_i/2m+ (1/2)mω2_zz2_i = kBTc/2 with evenly spaced (pi, zi) to fill a ring in phase space. The second class represents atoms with E = kBTc/4 while the third class shows an atom initially at the trap bottom with E = 0. The simulation yields that a linear rise of cloud oscillation according to Eq.5.3is given only during the first∼2 ms. This is also the timescale during which all trajectories stay in phase, and for which the excitation can be considered as coherent or reversible. For longer times, the rim of the cloud begins to lag behind the cloud center, given by the lower curvature and hence longer oscillation period for larger amplitudes. For t > 10 ms, the oscillations of the individual trajectories have dephased completely and the excitation of the cloud corresponds to an increased temperature rather than a coherent displacement.

An important characteristic is the existence of a maximum amplitude bmax for a

given excitation amplitude, a well known property of anharmonic oscillators [265]. Figure 5.15 (a) shows bmax(a) for an atom initially at the center of the trap. Only

for amplitudes a >50 nm the excitation suffices to kick atoms out of the trap. The upper bound is a consequence of the strong deformation of the trap close to the barrier. In our experiments, we observe coupling induced atom loss also for smaller amplitude. This is explained by the fact that the actual cloud extends up to the barrier and that atoms with energy close to U0 are lost either directly due to the

modulation of the barrier or excited within a few periods above the barrier.

Solving for bmax as a function of the trap frequency can be used to determine

the spectrum of 1D excitations. Figure 5.15 (b) shows spectra for two different cantilever amplitudes with similar atom-surface distance as in the experiments. The

Figure 5.14.: (a) Resonant excitation (ωm=ωz): Atomic oscillation vs excitation

time for a= 13 nm rms, d= 1.5 µm, andUs = 200UCP, resulting in = 0.13 and

δzt= 1.7 nm rms. Bright gray lines: trajectories of atoms with E =kBTc/2; dark

gray lines: atoms with E = kBTc/4; red line: atom with E = 0. Dephasing due

to the trap anharmonicity gives rise to a maximum amplitude bmax = 460 nm of the atoms. The blue line shows the linear rise in amplitude according to Eq. 5.3. Small panels show zooms into the green indicated areas. (b) Parametric excitation (ωm = 2ωz) ford= 1.7 µm and a= 70nm, resulting in δωz ≡ωzq = 408 Hz. The

5.8 Simulation of cloud excitation 117 0 20 40 60 0 100 200 300 400 500 600 700 cantilever amplitude a [nm] b max [nm] a) 4 6 8 10 12 0 100 200 300 400 500 600 700 ω_z,0/2π [kHz] b max [nm] b)

Figure 5.15.: (a) Maximum amplitude bmax as a function of cantilever amplitude

a. For bmax > 670 nm the atom is lost in the surface potential. (b) Maximum amplitude as a function of trap frequency, analogous to the measurements of chapter

5.7. Parameters are th = 20 ms, a= 50 nm (dark blue), a = 200 nm (light blue),

and varyingdbetween 1.5−1.9µm.

c.o.m. mode shows an asymmetry opposite to the experimental observation and has a width of∼300 Hz. As expected, it has a shape similar to a Duffing oscillator with softening anharmonicity.

We now study, to which extent coherent and reversible excitation of atomic motion is possible in the presence of anharmonicity. The condition that has to be met is that all trajectories remain in phase to a certain degree during a coupling interval. The degree of phase coherence then determines the fidelity of the c.o.m. excitation. Dephasing is directly related to the dependence of the oscillation frequency on the oscillation amplitude. Figure 5.16 a shows a calculation of the frequency shift ωcom(b)−ωz as a function of oscillation amplitude for a trap withωz,0/2π= 10 kHz

at d = 1.5 µm according to equation 2.77. To test the validity of Eq. 2.77 for large amplitudes, we evaluate the oscillation frequency of trajectories obtained by the numerical solution of Eq. 5.2 by Fourier transform. We find that the spectrum becomes substantially broadened for large amplitude, owing to non-uniform motion. The analytical prediction is very accurate for small amplitudes, but underestimates the shift for amplitudes close to the trap radius.

For an extended cloud of atoms, individual trajectories will differ in amplitude and will thus oscillate at a frequency deviating from the frequency for the c.o.m. coordinate of the cloud. The consequence is a phase spread of the trajectories. For a quantitative estimate we consider the difference of the accumulated phase of two trajectories with minimum and maximum amplitude bmin, bmax,

∆φ(b, t) = Z t 0 ωcom(bmax)dt0− Z t 0 ωcom(bmin)dt0, (5.20)

a) 0 100 200 300 400 −1 −0.8 −0.6 −0.4 −0.2 0 c.o.m. amplitude b [nm] ∆φ (b,T) [°] d= 1.5µm, ε = 0.13 d= 1.6µm, ε = 0.08 d= 1.7µm, ε = 0.05 d= 1.8µm, ε = 0.03 d= 1.9µm, ε = 0.02 d= 2.0µm, ε = 0.02 b)

Figure 5.16.: (a) Oscillation frequency shiftωcom(b)−ωzas a function of oscillation

amplitude calculated with Eq.2.77(red solid line). The color scale shows a Fourier spectrum calculated from trajectories simliar to those in Fig. 5.14 for a trap with ωz,0/2π= 10 kHz,d= 1.5 µm, th= 100 ms, andUs= 200×UCP. (b) Phase spread increase per cycle ∆φ(b,T) for various d and constant amplitudes bmax = b+rth

and bmin= max(0, b−rth) and rth= 150 nm.

where the two amplitudes are bmax

min = max(0, b±rth).

Figure 5.16 (b) shows the increase in phase spread per period as a function of the oscillation amplitude. For a typical coupling trap at d = 1.7 µm and a c.o.m. amplitude of the order of the cloud radius b = 150 nm, the per cycle phase spread amounts to ∆φ = 0.15◦, such that after a holding time th = 20 ms or equivalently 200 oscillations, a phase spread of 30◦has accumulated and the oscillation is already notably dephased. Note that due to its energy spread, a cloud at finite temperature yields a finite phase spread also for infinitesimal c.o.m. amplitude. Thus, also small amplitude oscillations show dephasing, and e.g. a collective single phonon excitation with b = p_~/2N mω2

z ∼ 10 nm dephases with ∆φ = 0.01−0.15

◦ _{per cycle. While}

larger atom-surface distance reduces the anharmonicity, the coupling strength is also reduced, and longer interaction time is necessary to excite the cloud to a given amplitude. From an evaluation of Eq.5.20for the phase spread accumulated during the ecxitation to a given amplitudebas a function of the coupling strength parameter , one finds ∆φ ∝1/. This shows that small atom-surface distance is advantageous despite large anharmonicity.