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Figure 3. Momentum distribution of the cloud for the excitation amplitudes A = 0.20, 0.25, 0.30, 0.40, 0.50, and 0.60119, panels (a—f) respectively, and holding times ranging from 20 to 90 ms. Increasing the excitation amplitude corresponds to larger energy input to the BEC, thus driving the system toward higher-momenta regions. For strong enough excitations, A > 0.5019 in this experimental setting, the system enters a turbulent regime with a particle cascade characterized by a power-law n(k) « k~ ° In panels (e,f), we plot a line corresponding to « k~*3 to guide the eye. Figure 3. Momentum distribution of the cloud for the excitation amplitudes A = 0.20, 0.25, 0.30, 0.40,

Figure 3 Momentum distribution of the cloud for the excitation amplitudes A = 0.20, 0.25, 0.30, 0.40, 0.50, and 0.60119, panels (a—f) respectively, and holding times ranging from 20 to 90 ms. Increasing the excitation amplitude corresponds to larger energy input to the BEC, thus driving the system toward higher-momenta regions. For strong enough excitations, A > 0.5019 in this experimental setting, the system enters a turbulent regime with a particle cascade characterized by a power-law n(k) « k~ ° In panels (e,f), we plot a line corresponding to « k~*3 to guide the eye. Figure 3. Momentum distribution of the cloud for the excitation amplitudes A = 0.20, 0.25, 0.30, 0.40,

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The expectation values of the density can be obtained by performing the Gaussian integrations, The equation above leads to the probability density function
The expectation values of the density can be obtained by performing the Gaussian integrations, The equation above leads to the probability density function
Figure 1. Momentum distribution of an unperturbed cloud (A = 0) held in the trap for a short time (thola = 20 ms) obtained through an optical absorption image following the release of the trap. The normalization is such that the integration over the plane yields one. In the right panel of Figure 2, we show the result of applying this transformation to the two-dimensional momentum distributions. For the remainder of this paper, we will only employ the reconstructed three-dimensional momentum distributions.
Figure 1. Momentum distribution of an unperturbed cloud (A = 0) held in the trap for a short time (thola = 20 ms) obtained through an optical absorption image following the release of the trap. The normalization is such that the integration over the plane yields one. In the right panel of Figure 2, we show the result of applying this transformation to the two-dimensional momentum distributions. For the remainder of this paper, we will only employ the reconstructed three-dimensional momentum distributions.
Figure 2. Momentum distribution of an unperturbed cloud (A = 0) held in the trap for different times (tholg =) from 20 to 115 ms. We show both (a) the angular average using the absorption images, and (b) the three-dimensional reconstruction of the momentum distributions using the inverse Abel transform, Equation (21). The profiles for different hold times are very similar, with small differences at high-momenta due to heating in the system. Figure 2. Momentum distribution of an unperturbed cloud (A = 0) held in the trap for different
Figure 2. Momentum distribution of an unperturbed cloud (A = 0) held in the trap for different times (tholg =) from 20 to 115 ms. We show both (a) the angular average using the absorption images, and (b) the three-dimensional reconstruction of the momentum distributions using the inverse Abel transform, Equation (21). The profiles for different hold times are very similar, with small differences at high-momenta due to heating in the system. Figure 2. Momentum distribution of an unperturbed cloud (A = 0) held in the trap for different
Figure 4. Entropy per particle calculated using Equation (16) for several excitation amplitudes as a function of the time held in the trap. The unperturbed BEC corresponds to a constant entropy per particle over time. Increasing the amplitude corresponds to higher values of the entropy per particle until the turbulent regime is reached, A > 0.509. The particle cascade that occurs at © 35 ms (see Section 4.1) is accompanied by a sudden increase in the entropy per particle, as seen for thold = 45 ms. Figure 4 shows the entropy per particle for different excitation amplitudes, ranging from no oerturbation to A = 0.609, for several values of tyo1g. The unperturbed cloud shows an approximately constant entropy per particle as a function of the time spent in the trap, as expected. For a relatively small excitation, A = 0.20.9, the entropy per particle is slightly higher than in the previous case, but with the same qualitative behavior. Increasing the excitation amplitude even further, A = 0.25 and ).30\19, increases the entropy of the system, but still gives an approximately constant behavior as a function of the time. For A = 0.409, the system is at the threshold of becoming turbulent: the entropy der particle is higher than previous values, and the slope is more inclined for the initial times. For nigher excitations, A > 0.509 ,turbulence is fully developed. For ty iq * 45 ms, the entropy per oarticle experiences a sudden increase, which occurs after the particle cascade was established.
Figure 4. Entropy per particle calculated using Equation (16) for several excitation amplitudes as a function of the time held in the trap. The unperturbed BEC corresponds to a constant entropy per particle over time. Increasing the amplitude corresponds to higher values of the entropy per particle until the turbulent regime is reached, A > 0.509. The particle cascade that occurs at © 35 ms (see Section 4.1) is accompanied by a sudden increase in the entropy per particle, as seen for thold = 45 ms. Figure 4 shows the entropy per particle for different excitation amplitudes, ranging from no oerturbation to A = 0.609, for several values of tyo1g. The unperturbed cloud shows an approximately constant entropy per particle as a function of the time spent in the trap, as expected. For a relatively small excitation, A = 0.20.9, the entropy per particle is slightly higher than in the previous case, but with the same qualitative behavior. Increasing the excitation amplitude even further, A = 0.25 and ).30\19, increases the entropy of the system, but still gives an approximately constant behavior as a function of the time. For A = 0.409, the system is at the threshold of becoming turbulent: the entropy der particle is higher than previous values, and the slope is more inclined for the initial times. For nigher excitations, A > 0.509 ,turbulence is fully developed. For ty iq * 45 ms, the entropy per oarticle experiences a sudden increase, which occurs after the particle cascade was established.
Figure 5. Entropy per particle as a function of the momentum for an unperturbed BEC (a) and a turbulent cloud (b). Notice that the same scale was employed in both plots. The turbulent BEC experiences a sudden increase in the entropy per particle, for tyojq = 45 ms, and it reaches thermal equilibrium for long times, corresponding to a more flat distribution of entropy among the momentum classes.
Figure 5. Entropy per particle as a function of the momentum for an unperturbed BEC (a) and a turbulent cloud (b). Notice that the same scale was employed in both plots. The turbulent BEC experiences a sudden increase in the entropy per particle, for tyojq = 45 ms, and it reaches thermal equilibrium for long times, corresponding to a more flat distribution of entropy among the momentum classes.
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Computer SciencePhysicsQuantum turbulenceMedicineBose-Einstein Condensate
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