## Lesson 2. Quantum dynamics. Dipole oscillations.

The main physical goal of this study is to investigate quantum dynamics of an ultra-cold cloud induced by a sudden displacement of the external parabolic trap potential at the mean-field level. Methodological goal is to learn how to modify parameters of the Hamiltonian and underlying TDSE.

Here we would like to recall that the one-orbital MCTDHB(M=1) theory is fully equivalent to the famous Gross-Pitaevskii mean-field approximation: MCTDHB(1)=GP.  So, we would like to study the dipole oscillations of a weakly-interaction ultra-cold atomic cloud in the harmonic (parabolic) trap. The displacement of the trap origin activates harmonic oscillations of the cloud with the trap frequency. This is pure mathematical result originating from the exact separation of the center-of-mass and relative motion. Here is the final movie which we are going to obtain: Technically we have to

I. Prepare the system in the ground state i.e., find a static solution of the TDSE and

II. Displace the trap and look for the time-dependent evolution of the initial state obtained above, which should reveal periodic oscillations with trap frequency

I. Static. Gross-Pitaevskii ground state of N=100 bosons in  V(x)=x*x/2, implying that the trap frequency is equal unity. To find ground state we have to relax the TDSE i.e., to use Relaxation.

0) Create new project from Template 0: click  How to create a new project from Template

Now we are going to define a new quantum system. Technically we have to specify/modify the following parameters: 1D, N=100 bosons, with contact inter-particle interaction W(x-x')=delta(x-x')*λ0 of strength λ0=0.01, trapped in V(x)=x*x/2

1) Change the number of bosons from N=10 to N=100 bosons: 2) Change the type of the interparticle interaction from the HIM W(x-x')=(x-x')2 to the contact, time-independent Dirac-delta V(x)=δ(x-x'): 3) Change the strength of the interparticle interaction from λ0=0.0555555 to λ0=0.001:C 4) Change Morb from M=6 MCTDHB(M=6) to M=1 MCTDHB(1)=GP: 5) Since we have changed the number of particle form N=10 to N=100 we also have to modify the initial wave function accordingly. The MCTDHB-Lab checks when it is possible the correctness of the input parameters and highlights with the red color the wrong ones. In the present case the wrong one is the CI part of the initial wave-packet - it should be |N>=|100> instead of |10>. Click "Delete"-button and, afterwards, click "Add configuration in Fock space". MCTDHB-Lab automatically inserts the correct one |100>. The red-colored highlighting frame will disappear. 6) Now we are ready to run MCTDHB(1)=GP job: So, we get the ground state of the system at the MCTDHB(1)=GP level.  At the last iteration you should get the numbers which are very similar to these:

Job->Relax. Forward     Iteration:     100     Time: [      0.000000 ->      9.900000 +       0.050000 ->     10.000000  ]

Input orbital energy E(t+0):       51.9640596793193623( CI Dim:         1)(ORB Dim:  1*       128=       128)

OUT  CI  energy E(t+tau):       51.9640596793193339     N =        100   l0*(N-1)= 0.9900000000E-01 kind of W(x-x'):0

Delta E     : +/-       -0.0000000043885464                  Error due to  dE:        0.0000000000000000

Error E     : +/-        0.0000010000000000

II. Quantum Dynamic - dipole oscillations

We are going to use the static solution obtained above as the initial state for the quantum dynamical study. In order to induce the evolution we suddenly displace the trap origin, i.e., from V(x)=x*x/2 to V(x)=(x-2.1)*(x-2.1)/2.

We continue to work in the same project, so technically we have to modify the Hamiltonian and TDSE again.

7) Change the type of the job from "Relaxation" to "Propagation Forward": 8) Change the origin of the harmonic trap from V(x)=x*x/2 to V(x)=(x-2.1)*(x-2.1)/2: 9) As we have mentioned above we are going to use the final state obtained in the previous run as the initial state for the propagation (dynamics). We use the possibility of reading the initial wave-function from binary files available after the first run. We take Psi_0 from *bin files at time-point t_0=10 (the last time point available after first run): 10) Run MCTDHB(1) propagation forward: That's all. What is left -  is to analyze the obtained solution - to plot figures and make movies:

10) First we want to plot the natural density at a given time-slice and animate it - create the movie with figures of all the available time-slices: Afterwards MCTDHB-Lab opens your default browser and plays the just created movie with ideal dipole oscillatios: 11) For the sake of convenience and to facilitate the ideal dipole oscillations we plot the same movie in so-called Minkowski-like space-time representation: 