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## Statistical met h od s for boson s

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**Statistical methodsfor bosons**Lecture 2. 9th January, 2012**Short version of the lecture plan: New version**Lecture 1 Lecture 2 Introductory matter BEC in extended non-interacting systems, ODLRO Atomic clouds in the traps; Confined independent bosons, what is BEC? Atom-atom interactions, Fermi pseudopotential; Gross-Pitaevski equation for extended gas and a trap Thomas-Fermi approximation Infinite systems: Bogolyubov-de Gennes theory, BEC and symmetry breaking, coherent states Time-dependent GPE. Cloud spreading and interference. Linearized TDGPE, Bogolyubov-de Gennes equations, physical meaning, link to Bogolyubov method, parallel with the RPA Dec 19 Jan 9 2**Reminder ofLecture I.**Offering many new details and alternative angles of view**Ideal quantum gases at a finite temperature**mean occupation number of a one-particle state with energy fermions bosons N N FD BE BEC? freezing out Aufbau principle 4**Trap potential**evaporation cooling Typical profile ? coordinate/ microns This is just one direction Presently, the traps are mostly 3D The trap is clearly from the real world, the atomic cloud is visible almost by a naked eye 5**Ground state orbital and the trap potential**400 nK level number 200 nK • characteristic energy • characteristic length 6**BEC observed by TOF in the velocity distribution**Qualitative features: all Gaussians wide vs.narrow isotropic vs. anisotropic 7**BUT **The non-interacting model is at most qualitative Interactions need to be accounted for**Importance of the interaction – synopsis**Without interaction, the condensate would occupy the ground state of the oscillator (dashed - - - - -) In fact, there is a significant broadening of the condensate of 80 000 sodium atoms in the experiment by Hau et al. (1998), The reason … the interactions experiment perfectly reproduced by the solution of the Gross – Pitaevski equation 9**Today: BEC for interacting bosons**Mean-field theory for zero temperature condensates Interatomic interactions Equilibrium: Gross-Pitaevskii equation Dynamics of condensates: TDGPE**Inter-atomic interaction**Interaction between neutral atoms is weak, basically van der Waals. Even that would be too strong for us, but at very low collision energies, the efective interaction potential is much weaker.**Are the interactions important?**• Weak interactions In the dilute gaseous atomic clouds in the traps, the interactions are incomparably weaker than in liquid helium. • Perturbative treatment That permits to develop a perturbative treatment and to study in a controlled manner many particle phenomena difficult to attack in HeII. • Several roles of the interactions • thermalization the atomic collisions take care of thermalization • mean field The mean field component of the interactions determines most of the deviations from the non-interacting case • beyond the mean field, the interactions change the quasi-particles and result into superfluidity even in these dilute systems 12**Fortunate properties of the interactions**• The atomic interactions in the dilute gas favor formation of long lived quasi-equilibrium clouds with condensate • Strange thing: the cloud lives for seconds, or even minutes at temperatures, at which the atoms should form a crystalline cluster. Why? • For binding of two atoms, a third one is necessary to carry away the released binding energy and momentum. Such ternary collisions are very unlikely in the rare cloud, however. • The interactions are almost elastic and spin independent: they only weakly spoil the separation of the hyperfine atomic species and preserve thus the identity of the atoms. • At the very low energies in question, the effective interaction is typically weak and repulsive … which enhances the formation and stabilization of the condensate. 13**Interatomic interactions**• For neutral atoms, the pairwise interaction has two parts • van der Waals force • strong repulsion at shorter distances due to the Pauli principle for electrons • Popular model is the 6-12 potential: • Example: • corresponds to ~12 K!! • Many bound states, too. 15**Interatomic interactions**The repulsive part of the potential – not well known The attractive part of the potential can be measured with precision Even this permits to define a characteristic length 16**Interatomic interactions**The repulsive part of the potential – not well known The attractive part of the potential can be measured with precision Even this permits to define a characteristic length rough estimate of the last bound state energy compare with 17**Experimental data**as 18**Experimental data**for “ordinary” gases (K) 5180 940 --- --- 73 --- nm 3.4 4.7 6.8 8.7 8.7 10.4 19**Scattering length, pseudopotential**Beyond the potential radius, say , the scattered wave propagates in free space For small energies, the scattering is purely isotropic , the s-wave scattering. The outside wave is For very small energies, , the radial part becomes just This may be extrapolated also into the interaction sphere (we are not interested in the short range details) Equivalent potential ("Fermi pseudopotential") 20**Experimental data**for “ordinary” gases VLT clouds (K) 5180 940 --- --- 73 --- nm 3.4 4.7 6.8 8.7 8.7 10.4 nm -1.4 4.1 -1.7 -19.5 5.6 127.2 as "well behaved”; decrease increase monotonous seemingly erratic, very interesting physics of Feshbach scattering resonances behind 21**II.The many-body Hamiltonian for**interacting atoms**Many-body Hamiltonian**At low energies (micro-kelvin range), true interaction potential replaced by an effective potential, Fermi pseudopotential**Many-body Hamiltonian**At low energies (micro-kelvin range), true interaction potential replaced by an effective potential, Fermi pseudopotential Experimental data for “ordinary” gases VLT clouds as**Many-body Hamiltonian**At low energies (micro-kelvin range), true interaction potential replaced by an effective potential, Fermi pseudopotential Experimental data for “ordinary” gases VLT clouds as**Many-body Hamiltonian**At low energies (micro-kelvin range), true interaction potential replaced by an effective potential, Fermi pseudopotential Experimental data for “ordinary” gases VLT clouds NOTES weak attraction ok weak repulsion ok weak attraction intermediate attraction weak repulsion ok strong resonant repulsion as**Many-body Hamiltonian: summary**At low energies (micro-kelvin range), true interaction potential replaced by an effective potential, Fermi pseudopotential Experimental data for “ordinary” gases VLT clouds NOTES weak attraction ok weak repulsion ok weak attraction intermediate attraction weak repulsion ok strong resonant repulsion as**III.Mean-field treatment of interacting atoms**In the mean-field approximation, the interacting particles are replaced by independent particles moving in an effective potential they create themselves**Many-body Hamiltonian and the Hartree approximation**We start from the mean field approximation. This is an educated way, similar to (almost identical with) the HARTREE APPROXIMATION we know for many electron systems. Most of the interactions is indeed absorbed into the mean field and what remains are explicit quantum correlation corrections self-consistent system**Many-body Hamiltonian and the Hartree approximation**HARTREE APPROXIMATION ~ many electron systems mean field approximation Most of the interactions absorbed into the mean field explicit quantum correlation corrections remain self-consistent system**Many-body Hamiltonian and the Hartree approximation**HARTREE APPROXIMATION ~ many electron systems mean field approximation Most of the interactions absorbed into the mean field explicit quantum correlation corrections remain self-consistent system ?**Many-body Hamiltonian and the Hartree approximation**HARTREE APPROXIMATION ~ many electron systems mean field approximation Most of the interactions absorbed into the mean field explicit quantum correlation corrections remain**Many-body Hamiltonian and the Hartree approximation**HARTREE APPROXIMATION ~ many electron systems mean field approximation Most of the interactions absorbed into the mean field explicit quantum correlation corrections remain self-consistent system**On the way to the mean-field Hamiltonian**ADDITIONAL NOTES**On the way to the mean-field Hamiltonian**ADDITIONAL NOTES First, the following exact transformations are performed particle density operator eliminates SI (self-interaction)**On the way to the mean-field Hamiltonian**ADDITIONAL NOTES Second, a specific many-body state is chosen, which defines the mean field: Then, the operator of the (quantum) density fluctuation is defined: The Hamiltonian, still exactly, becomes**substitute back**and integrate On the way to the mean-field Hamiltonian ADDITIONAL NOTES In the last step, the third line containing exchange, correlation and the self-interaction correction is neglected. The mean-field Hamiltonian of the main lecture results: • REMARKS • Second line … an additive constant compensation for double-counting of the Hartree interaction energy • In the original (variational) Hartree approximation, the self-interaction is not left out, leading to non-orthogonal Hartree orbitals • The same can be done for a time dependent Hamiltonian**Many-body Hamiltonian and the Hartree approximation**HARTREE APPROXIMATION particle-particle pair correlations neglected the only relevant quantity . . . single-particle density self-consistent system only occupied orbitals enter the cycle**Hartree approximation for bosons at zero temperature**Consider a condensate. Then all occupied orbitals are the same and Putting we obtain a closed equation for the order parameter: This is the celebrated Gross-Pitaevskii equation. This is a single self-consistent equation for a single orbital, the simplest HF like theory ever. 40**Hartree approximation for bosons at zero temperature**single self-consistent equation for a single orbital Consider a condensate. Then all occupied orbitals are the same and Putting we obtain a closed equation for the order parameter: This is the celebrated Gross-Pitaevskii equation. This is a single self-consistent equation for a single orbital, the simplest HF like theory ever. 41**Gross-Pitaevskii equation at zero temperature**single self-consistent equation for a single orbital Consider a condensate. Then all occupied orbitals are the same and Putting we obtain a closed equation for the order parameter: This is the celebrated Gross-Pitaevskii equation. 42**Gross-Pitaevskii equation at zero temperature**single self-consistent equation for a single orbital • Consider a condensate. Then all occupied orbitals are the same and • Putting • we obtain a closed equation for the order parameter: • This is the celebrated Gross-Pitaevskii equation. • has the form of a simple non-linear Schrödinger equation • concerns a macroscopic quantity • suitable for numerical solution. 43**Gross-Pitaevskii equation at zero temperature**single self-consistent equation for a single orbital • Consider a condensate. Then all occupied orbitals are the same and • Putting • we obtain a closed equation for the order parameter: • This is the celebrated Gross-Pitaevskii equation. • has the form of a simple non-linear Schrödinger equation • concerns a macroscopic quantity • suitable for numerical solution. The lowest level coincides with the chemical potential 44**Gross-Pitaevskii equation at zero temperature**single self-consistent equation for a single orbital • Consider a condensate. Then all occupied orbitals are the same and • Putting • we obtain a closed equation for the order parameter: • This is the celebrated Gross-Pitaevskii equation. • has the form of a simple non-linear Schrödinger equation • concerns a macroscopic quantity • suitable for numerical solution. The lowest level coincides with the chemical potential How do we know?? 45**Gross-Pitaevskii equation at zero temperature**single self-consistent equation for a single orbital • Consider a condensate. Then all occupied orbitals are the same and • Putting • we obtain a closed equation for the order parameter: • This is the celebrated Gross-Pitaevskii equation. • has the form of a simple non-linear Schrödinger equation • concerns a macroscopic quantity • suitable for numerical solution. The lowest level coincides with the chemical potential How do we know?? How much is it?? 46**Gross-Pitaevskii equation – "Bohmian" form**For a static condensate, the order parameter has CONSTANT PHASE. Then the Gross-Pitaevskii equation becomes Bohm's quantum potential the effective mean-field potential 47**Gross-Pitaevskii equation – variational interpretation**This equation results from a variational treatment of the Energy Functional It is required that with the auxiliary condition that is which is the GP equation written for the particle density (previous slide). 48**Gross-Pitaevskii equation – variational interpretation**This equation results from a variational treatment of the Energy Functional It is required that with the auxiliary condition that is which is the GP equation written for the particle density (previous slide). 49**Gross-Pitaevskii equation – variational interpretation**This equation results from a variational treatment of the Energy Functional It is required that with the auxiliary condition that is which is the GP equation written for the particle density (previous slide). LAGRANGE MULTIPLIER 50