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Statistical methods for bosons Lecture 2. 9th January, 2012 Short version of the lecture plan: New version Lecture 1 Introductory matter BEC in extended non-interacting systems, ODLRO Atomic clouds in the traps; Confined independent bosons, what is BEC? Lecture 2 Atom-atom interactions, Fermi pseudopotential; Gross-Pitaevski equation for extended gas and a trap Thomas-Fermi approximation Dec 19 Jan 9 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 2 Reminder of Lecture I. Offering many new details and alternative angles of view Ideal quantum gases at a finite temperature n e ( ) Boltzmann distribution mean occupation number of a oneparticle state with energy high temperatures, dilute gases fermions N FD n F bosons N 1 n e ( ) 1 1 BE e ( ) 1 n 1 e 1 T 0 T 0 T 0 Aufbau principle BEC? freezing out 1 ,1 , ,1 , 0 , B N , 0 ,0 , ,0 , vac 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 87 Rb a0 1 m =10 nK x / a0 x 0 ( x, y, z ) 0 x x 0 y y 0 z z 0 (u ) 1 a0 e u2 2 a 02 , N ~ 106 at. 2 2 1 1 1 a0 , E0 2 m 2 2 ma0 2 Mum a02 u 1 1 2 2 V (u ) m u 2 2 a0 2 • 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 1. 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. 2. 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. 3. 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 I. Interacting atoms Interatomic interactions For neutral atoms, the pairwise interaction has two parts • van der Waals force 1 6 r • strong repulsion at shorter distances due to the Pauli principle for electrons Popular model is the 6-12 potential: 12 6 U TRUE (r ) 4 r r Example: Ar =1.6 10-22 J =0.34 nm 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 U TRUE (r ) repulsive part - C6 r6 Even this permits to define a characteristic length "local kinetic energy" "local potential energy" 2 1 2m 2 6 C6 66 6 2mC6 2 1/ 4 16 Interatomic interactions The repulsive part of the potential – not well known The attractive part of the potential can be measured with precision U TRUE (r ) repulsive part - C6 r6 Even this permits to define a characteristic length "local kinetic energy" "local potential energy" 2 1 2m 2 6 C6 66 6 2mC6 2 1/ 4 rough estimate of the last bound state energy compare with kBTC collision energy of the condensate atoms 17 Experimental data as 18 Experimental data for “ordinary” gases (K) nm 5180 940 ----73 --- 3.4 4.7 6.8 8.7 8.7 10.4 1 a.u. length = 1 bohr 0.053 nm 1 a.u. energy = 1 hartree 3.16 10 05 K 19 Scattering length, pseudopotential Beyond the potential radius, say 3 6 , the scattered wave propagates in free space For small energies, the scattering is purely isotropic , the s-wave scattering. The outside wave is sin(kr 0 ) r For very small energies, k 0 , the radial part becomes just r as , as ... the scattering length This may be extrapolated also into the interaction sphere (we are not interested in the short range details) Equivalent potential ("Fermi pseudopotential") U (r) g (r ) 4 as g m 2 20 Experimental data for “ordinary” gases (K) 5180 940 ----73 --- VLT clouds nm as 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 "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 1 2 1 ˆ H pa V (ra ) 2 a 2m U (ra rb ) a b Many-body Hamiltonian 1 2 1 ˆ H pa V (ra ) 2 a 2m U (ra rb ) a b At low energies (micro-kelvin range), true interaction potential replaced by an effective potential, Fermi pseudopotential U (r ) g (r ) 4 as g m 2 , as ... the scattering length Many-body Hamiltonian 1 2 1 ˆ H pa V (ra ) 2 a 2m U (ra rb ) a b At low energies (micro-kelvin range), true interaction potential replaced by an effective potential, Fermi pseudopotential U (r ) g (r ) 4 as g m 2 , as ... the scattering length Experimental data for “ordinary” gases VLT clouds as Many-body Hamiltonian 1 2 1 ˆ H pa V (ra ) 2 a 2m U (ra rb ) a b At low energies (micro-kelvin range), true interaction potential replaced by an effective potential, Fermi pseudopotential U (r ) g (r ) 4 as g m 2 , as ... the scattering length Experimental data for “ordinary” gases VLT clouds as Many-body Hamiltonian 1 2 1 ˆ H pa V (ra ) 2 a 2m U (ra rb ) a b At low energies (micro-kelvin range), true interaction potential replaced by an effective potential, Fermi pseudopotential U (r ) g (r ) 4 as g m 2 , as ... the scattering length Experimental data for “ordinary” gases VLT clouds as NOTES weak attraction ok weak repulsion ok weak attraction intermediate attraction weak repulsion ok strong resonant repulsion Many-body Hamiltonian: summary 1 2 1 ˆ H pa V (ra ) 2 a 2m U (ra rb ) a b At low energies (micro-kelvin range), true interaction potential replaced by an effective potential, Fermi pseudopotential U (r ) g (r ) 4 as g m 2 , as ... the scattering length Experimental data for “ordinary” gases VLT clouds as NOTES weak attraction ok weak repulsion ok weak attraction intermediate attraction weak repulsion ok strong resonant repulsion 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 1 2 1 Hˆ pa V ( ra ) 2 a 2m U (ra rb ) a b We start from the mean 1 field approximation. Hˆ GP pa2 V ( ra ) VH ( ra ) const. This is an educated way, similar to (almost identical with) the a 2m 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 1 2 ˆ H GP pa V (ra ) VH (ra ) a 2m VH (ra ) drbU (ra rb ) n(rb ) g n(ra ) n(r ) n r 2 self-consistent system 1 2 p V ( r ) V ( r ) H r E r 2m 30 Many-body Hamiltonian and the Hartree approximation 1 2 1 Hˆ pa V ( ra ) 2 a 2m HARTREE APPROXIMATION 1 2 Hˆ GP U (ra rb ) a b ~ many electron systems pa V ( ra ) VH ( ra ) const. mean field approximation a 2m Most of the interactions absorbed into the mean field explicit quantum correlation corrections remain 1 2 ˆ H GP pa V (ra ) VH (ra ) a 2m VH (ra ) drbU (ra rb ) n(rb ) g n(ra ) n(r ) n r 2 self-consistent system 1 2 p V ( r ) V ( r ) H r E r 2m 31 Many-body Hamiltonian and the Hartree approximation 1 2 1 Hˆ pa V ( ra ) 2 a 2m HARTREE APPROXIMATION 1 2 Hˆ GP U (ra rb ) a b ~ many electron systems pa V ( ra ) VH ( ra ) const. mean field approximation a 2m Most of the interactions absorbed into the mean field explicit quantum correlation corrections remain 1 2 ˆ H GP pa V (ra ) VH (ra ) a 2m VH (ra ) drbU (ra rb ) n(rb ) g n(ra ) n(r ) n r 2 self-consistent system 1 2 p V ( r ) V ( r ) H r E r 2m ? 32 Many-body Hamiltonian and the Hartree approximation 1 2 1 Hˆ pa V ( ra ) 2 a 2m HARTREE APPROXIMATION 1 2 Hˆ GP U (ra rb ) a b ~ many electron systems pa V ( ra ) VH ( ra ) const. mean field approximation a 2m Most of the interactions absorbed into the mean field explicit quantum correlation corrections remain 1 2 ˆ H GP pa V (ra ) VH (ra ) a 2m VH (ra ) drbU (ra rb ) n(rb ) g n(ra ) n(r ) n r 2 1 2 p V ( r ) V ( r ) H r E r 2m 33 Many-body Hamiltonian and the Hartree approximation 1 2 1 Hˆ pa V ( ra ) 2 a 2m HARTREE APPROXIMATION 1 2 Hˆ GP U (ra rb ) a b ~ many electron systems pa V ( ra ) VH ( ra ) const. mean field approximation a 2m Most of the interactions absorbed into the mean field explicit quantum correlation corrections remain 1 2 ˆ H GP pa V (ra ) VH (ra ) a 2m VH (ra ) drbU (ra rb ) n(rb ) g n(ra ) n(r ) n r 2 self-consistent system 1 2 p V ( r ) V ( r ) H r E r 2m 34 ADDITIONAL NOTES On the way to the mean-field Hamiltonian 35 ADDITIONAL NOTES On the way to the mean-field Hamiltonian First, the following exact transformations are performed ˆ ˆ W V 1 1 2 ˆ H pa V (ra ) 2 a 2m a Uˆ U (ra rb ) a b Vˆ V (ra ) d 3 r V (r ) (r ra ) d 3r V (r ) nˆ (r ) a 1 ˆ U 2 a U (ra rb ) a b 1 2 3 3 d r d r ' U r r ' particle density operator r ra r ' rb a b TRICK!! d 3 r d 3 r 'U r r ' r ra r ' rb r r ' 2 a b SI eliminates nˆ (r' ) nˆ (r ) 1 (self-interaction) 1 Hˆ Wˆ d3 r V (r ) nˆ (r ) d 3 r d 3 r 'U r r ' nˆ (r ) nˆ (r' ) r r ' 2 36 ADDITIONAL NOTES On the way to the mean-field Hamiltonian Second, a specific many-body state is chosen, which defines the mean field: n( r ) nˆ ( r ) nˆ ( r ) Then, the operator of the (quantum) density fluctuation is defined: nˆ r n r nˆ r nˆ r nˆ r' nˆ r n r' n r nˆ r' nˆ r nˆ r' n r n r' The Hamiltonian, still exactly, becomes Hˆ Wˆ d 3 r V (r ) d 3 r' U r r ' n(r' ) nˆ (r ) 1 2 1 2 3 3 d r d r 'U r r ' n( r ) n( r' ) 3 3 d r d r 'U r r ' nˆ (r ) nˆ ( r' ) nˆ ( r ) r r ' 37 ADDITIONAL NOTES On the way to the mean-field Hamiltonian 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: Hˆ Wˆ d 3 r V (r ) d 3r' U r r ' n(r' ) nˆ (r ) 1 d r d r 'U r r ' n( r ) n( r' ) 2 1 2 3 3 VH r substitute back nˆ ( r ) ( r ra ) a and integrate 3 3 d r d r 'U r r ' nˆ (r ) nˆ (r' ) nˆ (r ) r r ' REMARKS • Second line … an additive constant compensation for doublecounting 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 38 Many-body Hamiltonian and the Hartree approximation HARTREE APPROXIMATION 1 2 Hˆ GP pa V ( ra ) VH ( ra ) const. a 2m 1 2 neglected particle-particle correlations Hˆ GPpair pa V (ra ) VH (ra ) the only relevant quantity . . . single-particle density a 2m VH (ra ) drbU (ra rb ) n(rb ) g n(ra ) n(r ) n r 2 self-consistent system only occupied orbitals enter the cycle 1 2 p V ( r ) V ( r ) H r E r 2m 39 Hartree approximation for bosons at zero temperature Consider a condensate. Then all occupied orbitals are the same and 2 1 2 p V ( r ) gN r 0 0 r E00 r 2m Putting This is a single self-consistent equation for a single orbital, theory (r ) ever. N 0 (r ) the simplest HF like we obtain a closed equation for the order parameter: 2 1 2 p V (r ) g r r r 2m This is the celebrated Gross-Pitaevskii equation. 40 Hartree approximation for bosons at zero temperature Consider a condensate. Then all occupied orbitals are the same and 2 1 2 p V ( r ) gN r 0 0 r E00 r 2m Putting This is a single self-consistent equation for a single orbital, theory (r ) ever. N 0 (r ) the simplest HF like we obtain a closed equation for the order parameter: 2 1 2 p V (r ) g r r r 2m This is the celebrated Gross-Pitaevskii equation. 41 Gross-Pitaevskii equation at zero temperature Consider a condensate. Then all occupied orbitals are the same and 2 1 2 p V ( r ) gN r 0 0 r E00 r 2m Putting (r ) N 0 (r ) we obtain a closed equation for the order parameter: 2 1 2 p V (r ) g r r r 2m This is the celebrated Gross-Pitaevskii equation. 42 Gross-Pitaevskii equation at zero temperature Consider a condensate. Then all occupied orbitals are the same and 2 1 2 p V ( r ) gN r 0 0 r E00 r 2m Putting (r ) N 0 (r ) we obtain a closed equation for the order parameter: 2 1 2 p V (r ) g r r r 2m 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 Consider a condensate. Then all occupied orbitals are the same and 2 1 2 p V ( r ) gN r 0 0 r E00 r 2m Putting (r ) N 0 (r ) we obtain a closed equation for the order parameter: The lowest level coincides with the chemical potential 2 1 2 p V (r ) g r r r 2m 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. 44 Gross-Pitaevskii equation at zero temperature Consider a condensate. Then all occupied orbitals are the same and 2 1 2 p V ( r ) gN r 0 0 r E00 r 2m Putting (r ) N 0 (r ) we obtain a closed equation for the order parameter: The lowest level coincides with the chemical potential 2 1 2 p V (r ) g r r r 2m 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. 45 Gross-Pitaevskii equation at zero temperature Consider a condensate. Then all occupied orbitals are the same and 2 1 2 p V ( r ) gN r 0 0 r E00 r 2m Putting (r ) N 0 (r ) we obtain a closed equation for the order parameter: The lowest level coincides with the chemical potential 2 1 2 p V (r ) g r r r 2m 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. 46 Gross-Pitaevskii equation – "Bohmian" form 2 n( r ) N | 0 ( r ) |2 ( r ) N [n ] N d 3 r ( r ) d 3 r n( r ) N 2 ( r ) N 0 ( r ) n( r ) ei For a static condensate, the order parameter has CONSTANT PHASE. Then the Gross-Pitaevskii equation 2 1 2 p V ( r ) g r r r 2m becomes 2 2m n( r ) n( r ) Bohm's quantum potential V ( r ) g n( r ) the effective mean-field potential 47 Gross-Pitaevskii equation – variational interpretation This equation results from a variational treatment of the Energy Functional E[ n ] d 3 r 2 ( n( r ) ) 2 d 3 r V ( r ) n( r ) d 3 r 2m It is required that E QP E[npressure ] min quantum E POT + + external potential 1 2 g n 2 (r ) E INT interaction with the auxiliary condition N [ n] N that is E[ n ] N [ n ] 0 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 2 1 3 2 2 E[n] d r ( n(r ) ) V (r )n(r ) g n (r ) 2 2m It is required that E[n] min with the auxiliary condition N [ n] N that is E[ n ] N [ n ] 0 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 2 1 3 2 2 E[n] d r ( n(r ) ) V (r )n(r ) g n (r ) 2 2m It is required that E[n] min with the auxiliary condition N [ n] N that is E[ n ] N [ n ] 0 LAGRANGE MULTIPLIER which is the GP equation written for the particle density (previous slide). 50 Gross-Pitaevskii equation – chemical potential This equation results from a variational treatment of the Energy Functional 2 1 3 2 2 E[n] d r ( n(r ) ) V (r )n(r ) g n (r ) 2 2m It is required that E[n] min with the auxiliary condition N [ n] N that is E[ n ] N [ n ] 0 which is the GP equation written for the particle density (previous slide). From there E[ n ] N [ n] 51 Gross-Pitaevskii equation – chemical potential This equation results from a variational treatment of the Energy Functional 2 1 3 2 2 E[n] d r ( n(r ) ) V (r )n(r ) g n (r ) 2 2m It is required that E[n] min with the auxiliary condition N [ n] N that is E[ n ] N [ n ] 0 which is the GP equation written for the particle density (previous slide). From there E[ n ] N [ n] chemical potential by definition in thermodynamic limit 52 Gross-Pitaevskii equation – chemical potential This equation results from a variational treatment of the Energy Functional 2 1 3 2 2 E[n] d r ( n(r ) ) V (r )n(r ) g n (r ) 2 2m It is required that E[n] min with the auxiliary condition N [ n] N that is E[ n ] N [ n ] 0 which is the GP equation written for the particle density (previous slide). From there E E[ n ] !!! N N [ n] 53 IV. Interacting atoms in a homogeneous gas Gross-Pitaevskii equation – homogeneous gas The GP equation simplifies 2 2 g r r r 2m For periodic boundary conditions in a box with V Lx Ly Lz 1 0 (r ) V ( r ) N 0 ( r ) N n V g r r r 2 ... GP equation g r gn 2 1 E 1 3 2 1 2 2 d r ( n ) V (r )n g n g n N N 2 2m 2 55 The simplest case of all: a homogeneous gas total energy E 12 gn 2 V internal pressure E 1 2 P 2 gn V 2 N E 12 g V 56 The simplest case of all: a homogeneous gas total energy E 12 gn 2 V internal pressure E 1 2 P 2 gn V 2 N E 12 g V 57 The simplest case of all: a homogeneous gas total energy E 12 gn 2 V If g 0, it would be internal pressure E 1 2 P 2 gn V 2 N E 12 g V n 0 and the gas would be thermodynamically unstable. 58 The simplest case of all: a homogeneous gas total energy E 12 gn 2 V If g 0, it would be internal pressure E 1 2 P 2 gn V 2 N E 12 g V n 0 and the gas would be thermodynamically unstable. 59 V. Condensate in a container with hard walls A container with hard walls A semi-infinite system … exact solution of the GP equation n 2 2m n( r ) n( r ) V ( r ) g n( r ) n( r ) 61 A container with hard walls A semi-infinite system … exact analytical solution of the GP equation n 2 2m n( r ) n( r ) V ( r ) g n( r ) n( r ) Particles evenly distributed Density high Density gradient zero Internal pressure due to interactions prevails 62 A container with hard walls A semi-infinite system … exact solution of the GP equation Particles pushed away from the wall Density low Density gradient large Quantum pressure prevails n 2 2m n( r ) n( r ) V ( r ) g n( r ) n( r ) Particles evenly distributed Density high Density gradient zero Internal pressure due to interactions prevails 63 A container with hard walls A semi-infinite system … exact solution of the GP equation Particles pushed away from the wall Density low Density gradient large Quantum pressure prevails 2 2m n( r ) n( r ) V ( r ) g n( r ) healing length 4 2 as 2 n 2m m 2 n 1 8 as n 12 n( r ) Particles evenly distributed Density high Density gradient zero Internal pressure due to interactions prevails 64 A container with hard walls A semi-infinite system … exact solution of the GP equation Particles pushed away from the wall Density low Density gradient large Quantum pressure prevails 2 2m n( r ) n( r ) V ( r ) g n( r ) healing length 4 2 as 2 n 2m m 2 n 1 8 as n 12 n( r ) Particles evenly distributed Density high Density gradient zero Internal pressure due to interactions prevails n(r ) n tanh x / 2 65 A container with hard walls no interactions Interactions gradually flatten out the density in the bulk Near the walls there is a depletion region about wide 66 VI. Interacting atoms in a parabolic trap Parabolic trap with interactions GP equation for a spherical trap ( … the simplest possible case) 2 2 1 2 2 m r g r r r 2 2m Where is the particle number N? ( … a little reminder) 3 3 d r ( r ) d r n(r ) N 2 68 Parabolic trap with interactions GP equation for a spherical trap ( … the simplest possible case) 2 2 1 2 2 m r g r r r 2 2m Where is the particle number N? ( … a little reminder) 3 3 d r ( r ) d r n(r ) N 2 Dimensionless GP equation for the trap r r a0 energy = energy 2 2 2 4 as 1 2 2 2 m a0 r r a0 r r a0 2 2 m 2ma0 r a0 r N a03 d r (r ) 3 2 1 2 8 as N 2 r r r r a0 69 Parabolic trap with interactions GP equation for a spherical trap ( … the simplest possible case) 2 2 1 2 2 m r g r r r 2 2m Where is the particle number N? ( … a little reminder) 3 3 d r ( r ) d r n(r ) N 2 Dimensionless GP equation for the trap r r a0 energy = energy 2 2 2 4 as 1 2 2 2 m a0 r r a0 r r a0 2 2 m 2ma0 2 2 N r a0 r 3 d 3r (r ) 1 a single dimensionless a0 parameter 2 8 as N 2 r r r r a0 70 Importance of the interaction Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" and the interaction together allow the condensate to form. It shrinks and becomes metastable. Onset of instability with respect to three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is EINT EKIN gNn N N 2 as a03 Na02 Nas a0 71 Importance of the interaction Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" acts against the interaction so as to form the condensate. Still, it shrinks and becomes metastable. Onset of three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is EINT EKIN gNn N N 2 as a03 Na02 Nas a0 72 Importance of the interaction Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" acts against the interaction so as to form the condensate. Still, it shrinks and becomes metastable. Onset of three particle recombination processes unlike in an extended homogeneous gas !! Quantitative The decisive parameter for the "importance" of interactions is EINT EKIN gNn N N 2 as a03 Na02 Nas a0 73 Importance of the interaction Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" acts against the interaction so as to form the condensate. Still, it shrinks and becomes metastable. Onset of three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is E INT E KIN Ngn N N 2as a03 Nas 2 4 Na0 a0 74 Importance of the interaction Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" acts against the interaction so as to form the condensate. Still, it shrinks and becomes metastable. Onset of three particle recombination processes Quantitative 2 4 a s The decisive g parameter for the "importance" of interactions is m E INT Ngn N 2as a03 Nas 2 E KIN N 4 Na0 a0 2 ma02 75 Importance of the interaction Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" acts against the interaction so as to form the condensate. Still, it shrinks and becomes metastable. Onset of three particle recombination processes Quantitative The decisive parameter for the "importance" of interactions is E INT E KIN Ngn N N 2as a03 Nas 2 4 Na0 a0 76 Importance of the interaction Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" acts against the interaction so as to form the condensate. Still, it shrinks and becomes metastable. Onset of three particle recombination processes can vary in a wide range Quantitative The decisive parameter for the "importance" of interactions is E INT E KIN Ngn N collective effect weak or strong depending on N N 2as a03 Nas 2 4 Na0 a0 1 weak individual collisions 77 Importance of the interaction Qualitative for g>0, repulsion, both inner "quantum pressure" and the interaction broaden the condensate. for g<0, attraction, "quantum pressure" acts against the interaction so as to form the condensate. Still, it shrinks and becomes metastable. Onset of three particle recombination processes can vary in a wide range Quantitative The decisive parameter for the "importance" of interactions is E INT E KIN Ngn N N 2as a03 Nas 2 4 Na0 a0 0 realistic value collective effect weak or strong depending on N 1 weak individual collisions 78 VIII. Variational approach to the condensate ground state Idea: Use the variational estimate of the condensate properties instead of solving the GP equation General introduction to the variational method VARIATIONAL PRINCIPLE OF QUANTUM MECHANICS The ground state and energy are uniquely defined by E Hˆ ' Hˆ ' for all ' H NS , ' ' 1 In words, ' is a normalized symmetrical wave function of N particles. The minimum condition in the variational form is Hˆ E1ˆ 0 equivalent with the SR Hˆ E HARTREE VARIATIONAL ANSATZ FOR THE CONDENSATE WAVE F. For our many-particle Hamiltonian, 1 2 1 ˆ H pa V (ra ) 2 a 2m U (ra rb ), U (r ) g (r ) a b the true ground state is approximated by the condensate of non-interacting particles (Hartree Ansatz, here identical with the symmetrized Hartree-Fock) r1 , r2 , , rp , , rN 0 r1 0 r2 0 rp 0 rN 85 General introduction to the variational method Here, 0 is a normalized real spinless orbital. It is a functional variable to be found from the variational condition E 0 ] 0 ] Hˆ 0 ] 0 with 0 ] 0 ] 1 0 0 1 Explicit calculation yields the total Hartree energy of the condensate E 0 ] 2 N d r 0 r N d r V r 0 r 2 3 2m 3 2 4 1 3 N N 1 g d r 0 r 2 Variation of energy with the use of a Lagrange multiplier: N 1E 0 ] 0 0 0 0 r , 0 0 r BY PARTS 2 2 3 4 3 3 3 d r 2 d r V r N 1 g d r 0 0 0 0 0 0 2m 2 This results into the GP equation derived here in the variational way: 2 1 2 p V (r ) N -1 g 0 r 0 r 0 r 2m eliminates self-interaction 86 Restricted search as upper estimate to Hartree energy Start from the Hartree energy functional E 0 ] 2 N d r 0 r N d r V r 0 r 2 3 2m 3 2 4 1 3 N N 1 g d r 0 r 2 Evaluate for a suitably chosen orbital having the properties normalization 0 | 0 1 variational parameters 0 (r ) f (r, 1, 2 , p ) Minimize the resulting energy function with respect to the parameters E 0 ] E f ( 1, 2 , n E ( 1, 2 , p ) 0 p )] E ( 1, 2 , p ) n 1, 2 , ,p This yields an optimized approximate Hartree condensate 87 Application to the condensate in a parabolic trap Reminescence: The trap potential and the ground state 400 nK level number 200 nK 87 Rb a0 1 m =10 nK x / a0 x 0 ( x, y, z ) 0 x x 0 y y 0 z z 0 (u ) 1 a0 e u2 2 a 02 , N ~ 106 at. 2 2 1 1 1 a0 , E0 2 m 2 2 ma0 2 Mum a02 ground state orbital 89 Rescaling the ground state for non-interacting gas SCALING ANSATZ The condensate orbital will be taken in the form 1 r2 2 2b 3 0 r A e , A b 2 1/ 4 It is just like the ground state orbital for the isotropic oscillator, but with a rescaled size. 90 Rescaling the ground state for non-interacting gas SCALING ANSATZ The condensate orbital will be taken in the form variational parameter b 1 r2 2 2b 3 0 r A e , A b 2 1/ 4 It is just like the ground state orbital for the isotropic oscillator, but with a rescaled size. This is reminescent of the well-known scaling for the ground state of the helium atom. 91 Scaling approximation for the total energy Introduce the variational ansatz into the energy functional 1 r2 2 2b 3 0 r A e , A b 2 1/ 4 92 Scaling approximation for the total energy Introduce the variational ansatz into the energy functional 1 r2 2 2b 3 dimension-less energy per particle E 0N E 0 r A e , A b 2 3 1 1 E 2 2 3 4 Variational estimate of the total energy of the condensate as a function of the parameter Variational parameter is the orbital width 1/ 4 dimension-less orbital size N 1 as 2 a0 b a0 self-interaction ADDITIONAL NOTES in units of a0 93 Scaling approximation for the total energy Variational ansatz: the GP orbital is a scaled ground state for g = 0 1 r2 2 2b 3 dimension-less energy per particle E 0N E 0 r A e A b 2 , 3 1 1 E 2 2 3 4 Variational estimate of the total energy of the condensate as a function of the parameter Variational parameter is the orbital width N 1 as 2 a0 1/ 4 dimension-less orbital size N 1 as 2 a0 b a0 self-interaction ADDITIONAL NOTES in units of a0 1 2(2 ) 3/ 2 94 Minimization of the scaled energy Variational ansatz: the GP orbital is a scaled ground state for g = 0 1 r2 2 2b 3 dimension-less energy per particle E 0N E 0 r A e , A b 2 3 1 1 E 2 2 3 4 Variational estimate of the total energy of the condensate as a function of the parameter Variational parameter is the orbital width 1/ 4 dimension-less orbital size N 1 as 2 a0 b a0 self-interaction ADDITIONAL NOTES in units of a0 minimize the energy E 0 5 2 0 95 Importance of the interaction: scaling approximation g 0 g 0 0 0 Variational estimate of the total energy of the condensate as a function of the parameter N 1 as 2 a0 Variational parameter is the orbital width The minimum of the E ( ) curve gives the condensate size for a given . With increasing , the condensate stretches with an 1/ 5 asymptotic power law min in agreement with TFA For 0.27 0, the condensate is metastable, below c 0.27, it becomes unstable and shrinks to a 'zero' volume. Quantum pressure no more manages to overcome the 96 attractive atom-atom interaction in units of a0 Now we will be interested in dynamics of the BE condensates described by Time dependent Gross-Pitaevskii equation Time dependent Gross-Pitaevskii equation Condensate dynamics is often described to a sufficient precision by the mean field approximation. It leads to the Time dependent Gross-Pitaevskii equation Intuitively, i t (r , t ) 2m V (r , t) (r , t ) g * (r , t ) (r , t ) (r , t ) 2 Today, I proceed following the conservative approach using the mean field decoupling as above in the stationary case Three steps: 1. Mean field approximation by decoupling 2. Assumption of the condensate macrosopic wave function 3. Transformation of the TD Hartree equation to TDGPE 101 1. Mean-field Hamiltonian by decoupling -- exact part 1 2 1 Hˆ pa V (t , ra ) 2 a 2m a U (ra rb ) a b 1 Hˆ Wˆ d 3r V (t , r ) nˆ ( r ) d 3 r d 3 r 'U r r ' nˆ ( r ) nˆ ( r' ) r r ' 2 t n(t , r ) nˆ ( r ) t t nˆ ( r ) t Hˆ Wˆ d 3r V (t , r ) d 3r' U r r ' n(t , r' ) nˆ ( r ) 1 2 1 2 3 3 d r d r 'U r r ' n( t , r )n( t , r' ) 3 3 d r d r 'U r r ' nˆ (t , r ) nˆ (t , r' ) nˆ ( r ) r r ' 102 1. Mean-field Hamiltonian by decoupling -- approximation Hˆ Wˆ d 3r V (t , r ) d 3r' U r r ' n(t , r' ) nˆ ( r ) 1 2 1 2 3 3 d r d r 'U r r ' n( t , r )n( t , r' ) 3 3 d r d r 'U r r ' nˆ (t , r ) nˆ (t , r' ) nˆ ( r ) r r ' Hˆ Wˆ d 3r V (t , r ) d 3r' U r r ' n(t , r' ) nˆ ( r ) 1 1 2 2 3 3 VH t , r d r d r ' U r r ' n ( t , r ) n ( t , r' ) 3 3 d r d r 'U r r ' nˆ ( r ) nˆ ( r' ) nˆ ( r ) r r ' 1 2 Hˆ GP pa V (t , ra ) VH (t , ra ) C (t ) a 2m 103 2. Condensate assumption & 3. order parameter Assume the Hartree condensate N (t ) ( ra , t ) a 1 Then n( r, t ) N | ( r, t ) |2 and a single Hartree equation for a single orbital follows i r, t 1 t 0 2m 2 p V ( r, t ) gN 0 r, t 0 r, t 2 define the order parameter r, t N ½0 r, t The last equation becomes the desired TDGPE: i t 2 1 2 r, t p V ( r, t ) g r, t r, t 2m Physical properties of the TDGPE i t (r , t ) 2m V (r , t) (r , t ) g * (r , t ) (r , t ) (r , t ) 1. 2 In the MFA, the present coincides with the order parameter defined from the one particle density matrix and (at zero temperature) all particles are in the condensate ( r, r; t ) ( r, t ) * ( r, t ) 2. TDGPE is a non-linear equation, hence there is no simple superposition principle 3. It has the form of a self-consistent Schrödinger equation: the same type of initial conditions, stationary states, isometric evolution 4. The order parameter now typically depends on both position and time and is complex. It is convenient to define the phase by (r , t ) n0 (r , t ) ei (r ,t ) 5. The MFA order parameter (macroscopic wave function) satisfies the continuity equation 105 Physical properties of the TDGPE time-independent Stationary state of the condensate for V r , t Make the ansatz (r , t ) ( r ) e i t / Then satisfies the usual stationary GPE 2 2m V r g r r r 2 This is a real-valued differential equation and has a real solution n0 r All other solutions are trivially obtained as (r ) n0 r ei 106 Physical properties of the TDGPE Continuity equation The TDGPE and its conjugate: (r , t ) 2m V (r , t) (r , t ) g (r , t ) ( r , t ) ( r , t ) i t (r , t ) 2m V (r , t) (r , t ) g * (r , t ) ( r , t ) ( r , t ) 2 i t * 2 * * * Just like for the usual Schrödinger equation, we get t n0 j0 0 2 n0 (r, t ) (r, t ) j0 (r, t ) 2mi * * ( r, t ) Only complex fields are able to carry a finite current (quite general) i ( r ,t ) Employ the form (r , t ) n0 (r , t ) e The current density becomes j0 (r, t ) n0 (r, t ) m (r, t ) n0 (r, t ) v0 (r, t ) 107 Physical properties of the TDGPE Continuity equation The TDGPE and its conjugate: (r , t ) 2m V (r , t) (r , t ) g (r , t ) ( r , t ) ( r , t ) i t (r , t ) 2m V (r , t) (r , t ) g * (r , t ) ( r , t ) ( r , t ) 2 i t * 2 * * * Just like for the usual Schrödinger equation, we get t n0 j0 0 2 n0 (r, t ) (r, t ) j0 (r, t ) 2mi * * ( r, t ) Only complex fields are able to carry a finite current (quite general) i ( r ,t ) Employ the form (r , t ) n0 (r , t ) e phase r, t dependent The current density becomes j0 (r, t ) n0 (r, t ) m (r, t ) n0 (r, t ) vs (r, t ) 108 Physical properties of the TDGPE Continuity equation The TDGPE and its conjugate: (r , t ) 2m V (r , t) (r , t ) g (r , t ) ( r , t ) ( r , t ) i t (r , t ) 2m V (r , t) (r , t ) g * (r , t ) ( r , t ) ( r , t ) 2 i t * 2 * * * Just like for the usual Schrödinger equation, we get t n0 j0 0 2 n0 (r, t ) (r, t ) j0 (r, t ) 2mi * * ( r, t ) Only complex fields are able to carry a finite current (quite general) i ( r ,t ) Employ the form (r , t ) n0 (r , t ) e superfluid phase velocity field gradient The current density becomes j0 (r, t ) n0 (r, t ) m (r, t ) n0 (r, t ) vs (r, t ) 109 Physical properties of the TDGPE Continuity equation The TDGPE and its conjugate: (r , t ) 2m V (r , t) (r , t ) g (r , t ) ( r , t ) ( r , t ) i t (r , t ) 2m V (r , t) (r , t ) g * (r , t ) ( r , t ) ( r , t ) 2 i t * 2 * * * Just like for the usual Schrödinger equation, we get t n0 j0 0 2 n0 (r, t ) (r, t ) j0 (r, t ) 2mi * * ( r, t ) Only complex fields are able to carry a finite current (quite general) i ( r ,t ) Employ the form (r , t ) n0 (r , t ) e superfluid phase velocity field gradient The current density becomes j0 (r, t ) n0 (r, t ) m (r, t ) n0 (r, t ) vs (r, t ) The superfluid flow is irrotational (potential): rot vs 0 But only in simply connected regions – vortices& quantized circulation!!! 110 Physical properties of the TDGPE Isometric evolution Three formulations The norm of the solution of TDGPE is conserved: t d r (r, t ) 0 2 t n0 j0 0 2 S t d r (r, t ) d S j0 (rS , t ) 0 This follows from the continuity eq. The condensate particle number is conserved (condensate persistent) N0 d r n0 (r, t ) const. Finally, isometric evolution 2 t d r (r, t ) const. 2 I avoid saying "unitary", because the evolution is non-linear 111 A glimpse at applications 1. Spreading of a cloud released from the trap 2. Interference of two interpenetrating clouds 1. Three crossed laser beams: 3D laser cooling 20 000 photons needed to stop from the room temperature braking force proportional to velocity: viscous medium, "molasses" For an intense laser a matter of milliseconds temperature measurement: turn off the lasers. Atoms slowly sink in the field of gravity simultaneously, they spread in a ballistic fashion the probe laser beam excites fluorescence. the velocity distribution inferred from the cloud size and shape 113 1. Three crossed laser beams: 3D laser cooling 20 000 photons needed to stop from the room temperature braking force proportional to velocity: viscous medium, "molasses" For an intense laser a matter of milliseconds temperature measurement: turn off the lasers. Atoms slowly sink in the field of gravity the probe laser beam excites fluorescence. the velocity distribution inferred from the cloud size and shape 114 1. Three crossed laser beams: 3D laser cooling 20 000 photons needed to stop from the room temperature braking force proportional to velocity: viscous medium, "molasses" For an intense laser a matter of milliseconds temperature measurement: turn off the lasers. Atoms slowly sink in the field of gravity simultaneously, they spread in a ballistic fashion the probe laser beam excites fluorescence. the velocity distribution inferred from the cloud size and shape 115 1. Three crossed laser beams: 3D laser cooling 20 000 photons needed to stop from the room temperature braking force proportional to velocity: viscous medium, "molasses" For an intense laser a matter of milliseconds temperature measurement: turn off the lasers. Atoms slowly sink in the field of gravity simultaneously, they spread in a ballistic fashion the probe laser beam excites fluorescence. the velocity distribution inferred from the cloud size and shape 116 1. Example of calculation Even without interactions and at zero temperature, the cloud would spread quantum-mechanically The result would then be identical with a ballistic expansion, the width increasing linearly in time The repulsive interaction acts strongly at the beginning and the atoms get "ahead of time". Later, the expansion is linear again From the measurements combined with the simulations based on the hydrodynamic version of the TDGPE, it was possible to infer the scattering length with a reasonable outcome Castin&Dum, PRL 77, 5315 (1996) To an outline of a simplified treatment 117 2. Interference of BE condensate in atomic clouds Sodium atoms form a macroscopic wave function Experimental proof: Two parts of an atomic cloud separated and then interpenetratin again interfere Wavelength on the order of hundreth of millimetre experiment in the Ketterle group. Waves on water 118 Basic idea of the experiment 1. Create a cloud in the trap 2.Cut it into two pieces by laser light 3. Turn off the trap and the laser 4. The two clouds spread, interpenetrate and interfere 119 2. Experimental example of interference of atoms Two coherent condensates are interpenetrating and interfering. Vertical stripe width …. 15 m Horizontal extension of the cloud …. 1,5mm 120 2. Calculation for this famous example It is an easy exercise to calculate the interference of two spherical non-interacting clouds Surprisingly perhaps, interference fringes are formed even taking into account the non-linearities – weak, to be sure To account for the interference, either the full Euler eq. including the Bohm quantum potential – OR TD GPE, which is non-linear: only a numerical solution is possible (no superposition principle) Theoretical contrast of almost 100 percent seems to be corroborated by the experiments, which appear as blurred mostly by the external optical transfer function A.Röhrl & al. PRL 78, 4143 (1997) 121 2. Calculation for this famous example It is an easy exercise to calculate the interference of two spherical non-interacting clouds Idealized theory Surprisingly perhaps, interference fringes are formed even taking into account the non-linearities – weak, to be sure Experim. profile Realistic theory To account for the interference, either the full Euler eq. including the Bohm quantum potential – OR TD GPE, which is non-linear: only a numerical solution is possible (no superposition principle) Theoretical contrast of almost 100 percent seems to be corroborated by the experiments, which appear as blurred mostly by the external optical transfer function A.Röhrl & al. PRL 78, 4143 (1997) 122 Linearized TDGPE Small linear oscillations around equilibrium i t (r , t ) 2m V (r) (r , t ) g * (r , t ) (r , t ) (r , t ) 2 I. Stationary state of the condensate 0 ( r , t ) ( r ) e i t / Stationary GPE 2 2m V (r ) g r r r 2 All solutions are obtained from the basic one as (r ) n0 r ei I 124 Small linear oscillations around equilibrium i t (r , t ) 2m V (r) (r , t ) g * (r , t ) (r , t ) (r , t ) 2 I. Stationary state of the condensate 0 ( r , t ) ( r ) e i t / Stationary GPE 2 2m V (r ) g r r r 2 All solutions are obtained from the basic one as (r ) n0 r ei II. Small oscillation about the stationary condensate (r , t ) 0 (r , t ) (r , t ) Substitute into the TDGPE and linearize (autonomous oscillations, not a linear response) 125 Small linear oscillations around equilibrium III. Linearized TD GPE for the small oscillation i t (r , t ) 2 m V (r) ( r , t ) 2 2 2 g 0 (r , t ) (r , t ) g 0 ( r , t ) 2 * ( r , t ) where 0 (r , t ) ei t / n0 r ei DROP THIS 126 Small linear oscillations around equilibrium III. Linearized TD GPE for the small oscillation i t (r , t ) 2 m V (r) ( r , t ) 2 2 2 g 0 (r , t ) (r , t ) g 0 ( r , t ) 2 * ( r , t ) where 0 (r , t ) ei t / n0 r ei DROP THIS 127 Small linear oscillations around equilibrium III. Linearized TD GPE for the small oscillation i t (r , t ) 2 m V (r) ( r , t ) 2 2 2 g 0 (r , t ) (r , t ) g 0 ( r , t ) 2 * ( r , t ) where 0 (r , t ) ei t / n0 r ei DROP THIS IV. Ansatz for the solution (r , t ) e i t / u (r )e i t v (r )e i t suggestive of Bogolyubov, excit. energy with respect to Substitute into the linearized TDGPE and its conjugate to obtain coupled diff. eqs. for u (r ), v (r ) (which are just Bogolyubov – de Gennes equations) 128 Small linear oscillations around equilibrium V. Bogolyubov – de Gennes equations 2 2m v r g r u r v r V (r ) 2 g r u r g r v r u r 2 V (r ) 2 g r 2m 2 (r , t ) e 2 i t / 2 * * u(r )e i t 2 v (r )e * i t precisely identical with the eqs. for the Bogolyubov transformation generalized to the inhomogeneous case (a large yet finite system with bound states). VI. Bogolyubov excitations of a homogeneous condensate For a large box, go over to the k- representation to recover the famous Bogolyubov solution for the homogeneous gas k ( r, t ) e i t / uk e i kr e i t v k e i kr e i t 129 Excitation spectrum DISPERSION LAW ( k ) ( k ) 2 asymptotically merge 2 (k ) 2m k 2 2 k gn GP 2 (k ) c k k c k 2 high energy region 2 2m 2 2 2 2 k gn gn k k 2 gn 2m 2m 2m 2 gn m quasi-particles are nearly just GP particles sound region quasi-particles are collective excitations cross-over k 4 mgn 2 2 defines scale for k in terms of healing length 2 2 2 m2 1 2 gn More about the sound part of the dispersion law Entirely dependent on the interactions, both the magnitude of the velocity and the linear frequency range determined by g Can be shown to really be a sound: V VV E c " " , mn BULK MODULUS 1: V1 (k ) c k c gN 2 E 2V p V , gn m p VE Even a weakly interacting gas exhibits superfluidity; the ideal gas does not. The phonons are actually Goldstone modes corresponding to a broken symmetry The dispersion law has no roton region, contrary to the reality in 4He The dispersion law bends upwards quasi-particles are unstable, can decay Small linear oscillations around equilibrium VII. Interpretation of the coincidence with Bogolyubov theory Except for Leggett in review of 2001, nobody seems to be surprised. Leggett means that it is a kind of analogy which sometimes happens. A PARALLEL Electrons (Fermions) Atoms (Bosons) Jellium Homogeneous gas Hartree Stationary GPE Linearized TD Hartree Linearized TD GPE equivalent with RPA equivalent with Bogol. plasmons Bogolons (sound) ne nJ , VH 0 n0 , VH gn0 132 The end Useful digression: energy units energy 1K 1eV 1K k B /J e / kB 1eV kB / e e/J s-1 kB / h e/h a.u. k B / Ha e / Ha s-1 a.u. h / kB Ha / k B h/e Ha / e h/J Ha / h h / Ha Ha / J energy 1K 1eV s-1 a.u. 1K 1.38 1023 8.63 10 05 2.08 1010 3.17 10 06 1eV 1.16 1004 1.60 1019 2.41 1014 3.67 10 02 s-1 4.80 1011 4.14 1015 6.63 1034 1.52 1016 a.u. 3.16 10 05 2.72 10 01 6.56 10 15 5.88 1021 134