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Statistical methods for bosons 19th Lecture 1. December 2012 by Bedřich Velický Short version of the lecture plan 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 Dec 19 Jan 9 Infinite systems: Bogolyubov-de Gennes theory, BEC and symmetry breaking, coherent states 2 The classes will turn around the Bose-Einstein condensation in cold atomic clouds • comparatively novel area of research •largely tractable using the mean field approximation to describe the interactions • only the basic early work will be covered, the recent progress is beyond the scope Nobelists The Nobel Prize in Physics 2001 "for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates" Eric A. Cornell Wolfgang Ketterle Carl E. Wieman 1/3 of the prize 1/3 of the prize 1/3 of the prize USA Federal Republic of Germany USA University of Colorado, JILA Boulder, CO, USA Massachusetts Institute of Technology (MIT) Cambridge, MA, USA University of Colorado, JILA Boulder, CO, USA b. 1961 b. 1957 b. 1951 4 I. Introductory matter on bosons Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable 6 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Cannot be labelled or numbered 7 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Cannot be labelled or numbered Permuting particles does not lead to a different state Two particles (x1 , x2 ) (x2 , x1 ) (x1 , x2 ) 8 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Permuting particles does not lead to a different state Two particles (x1 , x2 ) (x2 , x1 ) (x1 , x2 ) 9 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Permuting particles does not lead to a different state Two particles (x1 , x2 ) (x2 , x1 ) (x1 , x2 ) 2 (x2 , x1 ) 10 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Permuting particles does not lead to a different state Two particles (x1 , x2 ) (x2 , x1 ) (x1 , x2 ) 2 (x2 , x1 ) 2 1 1 1 fermions bosons antisymmetric symmetric 11 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Permuting particles does not lead to a different state Two particles (x1 , x2 ) (x2 , x1 ) (x1 , x2 ) 2 (x2 , x1 ) 2 1 1 1 fermions bosons antisymmetric symmetric "empirical half-integer spin integer spin fact" comes from nowhere 12 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Permuting particles does not lead to a different state Two particles (x1 , x2 ) (x2 , x1 ) (x1 , x2 ) 2 (x2 , x1 ) Finds justification in the relativistic quantum field theory 2 1 1 1 fermions bosons antisymmetric symmetric "empirical half-integer spin integer spin fact" comes from nowhere 13 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Permuting particles does not lead to a different state Two particles (x1 , x2 ) (x2 , x1 ) (x1 , x2 ) 2 (x2 , x1 ) Finds justification in the relativistic quantum field theory 2 1 1 1 fermions bosons antisymmetric symmetric "empirical half-integer spin integer spin fact" electrons photons comes from nowhere 14 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable Permuting particles does not lead to a different state Two particles (x1 , x2 ) (x2 , x1 ) (x1 , x2 ) 2 (x2 , x1 ) Finds justification in the relativistic quantum field theory 2 1 1 1 fermions bosons antisymmetric symmetric "empirical half-integer spin integer spin fact" electrons photons everybody knows our present concern comes from nowhere 15 Bosons and Fermions (capsule reminder) Independent particles (… non-interacting) basis of single-particle states ( complete set of quantum numbers) x x 16 Bosons and Fermions (capsule reminder) Independent particles (… non-interacting) basis of single-particle states ( complete set of quantum numbers) x x FOCK SPACE Hilbert space of many particle states basis states … symmetrized products of single-particle states for bosons … antisymmetrized products of single-particle states for fermions specified by the set of occupation numbers 0, 1, 2, 3, … for bosons 0, 1 … for fermions 17 Bosons and Fermions (capsule reminder) Independent particles (… non-interacting) basis of single-particle states ( complete set of quantum numbers) x x FOCK SPACE Hilbert space of many particle states basis states … symmetrized products of single-particle states for bosons … antisymmetrized products of single-particle states for fermions specified by the set of occupation numbers 0, 1, 2, 3, … for bosons 0, 1 , , , , p , n n1 , n2 , n3 , , np , 1 2 3 … for fermions n -particle state n Σn p 18 Bosons and Fermions (capsule reminder) Representation of occupation numbers (basically, second quantization) …. for fermions Pauli principle fermions keep apart – as sea-gulls , , , 3 , p , n n1 , n2 , n3 , , np , n -particle state n Σn p 0 0 , 0 ,0 , ,0 , 0-particle state vacuum 1p 0 , 0 ,0 , ,1 , 1-particle 0 , 1 ,1 , ,0 , 2-particle ( x) ( x ') ( x ') ( x) / 0 , 2, 0 , ,0 , 2-particle 1 ( x )1 ( x ') not allowed 1 F 2 1 ,1 , ,1 , 0 , p ( x ) 1 2 1 2 N -particle ground state 19 2 Bosons and Fermions (capsule reminder) Representation of occupation numbers (basically, second quantization) …. for fermions Pauli principle fermions keep apart – as sea-gulls , , , 3 , p , n n1 , n2 , n3 , , np , n -particle state n Σn p 0 0 , 0 ,0 , ,0 , 0-particle state vacuum 1p 0 , 0 ,0 , ,1 , 1-particle 0 , 1 ,1 , ,0 , 2-particle ( x) ( x ') ( x ') ( x) / 0 , 2, 0 , ,0 , 2-particle 1 ( x )1 ( x ') not allowed 1 F 2 1 ,1 , ,1 , 0 , p ( x ) 1 2 1 2 2 N -particle ground state in atoms: filling of the shells (Pauli Aufbau Prinzip) in metals: Fermi sea 20 Bosons and Fermions (capsule reminder) Representation of occupation numbers (basically, second quantization) …. for bosons princip identity bosons prefer to keep close – like monkeys , , , , p , n n1 , n2 , n3 , , np , n -particle state n Σn p 0 0 , 0 ,0 , ,0 , 0-particle state vacuum 1p 0 , 0 ,0 , ,1 , 1-particle 0 , 1 ,1 , ,0 , 2-particle ( x) ( x ') ( x ') ( x) / 0 , 2, 0 , ,0 , 2-particle ( x) ( x ') 1 B 2 3 N , 0 ,0 , ,0 , all on a single orbital p ( x) 1 2 1 2 1 1 N -particle ground state ( x1 ) ( x2 ) ( xN ) 1 1 1 21 2 Bosons and Fermions (capsule reminder) Representation of occupation numbers (basically, second quantization) …. for bosons princip identity bosons prefer to keep close – like monkeys , , , , p , n n1 , n2 , n3 , , np , n -particle state n Σn p 0 0 , 0 ,0 , ,0 , 0-particle state vacuum 1p 0 , 0 ,0 , ,1 , 1-particle 0 , 1 ,1 , ,0 , 2-particle ( x) ( x ') ( x ') ( x) / 0 , 2, 0 , ,0 , 2-particle ( x) ( x ') 1 B 2 3 N , 0 ,0 , ,0 , all on a single orbital p ( x) 1 2 1 2 1 1 N -particle ground state ( x1 ) ( x2 ) ( xN ) 1 1 1 Bose-Einstein condensate 22 2 Who are bosons ? • elementary particles • quasiparticles • complex massive particles, like atoms … compound bosons Examples of bosons bosons complex particles N conserved simple particles N not conserved elementary particles atomic nuclei photons atoms quasi particles phonons magnons 4 He, 7 Li, 23 Na, 87 Rb alkali metals excited atoms 24 Examples of bosons (extension of the table) bosons complex particles N conserved simple particles N not conserved elementary particles atomic nuclei photons atoms quasi particles phonons magnons 4 He, 7 Li, 23 Na, 87 Rb alkali metals excited atoms compound quasi particles ions excitons Cooper pairs molecules 25 Question: How a complex particle, like an atom, can behave as a single whole, a compound boson ESSENTIAL CONDITIONS 1) All compound particles in the ensemble must be identical; the identity includes o detailed elementary particle composition o characteristics like mass, charge or spin 2) The total spin must have an integer value 3) The identity requirement extends also on the values of observables corresponding to internal degrees of freedom 4) which are not allowed to vary during the dynamical processes in question 5) The system of the compound bosons must be dilute enough to make the exchange effects between the component particles unimportant and absorbed in an effective weak short range interaction between the bosons as a whole 26 Example: How a complex particle, like an atom, can behave as a single whole, a compound boson RUBIDIUM -- THE FIRST ATOMIC CLOUD TO UNDERGO BEC electron configuration 1s 2 2 s 2 2 p 6 3s 2 3 p 6 3d 10 4 s 2 4 p 6 4d 10 5s1 A 87 37Rb closed shells - spin compensation Z L0 1 [Kr]5s 2S 1 2 I 23 2 S 1 LJ S 1 2 J LS J SL, ,SL 1 2 • single element Z = 37 • single isotope A = 87 • single electron configuration 27 Example: How a complex particle, like an atom, can behave as a single whole, a compound boson RUBIDIUM -- THE FIRST ATOMIC CLOUD TO UNDERGO BEC • single element Z = 37 • single isotope A = 87 87 37Rb [Kr]5s1 2S 1 2 I 23 A • single electron configuration Z 37 electrons total electron spin S 12 total nuclear spin I 32 37 protons 50 neutrons • total spin of the atom decides F SI F SI , , S I 1, 2 Two distinguishable species coexist; can be separated by joint effect of the hyperfine interaction and of the Zeeman splitting in a magnetic field 28 Atomic radius vs. interatomic distance in the cloud http://intro.chem.okstate.edu/1314f00/lecture/chapter7/lec111300.html COMPARE rRb vs. 0.244 109 m vs. d n 13 10 1 21 3 m =0.1 106 m in the air at 0 C and 1 atm d n 13 5.7 10 1 25 3 m =3.3 109 m 29 II. Homogeneous gas of non-interacting bosons The basic system exhibiting the Bose-Einstein Condensation (BEC) original case studied by Einstein Plane waves in a cavity Plane wave in classical terms and its quantum transcription X X 0e i t k r , (k ), 2 / k , p k, ( p), h / p de Broglie wavelength Discretization ("quantization") of wave vectors in the cavity volume V Lx Ly Lz Lz periodic boundary conditions ky 2 , Lx 2 k ym m, Lx 2 k zn n Lx kx Cell size (per k vector) Lx Ly k (2 ) d / V k x Cell size (per p vector) p hd / V In the (r, p)-phase space d kV h d 31 Density of states ky IDOS Integrated Density Of States: How many states have energy less than kx Invert the dispersion law ( p) p( ) Find the volume of the d-sphere in the p-space d ( p ) Cd p d Divide by the volume of the cell ( ) d ( p( )) / p V d ( p( )) / h d DOS Density Of States: 2 d / 2 Cd (d / 2 1)! How many states are around per unit energy per unit volume 1 d D ( ) ( ) V d d d 1 d 1 d p ( ) d ( p ( ) / h) dCd h ( p( ) / h) d d 32 Ideal quantum gases at a finite temperature: a reminder mean occupation number of a oneparticle state with energy n e ( ) Boltzmann distribution high temperatures, dilute gases 33 Ideal quantum gases at a finite temperature mean occupation number of a oneparticle state with energy n e ( ) Boltzmann distribution high temperatures, dilute gases 34 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 1 e ( ) 1 bosons N n 1 e ( ) 1 BE n 1 e 1 35 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 bosons chemical potential N FD N fixes particle number N n 1 e ( ) 1 n 1 e ( ) 1 BE n 1 e 1 36 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 bosons N 1 n e ( ) 1 T 0 F 1 ,1 , ,1 , 0 , 1 BE n e ( ) 1 T 0 B N , 0 ,0 , 1 e 1 T 0 ,0 , vac 37 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 bosons N 1 n e ( ) 1 T 0 1 BE n e ( ) 1 T 0 1 e 1 T 0 Aufbau principle F 1 ,1 , ,1 , 0 , B N , 0 ,0 , ,0 , vac 38 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 bosons N 1 n e ( ) 1 T 0 1 BE n e ( ) 1 T 0 T 0 freezing out Aufbau principle F 1 ,1 , ,1 , 0 , 1 e 1 B N , 0 ,0 , ,0 , vac 39 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 n e ( ) 1 1 e 1 T 0 T 0 T 0 Aufbau principle ? freezing out 1 ,1 , ,1 , 0 , B N , 0 ,0 , ,0 , vac 40 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 n e ( ) 1 1 e 1 T 0 T 0 T 0 Aufbau principle BEC? freezing out 1 ,1 , ,1 , 0 , B N , 0 ,0 , ,0 , vac 41 Bose-Einstein condensation: elementary approach Einstein's manuscript with the derivation of BEC 43 A gas with a fixed average number of atoms Ideal boson gas (macroscopic system) p2 atoms: mass m, dispersion law ( p ) 2m system as a whole: volume V, particle number N, density n=N/V, temperature T. 44 A gas with a fixed average number of atoms Ideal boson gas (macroscopic system) p2 atoms: mass m, dispersion law ( p ) 2m system as a whole: volume V, particle number N, density n=N/V, temperature T. Equation for the chemical potential closes the equilibrium problem: N N (T , ) n( j ) j j 1 e ( j ) 1 45 A gas with a fixed average number of atoms Ideal boson gas (macroscopic system) p2 atoms: mass m, dispersion law ( p ) 2m system as a whole: volume V, particle number N, density n=N/V, temperature T. Equation for the chemical potential closes the equilibrium problem: N N (T , ) n( j ) j j 1 e ( j ) 1 Always < 0. At high temperatures, in the thermodynamic limit, the continuum approximation can be used: N V d 0 1 e ( ) 1 D ( ) N (T , ) 46 A gas with a fixed average number of atoms Ideal boson gas (macroscopic system) p2 atoms: mass m, dispersion law ( p ) 2m system as a whole: volume V, particle number N, density n=N/V, temperature T. Equation for the chemical potential closes the equilibrium problem: N N (T , ) n( j ) j j 1 e ( j ) 1 Always < 0. At high temperatures, in the thermodynamic limit, the continuum approximation can be used: N V d 0 1 e ( ) 1 D ( ) N (T , ) It holds N (T , 0) N (T ,0) For each temperature, we get a critical number of atoms the gas can accommodate. Where will go the rest? 47 A gas with a fixed average number of atoms Ideal boson gas (macroscopic system) p2 atoms: mass m, dispersion law ( p ) 2m system as a whole: volume V, particle number N, density n=N/V, temperature T. Equation for the chemical potential closes the equilibrium problem: N N (T , ) n( j ) j j 1 e ( j ) 1 Always < 0. At high temperatures, in the thermodynamic limit, the continuum approximation can be used: N V d 0 1 e ( ) 1 D ( ) N (T , ) This will be shown in a while It holds N (T , 0) N (T ,0) For each temperature, we get a critical number of atoms the gas can accommodate. Where will go the rest? 48 A gas with a fixed average number of atoms Ideal boson gas (macroscopic system) p2 atoms: mass m, dispersion law ( p ) 2m system as a whole: volume V, particle number N, density n=N/V, temperature T. Equation for the chemical potential closes the equilibrium problem: N N (T , ) n( j ) j j 1 e ( j ) 1 Always < 0. At high temperatures, in the thermodynamic limit, the continuum approximation can be used: N V d 0 1 e ( ) 1 D ( ) N (T , ) This will be shown in a while It holds N (T , 0) N (T ,0) For each temperature, we get a critical number of atoms the gas can accommodate. Where will go the rest? To the condensate 49 Gas particle concentration The integral is doable: 1 N (T ,0) V d D ( ) e 1 0 use the general formula 3 2 2mk BT 3 3 V 4 ( 2 ) ( 2 ) 2 h Riemann function 3 2 2mk BT V 2 2,612 2 h 50 Gas particle concentration 3 2 The integral is doable: 1 N (T ,0) V d D ( ) e 1 0 3 2 2mk BT 3 3 V 4 ( 2 ) ( 2 ) 2 h 2m D ( ) 2 2 h /2 Riemann function 3 2 2mk BT V 2 2,612 2 h 51 Bose-Einstein condensation: critical temperature Gas particle concentration 3 2 The integral is doable: 1 N (T ,0) V d D ( ) e 1 0 3 2 2mk BT 3 3 V 4 ( 2 ) ( 2 ) 2 h 2m D ( ) 2 2 h /2 Riemann function 3 2 2mk BT V 2 2,612 2 h CRITICAL TEMPERATURE the lowest temperature at which all atoms are still accomodated in the gas: N (Tc ,0) N 53 Critical temperature 3 2 The integral is doable: 1 N (T ,0) V d D ( ) e 1 0 3 2 2mk BT 3 3 V 4 ( 2 ) ( 2 ) 2 h 2m D ( ) 2 2 h /2 Riemann function 3 2 2mk BT V 2 2,612 2 h CRITICAL TEMPERATURE the lowest temperature at which all atoms are still accomodated in the gas: N (Tc ,0) N h2 Tc 4 mk B atomic mass 2 3 2 3 2 3 N h2 n 18 n 1,6061 10 0,52725 4 uk B M M 2,612V 54 Critical temperature CRITICAL TEMPERATURE the lowest temperature at which all atoms are still accomodated in the gas: Tc 2 h 4 mk B 2 3 2 3 2 3 N h n 18 n 0,52725 1,6061 10 4 uk B M M 2,612V 2 A few estimates: system M n TC He liquid 4 11028 3.54 K Na trap 23 11020 2.86 K Rb trap 87 11019 95 nK 55 Digression: simple interpretation of TC Rearranging the formula for critical temperature h2 Tc 4 mk B we get V N N 2,612 V 1 3 2 3 h mkBTc mean interatomic distance thermal de Broglie wavelength The quantum breakdown sets on when the wave clouds of the atoms start overlapping 56 de Broglie wave length for atoms and molekules 2 p Thermal energies small … NR formulae valid: 2 2mEkin m Mu ... at. (mol.) mass At thermal equilibrium Ekin 32 k BT thermal wave length 2 3mk B 2 3u k B 1 MT 9 2,5 10 1 MT Two useful equations Ekin 32 T /11600 eV K v v 2 158 T M 57 Ketterle explains BEC to the King of Sweden 58 Bose-Einstein condensation: condensate Condensate concentration 3 2 T N (TC ,0) nG BT = n for T TC V TC 3 3 2 T 2 T n nG nBE n n 1 TC TC 3 2 f r a c t i o n GAS T / TC 60 Condensate concentration 3 2 T N (TC ,0) nG BT = n for T TC V TC 3 3 2 T 2 T n nG nBE n n 1 TC TC 3 2 f r a c t i o n GAS T / TC 61 Condensate concentration 3 2 3 T N (TC ,0) 2 nG BT = n for T TC V TC 3 3 2 2 T T n nG nBE n n 1 TC TC f r a c t i o n GAS T / TC 62 Where are the condensate atoms? ANSWER: On the lowest one-particle energy level For understanding, return to the discrete levels. N N (T , ) n( j ) j j 1 e ( j ) 1 There is a sequence of energies 0 (0) 0 1 2 For very low temperatures, (1 0 ) 1 all atoms are on the lowest level, so that n0 N O (e (1 0 ) ) N 1 ( 0 ) e k BT 0 N 1 all atoms are in the condensate connecting equation chemical potential is zero on the gross energy scale 63 Where are the condensate atoms? Continuation ANSWER: On the lowest one-particle energy level TC For temperatures below all condensate atoms are on the lowest level, so that n0 N BE N BE all condensate atoms remain on the lowest level 1 ( 0 ) e k BT 0 N BE connecting equation 1 chemical potential keeps zero on the gross energy scale question … what happens with the occupancy of the next level now? 2 Estimate: 1 0 h 2 / m V 3 2 kBT kBT n0 O(V ), n1 O(V 3 ) .... much slower growth 0 1 64 Where are the condensate atoms? Summary ANSWER: On the lowest one-particle energy level The final balance equation for T N N (T , ) TC is 1 e ( 0 ) 1 V d 0 1 e ( ) 1 D ( ) LESSON: be slow with making the thermodynamic limit (or any other limits) 65 III. Physical properties and discussion of BEC Off-Diagonal Long Range Order Thermodynamics of BEC Closer look at BEC • Thermodynamically, this is a real phase transition, although unsual • Pure quantum effect • There are no real forces acting between the bosons, but there IS a real correlation in their motion caused by their identity (symmetrical wave functions) • BEC has been so difficult to observe, because other (classical G/L or G/S) phase transitions set on much earlier • BEC is a "condensation in the momentum space", unlike the usual liquefaction of classical gases, which gives rise to droplets in the coordinate space. • This is somewhat doubtful, especially now, that the best observed BEC takes place in traps, where the atoms are significantly localized • What is valid on the "momentum condensation": BEC gives rise to quantum coherence between very distant places, just like the usual plane wave • BEC is a macroscopic quantum phenomenon in two respects: it leads to a correlation between a macroscopic fraction of atoms the resulting coherence pervades the whole macroscopic sample 67 Closer look at BEC • Thermodynamically, this is a real phase transition, although unsual • Pure quantum effect • There are no real forces acting between the bosons, but there IS a real correlation in their motion caused by their identity (symmetrical wave functions) • BEC has been so difficult to observe, because other (classical G/L or G/S) phase transitions set on much earlier • BEC is a "condensation in the momentum space", unlike the usual liquefaction of classical gases, which gives rise to droplets in the coordinate space. • This is somewhat doubtful, especially now, that the best observed BEC takes place in traps, where the atoms are significantly localized • What is valid on the "momentum condensation": BEC gives rise to quantum coherence between very distant places, just like the usual plane wave • BEC is a macroscopic quantum phenomenon in two respects: it leads to a correlation between a macroscopic fraction of atoms the resulting coherence pervades the whole macroscopic sample 68 Closer look at BEC • Thermodynamically, this is a real phase transition, although unsual • Pure quantum effect • There are no real forces acting between the bosons, but there IS a real correlation in their motion caused by their identity (symmetrical wave functions) • BEC has been so difficult to observe, because other (classical G/L or G/S) phase transitions set on much earlier • BEC is a "condensation in the momentum space", unlike the usual liquefaction of classical gases, which gives rise to droplets in the coordinate space. • This is somewhat doubtful, especially now, that the best observed BEC takes place in traps, where the atoms are significantly localized • What is valid on the "momentum condensation": BEC gives rise to quantum coherence between very distant places, just like the usual plane wave • BEC is a macroscopic quantum phenomenon in two respects: it leads to a correlation between a macroscopic fraction of atoms the resulting coherence pervades the whole macroscopic sample 69 Closer look at BEC • Thermodynamically, this is a real phase transition, although unsual • Pure quantum effect • There are no real forces acting between the bosons, but there IS a real correlation in their motion caused by their identity (symmetrical wave functions) • BEC has been so difficult to observe, because other (classical G/L or G/S) phase transitions set on much earlier • BEC is a "condensation in the momentum space", unlike the usual liquefaction of classical gases, which gives rise to droplets in the coordinate space. • This is somewhat doubtful, especially now, that the best observed BEC takes place in traps, where the atoms are significantly localized • What is valid on the "momentum condensation": BEC gives rise to quantum coherence between very distant places, just like the usual plane wave • BEC is a macroscopic quantum phenomenon in two respects: it leads to a correlation between a macroscopic fraction of atoms the resulting coherence pervades the whole macroscopic sample 70 Closer look at BEC • Thermodynamically, this is a real phase transition, although unsual • Pure quantum effect • There are no real forces acting between the bosons, but there IS a real correlation in their motion caused by their identity (symmetrical wave functions) • BEC has been so difficult to observe, because other (classical G/L or G/S) phase transitions set on much earlier • BEC is a "condensation in the momentum space", unlike the usual liquefaction of classical gases, which gives rise to droplets in the coordinate space. • This is somewhat doubtful, especially now, that the best observed BEC takes place in traps, where the atoms are significantly localized • What is valid on the "momentum condensation": BEC gives rise to quantum coherence between very distant places, just like the usual plane wave • BEC is a macroscopic quantum phenomenon in two respects: it leads to a correlation between a macroscopic fraction of atoms the resulting coherence pervades the whole macroscopic sample 71 Closer look at BEC • Thermodynamically, this is a real phase transition, although unsual • Pure quantum effect • There are no real forces acting between the bosons, but there IS a real correlation in their motion caused by their identity (symmetrical wave functions) • BEC has been so difficult to observe, because other (classical G/L or G/S) phase transitions set on much earlier • BEC is a "condensation in the momentum space", unlike the usual liquefaction of classical gases, which gives rise to droplets in the coordinate space. • This is somewhat doubtful, especially now, that the best observed BEC takes place in traps, where the atoms are significantly localized • What is valid on the "momentum condensation": BEC gives rise to quantum coherence between very distant places, just like the usual plane wave • BEC is a macroscopic quantum phenomenon in two respects: it leads to a correlation between a macroscopic fraction of atoms the resulting coherence pervades the whole macroscopic sample 72 Off-Diagonal Long Range Order Beyond the thermodynamic view: Coherence of the condensate in real space Analysis on the one-particle level Coherence in BEC: ODLRO Off-Diagonal Long Range Order Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix. Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state: X X n double average, quantum and thermal = X n insert unit operator | X n X change the summation order define the one-particle density matrix Tr X = n 74 Coherence in BEC: ODLRO Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix. Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state: X X n double average, quantum and thermal = X n insert unit operator | X n X change the summation order define the one-particle density matrix Tr X = n 75 Coherence in BEC: ODLRO Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix. Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state: X X n double average, quantum and thermal = X n insert unit operator | X n X change the summation order define the one-particle density matrix Tr X = n 76 Coherence in BEC: ODLRO Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix. Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state: X X n double average, quantum and thermal = X n insert unit operator | X n X change the summation order define the one-particle density matrix Tr X = n 77 Coherence in BEC: ODLRO Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix. Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state: X X n double average, quantum and thermal = X n insert unit operator | X n X change the summation order define the one-particle density matrix Tr X = n 78 OPDM for homogeneous systems In coordinate representation (r , r ') r k nk k r' k 1 ei k ( r r ') nk V k • depends only on the relative position (transl. invariance) • Fourier transform of the occupation numbers • isotropic … provided thermodynamic limit is allowed • in systems without condensate, the momentum distribution is smooth and the density matrix has a finite range. CONDENSATE lowest orbital with k0 79 OPDM for homogeneous systems: ODLRO CONDENSATE lowest orbital with 1 3 k0 O(V ) 0 1 i k0 ( r r ') 1 (r r ') e n0 ei k ( r r ') nk V V k k0 coherent across the sample BE (r r ') FT of a smooth function has a finite range G (r r ') 80 OPDM for homogeneous systems: ODLRO CONDENSATE lowest orbital with 1 3 k0 O(V ) 0 1 i k0 ( r r ') 1 (r r ') e n0 ei k ( r r ') nk V V k k0 coherent across the sample BE (r r ') FT of a smooth function has a finite range G (r r ') DIAGONAL ELEMENT r = r' for k0 = 0 (0 ) BE (0 ) n nBE G (0 ) + nG 81 OPDM for homogeneous systems: ODLRO CONDENSATE lowest orbital with 1 3 k0 O(V ) 0 1 i k0 ( r r ') 1 (r r ') e n0 ei k ( r r ') nk V V k k0 coherent across the sample BE (r r ') FT of a smooth function has a finite range G (r r ') DIAGONAL ELEMENT r = r' (0 ) BE (0 ) nBE G (0 ) + nG DISTANT OFF-DIAGONAL ELEMENT | r - r' | |r r '| BE (r r ') nBE |r r '| G (r r ') 0 |r r '| nBE (r r ') Off-Diagonal Long Range Order ODLRO 82 From OPDM towards the macroscopic wave function CONDENSATE k0 O(V ) 0 lowest orbital with (r r ') 1 3 1 i k0 ( r r ') 1 e n0 ei k ( r r ') nk V V k k0 coherent across the sample FT of a smooth function has a finite range (r ) (r' ) dyadic 1 i k ( r r ') e nk V k k0 MACROSCOPIC WAVE FUNCTION (r ) nBE ei( k r+ ) , 0 an arbitrary phase • expresses ODLRO in the density matrix • measures the condensate density • appears like a pure state in the density matrix, but macroscopic • expresses the notion that the condensate atoms are in the same state • is the order parameter for the BEC transition 83 Capsule on thermodynamics Homogeneous one component phase: boundary conditions (environment) and state variables T P dual variables, intensities "intensive" S V N isolated, conservative open S V S P N isobaric isothermal T V N S P not in use grand T V T P N isothermal-isobaric not in use T P 85 Homogeneous one component phase: boundary conditions (environment) and state variables T P dual variables, intensities "intensive" S V N isolated, conservative The important four isothermal T V N grand T V The one we use presently T P N isothermal-isobaric 87 Digression: which environment to choose? THE ENVIRONMENT IN THE THEORY SHOULD CORRESPOND TO THE EXPERIMENTAL CONDITIONS … a truism difficult to satisfy For large systems, this is not so sensitive for two reasons • System serves as a thermal bath or particle reservoir all by itself • Relative fluctuations (distinguishing mark) are negligible Adiabatic system SB heat exchange – the slowest S B Real system medium fast process • temperature lag • interface layer Isothermal system the fastest S B Atoms in a trap: ideal model … isolated. In fact: unceasing energy exchange (laser cooling). A small number of atoms may be kept (one to, say, 40). With 107, they form a bath already. Besides, they are cooled by evaporation and they form an open (albeit non-equilibrium) system. Sometime, N =const. crucial (persistent currents in non-SC mesoscopic rings) 88 Thermodynamic potentials and all that Basic thermodynamic identity (for equilibria) dU T d S P dV d N For an isolated system, T T ( S ,V , N ) U S , etc. V ,N For an isothermic system, the independent variables are T, V, N. The change of variables is achieved by means of the Legendre transformation. Define Free Energy F U TS , U U (T ,V , N ), S S (T ,V , N ), d F S d T P dV d N S F T , etc. V ,N 89 Thermodynamic potentials and all that Basic thermodynamic identity (for equilibria) dU T d S P dV d N For an isolated system, T T ( S ,V , N ) U S , etc. V ,N For an isothermic system, the independent variables are T, V, N. The change of variables is achieved by means of the Legendre transformation. Define Free Energy F U TS , U U (T ,V , N ), S S (T ,V , N ), d F S d T P dV d N S F T V ,N , etc. New variables: perform the substitution everywhere; this shows in the Maxwell identities (partial derivatives) 90 Thermodynamic potentials and all that Basic thermodynamic identity (for equilibria) dU T d S P dV d N For an isolated system, T T ( S ,V , N ) U S , etc. V ,N For an isothermic system, the independent variables are T, V, N. The change of variables is achieved by means of the Legendre transformation. Define Free Energy F U TS , U U (T ,V , N ), S S (T ,V , N ), d F S d T P dV d N S F , etc. Legendre transformation: T V , N subtract the relevant product of conjugate (dual) variables New variables: perform the substitution everywhere; this shows in the Maxwell identities (partial derivatives) 91 Thermodynamic potentials and all that Basic thermodynamic identity (for equilibria) dU T d S P dV d N For an isolated system, T T ( S ,V , N ) U S , etc. V ,N For an isothermic system, the independent variables are T, S, V. The change of variables is achieved by means of the Legendre transformation. Define Free Energy F U TS , U U (T ,V , N ), S S (T ,V , N ), d F S d T P dV d N S F T , etc. V ,N 92 A table isolated system U internal energy isothermic system canonical ensemble F U TS free energy isothermic-isobaric system free enthalpy isothermic open system grand potential How comes Thus, S ,V , N dU T dS P dV dN microcanonical ensemble T ,V , N d F S dT P dV dN T , P, N U TS PV N d S dT V d P dN isothermic-isobaric ensemble T ,V , U TS N PV d S dT P dV Nd PV ? grand canonical ensemble is additive, V is the only additive independent variable. P V T , (T , ) V , (T , ) Similar consideration holds for (T , P) N 93 Grand canonical thermodynamic functions of an ideal Bose gas General procedure for independent particles Elementary treatment of thermodynamic functions of an ideal gas Specific heat Equation of the state Quantum statistics with grand canonical ensemble Grand canonical ensemble admits both energy and particle number exchange between the system and its environment. The statistical operator (many body density matrix) ˆ acts in the Fock space External variables are T ,V , . They are specified by the conditions ˆ ˆ U Hˆ Tr V sharp ˆ Nˆ N Nˆ Tr ˆ ln ˆ max S kB Tr Grand canonical statistical operator has the Gibbs' form ˆ ˆ ˆ Z 1 e ( H N ) ˆ ˆ Z ( , ,V ) Tr e ( H N ) e ( , ,V ) statistical sum ( , ,V ) kBT ln Z ( , ,V ) grand canonical potential 95 Grand canonical statistical sum for ideal Bose gas Recall ˆ ˆ Z ( , ,V ) Tr e ( H N ) e ( , ,V ) statistical sum = e ( E e N ) ( ) n n TRICK!! e n ( ) eigenstate label n e n ( ) n n with n N up to here trivial 1 1 e ( ) 96 Grand canonical statistical sum for ideal Bose gas Recall ˆ ˆ Z ( , ,V ) Tr e ( H N ) e ( , ,V ) statistical sum = e ( E e N ) ( ) n n TRICK!! e n ( ) eigenstate label n e n ( ) n n with n N up to here trivial 1 1 e ( ) 97 Grand canonical statistical sum for ideal Bose gas Recall ˆ ˆ Z ( , ,V ) Tr e ( H N ) e ( , ,V ) statistical sum = e ( E e N ) ( ) n n TRICK!! e Z ( , ,V ) ( ) eigenstate label n e n 1 1 e ( ) n ( ) n n with n N up to here trivial 1 1 e ( ) 1 e z activity fugacity 1 z e 98 Grand canonical statistical sum for ideal Bose gas Recall ˆ ˆ Z ( , ,V ) Tr e ( H N ) e ( , ,V ) statistical sum = e ( E e N ) ( ) n n TRICK!! e Z ( , ,V ) ( ) eigenstate label n e n 1 1 e ( ) n ( ) n n with n N up to here trivial 1 1 e ( ) 1 e z activity fugacity 1 z e 99 Grand canonical statistical sum for ideal Bose gas Recall ˆ ˆ Z ( , ,V ) Tr e ( H N ) e ( , ,V ) statistical sum = e ( E e N ) ( ) n n TRICK!! e Z ( , ,V ) ( ) eigenstate label n e n 1 1 e ( ) n ( ) n n with n N up to here trivial 1 1 e ( ) e z 1 activity fugacity 1 z e ( , ,V ) kBT ln Z ( , ,V ) +k T ln 1 z e grand canonical potential = +kBT ln 1 e ( ) B 100 Grand canonical statistical sum for ideal Bose gas Recall ˆ ˆ Z ( , ,V ) Tr e ( H N ) e ( , ,V ) statistical sum = e ( E e N ) ( ) n n TRICK!! e Z ( , ,V ) ( ) eigenstate label n e n 1 1 e ( ) n 1 e ( ) up to here trivial e z 1 activity fugacity 1 z e +k T ln 1 z e = +kBT ln 1 e ( ) n n N 1 ( , ,V ) kBT ln Z ( , ,V ) B ( ) n with grand canonical potential valid for - extended "ïnfinite" gas - parabolic traps just the same 101 Using the statistical sum A. (Re)deriving the Bose-Einstein distribution n Tr ˆ nˆ kBT kBT Z Z e Tr e k BT 1 nˆ ( Hˆ Nˆ ) ( Tr e ( ) nˆ Tr e nˆ ( ) nˆ ln Z ( , ,V ) ln 1 e 1 ( ) Tr e ( Hˆ Nˆ ) ) o.k. B. Pair correlations (particle number fluctuation) kBT n ( n n )2 n (1 n ) bunching For Fermions ..................... n (1 n ) anti-bunching For classical gas ................ n 102 Ideal Boson systems at a finite temperature n e ( ) Boltzmann distribution high temperatures, dilute gases fermions N FD n bosons N 1 n e ( ) 1 1 BE e ( ) 1 n 1 e 1 Equation for the chemical potential T closes 0the equilibrium problem: T 0 N N (T , ) n(? j) F 1 ,1 , ,1 , 0 , B j j e N , 0 ,0 , ,0 , 1 ( j ) 1 vac 103 Thermodynamic functions for an extended Bose gas For Born-Karman periodic boundary conditions, the lowest level is (k 0 ) 0 Its contribution has to be singled out, like before: (T , ,V ) ln(1 z ) V d ln(1 z e ) D ( ) N (T , ,V ) 0 z 1 V d 1 D ( ) 1 z z e 1 0 U (T , ,V ) V d 0 1 z e 1 activity e z fugacity 3D DOS 3 2 2m D ( ) 2 2 h D ( ) * PV 23 U 104 Specific heat of an ideal Bose gas CV U / N T V ,N A weak singularity … what decides is the coexistence of two phases 108 Specific heat: comparison liquid 4He vs. ideal Bose gas CV U / N T V ,N Dulong-Petit (classical) limit - singularity linear dispersion law Dulong-Petit (classical) limit weak singularity quadratic dispersion law 109 Isotherms in the P-V plane For a fixed temperature, the specific volume V / N can be arbitrarily small. By contrast, the pressure is volume independent …typical for condensation P vc AT 3 / 2 T vc / A 2 / 3 P BT 5 / 2 Pc BA5 / 3 vc 5 / 3 v vc N N BE 3/ 2 g3 / 2 (1) kBT V V P g5 / 2 (1) kBT 5/ 2 110 Compare with condensation of a real gas CO2 Basic similarity: increasing pressure with compression critical line beyond is a plateau Differences: at high pressures at high compressions 111 Compare with condensation of a real gas CO2 Basic similarity: increasing pressure with compression critical line beyond is a plateau Differences: at high pressures at high compressions no critical point Conclusion: Fig. 151 Experimental isotherms of carbon dioxide (CO2}. BEC in a gas is a phase transition of the first order 112 IV. Non-interacting bosons in a trap Useful digression: energy units energy 1K 1K k B /J 1eV kB / e s -1 kB / h 1eV e / kB e/J e/h s-1 h / kB h/e h/J energy 1K 1eV s-1 1K 1.38 1023 8.63 1005 2.08 1010 1eV 1.16 1004 1.60 1019 2.41 1014 s-1 4.80 1011 4.14 1015 6.63 1034 114 Trap potential (physics involved skipped) 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 115 Trap potential Parabolic approximation in general, an anisotropic harmonic oscillator usually with axial symmetry 1 2 1 1 1 2 2 2 2 H p m x x m y y m z2 z 2 2m 2 2 2 Hx H y Hz 1D 2D 3D 116 Ground state orbital and the trap potential 400 nK level number 200 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 , 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 117 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 118 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 119 Filling the trap with particles: IDOS, DOS 1D x E ( E ) int( E / ) E / D ( E ) '( E ) 1 For the finite trap, unlike in the extended gas, D ( E ) is not divided by volume !! 2D (E) x y 1 2 E 2 /( x y ) D ( E ) '( E ) E /( x y ) "thermodynamic limit" only approximate … finite systems better for small E Ex E y E const. kBTc meaning wide trap potentials 120 Filling the trap with particles 3D (E) 1 6 E 3 /( x y z ) D ( E ) '( E ) 12 E 2 /( x y z ) Estimate for the transition temperature particle number comparable with the number of states in the thermal shell N kBT 2D 3D Tc / kB 1 N2 Tc / kB 1 N3 For 106 particles, ( x y 1 )2 ( x y z 1 )3 • characteristic energy kBTc 102 121 Filling the trap with particles 3D (E) 1 6 E 3 /( x y z ) D ( E ) '( E ) 12 E 2 /( x y z ) Estimate for the transition temperature particle number comparable with the number of states in the thermal shell N kBT 2D 3D Tc / kB 1 N2 Tc / kB 1 N3 For 106 particles, kBTc 102 ( x y 1 )2 ( x y z 1 )3 • characteristic energy important for therm. limit 122 Exact expressions for critical temperature etc. The general expressions are the same like for the homogeneous gas. Working with discrete levels, we have N N (T , ) n( j ) j j 1 e ( j ) 1 and this can be used for numerics without exceptions. In the approximate thermodynamic limit, the old equation holds, only the volume V does not enter as a factor: N N (T , ) In 3D, Tc ( (3)) 1 e ( 0 ) 1 3 1 / kB V d 1 N3 0 1 e ( ) 1 0.94 / kB D ( ) 0 for T TC 1 N3 N BE N 1 (T / Tc )3 , T Tc 123 How sharp is the transition These are experimental data fitted by the formula N BE N 1 (T / Tc )3 , T Tc The rounding is apparent, but not really an essential feature 124 Seeing the condensate – reminder Without field-theoretical means, the coherence of the condensate may be studied using the one-particle density matrix. Definition of OPDM for non-interacting particles: Take an additive observable, like local density, or current density. Its average value for the whole assembly of atoms in a given equilibrium state: X X n double average, quantum and thermal = X n insert unit operator | X n X change the summation order define the one-particle density matrix Tr X = n 125 OPDM in the Trap • Use the eigenstates of the 3D oscillator • Use the BE occupation numbers = n ( x , y , z ), w 0, 1, 2, 3, 1 = ( E ) , = x y z e 1 E = E x E y E z = x x + x x + x x • Single out the ground state = 000 1 e 1 BEC 000 (000) 1 e ( E ) 1 TERM 126 OPDM in the Trap • Use the eigenstates of the 3D oscillator • Use the BE occupation numbers = n ( x , y , z ), w 0, 1, 2, 3, 1 = ( E ) , = x y z e 1 E = E x E y E z = x x + x x + x x Coherent component, be it condensate or not. ground / kB , itstate At Tout the contains • Single ALL atoms in the cloud = 000 1 e 1 BEC Incoherent thermal component, coexisting with the condensate. / kB, it freezes out At T and contains NO atoms 000 (000) 1 e ( E ) 1 TERM 127 OPDM in the Trap, Particle Density in Space The spatial distribution of atoms in the trap is inhomogeneous. Proceed by definition: n(r ) Tr (rop r ) Tr d r r ( r r ) r Tr r r r r r (r ) 1 e ( E ) 1 r 1 2 e ( E ) 1 Split into the two parts, the coherent and the incoherent phase n(r ) r r r BEC r r THERM r 128 Particle Density in Space: Boltzmann Limit We approximate the thermal distribution by its classical limit. Boltzmann distribution in an external field: Two directly observable characteristic lengths f B (r , p ) e ( W U ( r )) 1 3 a0 (a0 x a0 y a0 z ) nTHERM (r ) d 3 p f B (r , p ) e e U ( r ) R 1 m 2 T a0 k BT / 1 m(x2 x2 y2 y 2 z2 z 2 ) 2 m , a0 ( x y z ) For comparison: nBEC (r ) 0 x x 0 y y 0 z z 2 1 e 3 a0 x a0 y a0 z 2 1 2 y2 x2 z2 2 2 2 a0x a0 y a0z e 1 1 e 1 anisotropy given by analogous definitions of the two lengths for each direction 134 1 3 Real space Image of an Atomic Cloud • the cloud is macroscopic • basically, we see the thermal distribution • a cigar shape: prolate rotational ellipsoid • diffuse contours: Maxwell – Boltzmann distribution in a parabolic potential 135 Particle Velocity (Momentum) Distribution The procedure is similar, do it quickly: f ( p) p p = 1 p 000 000 ( p) p BEC p p THERM p e 1 e 1 (000) known f BEC ( p) ( p) f BEC ( p) 0 x px 0 y p y 0 z pz 2 p 2y px2 pz2 2 2 2 b0 x b0 y b0 z 1 e 1 , 1 e ( E ) 1 p 1 2 laborious 2 e p (000) 1 2 000 p e ( E ) 1 f THERM ( p) 1 2 e 1 b0 w a0 w 136 Thermal Particle Velocity (Momentum) Distribution Again, we approximate the thermal distribution by its classical limit. Boltzmann distribution in an external field: Two directly observable characteristic lengths f B (r , p ) e ( W U ( r )) 1 3 b0 (b0 x b0 y b0 z ) f THERM (r ) d 3 r f B (r , p ) e W B 1 m T b0 k BT / 1 m1 ( px2 p2y pz2 ) e 2 a0 , b0 Remarkable: f BEC ( p) 0 x px 0 y p y 0 z pz 2 2 e p 2y px2 pz2 2 2 2 b0 x b0 y b0 z 1 e 1 , 2 1 B b0 k BT / T e 1 R a0 k BT / T b0 w b0 a0 Thermal and condensate lengths in the same ratio a0for w positions and momenta 137 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 138 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 139 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 140 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 141 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 142 BEC observed by TOF in the velocity distribution 143 BEC observed by TOF in the velocity distribution Qualitative features: all Gaussians wide vs.narrow isotropic vs. anisotropic 144 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 146 Next time !!! The end 149 V. Interacting atoms Are the interactions important? In the dilute gaseous atomic clouds in the traps, the interactions are incomparably weaker than in liquid helium. 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 • the atomic collisions take care of thermalization • 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 151 Fortunate properties of the interactions 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 elastic and spin independent: they do not 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. 152 Fortunate properties of the interactions 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 elastic and spin independent: they do not 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. 153 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. 154 Interatomic interactions minimum vdW radius 1 6 2 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. 155 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/ 4 156 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 157 Scattering length, pseudopotential Beyond the potential radius, say propagates in free space 3 , the scattered wave 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 2 158 Experimental data as 159 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 h / kB h/e h/J h / Ha a.u. Ha / k B Ha / e Ha / h 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 160 Experimental data 1 a.u. = 1 bohr 0.053 nm for “ordinary” gases 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 NOTES weak attraction ok weak repulsion ok weak attraction intermediate attraction weak repulsion ok strong resonant repulsion "well behaved; monotonous increase seemingly erratic, very interesting physics of scattering resonances behind 161 Experimental data 1 a.u. = 1 bohr 0.053 nm 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 NOTES weak attraction ok weak repulsion ok weak attraction intermediate attraction weak repulsion ok strong resonant repulsion "well behaved; monotonous increase seemingly erratic, very interesting physics of scattering resonances behind 162 Experimental data 1 a.u. = 1 bohr 0.053 nm for “ordinary” gases 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 NOTES weak attraction ok weak repulsion ok weak attraction intermediate attraction weak repulsion ok strong resonant repulsion "well behaved; monotonous increase seemingly erratic, very interesting physics of scattering resonances behind 163 VI. Mean-field treatment of interacting atoms Many-body Hamiltonian and the Hartree approximation 1 2 1 ˆ H pa V (ra ) 2 a 2m U (ra rb ) a b 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 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 165 Hartree approximation 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. 166 Hartree approximation 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. 167 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. 168 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. 169 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. 170 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. 171 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. 172 Gross-Pitaevskii equation – "Bohmian" form For a static condensate, the order parameter has ZERO PHASE. Then ( r ) N 0 ( r ) n( r ) N [ n] N d 3 r ( r ) d 3 r n ( r ) N 2 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 173 Gross-Pitaevskii equation – "Bohmian" form For a static condensate, the order parameter has ZERO PHASE. Then ( r ) N 0 ( r ) n( r ) N [ n] N d 3 r ( r ) d 3 r n ( r ) N 2 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 174 Gross-Pitaevskii equation – "Bohmian" form For a static condensate, the order parameter has ZERO PHASE. Then ( r ) N 0 ( r ) n( r ) N [ n] N d 3 r ( r ) d 3 r n ( r ) N 2 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 175 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). 176 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). 177 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] 178 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 chemical potential E[ n ] N [ n] by definition 179 Interacting atoms in a constant potential The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies N n( r ) n const. V 2 2m n( r ) n( r ) V (r ) V const. V ( r ) g n( r ) g n V 184 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies N n( r ) n const. V (r ) V const. V 2 n( r ) V ( r ) g n( r ) 2m n( r ) g n V 185 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies N n( r ) n const. V (r ) V const. V 2 n( r ) V ( r ) g n( r ) 2m n( r ) g n V The repulsive interaction increases the chemical potential The repulsive interaction increases the chemical potential If g 0, it would be n 0 and the gas would be thermodynamically unstable. 186 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies N n( r ) n const. V (r ) V const. V 2 n( r ) V ( r ) g n( r ) 2m n( r ) g n V The repulsive interaction increases the chemical potential total energy E 12 gn2 V internal pressure E 1 2 P 2 gn V 187 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies N n( r ) n const. V (r ) V const. V 2 n( r ) V ( r ) g n( r ) 2m n( r ) g n V The repulsive interaction increases the chemical potential total energy E 12 gn2 V internal pressure E 1 2 P 2 gn V 2 N E 12 g V 188 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies N n( r ) n const. V (r ) V const. V 2 n( r ) V ( r ) g n( r ) 2m n( r ) g n V The repulsive interaction increases the chemical potential total energy E 12 gn2 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. 189 The simplest case of all: a homogeneous gas In an extended homogeneous system (… Born-Kármán boundary condition), the GP equation simplifies N n( r ) n const. V (r ) V const. V 2 n( r ) V ( r ) g n( r ) 2m n( r ) g n V The repulsive interaction increases the chemical potential total energy E 12 gn2 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. 190 VII. Interacting atoms 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 193 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 u 1 1 2 2 V (u ) m u 2 2 a0 2 • characteristic energy • characteristic length 194 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 195 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 2 196 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 2 3 d r (r ) 1 2 8 as N 2 r r r r a0 197 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 d3r (r ) 1 a single dimensionless a0 parameter 2 8 as N 2 r r r r a0 198 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), perfectly reproduced by the solution of the GP equation 199 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), perfectly reproduced by the solution of the GP equation guess: internal pressure due to repulsion competes with the trap potential 200 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 201 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 202 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 203 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 unlike in an extended interaction together allow the condensate to form. It homogeneous gas !! 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 204 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 E INT E KIN Ngn N N 2 as a03 Nas 2 4 Na0 a0 205 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 2 4 a s The decisive g parameter for the "importance" of interactions is m E INT Ngn N 2 as a03 Nas 2 E KIN N 4 Na0 a0 2 ma02 206 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 E INT E KIN Ngn N N 2 as a03 Nas 2 4 Na0 a0 207 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 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 2 as a03 Nas 2 4 Na0 a0 1 weak individual collisions 208 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 can vary in a wide range Quantitative The decisive parameter for the "importance" of interactions is E INT E KIN Ngn N N 2 as a03 Nas 2 4 Na0 a0 0 realistic value collective effect weak or strong depending on N 1 weak individual collisions 209 Variational method of solving the GPE for atoms 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 212 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 213 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 u2 2 1 0 (u ) e 2a 0 , a0 N ~ 106 at. 2 2 1 1 1 a0 , E0 2 m 2 2 ma0 2 Mum a02 ground state orbital 214 Variational estimate of the condensate properties 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. This is reminescent of the well-known scaling for the ground state of the helium atom. 215 Variational estimate of the condensate properties 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. 216 Importance of the interaction: scaling approximation Variational ansatz: the GP orbital is a scaled ground state for g = 0 1 r2 2 2b 3 0 r A e , A b 2 1/ 4 217 Importance of the interaction: scaling approximation 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 218 Importance of the interaction: scaling approximation 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/ 2 219 Importance of the interaction: scaling approximation 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 0 220 Importance of the interaction: scaling approximation g 0 g 0 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 in units of a0 221 Importance of the interaction: scaling approximation g 0 g 0 Variational estimate ofminimize the total energy the the eneofrgy condensate as a function of the parameter 5 2 0 N 1 as E 2 a0 0 is the orbital width in units of a0 Variational parameter 222 Importance of the interaction: scaling approximation g 0 g 0 Variational estimate of the total energy of the condensate as a function of the parameter 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 N 1 as 2 a0 in units of a0 223 Importance of the interaction: scaling approximation g 0 g 0 Variational estimate of the total energy of the condensate as a function of the parameter 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 N 1 as 2 a0 in units of a0 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 224 attractive atom-atom interaction The end
