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Cold atoms Lecture 1. 20. September 2006 Low temperature physics (borrowed from an undergraduate course) Existence absolutní nuly • Absolutní nula teploty pro ideální plyn definována vztahem 1 m 2 v 2 23 kBT a podmínkou nulové kinetické energie • Pro všechny další systémy se použije transitivnosti teploty pro tělesa v kontaktu (nultý zákon termodynamiky) • Absolutní nula není dostižitelná konečným procesem (3. zákon termodyn.) S 0, Cv 0, • Zvláštní jevy, makroskopické kvantové jevy, jako supravodivost, v blízkosti nuly. Ovšem co je „blízkost“ ? Vysokoteplotní supravodivost, život, … 3 Teploty ve vesmíru Stupnice nitra hvězd 106 - 108 K hvězdné atmosféry 103 - 104 K komety, planety … 101 - 102 K …. reliktní záření jako minimum mlhovina Bumerang (souhvězdí Kentaura) 2,72 K 1,15 K 4 Teploty ve vesmíru Stupnice nitra hvězd 106 - 108 K hvězdné atmosféry 103 - 104 K komety, planety … 101 - 102 K o …. reliktní záření jako minimum mlhovina Bumerang (souhvězdí Kentaura, objevena 1998, teplota určena 2003) Pozemský rekord -89,3 C183.75 K 2,72 K 1983 Antarktida stanice Vostok 1,15 K důvod: rychlá expanse plynů z centrální hvězdy 5 Nízké teploty v laboratoři (jen výběr !!) K 77 22 4,2 0,3 mK K Teplotní rekordy 1877 Pictet kapalný kyslík? 1895 von Linde kap. vzduch 1898 Dewar kapalný vodík 1905 von Linde kap. dusík 1908 Kamerlingh-Onnes kapalné helium odsávané helium 1933 paramagn. demagnet. 1951 H. London rozpouštěcí refrigerátor 1956 Kurti NDR (jaderná …) 1985 Hänsch laserové chlazení (princip) Objevy Teorie 1911 Kamerlingh-Onnes supravodivost kovů 1937 Kapica supratekutost Helia-4 1972 Osheroff supratekutost Helia-3 1986 Müller a Bednorz vysokoteplot. supravodivost nK 1924 Einstein BoseEinsteinova kondensace 1939 Landau teorie supratekutosti 1947 Bogoljubov teorie supratekutosti 1956 BCS * teorie supravodivosti 1975 Leggett teorie supratekutosti Helia-3 1995 Wieman, … Ketterle BEC v atomových parách *Bardeen, Cooper a Schrieffer pK 6 Naše hlavní téma K 77 22 4,2 0,3 mK K Teplotní rekordy 1877 Pictet kapalný kyslík? 1895 von Linde kap. vzduch 1898 Dewar kapalný vodík 1905 von Linde kap. dusík 1908 Kamerlingh-Onnes kapalné helium odsávané helium 1933 paramagn. demagnet. 1951 H. London rozpouštěcí refrigerátor 1956 Kurti NDR (jaderná …) 1985 Hänsch laserové chlazení (princip) Objevy Teorie 1911 Kamerlingh-Onnes supravodivost kovů odsávané helium 1937 Kapica supratekutost Helia-4 1972 Osheroff supratekutost Helia-3 1986 Müller a Bednorz vysokoteplot. supravodivost nK 1924 Einstein BoseEinsteinova kondensace 1939 Landau teorie supratekutosti 1947 Bogoljubov teorie supratekutosti 1956 BCS teorie supravodivosti 1975 Leggett teorie supratekutosti Helia-3 1995 Wieman, … Ketterle BEC v atomových parách *Bardeen, Cooper a Schrieffer pK 7 Nízké teploty v laboratoři (jen výběr !!) K 77 22 4,2 0,3 mK K Teplotní rekordy 1877 Pictet kapalný kyslík? 1895 von Linde kap. vzduch 1898 Dewar kapalný vodík 1905 von Linde kap. dusík 1908 Kamerlingh-Onnes kapalné helium odsávané helium 1933 paramagn. demagnet. 1951 H. London rozpouštěcí refrigerátor 1956 Kurti NDR (jaderná …) 1985 Hänsch laserové chlazení (princip) Objevy 1911 Kamerlingh-Onnes supravodivost kovů 1937 Kapica supratekutost Helia-4 1972 Osheroff supratekutost Helia-3 1986 Müller a Bednorz vysokoteplot. supravodivost nK pK pokrok na logaritmické škále Teorie 1924 Einstein BoseEinsteinova kondensace 1939 Landau teorie supratekutosti 1947 Bogoljubov teorie supratekutosti 1956 BCS * teorie supravodivosti 1975 Leggett teorie supratekutosti Helia-3 1995 Wieman, … Ketterle BEC v atomových parách *Bardeen, Cooper a Schrieffer 8 Chlazení jadernou adiabatickou demagnetisací NDR nuclear demagnetization refrigeration Te elektrony mřížkové kmity pevná látka TL L LS jádra jaderné spiny Princip NDR S I. TS S V rovnováze se teploty všech podsystémů vyrovnají. Spin-mřížková relaxace je pomalá! Můžeme proto generovat nerovnovážnou velmi nízkou spinovou teplotu I. KROK izotermická magnetizace Entropie s magnetickým polem klesá snižuje se orientační neuspořádanost B0 B0 II. KROK adiabatická demagnetizace Teplota a vnitřní energie klesají II. T 9 Kryostat, kde byla dosažena rekordní teplota 100 pK Helsinki University of Technology YKI, Low Temperature Group 2000 1. Předchlazení čerpáním helia 0,7 K 2. První stupeň: rozpouštěcí refrigerátor 3 mK 3. Druhý stupeň: NDR v mědi <0,1 mK 4. Třetí stupeň: NDR v samotném vzorku: monokrystal Rh <1 nK 10 Spinová magnetická susceptibilita monokrystalu rhodia Curie-Weissův zákon jaderné spiny v rhodiu … antiferomagnetické uspořádání 11 12 Introductory matter on bosons Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable 14 Bosons and Fermions (capsule reminder) independent quantum postulate Identical particles are indistinguishable 15 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 ) 16 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 ) 17 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 half-integer spin integer spin 18 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 19 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" electrons photons comes from nowhere 20 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" electrons photons everybody knows our present concern comes from nowhere 21 Bosons and Fermions (capsule reminder) Independent particles (… non-interacting) basis of single-particle states ( complete set of quantum numbers) x x 22 Bosons and Fermions (capsule reminder) Independent particles (… non-interacting) basis of single-particle states ( complete set of quantum numbers) x x FOCK SPACE 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 23 Bosons and Fermions (capsule reminder) Independent particles (… non-interacting) basis of single-particle states ( complete set of quantum numbers) x x FOCK SPACE 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 24 Bosons and Fermions (capsule reminder) Representation of occupation numbers (basically, second quantization) …. for fermions Pauli principle fermions keep apart – as sea-gulls , , , , 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 ') not allowed 1 F 2 1 ,1 , 3 ,1 , 0 , p ( x) 1 1 2 1 2 1 N -particle ground state N 25 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 -částicový základní stav ( x1 ) ( x2 ) ( xN ) 1 1 1 26 2 Examples of bosons bosons complex particles N conserved simple particles N not conserved elementary particles photons quasi particles phonons magnons atoms 4 He, 7 Li, 23 Na, 87 Rb alkali metals excited atoms 27 Examples of bosons (extension of the table) bosons complex particles N conserved simple particles N not conserved elementary particles photons quasi particles phonons magnons composite quasi particles atoms 4 He, 7 Li, 23 Na, 87 Rb alkali metals excited atoms ions molecules excitons Cooper pairs 28 Digression: How a complex particle, like an atom, can behave as a single whole, a boson ESSENTIAL CONDITION the identity includes characteristics like mass of charge, but also the values of observables corresponding to internal degrees of freedom, which are not allowed to vary during the dynamical processes in question. 29 Digression: How a complex particle, like an atom, can behave as a single whole, a boson ESSENTIAL CONDITION the identity includes characteristics like mass or charge, but also the values of observables corresponding to internal degrees of freedom, which are not allowed to vary during the dynamical processes in question. 30 Digression: How a complex particle, like an atom, can behave as a single whole, a boson ESSENTIAL CONDITION the identity includes characteristics like mass of charge, but also the values of observables corresponding to internal degrees of freedom, which are not allowed to vary during the dynamical processes in question. Rubidium 37 electrons total electron spin S 12 total nuclear spin I 37 protons 50 neutrons total spin of the atom 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 31 3 2 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 hd 32 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: 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 33 Ideal quantum gases at a finite temperature n e ( ) Boltzmann distribution high temperatures, dilute gases 34 Ideal quantum gases at a finite temperature n e ( ) Boltzmann distribution 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 high temperatures, dilute gases fermions N FD n bosons N 1 n e ( ) 1 T 0 F 1 ,1 , ,1 , 0 , 1 BE e ( ) 1 T 0 B N , 0 ,0 , ,0 , n 1 e 1 T 0 vac 36 Ideal quantum gases at a finite temperature n e ( ) Boltzmann distribution high temperatures, dilute gases fermions N FD n bosons N 1 n e ( ) 1 T 0 1 BE e ( ) 1 T 0 n 1 e 1 T 0 freezing out F 1 ,1 , ,1 , 0 , B N , 0 ,0 , ,0 , vac 37 Ideal quantum gases at a finite temperature n e ( ) Boltzmann distribution high temperatures, dilute gases fermions N FD n bosons N 1 n e ( ) 1 T 0 F 1 ,1 , ,1 , 0 , B 1 BE e ( ) 1 n 1 e 1 T 0 T 0 ? freezing out N , 0 ,0 , ,0 , vac 38 The Planck formula for black-body radiation Equilibrium radiation in a cavity Plane electromagnetic wave and its "photon" transcription two polarizations E E0 s e it kr , ck , 2 / k , p k , cp, h / p hc / CAVITY walls at temperature T emit and absorb radiation inside the cavity an equilibrium distribution of radiation … depends only on the temperature ENERGY DENSITY PER UNIT VOLUME (T , ) 2 n D ( ) polarization photon energy population of a mode DOS 40 Planck formula ENERGY DENSITY PER UNIT VOLUME (T , ) 2 n D ( ) polarization photon energy population of a mode n 1 e 1 DOS 4 D ( ) 3 3 2 ch d 8 3 W (T , ) 3 3 d c h e 1 d 8 h 3 W (T , ) 3 h d c e kBT 1 PLANCK FOMULA IN THE STANDARD FORM: our final result 41 Stephan-Boltzmann law ENERGY DENSITY PER UNIT VOLUME d 8 3 W (T , ) 3 3 d c h e 1 TOTAL ENERGY PER UNIT VOLUME 4 3 8 3 8 k W d (T , ) 3 3 d T 4 3 3B d c h 0 e 1 c h 0 e 1 0 W T 4 Stefan-Boltzmann law 8 k 15c h universal constant 5 4 B 3 3 4 15 42 Stephan-Boltzmann law ENERGY DENSITY PER UNIT VOLUME d 8 3 W (T , ) 3 3 d c h e 1 TOTAL ENERGY PER UNIT VOLUME 4 3 8 3 8 k W d (T , ) 3 3 d T 4 3 3B d c h 0 e 1 c h 0 e 1 0 W T 4 Stefan-Boltzmann law 8 k 15c h universal constant 5 4 B 3 3 4 15 43 Stephan-Boltzmann law ENERGY DENSITY PER UNIT VOLUME d 8 3 W (T , ) 3 3 d c h e 1 TOTAL ENERGY PER UNIT VOLUME 4 3 8 3 8 k W d (T , ) 3 3 d T 4 3 3B d c h 0 e 1 c h 0 e 1 0 W T 4 Stefan-Boltzmann law 8 k 15c h universal constant 5 4 B 3 3 4 15 44 Stephan-Boltzmann law ENERGY DENSITY PER UNIT VOLUME d 8 3 W (T , ) 3 3 d c h e 1 TOTAL ENERGY PER UNIT VOLUME 4 3 8 3 8 k W d (T , ) 3 3 d T 4 3 3B d c h 0 e 1 c h 0 e 1 0 W T 4 Stefan-Boltzmann law 8 k 15c h universal constant 5 4 B 3 3 4 15 45 Stephan-Boltzmann law ENERGY DENSITY PER UNIT VOLUME d 8 3 W (T , ) 3 3 d c h e 1 TOTAL ENERGY PER UNIT VOLUME 4 3 8 3 8 k W d (T , ) 3 3 d T 4 3 3B d c h 0 e 1 c h 0 e 1 0 W T 4 Stefan-Boltzmann law 8 k 15c h universal constant 5 4 B 3 3 4 15 46 Bose-Einstein condensation: elementary approach Einstein's manuscript with the derivation of BEC 48 What is the nature of BEC? With lowering the temperature, the atoms of the gas lose their energy and drain down to the lowest energy states. There is less and less of these: N ( E k BT ) const T 3/ 2 A given amount N of the atoms becomes too large starting from a critical temperature. Their excess precipitates to the lowest level, which becomes macroscopically occupied, i.e., it holds a finite fraction of all atoms. This is the BE condensate. At the zero temperature, all atoms are in the condensate. Einstein was the first to realize that and to make an exact calculation of the integrals involved. 3 2 3 2 mk T B 3 3 2 N G (T ) V 4 ( 2 ) ( 2 ) BT 2 h 49 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. 50 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 51 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 , ) 52 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? 53 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? 54 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 55 Condensate concentration 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 use the general formula /2 Riemann function 3 2 2mk BT V 2 2,612 2 h 56 Condensate 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 57 Condensate 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 58 Condensate 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 h2 Tc 4 mk B atomic mass 2 3 2 3 2 3 N h2 n 19 n 8,0306 10 0,52725 4 uk B M M 2,612V 59 Condensate concentration 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 19 n 0,52725 8,0306 10 4 uk B M M 2,612V 2 A few estimates: system M n TC He liquid 4 21028 1.47 K Na trap 23 21020 1.19 K Rb trap 87 21017 3.16 nK 60 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 mk BTc mean interatomic distance thermal de Broglie wavelength The quantum breakdown sets on when the wave clouds of the atoms start overlapping 61 de Broglie wave length for atoms and molekules 2 p Thermal energies small … NR formulae valid: 2 2mEkin m Au ... at. (mol.) mass At thermal equilibrium Ekin 32 k BT thermal wave length 2 1 1 2,5 109 3u k B AT AT Two useful equations Ekin 32 T /11600 eV K v v 2 158 T A 62 Ketterle explains BEC to the King of Sweden 63 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 64 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 65 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 k BT k BT n0 O(V ), n1 O(V 3 ) .... much slower growth 0 1 66 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) 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 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' (0 ) BE (0 ) 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 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 84 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 ? why bother? • 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 ? what is it? ? how? 85 F.Laloë: Do we really understand Quantum mechanics, Am.J.Phys.69, 655 (2001) 86 The end Problems Some problems are expanding on the presented subject matter and are voluntary… (*) The other ones are directly related to the theme of the class and are to be worked out within a week. The solutions will be presented on the next seminar and posted on the web. (1.1*) Problems with metastable states and quasi-equilibria in defining the temperature and applying the 3rd law of thermodynamics (1.2*) Relict radiation and the Boomerang Nebula (1.3) Work out in detail the integral defining Tc (1.4) Extend the resulting series expansion to the full balance equation (BE integral) (1.5) Modify for a 2D gas and show that the BE condensation takes never place (1.6) Obtain an explicit procedure for calculating the one-particle density matrix for an ideal boson gas [difficult] 89