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Cold atoms 21st Lecture 7. February, 2007 Preliminary plan/reality in the fall term Lecture 1 … Something about everything (see next slide) The textbook version of BEC in extended systems Sep 22 Lecture 2 … thermodynamics, grand canonical ensemble, extended gas: ODLRO, nature of the BE phase transition Oct 4 Lecture 3 … atomic clouds in the traps – independent bosons, what is BEC?, "thermodynamic limit", properties of OPDM Oct 18 Lecture 4 … atomic clouds in the traps – interactions, GP equation at zero temperature, variational prop., chem. potential Nov 1 Lecture 5 … Infinite systems: Bogolyubov theory Nov 15 Lecture 6 … BEC and symmetry breaking, coherent states Nov 29 Lecture 7 … Time dependent GP theory. Finite systems: BEC theory preserving the particle number 2 Recapitulation Offering many new details and alternative angles of view BEC in atomic clouds Nobelists I. Laser cooling and trapping of atoms The Nobel Prize in Physics 1997 "for development of methods to cool and trap atoms with laser light" Steven Chu Claude CohenTannoudji William D. Phillips 1/3 of the prize 1/3 of the prize 1/3 of the prize USA France USA Stanford University Stanford, CA, USA Collège de France; École Normale Supérieure Paris, France National Institute of Standards and Technology Gaithersburg, MD, USA b. 1948 b. 1933 (in Constantine, Algeria) b. 1948 5 Doppler cooling in the Chu lab 6 Doppler cooling in the Chu lab atomic cloud 7 Nobelists II. BEC in atomic clouds 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 8 Trap potential evaporation cooling Typical profile ? coordinate/ microns This is just one direction Presently, the traps are mostly 3D The trap is clearly from the real world, the atomic cloud is visible almost by a naked eye 9 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 length10 BEC observed by TOF in the velocity distribution 11 Ketterle explains BEC to the King of Sweden 12 Simple estimate of TC (following the Ketterle slide) The quantum breakdown sets on when the wave clouds of the atoms start overlapping mean interatomic distance V N 1 3 thermal de Broglie wavelength h mk BTc Critical temperature 2 ESTIMATE TRUE EXPRESSION Tc h mk B N V h2 Tc 4 mk B 2 3 N 2,612 V 2 3 13 Interference of atoms Two coherent condensates are interpenetrating and interfering. Vertical stripe width …. 15 m Horizontal extension of the cloud …. 1,5mm 14 Today, we will be mostly concerned with the extended (" infinite" ) BE gas/liquid Microscopic theory well developed over nearly 60 past years Interacting atoms 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 17 Many-body Hamiltonian 1 2 1 ˆ H pa V (ra ) 2 a 2m U (ra rb ) a b True interaction potential at low energies (micro-kelvin range) replaced by an effective potential, Fermi pseudopotential U (r ) g (r ) 4 as g m 2 , as ... the scattering length Experimental data 18 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 20 Gross-Pitaevskii equation at zero temperature Consider a condensate. Then all occupied orbitals are the same and we have a single self-consistent equation for a single orbital 2 1 2 p V (r ) gN 0 r 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. 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 21 Gross-Pitaevskii equation – homogeneous gas The GP equation simplifies 2 2 g r r r 2m For periodic boundary conditions in a box with V Lx Ly Lz 1 0 (r ) V ( r ) N 0 ( r ) N n V g r r r 2 ... GP equation g r gn 2 1 E 1 3 2 1 2 2 d r ( n ) V (r )n g n g n N N 2 2m 2 22 Field theoretic reformulation (second quantization) Purpose: go beyond the GP approximation, treat also the excitations Field operator for spin-less bosons Definition by commutation relations (r ), † (r' ) (r r' ), (r ), (r' ) 0, † ( r ), † ( r' ) 0 basis of single-particle states ( complete set of quantum numbers) r r , ... single particle state r r decomposition of the field operator r r a , a " " d3 * r r † r * r a† commutation relations a , a† , a , a 0, a† , a† 0 24 Action of the field operators in the Fock space basis of single-particle states , ... single particle state r r r r FOCK SPACE F space of many particle states basis states … symmetrized products of single-particle states for bosons specified by the set of occupation numbers 0, 1, 2, 3, … , , , , p , n n1 , n2 , n3 , , np , 1 2 3 n -particle state n Σn p a †p n1 , n2 , n3 , , np , n p 1 n1 , n2 , n3 , a p n1 , n2 , n3 , , np , n p n1 , n2 , n3 , , n p 1, , n p 1, 25 Action of the field operators in the Fock space Average values of the field operators in the Fock states Off-diagonal elements only!!! The diagonal elements vanish: n1 , n2 , n3 , , np , n1 , n2 , n3 , , np , a p n1 , n2 , n3 , , np , , n p 1, n p n1 , n2 , n3 , 0 Creating a Fock state from the vacuum: n1 , n2 , n3 , , np , p 1 np a ! † p np vac 26 Field operator for spin-less bosons – cont'd Important special case – an extended homogeneous system Translational invariance suggests to use the Plane wave representation (BK normalization) r V 1/ 2 ei kr ak , ak V 1/ 2 d3 r e i kr r † r V 1/ 2 e i kr ak† V 1/ 2 ei kr a-k† The other form is made possible by the inversion symmetry (parity) important, because the combination u ak + v a-k† corresponds to the momentum transfer by k Commutation rules do not involve a - function, because the BK momentum is discrete, albeit quasi-continuous: ak , ak'† kk' , ak , ak' 0, ak† , ak'† 0 27 Operators Additive observable X Xj X d3 r d3 r' † (r ) r X r' (r' ) General definition of the one particle density matrix – OPDM X d 3 r d 3 r' † (r ) r X r' (r' ) d 3 r d 3 r' r X r' † (r ) (r' ) d 3 r d 3 r' r X r' r' r Tr X r' r Particle number N 1OP, j N a† a Momentum P pj N d3 r † (r ) (r ) P d3 r † (r ) i (r ) P k a† a 28 Hamiltonian 1 2 H pa V (ra ) single-particle additive a 2m 1 U (ra rb ) two-particle binary 2 a b d 3 r † (r ) 2 m V (r ) ( r ) 2 12 d 3 r d 3 r' † (r ) † (r' )U (r r' ) (r' ) (r ) 29 Hamiltonian 1 2 H pa V (ra ) single-particle additive a 2m acts in the N -particle sub-space H N F 1 U (ra rb ) two-particle binary 2 a b d 3 r † (r ) 2 m V (r ) ( r ) 2 acts in the whole Fock space F 12 d 3 r d 3 r' † (r ) † (r' )U (r r' ) (r' ) (r ) 30 Hamiltonian 1 2 H pa V (ra ) single-particle additive a 2m acts in the N -particle sub-space H N F 1 U (ra rb ) two-particle binary 2 a b d 3 r † (r ) 2 m V (r ) ( r ) 2 acts in the whole Fock space F 12 d 3 r d 3 r' † (r ) † (r' )U (r r' ) (r' ) (r ) but Particle number conservation H ,N 0 Equilibrium density operators and the ground state (ergodic property) P P (H ), N , P 0 31 On symmetries and conservation laws Hamiltonian is conserving the particle number Particle number conservation H ,N 0 Equilibrium density operators and the ground state (ergodic property) P P (H ), N , P 0 Typical selection rule (r ) Tr (r )P 0 is a consequence: (similarly 0, † 0, ) Proof: 0 Tr [N , P ] Tr P [ , N ] Tr P Tr A[ B, C ] Tr C[ A, B ] [ ( x), d x ' † ( x ') ( x ')] d x '( † ( x ')[ ( x), ( x ')] [ ( x), † ( x ')] ( x ')) ( x) QED Deeper insight: gauge invariance of the 1st kind 33 Gauge invariance of the 1st kind Particle number conservation H ,N 0 Equilibrium density operators and the ground state (ergodic property) P P (H ), N , P 0 Gauge invariance of the 1st kind H ,N 0 ei N H e i N H unitary transform The equilibrium states have then the same invariance property: N , P 0 e i N P ei N P Selection rule rederived: Tr P Tr e i N P ei N Tr ei N e i N P ei Tr P (1 ei ) Tr P 0 Tr ( r )P ( r ) 0 34 Hamiltonian of a homogeneous gas H a 1 2 1 pa V 2m 2 U (r a b a rb ), V const. d 3 r † (r ) 2 m V (r ) 12 d 3 r d 3 r' † (r ) † (r' )U ( r r' ) ( r' ) ( r ) 2 ToTranslationally study the symmetry properties Hamiltonian invariant system … howof to the formalize (and to learn more about the gauge invariance) Proceed in three steps … T † (a )HT (a ) H, a R3 ... translation vector in the direction reverse to that for the gauge invariance Constructing the unitary operatorT T (a ) (r1 , r2 , r3 , e i p a / (r1 , rp , (a ) rN ) (r1 a, r2 a, r3 a, rN ) e i p a / (r1 , T (a ) e i P a / rp a , rN ) e iP a / (r1 , rN a ) rN ) ... compare O ( ) e i N [N ,H ]=0 Infinitesimal translation H T † (a )HT (a ) e iP a / H e iP a / H i/ P aH i/ HP a O( a 2 ) [H , P ]a 0 [H , P x, y , z ] 0 ... momentum conservation 35 Hamiltonian of the homogeneous gas H a 1 2 1 pa V 2m 2 U (r a b a rb ), V const. d 3 r † (r ) 2 m V (r ) 12 d 3 r d 3 r' † (r ) † (r' )U ( r r' ) ( r' ) ( r ) 2 Translationally invariant system … how to formalize (and to learn more about the gauge invariance) † T (a )H T (a ) H, a R3 ... translation vector Constructing the unitary operatorT T (a ) (r1 , r2 , r3 , e i p a / (r1 , rp , (a ) rN ) (r1 a, r2 a, r3 a, rN ) e i p a / (r1 , T (a ) e i P a / rp a , rN ) e iP a / (r1 , rN a ) rN ) ... compare O ( ) e i N [N ,H ]=0 Infinitesimal translation H T † (a )HT (a ) e iP a / H e iP a / H i/ P aH i/ HP a O( a 2 ) [H , P ]a 0 [H , P x, y , z ] 0 ... momentum conservation 36 Hamiltonian of the homogeneous gas H a 1 2 1 pa V 2m 2 U (r a b a rb ), V const. d 3 r † (r ) 2 m V (r ) 12 d 3 r d 3 r' † (r ) † (r' )U ( r r' ) ( r' ) ( r ) 2 Translationally invariant system … how to formalize (and to learn more about the gauge invariance) † T (a )H T (a ) H, a R3 ... translation vector Constructing the unitary operatorT (a ) T (a ) (r1 ,in r2 ,the r3 , one-particle rp , rN ) orbital (r1 a, rspace Translation 2 a , r3 a , i p a / rp a , rN a ) i p a / iP an/ e (r1 , rN ) 1e ( r , r ) e rN /) 1 i pa ( r1 , i pa 1 N n T (a ) (r ) (r a ) (a ) (r ) (r ) e i N ( r ) i P a / n ! compare O ( ) e Tn !(a ) e ... [N ,H ]=0 Infinitesimal translation H T † (a )HT (a ) e iP a / H e iP a / H i/ P aH i/ HP a O( a 2 ) [H , P ]a 0 [H , P x, y , z ] 0 ... momentum conservation 37 Hamiltonian of the homogeneous gas H a 1 2 1 pa V 2m 2 U (r a b a rb ), V const. d 3 r † (r ) 2 m V (r ) 12 d 3 r d 3 r' † (r ) † (r' )U ( r r' ) ( r' ) ( r ) 2 Translationally invariant system … how to formalize (and to learn more about the gauge invariance) † T (a )H T (a ) H, a R3 ... translation vector Constructing the unitary operatorT T (a ) (r1 , r2 , r3 , e i p a / (r1 , rp , (a ) rN ) (r1 a, r2 a, r3 a, rN ) e i p a / (r1 , T (a ) e i P a / rp a , rN ) e iP a / (r1 , rN a ) rN ) ... compare O ( ) e i N [N ,H ]=0 Infinitesimal translation H T † (a )HT (a ) e iP a / H e iP a / H i/ P aH i/ HP a O( a 2 ) [H , P ]a 0 [H , P x, y , z ] 0 ... momentum conservation 38 Hamiltonian of the homogeneous gas H a 1 2 1 pa V 2m 2 U (r a b a rb ), V const. d 3 r † (r ) 2 m V (r ) 12 d 3 r d 3 r' † (r ) † (r' )U ( r r' ) ( r' ) ( r ) 2 Translationally invariant system … how to formalize (and to learn more about the gauge invariance) † T (a )H T (a ) H, a R3 ... translation vector Constructing the unitary operatorT T (a ) (r1 , r2 , r3 , e i p a / (r1 , rp , (a ) rN ) (r1 a, r2 a, r3 a, rN ) e i p a / (r1 , T (a ) e i P a / rp a , rN ) e iP a / (r1 , rN a ) rN ) ... compare O ( ) e i N [N ,H ]=0 Infinitesimal translation H T † (a )HT (a ) e iP a / H e iP a / H i/ P aH i/ HP a O( a 2 ) [H , P ]a 0 [H , P x, y , z ] 0 ... momentum conservation 39 Hamiltonian of the homogeneous gas H a 1 2 1 pa V 2m 2 U (r a b a rb ), V const. d 3 r † (r ) 2 m V (r ) 12 d 3 r d 3 r' † (r ) † (r' )U ( r r' ) ( r' ) ( r ) 2 Translationally invariant system … how to formalize (and to learn more about the gauge invariance) † T (a )H T (a ) H, a R3 ... translation vector Constructing the unitary operatorT T (a ) (r1 , r2 , r3 , e i p a / (r1 , rp , (a ) rN ) (r1 a, r2 a, r3 a, rN ) e i p a / (r1 , T (a ) e i P a / rp a , rN ) e iP a / (r1 , rN a ) rN ) ... compare O ( ) e i N [H ,N ]=0 Infinitesimal translation H T † (a )HT (a ) e iP a / H e iP a / H i/ P aH i/ HP a O( a 2 ) [H , P ]a 0 [H , P x, y , z ] 0 ... momentum conservation 40 Summary: two symmetries compared Gauge invariance of the 1st kind Translational invariance universal for atomic systems specific for homogeneous systems O † ( )HO ( ) H, 0, 2 T † (a )HT (a ) H, a R3 O ( ) e i N T (a ) e i P a / global phase shift of the wave function global shift in the configuration space [H , P x , y , z ] 0 [H ,N ]=0 particle number conservation N , P 0 e iN Pe iN P total momentum conservation P , P 0 e i Pa Pe i Pa for equilibrium states for equilibrium states selection rules selection rules † 0 unless there are as many ak ak' † as . P † ak'' 0 unless the total momentum transfer 41 k k ' k '' is0zero Hamiltonian of the homogeneous gas In the momentum representation H 2m k 2 ak† ak 12 V 1 U q ak† q ak'† q ak' ak 2 kk'q U k d3 r e i kr U r k' q k q k' k Momentum conservation k + q + k' q k k' = 0 Particle number conservation a † a †a a 42 Hamiltonian of the homogeneous gas For the Fermi pseudopotential In the momentum representation U q U 0 U ( g ) H 2m k 2 ak† ak 12 V 1 U q ak† q ak'† q ak' ak 2 kk'q U k d3 r e i kr U r k' q k q k' k Momentum conservation k + q + k' q k k' = 0 Particle number conservation a † a †a a 43 Bogolyubov method Originally, intended and conceived for extended (rather infinite) homogeneous system. Reflects the 'Paradoxien der Unendlichen' Basic idea Bogolyubov method is devised for boson quantum fluids with weak interactions – at T=0 now no interaction weak interaction g 0 g0 N N BE a0† a0 1 N N BE ak† ak N BE 1 k 0 The condensate dominates, but some particles are kicked out by the interaction (not thermally) Strange idea introduced by Bogolyubov N 0 a0† a0 1 a0† a0 a0† a0 a0 a0† 1 like c-numbers a0 N 0 , a0† N 0 N N 0 ak† ak ... mixture of c-numbers and q -numbers k 0 45 Basic idea Bogolyubov method is devised for boson quantum fluids with weak interactions – at T=0 now no interaction weak interaction g 0 g0 N N BE a0† a0 1 N N BE ak† ak N BE 1 k 0 The condensate dominates, but some particles are kicked out by the interaction (not thermally) Strange idea introduced by Bogolyubov N 0 a0† a0 1 a0† a0 a0† a0 a0 a0† 1 like c-numbers a0 N 0 , a0† N 0 N N 0 ak† ak ... mixture of c-numbers and q -numbers k 0 46 Approximate Hamiltonian Keep at most two particles out of the condensate, use a0 N0 , a0† N0 H 2m k 2 ak† ak 12 V 1 U q ak† q , ak'† q ak' ak 2 kk'q † k k UN 2V0 a a 4a a ak a k UN 02 2V † k k UN 2V a a 2a a ak a k UN 2 2V 2m k a a 2 k 2 2m k 2 2 aa k k 0 k 0 † † k k † † k k † k k † k k k k k k k k condensate particle 47 Approximate Hamiltonian Keep at most two particles out of the condensate, use a0 N0 , a0† N0 H 2m k 2 ak† ak 12 V 1 U qq ak† q , ak'† q ak' ak 2 kk'q † k k UN 2V0 a a 4a a ak a k UN 02 2V † k k UN 2V a a 2a a ak a k UN 2 2V 2m k a a 2 k 2 2m k 2 2 aa k k 0 k 0 † † k k † † k k † k k † k k k k k k k k condensate particle 48 Approximate Hamiltonian Keep at most two particles out of the condensate, use a0 N0 , a0† N0 H 2m k 2 ak† ak 12 V 1 U q ak† q , ak'† q ak' ak 2 kk'q † k k UN 2V0 a a 4a a ak a k UN 02 2V † k k UN 2V a a 2a a ak a k UN 2 2V 2m k a a 2 k 2 2m k 2 2 aa k k 0 k 0 † † k k † † k k † k k † k k k k k k k k condensate particle 49 Approximate Hamiltonian Keep at most two particles out of the condensate, use a0 N0 , a0† N0 H 2m k 2 ak† ak 12 V 1 U q ak† q , ak'† q ak' ak 2 kk'q † k k UN 2V0 a a 4a a ak a k UN 02 2V † k k UN 2V a a 2a a ak a k UN 2 2V 0th order condensate 2m k a a 2 2 2m k 2 2 aa 3rd & 4th order neglected k k k 0 k 0 † † k k † † k k † k k † k k 2nd order pair excitations k 1st order is zero violates k-conservation k k k k k condensate particle 50 Approximate Hamiltonian Keep at most two particles out of the condensate, use a0 N0 , a0† N0 H 2m k 2 ak† ak 12 V 1 U q ak† q , ak'† q ak' ak 2 kk'q † k k UN 2V0 a a 4a a ak a k UN 02 2V † k k UN 2V a a 2a a ak a k UN 2 2V 2m k a a 2 k 2 2m k 2 2 aa k k 0 k 0 † † k k † † k k † k k † k k k k k k k k condensate particle 51 Approximate Hamiltonian Keep at most two particles out of the condensate, use a0 N0 , a0† N0 H 2m k 2 ak† ak 12 V 1 U q ak† q , ak'† q ak' ak 2 use N 0 N ak† ak k 0 kk'q † k k UN 2V0 a a 4a a ak a k UN 02 2V † k k UN 2V a a 2a a ak a k UN 2 2V 2m k a a 2 2 2m k 2 2 aa k 0 k 0 † † k k † † k k † k k † k k The idea: replace the unknown condensate occupation by the known particle number neglecting again higher than pair excitations k k k k k k k k condensate particle 52 Approximate Hamiltonian Keep at most two particles out of the condensate, use a0 N0 , a0† N0 H 2m k 2 ak† ak 12 V 1 U q ak† q , ak'† q ak' ak 2 use N 0 N ak† ak k 0 kk'q † k k UN 2V0 a a 4a a ak a k UN 02 2V † k k UN 2V a a 2a a ak a k UN 2 2V 2m k a a 2 2 2m k 2 2 aa k 0 k 0 † † k k † † k k † k k † k k The idea: replace the unknown condensate occupation by the known particle number neglecting again higher than pair excitations k k k k k k k k condensate particle 53 Approximate Hamiltonian Keep at most two particles out of the condensate, use a0 N0 , a0† N0 H 2m k 2 ak† ak 12 V 1 U q ak† q , ak'† q ak' ak 2 use N 0 N ak† ak k 0 kk'q † k k UN 2V0 a a 4a a ak a k UN 02 2V † k k UN 2V a a 2a a ak a k UN 2 2V 2m k a a 2 k 2 2m k 2 2 aa k k 0 k 0 † † k k † † k k † k k † k k k k k k k k condensate particle 54 Bogolyubov transformation Last rearrangement H 1 2 2m k 2 2 gn a a a a k † k k † k gn 2 a a † † k k ak a k k gN 2 2V anomalous mean field Conservation properties: momentum … YES, particle number … NO NEW FIELD OPERATORS notice momentum conservation!! bk uk ak vk a† k ak uk bk vk b†k b†k vk ak uk a† k a† k vk bk uk b†k requirements New operators should satisfy the boson commutation rules bk , bk'† kk' , bk , bk' 0, bk† , bk'† 0 iff uk2 vk2 1 In terms of the new operators, the anomalous terms in the Hamiltonian have to vanish 55 Bogolyubov transformation Last rearrangement H 1 2 2m k 2 2 gn a a a a k † k k † k gn 2 a a † † k k ak a k k gN 2 2V anomalous mean field Conservation properties: momentum … YES, particle number … NO NEW FIELD OPERATORS notice momentum conservation!! bk uk ak vk a† k ak uk bk vk b†k b†k vk ak uk a† k a† k vk bk uk b†k requirements New operators should satisfy the boson commutation rules bk , bk'† kk' , bk , bk' 0, bk† , bk'† 0 iff uk2 vk2 1 In terms of the new operators, the anomalous terms in the Hamiltonian have to vanish 56 Bogolyubov transformation – result Without quoting the transformation matrix (k ) bk†bk b†k b k H 12 independent quasiparticles (k ) 2 k gn 2m 2 2 gN 2 higher order constant 2V ground state energy E gn 2m k 2 2m k 2 2 gn 2 2 2 57 Bogolyubov transformation – result Without quoting the transformation matrix H 1 2 (k )b b gN 2 2V † k k ind. quasi-particles (k ) 2 k gn 2m 2 2 ground state energy E gn 2m k 2 asymptotically merge 2 2m (k ) 2 2 k 2m k k higher order constant 2 2 2 2 k 2 gn 2m high energy region k gn 2 quasi-particles are nearly just particles GP (k ) c k c gn m cross-over k sound region 4mgn 2 defines scale for k quasi-particles are collective excitations 58 More about the sound part of the dispersion law Entirely dependent on the interactions, both the magnitude of the velocity (k ) c k and the linear frequency range determined by g Can be shown to really be a sound: V VV E c " " , mn 2 gN E 2V c gn m Even a weakly interacting gas exhibits superfluidity; the ideal gas does not. The phonons are actually Goldstone modes corresponding to a broken symmetry The dispersion law has no roton region, contrary to the reality in 4He The dispersion law bends upwards quasi-particles are unstable, can decay 59 Particles and quasi-particles At zero temperature, there are no quasi-particles, just the condensate. Things are different with the true particles. Not all particles are in the condensate, but they are not thermally agitated in an incoherent way, they are a part of the fully coherent ground state ak† ak vk bk uk b†k uk b k vk bk† vk2 0 The total fraction of particles outside of the condensate is N N0 8 3/ 2 1/ 2 as n N 3 as3 n square root of the gas parameter is the expansion variable 60 What is the Bogolyubov approximation about The results for various quantities are 8 N 0 N 1 as3/ 2 n1/ 2 3 gn 128 3/ 2 1/ 2 E N 1 as n 2 15 gn 1 as3/ 2 n1/ 2 3 [BG] [GP] 1 as3/ 2 n1/ 2 general pattern 32 square root of the gas parameter as3 n is the expansion variable The Bogolyubov theory is the lowest order correction to the mean field (GrossPitaevskii) approximation It provides thus the criterion for the validity of the mean field results It is the simplest genuine field theory for quantum liquids with a condensate 61 Trying to understand the Bogolyubov method Notes to the contents of the Bogolyubov theory The first consistent microsopic theory of the ground state and elementary excitations (quasi-particles) for a quantum liquid (1947) The quantum condensate turns into the classical order parameter in the thermodynamic limit N , V , N / V n const. The Bogolyubov transformation became one of the standard technical means for treatment of "anomalous terms" in many body Hamiltonians (…de Gennes) Central point of the theory is the assumption a0 N0 , a0† N0 Its introduction and justification intuitive, surprisingly lacks mathematical rigor. Two related problems: lowering operator a0 G, N H N 1 ? gauge symmetry, s. rule G, N a0 G, N N 0 a0 0 Additional assumptions: something of a crutch/bar to study of finite systems homogeneous system infinite system Infinity as a problem: philosophical, mathematical, physical 63 ODLRO in the Bogolyubov theory Basic expressions for the OPDM for a homogeneous system r' r † (r ) (r' ) V 1 e i kr ak† ei k'r' ak' V 1 ei k'r' e i kr ak† ak' V 1 ei k'r' e i kr ak† ak kk' k ,k ' by definition transl. invariance k ,k ' 64 Off-diagonal long range order ODLRO in the Bogolyubov theory One particle density matrix Basic expressions for the OPDM for a homogeneous system r' r † (r ) (r' ) V 1 e i kr ak† ei k'r' ak' V 1 ei k'r' e i kr ak† ak' V 1 ei k'r' e i kr ak† ak kk' k ,k ' by definition transl. invariance k ,k ' 65 ODLRO in the Bogolyubov theory Basic expressions for the OPDM for a homogeneous system r r' † (r' ) (r ) V 1 e i k'r' ak'† ak ei kr V 1 ei kr e i k'r' ak'† ak V 1 ei kr e i k'r' ak† ak kk' k ,k ' by definition transl. invariance k ,k ' General expression for the one particle density matrix with condensate (r , r ') V 1 ei k ( r r ') nk k0 0, n0 N 0 k V 1 ei k0 ( r r ') n0 V 1 ei k ( r r ') nk coherent across the sample (r ) (r' ) dyadic k k0 FT of a smooth function has a finite range V 1 ei k ( r r ') nk N 0 i i k0 r (r ) e e V an arbitrary phase k k0 66 ODLRO in the Bogolyubov theory Basic expressions for the OPDM for a homogeneous system r r' † (r' ) (r ) V 1 e i k'r' ak'† ak ei kr V 1 ei kr e i k'r' ak'† ak V 1 ei kr e i k'r' ak† ak kk' k ,k ' by definition transl. invariance k ,k ' General expression for the one particle density matrix with condensate (r , r ') V 1 ei k ( r r ') nk k0 0, n0 N 0 k V 1 ei k0 ( r r ') n0 V 1 ei k ( r r ') nk coherent across the sample (r ) (r' ) dyadic k k0 FT of a smooth function has a finite range V 1 ei k ( r r ') nk N 0 i (r ) e V an arbitrary phase k k0 67 ODLRO in the Bogolyubov theory Basic expressions for the OPDM for a homogeneous system r r' † (r' ) (r ) V 1 e i k'r' ak'† ak ei kr by definition V 1 ei kr e i k'r' ak'† ak V 1 ei kr e i k'r' ak† ak kk' k ,k ' transl. invariance k ,k ' General expression for the one particle density matrix with condensate (r , r ') V 1 ei k ( r r ') nk k0 0, n0 N 0 k V 1 ei k0 ( r r ') n0 V 1 ei k ( r r ') nk coherent across the sample (r ) (r' ) dyadic N 0 i (r ) e V an arbitrary phase k k0 FT of a smooth function has a finite range V 1 ei k ( r r ') nk k k0 Interpretation in the Bogolyubov theory – at zero temperature (r , r ') V 1/ 2 a0 V 1/ 2 a0† V 1 ei k ( r r ') vk2 k k0 Rich microscopic content hinging on the Bogolyubov assumption 68 Three methods of reformulating the Bogolyubov theory In the original BEC theory … no need for non-zero averages of linear field operators Why so important? … microscopic view of the condensate phase quasi-particles and superfluidity basis for a perturbation treatment of Bose fluids We shall explore three approaches having a common basic idea: relax the particle number conservation work in the thermodynamic limit 69 Three methods of reformulating the Bogolyubov theory In the original BEC theory … no need for non-zero averages of linear field operators Why so important? … microscopic view of the condensate phase quasi-particles and superfluidity basis for a perturbation treatment of Bose fluids We shall explore three approaches having a common basic idea: relax the particle number conservation work in the thermodynamic limit I explicit construction of the classical part of the field operators Pitaevski in LL IX (1978) II the condensate represented by a coherent state Cummings & Johnston (1966) Langer, Fisher & Ambegaokar (1967 – 1969) III spontaneous symmetry breakdown, particle number conservation violated Bogolyubov (1960) Hohenberg&Martin (1965) P W Anderson (1983 – book) 70 I. explicit construction of the classical part of the field operators Quotation from Landau-Lifshitz IX 72 Quotation from Landau-Lifshitz IX … that part of the operators, which changes the condensate particle number by 1, we have, then, by definition the symbols denoting two "identical" states, differing only by the number of the particles in the system, and is a complex number. These statements are strictly valid in the limit The definition of the quantity has thus to be written in the form the limiting transition is to be performed at a given fixed value of the liquid density 73 Quotation from Landau-Lifshitz IX … that part of the operators, which changes the condensate particle number by 1, we have, then, by definition the symbols denoting two "identical" states, differing only by the number of the particles in the system, and is a complex number. These statements are strictly valid in the limit The definition of the quantity has thus to be written in the form the limiting transition is to be performed at a given fixed value of the liquid density If the operators are represented in the form then their remaining ("supercondensate") parts transform the state to states which are orthogonal to it, that is, the matrix elements are In the limit , the difference between the states vanishes entirely and in this sense the quantity comes the mean value of the operator over this state. and be- 74 Quotation from Landau-Lifshitz IX … that part of the operators, which changes the condensate particle number by 1, we have, then, by definition the symbols denoting two "identical" states, differing only by the number of the particles in the system, and is a complex number. These statements are strictly valid in the limit The definition of the quantity has thus to be written in the form the limiting transition is to be performed at a given fixed value of the liquid density If the operators are represented in the form then their remaining ("supercondensate") parts transform the state to states which are orthogonal to it, that is, the matrix elements are In the limit , the difference between the states vanishes entirely and in this sense the quantity comes the mean value of the operator over this state. and be- 75 II. the condensate represented by a coherent state Reformulation of the Bogolyubov requirements Bogolyubov himself and his faithful followers never speak of the many particle wave function. Looks like he wanted a0 , N 0 ei , so that a0 The ground state This is in contradiction with the selection rule, a0 0 The above eigenvalue equation is known and defines the ground state to be a coherent state with the parameter 77 Reformulation of the Bogolyubov requirements Bogolyubov himself and his faithful followers never speak of the many particle wave function. Looks like he wanted a0 , N 0 ei , so that a0 The ground state This is in contradiction with the selection rule, a0 0 The above eigenvalue equation is known and defines the ground state to be a coherent state with the parameter HISTORICAL REMARK The coherent states (not their name) discovered by Schrödinger as the minimum uncertainty wave packets, obtained by shifting the ground state of a harmonic oscillator. These states were introduced into the quantum theory of the coherence of light by Roy Glauber (NP 2005). Hence the name. The uses of the coherent states in the many body theory and quantum field theory have been manifold. 78 About the coherent states OUR BASIC DEFINITION a0 , N 0 ei , a0 If a particle is removed from a coherent state, it remains unchanged (cf. the Pitaevskii requirement above). It has a rather uncertain particle number, but a reasonably well defined phase General coherent state e 2 /2 e a0† Condensate vac N 0 ei a0 a0† a0 n0 N 0 2 a0† a0 a0† a0 4 n0 2 n02 N 02 N 0 n0 N 0 N0 79 New vacuum and the shifted field operators Does all that make sense? Try to work in the full Fock space F rather in its fixed N sub-space HN This implies using the "grand Hamiltonian" H N Let us define the shifted field operator b0 a0 , b0† a0† * b0 , b0† 1, b0 0 ... new vacuum What next? Example: BE system without interactions – ideal Bose gas H N 2 m k 2 ak† ak a0† a0 0 2 for 0 Here, is a true eigenstate, coincides with the previous result for the particle number conserving state B N . 0 , 0 ,0 , ,0 , Two different, but macroscopically equivalent possibilities. 80 L1: Thermodynamics: 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) 81 L1: Thermodynamics: 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) 82 L1: Thermodynamics: 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) 83 L1: Homogeneous one component phase: boundary conditions (environment) and state variables T P dual variables, intensities "intensive" SV 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 84 L1: Homogeneous one component phase: boundary conditions (environment) and state variables T P dual variables, intensities "intensive" SV S V N isolated, conservativ conservativee sharp boundary isolated/open open S V At zero temperature the two coincide TVN T=0 isothermal S=0 (Third Principle) S P N isobaric S P not in use grand T V T P N isothermal-isobaric not in use T P 85 New vacuum and the shifted field operators Does all that make sense? Try to work in the full Fock space F rather in its fixed N sub-space HN This implies using the "grand Hamiltonian" H N Let us define the shifted field operator b0 a0 , b0† a0† * b0 , b0† 1, b0 0 ... new vacuum What next? Example: BE system without interactions – ideal Bose gas H N 2 m k 2 ak† ak a0† a0 0 2 for 0 Here, is a true eigenstate, coincides with the previous result for the particle number conserving state B N . 0 , 0 ,0 , ,0 , Two different, but macroscopically equivalent possibilities. 86 New vacuum and the shifted field operators Does all that make sense? Yes: work in the full Fock space F rather than in its fixed N sub-space HN This implies using the "grand Hamiltonian" H N Let us define the shifted field operator b0 a0 , b0† a0† * b0 , b0† 1, b0 0 ... new vacuum What next? Example: BE system without interactions – ideal Bose gas H N 2 m k 2 ak† ak a0† a0 0 2 for 0 Here, is a true eigenstate, coincides with the previous result for the particle number conserving state B N . 0 , 0 ,0 , ,0 , Two different, but macroscopically equivalent possibilities. 87 New vacuum and the shifted field operators Does all that make sense? Yes: work in the full Fock space F rather than in its fixed N sub-space HN This implies using the "grand Hamiltonian" H N Let us define the shifted field operator b0 a0 , b0† a0† * b0 , b0† 1, b0 0 ... new vacuum What next? … is this coherent state able to represent the condensate? Test example: ideal Bose gas – limit of a BE system without interactions H N 2 m k 2 ak† ak a0† a0 0 2 for 0 Here, is a true eigenstate, coincides with the previous result for the particle number conserving state B N. 0 , 0 ,0 , ,0 , Two different, but macroscopically equivalent possibilities. 88 General case: the approximate vacuum H d3 r † (r ) 2m V (r ) (r ) 12 d3r d3 r' † (r ) † (r' )U (r r' ) (r' ) (r ) 2 ● Non-zero interaction, repulsive forces make the lowest energies N dependent g r r ' ● Inhomogeneous, possibly finite, system with a confinement potential ● There is no privileged symmetry related basis of one-particle orbitals 89 General case: the approximate vacuum H d3 r † (r ) 2m V (r ) (r ) 12 d3r d3 r' † (r ) † (r' )U (r r' ) (r' ) (r ) 2 Trial function … a coherent state a marked generalization!! r r g r r ' We should minimize the average grand energy H N d r (r ) V (r ) (r ) 3 * 2 2m 12 d 3r d 3r' *(r ) (r )U (r r' ) * (r' ) ( r' ) This is precisely the energy functional of the Hartree type we met already and the Euler-Lagrange equation is the good old Gross-Pitaevski equation 2 2 V (r ) g r r r 2m with the normalization condition N [n] N d3 r (r ) 2 90 General case: the approximate vacuum H d3 r † (r ) 2m V (r ) (r ) 12 d3r d3 r' † (r ) † (r' )U (r r' ) (r' ) (r ) 2 Trial function … a coherent state a marked generalization!! r r g r r ' We should minimize the average grand energy H N d r (r ) V (r ) (r ) PROPERTIES OF THE 3 2 * 2m COHERENT 12 d 3GROUND r d 3r' *(STATE r ) (r )U (r r' ) * (r' ) ( r' ) ● energy Groundfunctional state in the mean field type we met already This is precisely the of the Hartree approximation and the Euler-Lagrange equation is the good old Gross-Pitaevski equation ● 2All particles at zero temperature 2 in the V (condensate r ) g r r r are 2 2m r N [n] N d ● condition r n with the normalization 3 r (r ) 2 91 General case: perturbative expansion Define the shifted field operators and the condensate as the new vacuum r r r , † r † r * r r , † r' r r' etc., r 0 Expansion parameter … concentration of the supra-condensate particles deviation from the MF condensate = -operators If we keep only the terms not more than quadratic in the new operators, the resulting approximate Hamiltonian becomes 92 General case: perturbative expansion Define the shifted field operators and the condensate as the new vacuum r r r , † r † r * r r , † r' r r' etc., r 0 Expansion parameter … concentration of the supra-condensate particles deviation from the MF condensate = -operators If we keep only the terms not more than quadratic in the new operators, the resulting approximate Hamiltonian becomes d r (r ) 2 m V (r ) gn d r (r ) 2 m V (r ) (r ) H N d 3r *(r ) 2 m V (r ) 12 gnBE (r ) ( r ) 2 3 † 2 BE 3 † (r ) ( r ) h.c. 2 2 d 3r nBE (r ) † (r ) † (r ) 4 † (r ) ( r ) ( r ) ( r )} g Here (see in a moment ) nBE r r 2 93 General case: perturbative expansion d r (r ) 2 m V (r ) gn d r (r ) 2 m V (r ) (r ) H N d 3r *(r ) 2 m V (r ) 12 gnBE (r ) ( r ) 2 3 † 2 BE 3 † (r ) ( r ) h.c. 2 2 d 3r nBE (r ) † (r ) † (r ) 4 † (r ) ( r ) ( r ) ( r )} g 94 General case: perturbative expansion d r (r ) 2 m V (r ) gn d r (r ) 2 m V (r ) (r ) H N d 3r *(r ) 2 m V (r ) 12 gnBE (r ) (r ) 2 3 † 2 BE 3 † (r ) (r ) h.c. 2 2 d 3r nBE (r ) † (r ) † (r ) 4 † (r ) (r ) (r ) (r )} g 1. The linear part must vanish to have minimum at 0 . This is identical with the Gross-Pitaevskii equation and justifies the identification nBE r r 2 95 General case: perturbative expansion 2 H N d 3r 12 gnBE (r ) d r (r ) 2 m V (r ) (r ) d 3r †(r ) 2 m V (r ) gnBE (r ) (r ) h.c. 2 3 † 2 2 d 3r nBE (r ) † (r ) † (r ) 4 † (r ) (r ) (r ) (r )} g 1. The linear part must vanish to have minimum at 0 . This is identical with the Gross-Pitaevskii equation and justifies the identification nBE r r 2. 2 The zero-th order part simplifies – substitute from the GPE 96 General case: perturbative expansion 2 H N d 3 r 12 gnBE (r ) d r (r ) d 3 r †(r ) 2 m V (r ) gnBE (r ) ( r ) h.c. 3 † 2 2 2m V (r ) (r ) 2 d 3 r nBE (r ) † (r ) † (r ) 4 † (r ) ( r ) ( r ) (r )} g 1. The linear part must vanish to have minimum at 0 . This is identical with the Gross-Pitaevskii equation and justifies the identification nBE r r 2 2. The zero-th order part simplifies – substitute from the GPE 3. There remains to eliminate the anomalous terms from the quadratic part: Bogolyubov transformation 97 General case: the Bogolyubov transformation A simple method: use EOM for the field operators i t (r , t ) (r , t ),H N 2 m V (r ) 2 gnBE (r ) ( r , t ) gnBE ( r ) †( r , t ) 2 i t †(r , t ) †( r , t ),H N 2 m V (r ) 2 gnBE (r ) †(r , t ) gnBE ( r ) ( r , t ) 2 These are linear eqs. To find their mode structure, make a linear ansatz iEk t bk†vk* ( r )e iEk t iEk t bk†uk* ( r )e iEk t (r , t ) bk uk (r )e † (r , t ) bk vk (r )e Reminescent of the old u and v for infinite system. Substitute into the EOM and separate individual frequencies. Bogolyubov – de Gennes eqs. are obtained. 98 General case: the Bogolyubov transformation Bogolyubov – de Gennes eqs. v Ek uk 2 m V 2 gnBE uk gnBE vk* Ek vk 2 2 m V 2 gnBE 2 k gnBE uk* Strange coupled “Schrödinger” equations. Strange orthogonality relations: 3 * * d r u u v k k v (r ) k It is our choice to normalize to unity 3 d r uk v vk u (r ) 0 The definition of the QP field operators can be inverted (r ) bk uk (r ) bk† vk* (r ) † (r ) bk vk (r ) bk†uk* (r ) d r v u † ( r ) bk d 3 r uk* vk* † ( r ) bk† 3 k k 99 General case: the Bogolyubov transformation These field operators satisfy the correct commutation rules bk , b† k , bk , b 0, bk† , b† 0 Finally, H N d r 3 gn 12 2 BE (r ) E b b d r vk (r ) † k k k 3 2 a neat QP form of the grand Hamiltonian. 100 Detail: the mean-field for a homogeneous system Before: minimize the energy functional with fixed particle number N, find the chemical potential afterwards Now: minimize the grand energy functional with fixed chemical potential, find the average particle number in the process 101 Detail: the mean-field for a homogeneous system Before: minimize the energy functional with fixed particle number N, find the chemical potential afterwards Now: minimize the grand energy functional with fixed chemical potential, find the average particle number in the process Homogeneous system: order parameter ( r ) const. N 0 /V n H N d r (r ) 2 m V (r ) (r ) 3 * 2 12 d 3 r d 3 r' *(r ) ( r ) g r r ' * ( r' ) ( r ) V energy per unit volume 2 * average particle density N d r (r ) (r ) V 3 4 12 g 2 n 102 Detail: the mean-field for a homogeneous system The GP equation reduces from differential to an algebraic one: x 0, x x 2 x 12 g 4 x3 0, n 2 min max 0, min g , min 12 g 4 min 2 2g g 103 Detail: the mean-field for a homogeneous system The GP equation reduces from differential to an algebraic one: x 0, x x 2 x 12 g 4 x3 0, n 2 min max 0, min 2g ref 4 min 12 g 2 g choose ref ; ref ref / g g ref g , 4 Plot in relative units ref min locus of the minima 104 III. broken symmetry and quasi-averages Zero temperature limit of the grand canonical ensemble 1 Z e P E N Z 1 N e N N H N Z 1 0 N e E0 N N 0N 0N 0N Picks up the correct ground state energy, all ground states are taken with equal statistical weight NORMAL NON-DEGENERATE GROUND STATE SYSTEMS "ANOMALOUS" DEGENERATE GROUND STATE 106 Degenerate ground state NORMAL NON-DEGENERATE GROUND STATE SYSTEMS "ANOMALOUS" DEGENERATE GROUND STATE Characterized by a classical order parameter … macroscopic quantity Typical cause: a symmetry degeneracy Everything depends on the system characteristic parameters Ginsburg – Landau phenomenological model E a 2 b 4 a0 a0 stable equilibrium metastable equilibrium non-degenerate degenerate 107 Degenerate ground state NORMAL NON-DEGENERATE GROUND STATE SYSTEMS "ANOMALOUS" DEGENERATE GROUND STATE Characterized by a classical order parameter … macroscopic quantity Typical cause: a symmetry degeneracy Everything depends on the system characteristic parameters Ginsburg – Landau phenomenological model E a 2 b 4 spontaneous symmetry breaking a0 a0 stable equilibrium metastable equilibrium non-degenerate degenerate 108 Rovnovážná struktura molekul AB3 BF3 NH3 F N H B H F H F N U U rovina F h rovina H h U adiabatická potenciální energie 109 Equilibrium structure of the AB3 molekules BF3 NH3 F H B F N H H F N U U F plane h H plane h U adiabatic potential energy stable equilibrium non-degenerate ground state metastable equilibrium degenerate ground state 110 Equilibrium structure of the AB3 molekules BF3 NH3 F H B F N H H F N U U F plane h H plane h U adiabatic potential energy stable equilibrium non-degenerate ground state metastable equilibrium degenerate ground state 111 Equilibrium structure of the AB3 molekules F BF3 Ammonia molecule pyramidal molecule. B two minima of F potential energy F separated by a barrier. U Different from a typical extended system: Small system: quantum barrier & tunneling rovina F h NH3 N H H H N U H plane Discrete symmetry broken: U adiabatická potenciální energie discrete set of equivalent equilibria states h 112 Broken continouous symmetries in extended systems Three popular cases System Isotropic ferromagnet Atomic crystal lattice Bosonic gas/liquid Hamiltonian Heisenberg spin Hamiltonian Distinguishable atoms with int. Bosons with short range interactions Symmetry 3D rotational in spin space Translational Global gauge invariance Order parameter homogeneous magnetization periodic particle density macroscopic wave function Symmetry breaking field external magnetic field "empty lattice" potential particle source/drain Goldstone modes magnons acoustic phonons sound waves For a nearly exhaustive list see the PWA book of 1983 113 Bose condensate - degeneracy of the ground state The coherent ground state mean field energy order parameter mf ground state E 2 N 0 ei e 2 1 2 e 4 12 g any from 0,2 N0 ei a0 degeneracy genuinely different for different vac Selection rule a0 E ei 0 a0 d a0 0 average over all degenerate states "Mexican hat" 114 Symmetry breaking – removal of the degeneracy The coherent ground state mean field energy order parameter mf ground state E 2 N 0 ei e 2 1 2 e 4 12 g any from 0,2 N0 ei a0 H N - a0† ei a0 e i E degeneracy genuinely different for different vac Symmetry broken by a small perturbation picking up one H N particle number NOT conserved "Mexican hat" 115 Symmetry breaking – removal of the degeneracy The coherent ground state mean field energy order parameter mf ground state E 2 N 0 ei e 2 1 2 e 4 12 g any from 0,2 N0 ei a0 H N - a0† ei a0 e i E degeneracy genuinely different for different vac Symmetry broken by a small perturbation picking up one H N particle number NOT conserved For 0 one particular phase selected "Mexican hat" 116 How the symmetry breaking works – ideal BE gas Without interactions, the ground level is uncoupled from the excited levels: H N - a0† ei a0 e i i a a - a e a0 e † 0 0 † 0 i a0† a0 - a0† ei a0 e i 2m k 2 ak† ak 2 k 0 The control parameter is the chemical potential , but it will be adjusted to yield a fixed average particle number in the condensate. Transformation: a0† a0 a0† ei a0 e i a0† a0 ei a0† e i a0 a0† a0 a0† * a0 a0† * a0 * b0†b0 * 117 How the symmetry breaking works – ideal BE gas Now we determine the many-body ground state b0†b0 * E , b0 a0 , 1 ei , 0 The lowest energy corresponds to b0 0, i.e. a0 ... coherent state * a0† a0 N 0 , N 0 ei E * N 0 / N 0 The control parameter is the chemical potential , but it will be adjusted to yield a fixed average particle number in the condensate. Infinitesimal symmetry breaking field 0 0 with N 0 fixed: 00 E 0 N 0 ei , fixed 118 How the symmetry breaking works – ideal BE gas Now we determine the many-body ground state b0†b0 * E , b0 a0 , 1 ei , 0 The lowest energy corresponds to b0 0, i.e. a0 ... coherent state * a0† a0 N 0 , N 0 ei E * N 0 / N 0 The control parameter is the chemical potential , but it will be adjusted to yield a fixed average particle number in the condensate. Infinitesimal symmetry breaking field 0 0 with N 0 fixed: 00 E 0 N 0 ei , fixed • The coherent state is the exact ground state for the ideal BE gas • The order parameter picks up the phase from the perturbing field • The order of limits: first 0, only then the thermodynamic limit N0 . 119 The end