Advanced Quantum Mechanics. Part 1.

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1 Advaced Quatum Mechaics. Part 1. Partial otes of F. Chev's course ad D. Papoular's tutorials. 1. Idistiguishable particules i "first quatizatio" First lecture video here. Will probabl sta uredacted. Smmetrizatio postulate, bosoic ad fermioic subspaces. Exemples : origi of magetism ad o-iteractig bosos buchig. Tutorials explore a major cosequece : iterferece effects for two idetical bosos/fermios! boso buchig ad Pauli exclusio. 2. The framework of "secod quatizatio" 2.1. The Fock space ad creatio/aihilatio operators A realizatio of creatio/aihilatio operators usig first quatizatio formalism Properties, Chage of basis i Fock space At first glace, we'd expect that the expectatio value h jaj i of a creatio/aihilatio operator a is alwas zero. Ideed, if j i is a N-particules state, the a N part a N part / hn partjn 1 parti However, if j i is a mixed-umber-of-particules state i.e. a combiatio of N ad N +1 particules states), the it is totall possible that h jaj i/. Ideed, aihilatio ad creatio operators have o-zero matrix elemets ol betwee sectors of fixed umber of particules. I statistical mechaics terms, we'll see that this is possible ol i a o-fixed N esemble e.g. grad caoical), ad ot i the caoical esemble.. Please report errors ad tpos to A help eve clea scaed otes) would be appreciated. Git repo. Covetios here : operators are i bold, ad 3D vectors are writte like r~. Ati-commutators are writte [; ] +. 1

2 Field operators Whe we stud particles i space, a usefull basis is the positio represetatio basis jr~ ; mi) r~ ;m where m stad for the iteral degrees of freedom, such as spi). The wavefuctio of a sigle particule is r~ ; m) hr~ ; mj i. The associated creatio/aihilatio operators are writte mr~) : a jr~ ;mi creates a particle at r~ with iteral d.o.f. m ad mr~) : a jr~ ;mi destos a particle at r~ with iteral d.o.f. m which we call field operators. This otatio is historical ad quite cofusig because there is othig like " r~) h j r~)j i". As is, because we are dealig with a cotious basis hr~ jr~ i r~ r~ )), ad thus a cotiuous famill of operators, a justificatio is eeded, or at least a sait check. Now let's drop m for clarit. A other importat basis is the mometum represetatio basis jp~ i) p~. I free space r~ 2 R 3 ), both basis are a bit pathological, ad i particular jp~ i ca't be ormalized. However, if we work i a box of size L deoted r~ 2 L 3 ad with periodic boudar coditios), ow jp~ i) p~ becomes a discrete basis with a p~ ~k ~ 2 ~ 2 L 3, for which creatio ad aihilatio operators a p~ ad a p~ are well defied, which create/aihilate a p~-mode. Usig???) ad the positio represetatio of plae waves hr~ jp~ i p 1 L 3 e ip~ r~ /~, we have ad jr~ i hp~ jr~ i jp~ i 1 p e ip~ r~ /~ jp~ i! r~) 1 p p~ p~ L 3 L 3 r~) p 1 L 3 p~ e+ip~ r~ /~ a p~ p~ e ip~ r~ /~ a p~ 1) Thus, field operators are well defied i a box. We ca alwas take the limit L! 1 at the ed of our calculatios to recover free space. As usual, we switch betwee positio ad mometum represetatio usig Fourier trasform. Let's check commutatios relatios usig caoical commutatio relatios for a p~ 's, first for bosos : r~); r~ ) :::) a p~ ; a p~ ; r~); r~ ) :::) a p~ ; a p~ p~ ;p~ p~ ;p~ ad Fiall, r~); r~ ) 1 L 3 [ p~ p~ ~k ~ ] 1 L 3 p~ z p~ ;p~ { { z z} { p~ e +ip~ r~ /~ e ip~ r~ /~ a p~ ; a p~ k~ 2 2 L 3 eik~ r~ r~ ) 1 L 3 ~ 2 3 e 2i~ r~ r~ L 1 L 3 3) r~ r~ r~); r~ ) r~); r~ ) ad r~); r~ ) 3) r~ r~ ) for bosos, ad the same relatios with ati-commutators for fermios. Reciprocall, b iverse Fourier trasform or performig a chage of basis jp~ i R dr~ hr~ jp~ i jr~ i), a d p~ 3 r~ p e ip~ r~ /~ d r~) ad a p~ 3 r~ p e ip~ r~ /~ r~) 2) L 3 L 3 L 2

3 2.2. Oe-bod operators Traslatig from first quatizatio to secod quatizatio [ todo ] Mometum operator ad kietic eerg B defiitio, the plae-wave basis jp~ i) p~ is the eigebasis of the mometum operator p~, that is p~ p~ jp~ ihp~ j or ~ ~ k jk ~ ihk ~ j p~ ~ k Thus, the total mometum operator is simpl P ~ p~ p~ a p~ a p~ or ~ ~ k a ~ k a~ k ~ k We'd wat to express it as a fuctio of field operators r~), r~). Usig the Fourier relatios 2), P~ ~ ~ k d3 r~ p e ik~ r~ r~) d3 r~ p e ik~ r~ r~ ) L k~ 3 L 3 L 3 L 3 ~ d 3 r~ d 3 r~ r~) r~ ) 1 k~ L e ik ~ r~ r~ ) 3 At the limit L! 1, the ill-defied) sum becomes a Riema itegral i 1D to simplif) : 1 k e ikx x ) 1!! 1 dk k e ikx x) 1 1 L k2 2 L!1 L 2 1 / 2i e ikx x ) L 1 L k / 2 L which could be writte as a might derivative of the Dirac delta) so that, uder the assumptio that the field vaishes quickl eough for the itegratio-b-parts-bracket to vaish, P ~ ~ d 3 r~ d 3 r~ 1 d 3~ k 1 r ~ 2) 3 i r~ eik~ r~ r~ ) r~) r~ ) [ it. b parts ] d 3 r~ d 3 r~ 1 d 3~ k e ik~ r~ r~ ) r~) ~ r ~ 2) 3 i r~ r~ ) d 3 r~ d 3 r~ r~) ~ i r~ r~ r~ d ) 3~ k e ik~ r~ r~ ) 2) 3 which fiall gives ote that this is still valid i the box L 3 rather tha R 3 ) : P~ k~ 3) r~ r~ ) d 3 r~ r~) i~ r~ ) r~) 3) The kietic eerg operator for a sigle particle is p~ 2 2m, which is diagoal i the jp~ i) p~ basis, thus the total kietic eerg operator actig i Fock space is K p~ 2 2m a p~ a similar p~ d 3 r~ ~2 r ~ 2m r~) r~ r~) it. b d 3 r~ r~) ~2 parts 2m r2 r~) p~ Traslatio operator The traslatio operator actig i the co space is defied b 8r~ ; T a~ r~) T a~ 1 r~ + a~) ad T a~ ji ji 4) This relatio ca b uderstood as such : traslatig b a~ ad creatig a particule at r~ from where we are is just like creatig a particule i r~ + a~ from the origial positio. 3

4 Let's check that P~ is ideed the geerator of traslatios, that is Ta~ e ip~ a~ /~. [ todo ] Local operators If the sigle-particule operator b is ol a fuctio of r~, i.e. b fr~ ), the it is diagoal i the positio basis b R dr~ fr~) jr~ ihr~ j) ad the associated oe-bod operator actig i Fock space ca be writte B dr~ fr~) r~) r~) The simplest example is the desit operator r~) where b r~ )) : r~) : r~) r~) It as the importat propert that, o a domai D of space, N D : R dr~ r~) has a iteger spectrum, D ad thus ca be iterpreted as the umber of particules i the domai. A aother importat example is that of a exteral potetial, which ca be expressed i secod quatizatio a a oe-bod operator No-iteractig particules [ todo ] 2.3. Two-bod operators [ todo ] C N) i<j c i;j C 1 2 ijk` i; j c k; ` a i a j a k a` 2.4. Evolutio of quatum fields [ todo ] i~ du dt H U ) i~ dat) dt At); H The issue is : caoical commutatio relatios are expressed i the Scrödiger picture, we have to express them i the Heiseberg picture. For same-time commutatio relatios, it is simple : At differet times, it is more complicated, ad we will ot derive them. However, the are useful, e.g. for time correlatio fuctio Quatizatio of a classical field [ todo ] 4

5 3. Correlatios ad quatum coherece The eigestates of sigle-bod hamiltoias are simpl products of sigle-particle states. This is ot ver iterestig, or useful to model real sstems, which are described b ma-bod hamiltoias ad where particules iteract with each other. Eigestates are ot amore simple products because iteractios etagle particles. The secod quatizatio framework gives us the tools to stud these correlated states, i particular correlatio fuctios, which will eable us to measure spatial correlatios First order correlatio fuctio Take a sigle-particule 1 observable b. I the positio basis, the sigle-bod operator actig i Fock space is B d 3 r~ d 3 r~ r~ b r~ r~ ) r~) Its expectatio value is B d 3 r~ d 3 r~ r~ b r~ r~ ) r~) b liearit of hi, ad ca be obtaied as well as a other expectatio value of a sigle-bod observable) as soo as we kow the first order correlatio fuctio g 1) r~ ; r~) : r~ ) r~) 5) which depeds, of course, o the state of the sstem let's recall that hi h jj i for a pure state j i, ad hi tr ) for a desit matrix, where tr tr F is the trace o the whole Fock space). But more importatl, the first order correlatio fuctio allows us to stud coherece : it tells us how far we ca go before matter wave becomes icoheret Geeral properties of g 1) The correlatio fuctio take at idetical poits r~ r~ is simpl the desit at poit r~ : g 1) r~ ; r~) : r~) r~) r~) 6) It ca be bouded b the desit : p jg 1) r~ ; r~ )j 6 r~) r~ ) 7) Ideed, A; B 7! AjB) : tr A B) def ha Bi where is the desit matrix of the state) defies a hermitia scalar product o p the vector space of operators 2, ad we ca the appl the Cauch-Schwarz iequalit jajb)j 6 AjA) B jb) to A r~ ) ad B r~) : b defiitio of expectatio values, it ields r~ )j r~)) q h r~ ) r~)i 6 h r~ ) r~ )i h r~) r~)i g 1) r~ ;r~) r~ ) r~) 1. Spi-less particule. Ca be easil geeralized with spis b addig P ad cosiderig r~)'s. 2. Let us prove that. It is obviousl sesquiliear b liearit of the trace. Secod, it is positive : if we write the trace i a eigebasis ji) of, of eigevalues ), the AjA) tr A A) A A A A [ closure idetit ] A m ma m A 2 > ;m ;m because eigevalues of a desit matrix are alwas positive. Third, is is smmetric simpl due to cclicit of the trace ad hermitiait of : AjB) tr A B) tr A B) ccl. tra B ) tr B A) BjA). 5

6 Because it is a scalar product, g 1) r~ ; r~) r~ )j r~) r~)j r~ ) g 1) r~ ; r~ ). It is thus the hr~ jjr~ i-elemet of the hermitia operator 1) R dr~ R dr~ g 1) r~ ; r~ ) jr~ ihr~ j which acts o the sigle-particule Hilbert space H 1). Moreover, this operator is positive 3. Appart from the fact that its trace is tr H 1) 1) ) R dr~ g 1) r~ ; r~) R N the total umber of particles i the state whe it is a state with well defied umber of particules) ad ot 1, it looks ver like a desit matrix o H 1). We'll call 1) the oe-bod desit matrix. I the special case of a uiform sstem, g 1) r~ ; r~ ) is a fuctio of r~ r~ r~ ol, g 1) r~). As a cosequece, the desit r~) g 1) r~ ; r~) is costat, ad jg 1) r~)j 6. Ideed, the state of a uiform sstem is ivariat b traslatios, so that 8a~ ; T a~ T a~, 8a~ ; [; T a~ ] where T a~ is the traslatio operator 4). The, usig the relatio T a~ r~) T a~ 1 r~ + a~), g 1) r~ ; r~ ) tr r~) r~ ) [ iject T r~ 1 T r~ 1 ] tr T r~ 1 T r~ r~) r~ ) [ traslatio of crea/aihil ops ] tr T r~ 1 r~ r~) r~ r~) T r~ [ cclicit ] tr T r~ T r~ 1 r~ r~) r~ r~) [ traslatioal ivariace ] tr ~ ) r~ r~) g 1) ~ ; r~ r~) Equivaletl, T a~ 1) 1) T a~ 1) 1), where T a~ 1) is the usual sigle-particule traslatio operator More geeral formulatio : Oe-bod desit matrix We saw that the first-order correlatio fuctio g 1) r~ ; r~) is the hr~ jjr~ i matrix elemet of the oe-bod desit matrix 1) which acts o the sigle-particule Hilbert space H 1), ad which is ver much like a desit matrix i H 1) except that its trace is ot 1. Let's defie it i geeral terms Defiitio It ca be defied through a other basis ji), with creatio/aihilatio operators a, a. To fid this defiitio, we agai look at the excpectatio value of the oe-bod observable B P ; hjbj ia a : B hjbj i a a hjbj i a a tr b 1) ; ; with 1) a a j ihj, 1) 1) : a a tr F a a ; The expectatio value of a oe-bod operator B ca be writte hbi tr B ) tr H 1) 1) b 9) 8) 3. 8' 2 H 1), h'j 1) j'i [ sesquiliearit ] dr~ dr~ g 1) r~ ; r~ ) h'jr~ ihr~ j'i dr~ dr~ r~)j r~ ) 'r~) 'r~ ) dr~ r~) 'r~) dr~ r~ ) 'r~ ) AjA) > with A dr~ r~) 'r~) b positivit of the scalar product defied above. This ca also be see usig the trace directl. 4. Refer to the Codesed Matter Theor course b Beoît Douçot for more details i the case of fermios. It is show that whe satisfies Wick's theorem ad is ot codesed ha i a j i ha i a j i ) the is full determied b 1) as a thermal distributio with a effective hamiltoia : the occupatio probabilit of a give state is the Fermi-Dirac distributio. 6

7 I geeral, however, the oe-bod desit matrix does ot cotai eough iformatio to compute the expectatio of two-bod operators, such as iteractio terms. It is thus of limited use. We see that the diagoal terms 1) ha a i h i gives the populatio i the state ji. For a pure state, j ih j ad we have 1) tr F j ih j a a a a Ideal gas states Let's take a N-particles state js i j 1 : 1 ; :::; k : k i, where P i i N. For fermios, this is a Slater determiat, ad 1 ; :::; k 2 f; 1g. These states o well defied are also called Fock space basis states. These states aturall appears as eigestates of a oe-bod separable) hamiltoia H P P a a where ji) is the eigebasis of the sigle-particule hamiltoia h, that is h ji ji. Let's compute its oe-bod desit matrix 1) : 1) a a js ihs j S a a S a j 1 : 1 ; :::; k : k i if 2/ f 1 ;:::; k g a j 1 : 1 ; :::; k : k i if 2/ f 1 ;:::; k g I the case where i ; j 2 f 1 ; :::; k g, b applig destructio operators, we get 1) p i 1 : 1 ; :::; i : i 1; :::; k : k p j 1 : 1 ; :::; j : j 1; :::; k : k ad it is clear that, b orthoormalit of the Fock basis, it is zero if i / j, ad oe if i j : 1) 2f1 ;:::; k g i so 1) js i i i j i ih i j 1) We recover, i this particular case, that tr H 1) 1) ) P i i N. Back to the first-order correlatio fuctio, if we deote the wave-fuctio of a basis states ji b r~) : hrji, k g 1) js i r~ ; r~ ) i1 i hr~ j i ih i jr~ i i1 k i r~) r~ ) 11) Oe-bod desit matrix i first quatizatio [ TD Papoular ] Let's costruct 1) b importig it from first quatizatio, for a sstem of N idetical particules. [ def ad sm properties of i H N) ] As before, let's take a sigle-particule observable b actig i H 1), ad its correspodig N-particules operator B N), which acts i the N-particules Hilbert space H N), ad which is defied b N B N) i1 N b i) i1 1 ::: 1 b 1 ::: 1 {z }}} positio i 7

8 Its expectatio value o a state is 5 B N) def N tr H N) B N) i1 tr H N ) b i) tr b 1) [ is full smmetric ] N tr H N) b 1) [ tr 1:::N is the trace o all but the first cop of H 1) ] N tr 1 tr2:::n b 1) ) [ b 1) b 1 ::: acts ol o the first cop of H 1) ] N tr 1 tr2:::n 1 ::: 1) b tr 1 1) b where the oe-bod desit matrix is defied as a partial trace o all but the first cop of the Hilbert space 1) N tr 2:::N ) It acts o H 1) ad a expectatio value o a oe-bod operator is give b B N) tr H 1) 1) b, i direct aalog with 9). Usig this aalog, which is reall a idetit for states of well defied umber of particules, if we take b jih j, the the Fock space operator is simpl B Fock) a a as usual, ad 1) tr H 1) 1) jih j B N) B Fock) a a : thus justifig the defiitio of 1) i secod quatizatio i caoical esemble, the sum over P i ) is costraied b P i i N so h i i ca't be 3.3. From full coheret state to thermal cloud Mometum distributio The distributio of speed ad the correlatio fuctio are two importat quatities of a fluid. Let's see how the are liked i the case of a homogeeous sstem. The umber of particules with mometum p~ is give b N p~ hn p~ i ha p~ a p~ i To express it as a fuctio of g 1) r~), we use the relatio a p~ R d 3 r~ p 3 e ip~ r~ /~ r~) : h R ~ r~ + r~, r~ r~ r~ 2 d N p~ 3 r~ p 3 e ip~ r~ /~ d r~) 3 r~ p 3 e ip~ r~ /~ r~ ) L L 1 d 3 r~ d 3 r~ h r~) r~ )i e ip~ r~ r~ )/~ L 3 i 1 d 3 R ~ L 3 1 L g 1) r~ ;r~ ) g 1) r~ r~) d 3 r ~ g 1) ip~ r~ /~ r~) e 5. More details o the smetr argumet : b ca be diagoalized as b P b jbihbj, which gives b i) P b b ji: bihi: bj where ji: bi ::: jbi ::: The, hi: bjji: bi h1: bjj1: bi b smmetr of the desit matrix uder the permutatio 1 $ i. Thus, tr b i) P b b tr ji: bihi: bj P b b hi: bjji: bi P b b h1: bjj1: bi tr b 1) 8

9 so the mometum distributio is the Fourier trasform of the first order correlatio fuctio : N p~ d 3 r~ g 1) r~) e ip~ r~ /~ 12) The usual properties of Fourier trasform thus applies, ad i particular the scalig propert :! the sstem is correlated over a log distace, mometa are arrowl distributed! the mometum distributio is broad, correlatios are ol over a short distace Thermal cloud I a thermal cloud, N p~ is a smooth fuctio of p~, that is there is o macroscopic occupatio of a level. Ofte, N p~ is a usual Bose-Eistei or Fermi-Dirac distributio. At high eough temperature, its width is of the order of p 2 /2m k B T, so the width of g 1), the coherece legth, dimiishes as T icreases Full coheret state Let's look at the opposite situatio whe we have N bosos codesed i the same state j'i : B usig 1) 6, the oe-bod desit matrix is the j i jn: 'i a ' ) N j;i 1) N j'ih'j This ca also be see lookig as the expectatio value of a oe-bod operator B : B bi h'jbj'i N h'jbj'i tr N j'ih'j b tr 1) b i i I terms of the correlatio fuctios, we simpl have with 'r~) hrj'i the wave-fuctio) g 1) r~ ; r~) hr~ j 1) jr~ i N 'r~ ) 'r~) ad g 1) r~ ; r~) r) N j'r~)j 2 I geeral, we are betwee a thermal ad a full coheret state. If we begi with a coheret state, it thermalizes ad the evolutio of the correlatio fuctio looks like this : 6. Let's give a boiled-dow proof of 1) usig 1) a a b takig a basis cotaiig j'i : 1) a a j ih j tr jn: 'ihn: 'j a a hn: 'ja a jn: 'i, which is clearl zero for / ' destroig a o-occupated mode ), ad ' 1) ' hn: 'j' jn: 'i N. Thus, 1) P 1) ; jih j N j'ih'j. 9

10 Fermi sphere The free gas oe-particle hamiltoia is h p~ 2 2m, for which plae waves jk~ i are eigestates of eigevalues ~ 2 k 2 2m, with k~ 2 2 L d. The oe-bod hamiltoia i secod quatizatio is the H P ~ 2 k 2 ~ a k 2m ~ a k ~. k The N-particules fermioic groud state is the Fermi sphere j i Y kk ~ k6k F a k ~ where k F is the radius of the Fermi sphere, such that it cotais N vectors of 2 L d, that is k d1) F ad k d3) F 6 2 ) 1/ 3 with N /L d we assume that our fermios has o iteral D.O.F.). j;i Let's compute the first-order correlatio fuctio. Usig 11), g 1) r~ ; r~ ) kk ~ k6k F hr~ jk ~ ihk ~ jr~ i kk ~ k6k F 1 p e ik~ r~ 1 p L d L d e ik~ r~ 1 L d kk ~ k6k F e ik~ r~ r~ ) I the ifiite-size sstem limit L; N! 1 at fixed, the sum which depeds ol o r~ r~ ) becomes g 1) r~) dk ~ 1 kk ~ k6k F 2/ L ) d e ik~ r~ L d 8 >< +k F dk 2 eikr sik F r) 1 k F 2 i r sick 2 Fr) i 1D >: i 3D [ figure ] We could have computed this result directl usig the Fourier trasform relatio 12) betwee N p~ the Fermi sphere here) ad g 1) r~). Cotrar to the bosoic case, the fermioic groud state has, eve at zero temperature, a fiite coherece legth of the order of k F 1. Note that this g 1) r~) is sometimes egative. This is simpl a cosequece of the Fermi sphere shape, ad should ot be iterpreted as a cosequece of Pauli exclusio we should look at the secod-order correlatio fuctio for that), as it is totall possible pour bosos too. 1

11 3.4. Bose-Eistei codesatio ad the Perose/Osager criterio We'll ow use g 1) r~) to quatif coherece i a geeral, possibl iteractig sstem, which displas Bose- Eistei codesatio. I a o-iteractig sstem separable, oe-bod hamiltoia), the eigestates are simple smmetrized or ati-smmetrized) products of oe-particule eigestates. I a iteractig sstem, oe-particule eigestates become etagled b the iteractios ad the state of the sstem is ot amore a simple product. However, eigevectors of the oe-bod desit matrix 1) ca sta simple. If our sstem is traslatioall ivariat [ 1) ; T a~ ] ), the we kow that plae waves jk ~ i are eigevectors of 1), of eigevalues k ~ which gives the populatio i the states jk ~ i, ad such that tr 1) ) P k ~ k ~ N is the total umber of particules). Agai, the full desit matrix is more complicated tha a product of plae waves. Let's see what happes i the thermodamic limit, where both the quatizatio volume ad the umber of particule become large L 3! 1, N! 1) at costat 7 desit N /. I this limit, the wavevector k ~ 2 2 L 3 becomes cotiuous. The geeric behavior is that k ~ is costat, give b the Fermi-Dirac or Bose-Eistei distributio k ~ g k ~ /e k~) 1), ad becomes a smooth regular fuctio k ~ ) at the thermodamic limit, i.e. the populatio fractio k ~ N! microscopic occupatio ol of idividual states). The umber of particules ca the be writte N ~ k k ~ d! 3~ k k~ d ) 3~ k k ~ ) 3~ k 2) 3 ; itesive 13) However, uder a fiite critical temperature, it is well kow that this descriptio breaks for bosoic gases for a ideal 3D bosoic gas, usig the above itegral, R 1 d )/e ) 1), does ot admit a solutio for < whe 3 T > 3 / 2), that is at T <T be ) 8, ad this is because the Bose-Eistei populatio for the fudametal eerg level diverges whe the chemical potetial! 1 ' 1/ e )! 1). 1 This is codesatio, or macroscopic occupatio of a sigle idividual state k ~ usuall k ~ ) : k ~ /N!, tpicall k ~ stas o the order of N extesive) ad we call k ~ /N the codesed fractio. We ca o loger write the itegral 13). It is importat to uderstad that it is ot a idividual state i the sese that we could talk about the wave-fuctio of a give particule because it is a iteractig sstem), but it simpl refers to the eigestates ad eigevalues of the oe-bod desit matrix. I a sstem where there is codesatio, some states stas microscopicall occupied, ad we write k ~ k ~ k ~ ;k ~ + ~ k ~ where ~ k ~ describe microscopic occupatio ol still a Bose-Eistei or Fermi-Dirac distributio, or somethig more exotic), which becomes a smooth fuctio ~k ~ ) i the ifiite-size limit, while k ~ ;k ~ becomes a Dirac delta 9 : k ~ ) 2) 3 k ~ k ~ ) + ~k ~ ) 14) 7. We're ot iterested i the costat N limit, where the sstem dilutes ad iteractios vaish. 8. Refer to our L3/M1 statistical phsics course or to Phsique Statistique, C. TEIER ad G. ROU, p ~ k ;k ~!! L!1 3~ k k ~ ~ k ) 2 /L) 3 k ~ ~ k ) so that ~ k ~ k ;k ~!! L!1 "sigularisé" le terme ~ k. k ~ 2)3 k ~ k ~ ). O dit que l'o a 11

12 where lim ~ k / is the codesed fractio desit. The we have 1, at the limit, d + 3~ k ~k ~ ) 15) 2) 3 ad we ca ivert the relatio with the correlatio fuctio ~ k R d 3 r~ g 1) r~) e ik~ r~ 12) to obtai g 1) d r~) + 3~ k ~k ~ ) e ik~ r~ 16) 2) 3 The first order correlatio fuctio the looks like : It goes from the total desit at r~ dow to the fiite value the populatio desit of the idividual state k ~ ) over a distace which is the coherece legth of the sstem. This idicates log-rage order : just like spis i a ferromaget are aliged ad thus correlated) over a log distace, it is here the phase which exhibits log-rage order. I a sese, all the particles i the codesate share the same state, ad thus the same phase. We'll see that this log-rage phase coherece ca be mesured b a sort of iterferometr. Agai, if we have a full codesed state, ad the correlatio fuctio is costat. I a real sstem, here liquid helium, the correlatio fuctio has ideed this behavior : Figure. First order correlatio fuctio i liquid He, computed b Quatum Mote Carlo methods Pollock ad Ceperle, 1987). Ulike a ideal bosoic gas which full codesates at T, ol a small fractio 7%) of He atoms are actuall codesed Measuremet of the first order correlatio fuctio How ca we measure i practice coherece properties? Depedig o the commuit, there are ma differet was, but at the ed, it alwas boils dow to some sort of iterferometr. Here, it will be illustated 1. Note that P + R is still a approximatio of real sstems because of their fiite size. We could add correctio terms i the desit of states ) takig ito accout this fact, e.g. 2~! ) /~!) /~! for a harmoic trap. 12

13 from the viewpoit of atomic phsics, but similar methods exist i quatum optics, Youg's slits experimet i ma bod sstems To measure g 1), we usuall perform a kid of Youg's slits experimet, that is make the sstem iterfer with itself at two differet poits r~ 1 ad r~ 2. Let's first recall how a delocalized wave-packet j )i jr ~ 1i + jr~ 2 i p 2 describig a particule goig through two slits r~ 1 ad r~ 2, will evolve freel at time t ad iterfer with itself. B defiitio of Gree's fuctio for the Scrödiger equatio, hr~ j Ut) jr~ i Gr~ ; r~ ; t), thus the wavefuctio at time t is r~ ; t) hr~ j Ut)j )i 1 p 2 Gr~ ; r~ 1 ; t) + Gr~ ; r~ 2 ; t) This sum of two wavelets creates iterfereces whe we look at the probabilit desit : r~ ; t) j r~ ; t)j 2 / jgr~ ; r~ 1 ; t)j 2 + jgr~ ; r~ 2 ; t)j 2 + Re Gr~ ; r~ 1 ; t) Gr~ ; r~ 2 ; t) Now, if the iitial wave-fuctio is ot delocalized o ol 2 poits but r~ ; t) dr~ Gr~ ; r~ ; t) r~ ) iterferece term r~ ; t) r~ ; t) r~ ; t) dr~ dr~ Gr~ ; r~ ; t) Gr~ ; r~ ; t) r~ ) r~ ) We wat to geeralize this result to ma-bod sstems. the evolutio of the field operator is the same tha the evolutio of a wavefuctio : r~ ; t) d 3 r~ Gr~ ; r~; t) r~) so cohérece slits e a et b : r~ ; t) g 1) sx1 a) x 1 ; x 2 ) sx 1 b) 1 g g 1 sx2 a) sx 2 b) 4. Quadratic hamiltoias Motivatio : quadratic hamiltoias are simple eough to be diagoalized. We will show that the look like o-iteractig sstems. However, the ca be used as approximate models of real-world iteractig sstems, describig excitatios i such sstems. 13

14 4.1. Geeral framework The hamiltoia of o-iteractig particles H P i h i which is a oe-bod operator actig o H N ) ca be writte, actig o the Fock space F ad i a basis ji) of H 1 we wat, as H ; A a a It is quadratic i the sese that it ivolves ol biar products of creatio ad aihilatio operators, i particular a a pairs, which coserve the umber of particules. Let us add other tpes of quadratic terms, such as a a. It obviousl does ot coserve the umber of particules. This ca be useful to describe sstems, for exemple light photos are ofte ot coserved) or sstems i cotact with a reservoir the excitatios i a BEC or a codesate, electros i the BCS theor for supercoductors...). However, P ; B a a is clearl ot hermitia, so we also eed to add its hermitia cojugate to the hamiltoia. A quadratic hamiltoia thus takes the geeral form H ; A a a + ; B a a + B a a 17) This hamiltoia will tpicall be a expesio for small excitatios i.e. where most particules are i a codesed state, the reservoir, ad a small fractio of particules are i a excited state) of a more complex hamiltoia describig the full sstem. This expesio is called a Bogoliubov theor. I this framework, a a ad a a creates/aihilates excitatios. The expasio of the BCS hamiltoia for attractig fermios explais supercoductivit ad the properties of supercoductors this was the iitial motivatio of Bogoliubov). For Bose-Eistei codesates, this explais superfluidit.! Awa, the goal is to diagoalize this geeral hamiltoia, so as to express it as a sum of o-iteractig excitatios, after a appropriate Bogoliubov trasformatio a; a 7! b; b. As we'll see, this procedure is somewhat differet for the bosoic or fermioic cases Field evolutio We're iterested i the evolutio of field operators, which are expressed with a 's ad a 's. The evolutio of the a 's is give b i~ da dt a ; H There are three tpes of commutators to compute. First, a ; a a This is obvious for bosos. For fermios, usig aticommutatio, a a a a a a a a a two permutatios) so a a a a a a is also zero. Secod, usig the well kow Leibiz-tpe idetit 11 ad caoical commutatio relatios, 8 a ; a < a ; a a + a a ; a a + a for bosos a : a ; a + a a a ; a + a a for fermios Third, a ; a a a i both cases 8 < a ; a a + a a ; a a + a for bosos : a ; a + a a a ; a + a a for fermios a a i both cases if we defie for resp. bosos/fermios 11. See 14

15 Fiall i~ da dt ; A a + ; B a a ) + B A a + B a B a so i~ da dt + B B ) : C a A a + C a with C b reamig! B + B > for bosos B B > for fermios This equatio mixes aihilatio ad creatio operators, so we have to write evolutio equatio for a 's too. We ca reproduce the above procedure, but simpl takig the hermitia cojugate is quicker : i~ da dt A a C a We ca collect all these liear equatios i matrix form : " # " #" i~ d [a ] A C [a ] dt [a ] C A [a ] : t) : L # l dimh 1 ) l dimh 1 ) 18) Here 12, t) is a vector of operators actig i F, ad L is a 2d 2d matrix where d dimh 1 )) actig o the space of operator-vectors, or alterativel, actig o H 1 H 1. We will alterate betwee the space of operator-vectors A, whose elemets will be deoted ) ad simple vectors of H 1 H 1 deoted ). Liear algebra o A works as usual as log as it does ot ivolve products of operators. The formal solutio of 18) is t) e it ~ L ) 19) To solve this, we must diagoalize L. Recall that our goal is to diagoalize the hamiltoia H. What the above equatio suggests is that this ca be doe through the diagoalizatio of L. It is the importat to stud importat the geeral properties of L, ad chiefl the smmetr properties. At the ed of the da, we'll discover that the eigevectors of L ca be recogized as quasi-particules Smmetr properties of L Let's see if L is hermitia or ot. Usig hermitiait of A for H P A a a to be hermitia), " # " # A C L A C A C A C C A > C > A From the defiitio C B B > let's recall that for resp. bosos ad fermios), C > B > B B > + B) C that is C is smmetric for bosos ad atismmetric for fermios. Thus, L A C L for fermios C A ot L! for bosos 2) 12. I F. Chev's otatios t) is ^ t) ad is t). But I do't like hats o operators, sorr. 15

16 I order to fid out more smmetr properties, we ca uitar evolutio uitarit of operators a t) ad a t) i Heiseberg picture), ad the importat cosequece that caoical commutatio relatios for a t)'s ad a t)'s must be preserved. Let's expad 19) at first order i t dt : t) e idtl/~ ) 1 i~ dt L ) thus i t) i ) i dt L ~ ik k ) We must the have i ); j ) i t); j t) h i ) i dt L ~ ik k ); j ) i dt L ~ jl l ) i ); j ) i ~ dt L ik k ); j ) i ~ dt L jl i ); l ) + dt2 ::: so [ i ); j )] cacels ad we are left with i so that L ik k ); j ) ) J kj + L jl {z} L lj > i ); l ) ) J il L J ) + J ) L > L ik J ) kj + J ) > il L lj [ todo : J 1 ] We could have derived this propert directl from the defiitio of L ad our computatio of L 2) : L > J ) L J ) A C C A C A A C J ) L 1 A C C A For bosos, the fact that J L + L > J with J J ) 1 d 1 d C A A C J ; ) 2 1 L J L > J makes L a smplectic matrix rather tha a hermitia matrix for the case of fermios). This propert will ao us a bit for the diagoalizatio, ad it is actuall liked to classical mechaics. Smplectic smmetr i classical mechaics [ todo ] Lié au fait que les équatios de Hamilto sot q_ ; i Wh do we have this coectio for bosos ol? Well, this is ot surprisig : we saw with the exemple of the vibratig strig) that whe we quatif a classical field, we automaticall get a bosoic theor. The properties we derived above are valid for ever liear evolutio of creatio ad aihilatio operators. 16

17 Geeral liear evolutio of creatio ad aihilatio operators A geeral liear evolutio ca be writte " # " i~ d [a ] dt A C a B D #" # [a ] a ) i~ d t) L t) with above otatios dt Let's see what costraits uitar evolutio puts o A, B, C ad D. First, we ca use that the hermitia cojugate relatio betwee creatio ad aihilatio operators. The two halfs of the equatio reads i~ d dt a A a + C a ad i~ d dt a B a + D a These equatios are simpl hermitia cojugates of each other, ad i~ d dt a A a + C a A a + C a so that A B ad C D, that is B C ad D A ) L A C C A As we saw, eforcig uitar evolutio coservatio of caoical commutatio relatios for a t)'s ad a t)'s) gives " # L J ) + J ) L > here A C 1 1 A C A + > C > C > A > The 1; 1) matrix reads C + C >, ad the 1; 2) matrix reads A + A > ), that is C > C ad A A This result shows how geeral the hamiltoia H P A a a + P B a a + B a a is Rewritig the hamiltoia Recall that our goal is to diagoalize the quadratic hamiltoia H. This will be doe through the diagoalizatio of L. Ideed, if we see H as a quadratic form of a 's ad a 's hermitic form to be precise), L or rather I ) L) is the matrix represetatio of this form, meaig that the hamiltoia ca be recast as 13 H 1 2 I ) L + 1 tra) with 2 I) 21) Ideed careful with the order of operators here!), h i " #" # J ) L [a A C [a ] ] [a ] C A [a ] h i [a A [a ] [a ] ] + C [a ] C [a ] A [a ] implicit P ) a A a + a C a a C a a A a We see that it almost looks like our hamiltoia 17), except the operators of the last term are ot i the right order. To chage this, we use ati-)commutatio relatios to rewrite first ad fourth terms : a A a a A a A a a A a ; a a a [ A A > A > A hermitia) ] A a a A + 2 A a a tra) [ $ dumm idices, 2 1 ] 2 A a a tra) 13. F. Chevr used the otatio > ) i place of to make clear that we make it a lie vector with all elemet-operators cojugated. However, the otatio alread does both : traspose ad cojugate, so I do't fid ecessar to use >. 17

18 Similarl, the secod ad third terms ca be rewritte usig ati-)commutatio relatios : a C a a C a B B ) a a B B ) a a B B ) a a + B [ [a ; a ] ) a a a a ] B a a + B a a + B [ $ dumm idices ] 2 B a a + 2 B a a Collectig all the terms, we have what we wat : 1 B ) a a a a + B a a 2 I ) L A a a + B a a + B a a tra) H tra) 2 2 Is is ow clear that diagoalizig the hamiltoia, diagoalizig its matrix represetatio L Bogoliubov theor for fermios Let's fiall work out the diagoalizatio of the quadratic hamiltoia H i the simplest case of fermios. Ideed, L is hermitia so it ca be directl diagoalized : L with f g eigebasis of size 2dimH 1 of eigevalues 2 R 22) At the ed of the da, the eigevectors 2 H 1 H 1 will represet vectors of creatio operators first cop of H 1 ) ad aihilatio operators secod cop of H 1 ), so it is usefull to write them u with u ; v 2 H 1 v With these otatios, the hermitia scalar product o H 1 H 1 is simpl u u j ) huju i + hvjv i where ; v v The eigevectors are chose ormalised : j m ) m. 23) Dualit Let's ow exploit the smmetr properties of L b writig explicitl the 2dimH 1 eigevector equatios with the previous otatios : L ) A C C A [ complex-cojugate ] ) A C C A u u [ switch lies ] ) C u A v v v v A u + C v u ) A C C A v u u u v v v u 2 R) so that for a eigevector, we automaticall have a other eigevector 14 L with : J + v u ) L v u v the dual eigevector of of eigevalue 24) We'll ow assume, to lighte otatios, that there are o ull-eigevalues. This would ot chage much the fial result, but ull-eigevalues ca have ver importat cosequeces that we'll explore later i particular Goldstoe excitatios, see??). 14. Alterative proof : usig the fact that L J L > J with J 1 1 v v J L u L L u J L > J u v J L ) J ) J J 2 1) ad the hermitiait of L so L > L ), u J v v u u 18

19 Hece, the spectrum of L ca be actuall separated as EigvalL) + ; 2F + 25) with F + f : > g ad F f : < g fg 2F + the idices of positive ad egative eigevalues, which are related b the dualit 7! BogoliubovValati trasformatio We're ow read to come back to our vector of creatio ad aihilatio operators " # [a ] [a ] ad diagoalize the hamiltoia. Elemets of this vector are operators, but as log as we do't multipl operators e.g. o quadratic relatios i ), we ca treat them as scalar ad the rules of liear algebra appl as usual. We ca thus maipulate as a vector of H 1. Let's ow ivoke the geius of Bogoliubov we'll see later what were its motivatios i the discussio about BECs), who itroduced i 1958 the trasformatio b : j) hu j[a ]i + hvj[a ]i u; a + v ; a 26) The b ; b 2F are the fermioic aihilatio ad creatio operators. Ideed, + b u; a + v ; a ad we check the caoical commutatio relatios 8; m 2 F + ; b ; b m + m ad b ; b m + b ; b m + which follow from orthogoalit of the basis of eigevectors : b ; b m + u a + v a ; u m a + v m a + ; u u m a ; a + + u v m a ; a + + v u m a ; a + + v v m a ; a + ; u u m + v v m ad similarl b ; b m hu ju m i + hv jv m i j m ) m + ::: u v m + v u m hu jv m i + hv ju m i j m ) m because ; m 2 F + ad F + \ F?, so it is impossible to have m 2 F. Note that i out set fb g 2F +, we have jf + j dim H 1 operators, that is the same umber of fa g operators, which is ecessar if we wat to iverse the trasformatio. For these two reasos, we must limit ourselves to the set F + or F ) of positive eigevalues. 19

20 So what are b ; b the? b j) so we have a particule-hole dualit v u v; a + u ; a b b b ; b b 27)! Creatig a dual mode of eigevalue ) is detroig the mode of eigevalue ). This allows us to iterpret 's as eergies create destro ), justifies the term "hole" for dual modes, ad allows us to express everthig as a fuctio of positive eergies modes F + ol. From its defiitio, b mixes a 's ad a 's, so the clearl do ot create xor destro particules of the ji) basis, thus do ot eve coserve the well-defiedess-umber-of-particules. Rather, if we call a 's "particules", the do create what we call quasi-particules, which will ofte be excitatios of a sstem just as spi waves/magos, phoos, excitatios of a Fermi sea... are quasi-particules). Recall that our goal is to re-write the hamiltoia H. To do this, we have to ivert the Bogoliubov trasformatio to get back a 's as a fuctio of b 's. As it is a liear trasformatio b j) 26), it ca be simpl iverted : b j) 28) usig the closure relatio i the Hilbert space H 1 H 1 ; j) for the orthoormal basis f g. Now, a is the th compoet of, ad the th compoet of u is u v ;, so a u ; b u ; b + u ; b 2F + 2F As we said, we wat to express everthig as fuctio of positive eerg modes, so usig particulehole dualit 24) ad 27), P u 2F ; b P u 2F + ; b P v 2F + ; b, so we fiall have the aihilatio ad creatio operators of the particules as a fuctio of quasi-particules' oes : a 2F + u ; b + v ; b ad a 2F + u ; b + v ; b 29) Mathematicall, the Bogoliubov trasformatio is thus a isomorphism of the algebra of caoicall aticommutig operators : Bogoliubov trasformatio b u ;v 's a ; a fermioic ca. op. $ $ b ; b 2F fermioic ca. op. +! particule-hole dualit b ; b 2F fermioic ca. op. Note that the Bogoliubov trasformatio has much broader use, for example i quatum field theor where is describes particules ad ati-particules of opposite charges from two free fields. It is a ke igrediet to uderstad the Uruh radiatio 15 ad its aalog for black hole horizos, the Hawkig radiatio Bogoliubov hamiltoia To recast our quadratic hamiltoia H, we simpl have to re-iject 29) ito 17) ad expad. But we alread did all the hard work i sectio 4.1.3, so let's use 21) directl. We eed to compute L. For 15. Where a uiforml acceleratig oberver perceives the Mikowski vacuum of a free field as a thermal bath of temperature k B T a ~/2c. Much love to R. Paretai. 2

21 that 16, we use the iverse Bogoliubov trasform P b 28) : L m m b m ) L b b m b m L ;m m m j ) m b b b b + b b usig L's spectrum smmetr 2F + 2F + b b b b usig p-h dualit 24),27) 2F + b b b ; b b + b 2 b b 2F + 2F + 2F + So from H 1 2 L + 1 tra) we get the fial diagoalized hamiltoia 2 1 H E + 2F + b b + 2F 2F + which is a sum of idepedat modes/quasi-particules of eergies > a set of idepedat harmoic oscillators if ou wat). Agai, this is remarquable that all quadratic hamiltoias ca be diagoalized, but ote this excludes exact iteractig hamiltoias where two-bod operators appear. However, as we'll see, this icludes mea-field treatemets of weakl iteractig theories, where these quasi-particules are excitatios, so it is still of great use. The costat E is b defiitio the groud state eerg, where the groud state j i is defied b 8 2 F + ; b j i cotais o quasi-particules). It ca be computed explicitel 17 : E 2F + ku k Illustratio : the ideal Fermi gas [ todo ] : cours + TD + quelques revois vers le cours de Douçot 16. Alterative demo : L ca be writte i diagoal form L P, so L j ) j) b b b defiitio of b j). 17. Let's compute the groud state eerg. tra) is the trace of the upper left block of L, ad because " # A C C A L " # 2 3 u h i u v v 4 u u u v 5 v u v v we have tra) tr u u tr u u hu ju i ku k 2 + ku k 2 2F + kv k 2 But 's are ormalized b defiitio, so 1 j ) ku k 2 + kv k 2, thus ku k 2 kv k kv k 2, thus fiall E def 1 2 tra) kv k 2 ) + kv k 2 2F + 2F + 2F + 21

22 4.3. Bogoliubov theor for bosos We'll ow treat the case of bosos, which is a bit more complex tha for fermios. Ideed, the matrix L we have to diagoalize is ot loger hermitia, so we have o guaratee it is actuall diagoalizable. The o-diagoalizabilit has importat cosequeces as we'll see later o Bose-Eistei codesates. [ todo ] 4.4. Applicatio : The weakl iteractig Bose-Eistei codesate Mea-field, "classical" theor : the Gross-Pitaevskii equatio Bogoliubov expasio : a semi-classical model Bogoliubov spectrum of excitatios Goldstoe's theorem for a Bose-Eistei codesate ad the phoo brach The Gross-Pita eq has U1) sm! The groud state is defied up to a phase! there is spotaeous smmetr breakig the sstem chooses a phase 22