Strongly correlated systems in atomic and condensed matter physics. Lecture notes for Physics 284 by Eugene Demler Harvard University

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1 Strongly correlated systems n atomc and condensed matter physcs Lecture notes for Physcs 84 by Eugene Demler Harvard Unversty September, 014

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3 Chapter 7 Bose Hubbard model 7.1 Qualtatve arguments We consder spnless bosonc atoms n an optcal lattce wth repulsve nteracton between atoms[11]. They can be descrbed by the Bose Hubbard model[9]. H = t b b j + U n (n 1) µ n (7.1) j We have on-ste nteracton only snce partcles have contact nteracton and and we assume tght-bndng lmt. Away from the tght-bndng regme, and especally for shallower lattces, one may need to nclude non-local nteractons. In the presence of confnng potental we also need to nclude H pot = V (r )n (7.) Parameters t and U can be controlled by selectng the strength of the optcal lattce and tunng the scatterng length wth magnetc feld. We dscuss two lmtng cases frst. Weak nteractons regme t >> U. Atoms condense nto the state of the lowest knetc energy Ψ SF = 1 N! (b k=0 )Nat 0 c e N atb k=0 0 (7.3) Here 0 s the vaccum state wth no partcles. State (6.3) s a superflud state. We can also wrte b k=0 = 1 b (7.4) Nstes to wrte Ψ SF = c Natoms N b e stes (7.5) 3

4 4 CHAPTER 7. BOSE HUBBARD MODEL Fgure 7.1: Bose Hubbard model wth t = 0. Level crossngs between states wth dfferent atom numbers. Strong nteractons regme t << U. Interacton term n the Hamltonan plays the domnant role. As a frst step let us take t = 0. H U + H µ = U n (n 1) µ n (7.6) Dfferent stes are now decoupled. Egenstates n each well have a well defned number of partcles. For an ndvdual well we can wrte n = 0 E = 0 n = 1 E = µ n = E = µ + U n = 3 E = 3µ + 3U n = 4 E = 4µ + 4U n E n = nµ + U n(n 1) (7.7) The ground state has n bosons when (n 1) U < µ < nu (7.8) There are level crossng between states wth wth dfferent nteger fllngs for µ = nu. Away from level crossngs these states have a gap. Hence they should be stable aganst small changes n the Hamltonan, such as small tunnelng. These are nsultng Mott states.

5 7.. GUTZWILLER VARIATIONAL WAVEFUNCTIONS 5 7. Gutzwller varatonal wavefunctons To descrbe transtons between the superflud and nsulatng states we use Gutzwller varatonal wavefuncton Ψ G = ( ) f 0 + f 1 b b b + f n + + f n 0 (7.9) n! Ths wavefuncton requres normalzaton condton f n = 1 (7.10) n Our justfcaton for usng the Gutzwller ansatz s that a factorzable wavefuncton works n both extreme lmts: deep n the superflud state and deep n the Mott state. It s natural to assume that ths wavefuncton wll work near the transton pont as well. It turns out that ths wavefuncton s qute accurate n d=3, but n lower dmensons t works only qualtatvely. The transton pont obtaned from the MC analyss s suffcently dfferent from the one predcted by the Gutzwller ansatz. Ths s not surprsng snce ths s essentally the mean-feld analyss whch becomes ncreasngly better n hgher dmensons. We need to mnmze the wavefuncton (6.9) wth respect to f n subject to the constrant (6.10). Interacton energy H U + H µ = µ f 1 + (µ + U) f + + ( nµ + U n(n 1)) f n + (7.11) Knetc energy H t = zt f 0 f 1 + f 1 f + 3f f 3 + (7.1) Interacton energy favors a fxed number of partcles per well, whle the knetc energy favors a coherent superposton of the number states. To understand the transton let us consder the stablty of the Mott state wth n partcles (at a fxed densty). We take the Gutzwller wavefuncton wth f n = (1 α ) 1/ f n 1 = f n+1 = α (7.13) here α s assumed to be small and all other f are zero. We expand the energy up to the second order n α H t = zt α (1 α ) 1/ n + α (1 α ) 1/ n + 1 ztα n + n + 1 (7.14) We take the mddle of the Mott plateau µ = U(n 1/) (7.15)

6 6 CHAPTER 7. BOSE HUBBARD MODEL Fgure 7.: Phase dagram of the Bose Hubbard model accordng to the Gutzwller varatonal wavefuncton. We have for the relvant number states H U + H µ n = U n(n 1) U(n 1/)n = U n H U + H µ n 1 = U (n 1)(n ) U(n 1/)(n 1) = U, (n 1) Thus we fnd H U + H µ n+1 = U (n 1)n U(n 1/)(n + 1) = U, (n 1)(7.16) E tot (α) = z tα n + n U α + (7.17) The Mott to Superflud transton takes place when the coeffcent n front of α becomes negatve. For large n ths corresponds to U = 4nzt (7.18) Note that the Mott state exsts only for nteger fllng factors. For n = N + ɛ, even when N atoms become localzed, ɛ makes a superflud state even for the smallest values of tunnelng. Confnng parabolc potental acts as a cut through the phase dagram. Hence n a parabolc potental we fnd a weddng cake structure of shells.

7 7.3. EXPERIMENTS ON THE SUPERFLUID TO MOTT TRANSITION IN OPTICAL LATTICES7 Fgure 7.3: Densty dstrbuton of atoms n an optcal lattce n the Hubbard regme. Incompressble Mott states gve rse to the flat plateaus. Compressble superflud shells make regons where the densty s changng smoothly. 7.3 Experments on the superflud to Mott transton n optcal lattces Superflud to Mott nsulator transton for ultracold atoms n optcal lattces was frst demonstrated by Grener et al. [1] (see fg. 6.4). They use TOF experments to measure occupaton numbers n momentum space n k = b k b k, where k s the physcal momentum. In the SF state we have macroscopc occupaton of the state wth the lowest knetc energy. In a lattce a state wth the lowest knetc energy s a state wth quas-momentum zero. Quas-momentum dffers from the physcal momentum by Bragg reflectons. So n the SF state atoms should exhbt macroscopc occupaton of states wth momenta equal to recprocal lattce vectors. One can also understand ths result as constructve nterference from a perodc array of coherent sources. Fnte sze of wavefunctons n ndvdual wells determnes how many Bragg peaks can be observed. In the Mott state one occupes all quasmomenta so expanson mages do not have sharp peaks. We can also say that we no longer have coherent sources so nterference s lost. There has been a certan controversy regardng fnte wdth of the peaks. It was suggested that ths was due to the hgh temperature n the system, thus experments could not be consdered a demonstraton of the quantum phase transton. Detaled analyss showed that ths can be explaned by the fnte TOF expanson tme []. Consderable effort was also dedcated to observng the weddng cake structure n a parabolc potental. Recent experments reached sngle ste resoluton and provded convncng demonstraton of the ncompressble Mott plateaus and the weddng cake structure [4, 8, 3] (see fg.6.5). Note that these experments can be used to measure not only the average number of partcles, but also fluctuatons. In the superflud state we expect to see much large fluctuatons snce they correspond to a superposton of several number states.

8 [ e χ cos θ ( e φ sn( γ ) n 1 + e φ cos( γ ) n + 1 ) + e χ sn θ n ] 8 CHAPTER 7. BOSE HUBBARD MODEL Fgure 7.4: Demonstraton of the superflud to nsulator transton wth bosons n an optcal lattce[1]. These TOF experments measure occupaton number n momentum space n k = b k b k. In the superflud state there s macroscopc occupaton of the state wth the lattce quasmomentum equal to zero. In TOF ths shows up as peaks for momenta that correspond to recprocal lattce vectors. In the Mott state all quasmomenta are occuped. 7.4 Collectve modes n Bose Hubbard model Superflud state In the superflud phase we have an order parameter b = Φ e φ. We expect two types of collectve mode: phase fluctuatons correspond to gapless Bogolubov-lke exctatons. Ths s the Goldstone mode of the spontaneously broken symmetry. Fluctuatons of the ampltude of the order parameter correspond to a massve Hggs lke mode (see fg. 6.6). We dscuss smple analyss that llustrates the appearance of these modes. We consder a varatonal wavefuncton n whch we keep states wth n 1, n, and n + 1 partcles only. Ψ = We are nterested n states wth the average number of atoms per ste n, hence γ should be close to π/4. We ntroduce γ = π 4 σ. Truncatng the Gutzwller wavefuncton to only three Fock (number) states s a reasonable approxmaton close to the superflud to Mott transton, where number fluctuatons are small. For smplcty, we wll also assume that n s large so we can neglect the dfference between n and n ± 1. Frst we do the mean-feld analyss for the wavefuncton (6.19).The mnmum energy state requres χ = 0, φ arbtrary but unform, and σ = 0 when the densty (7.19)

9 7.4. COLLECTIVE MODES IN BOSE HUBBARD MODEL 9 Fgure 7.5: Weddng cake structure n a parabolc potental[3]. In these experments the number of partcles n ndvdual wells s measured modulo two. In other words two partcles appear as zero, three partcles appear as one. Fgures show both the average number of partcles and the varance. The varance s large n the superflud shells. s precsely n. Energy as a functon of θ E = nzt sn θ + U cos θ (7.0) Mnmzng wth respect to θ we fnd that when U > 4nzt we have a Mott state wth θ = π/, when U < 4nzt we fnd a superflud state wth cos θ 0 = U/4nzt (7.1) Our goal s to project dynamcs nto the varatonal state (6.19) and analyze collectve modes. Let us consder the Lagrangan defned as L = Ψ(t) ( ) t + H Ψ(t) (7.) If Ψ(t) was arbtrary, by fndng an extremum of L wth respect to Ψ(t) we would recover the Schroednger equaton. To fnd projected dynamcs we assume that wavefunctons n (6.) are lmted to the class of wavefunctons descrbed by equaton (6.19) and look for the extremum of L [10, 6]. All parameters n (6.19) are functons of tme and can be dfferent at dfferent stes. To fnd the extremum of L one can wrte equatons of moton for ndvdual varables d t q L = q L, where q stands for all varables n (6.19). Snce we are nterested n collectve modes only, we consder small fluctuatons around the eulbrum state (6.1) and expand up to quadratc order n σ, χ, δθ = θ θ 0. Moreover we are nterested n the long wavelength behavor of collectve modes. Thus our plan s to get the contnuum lmt of L frst and then wrte equatons of moton. We have Ψ(t) ( ) t Ψ(t) = [ cos θ 0 χ sn θ 0 δθ χ + sn θ 0 σ φ ] (7.3)

10 10 CHAPTER 7. BOSE HUBBARD MODEL Fgure 7.6: Schematc llustraton of the orgn of two modes n the superflud state. A state of broken symmetry can be represented as resdng n the trough of the Mexcan hat potental. Fluctuatons of the phase of the order parameter correspond to movng along the bottom of the trough. They gve rse to the gapless Bogolubov mode. Fluctuatons n the magntude of the order parameter ( uphll wth respect to the potental) correspond to the gapped ampltude (Hggs) exctaton. The part of (6.) wth the Hamltonan s more subtle so let us dscuss t n more detals Ψ(t) H kn Ψ(t) = nt j sn θ cos θ sn θ j cos θ j { } [cos γ e (φ χ) + sn γ e (φ+χ) ][cos γ j e (φj χj) + sn γ j e (φj+χj) ] + c.c. (7.4) We have sn θ cos θ sn θ j cos θ j = 1 4 sn(θ 0 + δθ ) sn(θ 0 + δθ j ) 1 4 sn θ sn 4θ 0(δθ + δθ j ) + 1 cos 4θ 0(δθ + δθ j ) 1 cos θ 0 (δθ δθ j ) (7.5) and [ cos( π 4 + σ )e (φ χ) + sn( π 4 + σ )e (φ+χ) ][cos( π 4 + σ j)e (φj χj) + sn( π 4 + σ j)e (φj+χj) ] + c.c. 1 ( σ φ χ + φ + σ χ ) ( σ j φ j χ j φ j + σ j χ j ) + c.c. = 4 [χ + χ j + σ + σ j + (φ φ j ) ] (7.6)

11 7.4. COLLECTIVE MODES IN BOSE HUBBARD MODEL 11 Hence Ψ(t) H kn Ψ(t) = N stes nzt sn θ 0 nt sn 4θ 0 ( δθ + δθ j ) nt cos 4θ 0 (δθ + δθj ) + nt cos θ 0 (δθ δθ j ) j j + nt sn θ 0 [χ + χ j + σ + σj + (φ φ j ) ] (7.7) j Here z s the coordnaton number. And we also fnd Ψ(t) H U + H µ Ψ(t) = U cos θ = U cos(θ 0 + δθ ) + const 4 = N stesu cos θ 0 U 4 sn θ 0 δθ U cos θ 0 δθ (7.8) To analyze long wavelength exctatons we take the contnuum lmt of ths expresson. Ths means (σ + σj ) = z a d dx d σ (x) j (φ φ j ) = j 1 a d j dx d ( φ(x)) (7.9) Here a s a lattce constant whch we wll set equal to one. Thus we fnd for the relevant part of the lagrangan L = d d x ( sn θ 0 δθ χ + sn θ o σ φ) ( nzt cos 4θ 0 + U cos θ 0 ) d d x ( δθ ) + nt cos θ 0 d d x ( δθ ) + znt sn θ 0 d d x [ χ + σ ] + nt sn θ 0 d d x ( φ ) (7.30) Note that terms lnear n fluctuatng felds canceled because we expanded around a state that satsfes the mean-feld mnmzaton procedure. We also omtted the frst term n (6.3) snce t corresponds to the ntegral of the tme dervatve and does not change equatons of moton. Equatons of moton separate nto two coupled pars. The frst par corresponds to the usual superflud hydrodynamcs: contnuty equaton and Josephson relaton between the rate of phase wndng and the change n the chemcal potnetal sn θ 0 σ = nt sn θ 0 φ sn θ 0 φ = znt sn θ 0 σ (7.31)

12 1 CHAPTER 7. BOSE HUBBARD MODEL The second par of equatons s less famlar sn θ 0 δ θ = znt sn θ 0 χ sn θ 0 χ (nzt cos 4θ 0 + U cos θ 0)δθ 4nt cos θ 0 δθ = 0 (7.3) From equatons (6.31) we fnd φ = 4z(nt) cos 4 θ 0 φ (7.33) Ths equaton descrbes the phase mode ( = Bogolubov mode = Goldstone mode) wth ω k = v k. From equatons (6.3) we obtan wth θ = ω 0θ + α θ (7.34) ω 0 = (4nzt) U 3 (7.35) Ths corresponds to a massve ampltude (Hggs) mode ω k = ω 0 + αk. Note that at the transton pont nto the Mott phase U = 4znt and the energy of the ampltude mode goes to zero. Such mode softenng s expected genercally at a contnuous quantum phase transton. Analyss presented above can be easly extended away from the long-wavelength lmt. The full spectrum of collectve modes s shown n fgure Mott state In the nsulatng state we fnd partcle- and hole-lke collectve modes (see fg. 6.7). If we take mddle of the Mott plateau wth µ 0 = U(n 1 ) we fnd that energes of these exctatons are E k = U nt(cos k x + cos k y + cos k z ). Ths s the partcle-hole symmetrc case. At the transton pont nto the SF state the energy of both p- and h-lke exctatons goes to zero. So the SF state can be thought of as a result of Bose condensaton of p- and h-lke exctatons Away from the p-h symmetrc case, when µ = µ 0 + δµ energes of partcleand hole-lke exctatons dffer E {p,h} k = U nt(cos k x + cos k y + cos k z ) δµ. The spectrum of exctatons n the Mott state s shown n fgure Probng collectve modes The frst experments probng collectve modes n an optcal lattce were lattce modulaton experments performed by Stoeferle et al [13] (see fg. 6.10). In the Mott state the system can only be excted by creatng partcle-hole exctatons, whch requres energy U. Thus we see peaks at fnte energy. In the superflud state we have gapless exctatons so one would navely expect that we should se response at low frequences. But ths s not what we see n experments. There are two mportant factors. Frstly, lattce modulaton s a translatonally

13 7.4. COLLECTIVE MODES IN BOSE HUBBARD MODEL 13 Fgure 7.7: Schematc representaton of exctatons n the Mott state. Holelke exctaton (top) corresponds to a mssng atom. Partcle-lke exctaton (bottom) corresponds to an extra atom. nvarant perturbaton, so t can only create exctatons wth the net momentum equal to zero. So lattce modulaton can excte an ampltude mode at k = 0 or a par of the Bogolubov exctatons wth the opposte momenta ( see fg. 6.9). Secondly, t can be shown [5, 6] that lattce modulaton does not couple to Bogolubov exctatons n the long wavelength lmt. Roughly the argument s that Bogolubov modes correspond to densty fluctuatons, whereas modulaton of tunnelng does not couple to the densty. Thus even n the SF state the peak of the absorpton spectrum s at fnte frequences: ths s the combnaton of the ampltude mode and pars of Bogolubov exctatons from near the zone boundary (phase space also favors exctng large q modes). Note that these experments do not show mode softenng at the transton. Close to the SF/Mott transton the energy of the ampltude mode becomes smaller but ts couplng to lattce modulaton n the long wavevlength lmt becomes suppressed. Ths s not surprsng snce at the pont of the SF/Mott transton the ampltude mode s essentally the same as the phase mode (when there s no expectaton value of the order parameter we can not separate longtudnal and transverse fluctuatons). Expermental observaton of the ampltude mode would be really exctng. Especally f we could see the mode softenng to demonstrate the basc feature expected at the quantum phase transton. A queston of the dampng of the ampltude mode s not resolved theoretcally. It s expected that the mode remans underdamped n d=3 but may be overdamped n lower dmensons. Another possble probe of the ampltude mode would be to change parameters n the SF state and observe oscllatons of the order parameter. Ths would appear as oscllatons n the number fluctuatons as measured by Bakr et al. [8]. The man dffculty of such experments would be nhomogeneous confnng potental. Dfferent parts of the system would oscllate at dfferent frequences so the net oscllatons could be strongly suppressed. Expermental resoluton also requres fnte strength of change n the parameters whch quckly takes us outsde of the harmonc theory.

14 14 CHAPTER 7. BOSE HUBBARD MODEL Fgure 7.8: Collectve modes n the Bose Hubbard model. Fgure (a) and (c) show the dsperson of hole- and partcle lke exctatons for dfferent values of the nteracton and the chemcal potental. Fgures (b) and (d) show dspersons of the ampltude and the phase modes. Fgure (e) shows the the change n the spectrum across the SF/Mott transton as the system s tuned through the tp of the Mott lobe (partcle-hole symmetrc case). Fgure (f) shows a change n the spectrum across the SF/Mott transton when the Mott lobe boundary s crossed through ts sde (ths s the p-h asymmetrc case). Fgure taken from [5]. 7.5 Extended Hubbard models One can also consder extensons of the Hubbard model to non-local nteractons. For example, f we add nearest-neghbor nteractons we have H = t b b j + U n (n 1) + V n n j µ j j n (7.36) Hamltonan (6.36) allows a new type of an nsulatng state: a checkerboard state shown n fg6.11. Ths state breaks lattce translatonal symmetry. Transton from the superflud to the checkerbaord phase nvolves gong from one spontaneously broken symmetry to another (number conservaton n the SF to translatonal symmetry n the CB). Thus t s natural to expect the appearance of the ntermedate phase where both are broken. Ths s called the supersold phase. Ths phase has been predcted from the Gutzwler analyss [14] (see fg. 6.1) and verfed n the Monte-Carlo calculatons [1]. Such phases are expected to be relevant for polar molecules n optcal lattces[7].

15 7.6. PROBLEMS FOR CHAPTER?? 15 Fgure 7.9: Theoretcal analyss of the energy absorpton spectrum n lattce modulaton experments n the bosonc Hubbard model. Lattce modulaton can excte the ampltude mode at k = 0 or a par of Bogolubov modes wth the opposte momenta (so that the net momentum deposted nto the system s zero). Bogolubov modes wth large momenta are predomnantly excted. Hence absorpton spectrum s domnated by the fnte energy peaks even n the superflud state. Deeply nto the superflud regme the ntensty of the ampltude ( Hggs ) mode peak s strongly reduced. Fgure taken from [6]. 7.6 Problems for Chapter 6 Problem 1 In ths problem you wll analyze effects of nteractons on the Bloch dynamcs of Bose-Ensten condensates n one dmensonal optcal lattces. Hamltonan of the system s gven by H = J b l b k + U l lk n l (n l 1) df l ln l (7.37) here J s the hoppng, U s the nteracton strength, d the lattce perod, F magntude of the statc force. a) Show that Hamltonan (6.37) leads to a lattce verson of the GP equaton ḃl = J (b l+1 + b l 1 ) + U b l b l df lb l (7.38) b) Introduce new varables b k = 1 l=l e kl ωblt b l (7.39) L l=1 where ω B = df s the Bloch frequency. Show that GP equatons (6.38) can be

16 16 CHAPTER 7. BOSE HUBBARD MODEL Fgure 7.10: Lattce modulaton experments across the SF/Mott transton. Fgure taken from [13]. wrtten as ḃk = Jcos(dk ω B t)b k + U b k1 b k L b k3 δ(k k 1 + k k 3 ) (7.40) k 1,k,k 3 Equaton (6.40) allows a trval soluton ( b 0 (t) = exp J df sn(ω Bt) UN ) 0 dl t (7.41) What s the physcal nterpretaton of ths soluton? c) By lnearzng equatons (6.38) around the mean-feld soluton (6.41) we obtan ḃ+k = Jcos(dk ω B t)b +k + U L b 0 b +k + U dl b 0b k ḃ k = Jcos(dk ω B t)b k + U L b 0 b k + U dl b 0b +k (7.4) Use these equatons to calculate the decay rate of Bloch oscllatons Perform numercal analyss for U = 0.4. Calculate the decay rate as a functon of kd and F. Hnt : Introduce the Floquet matrx V = T t exp[ U TB 0 ( 1 f(t) f (t) 1 where T t denotes tme orderng, T B = π/ω B, and ) dt] (7.43) f(t) = exp( J df [1 cos(dk)] sn(ω Bt)) (7.44)

17 7.6. PROBLEMS FOR CHAPTER?? 17 Fgure 7.11: Checker-Board state. Insulatng state that appears for nearest neghbor nteractons. Unlke the Mott state t has a spontaneously broken symmetry: lattce translatonal symmetry. Consder the maxmal egenvalue of the Floquet matrx V and relate t to the soluton of the form b ±k (t) exp(νt). d) More dffcult problem. One can use Feshbach resonance to change the contact nteracton. When the s-wave scatterng length s tuned to zero one s stll left wth magnetc dpolar nteractons. In ths problem you need to analyze the resdual value of the decay rate of Bloch osclaltons due to dpolar nteractons. To solve ths problem you also need to nclude transverse degrees of freedom. For smplcty assume layers to be nfnte, so exctatons can be characterzed by the n-plane momentum q. Use the followng form of effectve dpolar nteractons: Intralayer nteracton V 0 (q) = U d W π 3U d W F ( q ) (7.45) π Interlayer nteracton for layers l lattce constant apart from each other V l (q) = 3U d q exp{ q ld} (7.46) Here W s the thckness of ndvdual layers, U d s the strength of dpolar nteracton, q s the n-plane momentum, F (x) = π W q W q [1 Erf( )]e q W /, where Erf(x) s the error functon. These expresson assume that dpolar moments are perpendcular to the planes and take nto account fnte wdth of ndvdual layers. Formula (6.46) apples when W << d. Problem Consder Hubbard model wth nonlocal nteractons H = t b b j µ n + U n (n 1) + V 1 n n j + V j j k n n k (7.47)

18 18 CHAPTER 7. BOSE HUBBARD MODEL Fgure 7.1: Phase dagram of the extended Hubabrd model wth on-se and nearest neghbor nteractons. Sol densotes the checkerboard phase, Ssol denotes the supersold. Fgure taken from [14]. In the lmt when U s large we can keep states wth occupatons 0 and 1 only. Ths s known as the lmt of hard-core bosons. a) Show that n ths lmt Hamltonan (6.47) can be mapped to the spn model H = t j (S x S x j + S y Sy j ) h S z + V 1 S z Sj z + V S z Sk z (7.48) j k Dscuss the relaton between varous spn ordered states of (6.48) and nsulatng/superflud/supersold states of orgnal bosons. b) Use Cure-Wess type mean feld approach to study the phase dagram of (6.48). Assume V = 0. Keepng V 1 fxed plot the phase dagram as a functon of µ and t. c) Extend analyss of part b) to fnte V. Problem 3 In ths problem you wll consder collapse and revval experments wth (spnless) bosonc atoms n an optcal lattce (M. Grener et al. (00)). The system s prepared n a superflud state. You can take ths ntal state to be a product

19 7.6. PROBLEMS FOR CHAPTER?? 19 of coherent states for ndvdual wells Ψ(t = 0) = α α = e α / n α n n! n (7.49) where n s a Fock states wth n atoms n a well. At t = 0 the strength of the optcal lattce potental s suddenly ncreased to a very large value, so that dfferent wells become completely decoupled. You can take the Hamltonan n ths regme to be H = U n (n 1) (7.50) After the system evolves wth the Hamltonan (6.50) durng tme t, the TOF measurement s performed: both the perodc and parabolc confnng potentals are removed, atoms expand freely, and mage of the cloud s taken after long expanson. a) Show the ampltude of nterference peaks n the TOF mages collapses after some tme t then revves, and then ths cycle contnues. Calculate both collapse and revval tmes. b) Show that half-way between revvals the system goes through the so-called cat state, n whch b = 0 but b 0.

20 0 CHAPTER 7. BOSE HUBBARD MODEL

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