8. Superfluid to Mott-insulator transition

Transcription

1 8. Superflud to Mott-nsulator transton

2 Overvew Optcal lattce potentals Soluton of the Schrödnger equaton for perodc potentals Band structure Bloch oscllaton of bosonc and fermonc atoms n optcal lattces Wanner functons Bose-Hubbard Hamltonan Superflud to Mott-nsulator transton

3 Optcal dpole traps Potental: U Dp ε c () r = d E = Re( α ) I( r) I () r t 0 Γ Δ ϕ Loss rate: γ sc P ω 1 ε c abs () r = = Im( α ) I( r) I () r 0 Γ Δ 2 abs ω 0 ω d: nduced dpole moment, α: polarzablty Γ: Dpole matrx element between ground and excted states, Δ=ω ω 0 : detunng Δ<0 : red detuned attractve potental Δ>0 : blue detuned repulsve potental Gaussan beam: I() r = I e 0 2 2r 2 w

4 Optcal lattces Optcal standng wave felds produce perodc potentals wth lattce constants half of the laser s wavelenght (λ ~ 800 nm 10 μm). π I x = I 2 2 ( ) 0 sn x + ϕ λlaser a = λ 2 laser 1-D 2-D 3-D

5 Band model Solutons of the Schrödnger equaton n perodc potentals SE: Ansatz: Comparng coeffcents yelds condtonal equatons for parameters C(k) For a specfc k [-k 0 /2, k 0 /2] the wavefuncton ψ k contans also wavevectors k+nk 0. In order to fnd energes and egenstates for a gven k, the equaton system (*) has to be solved for all k [k+nk 0 ]. Note: k> s not an egenstate of the Hamltonan.

6 Band model energes and egenstates system

7 Band model example

8 Dynamcs n a lattce potental Phase dfference between neghborng lattce stes ϕ j ϕ = ( E E ) j t Bloch-Oscllaton (1D lattce) Δφ = 0 Nonlnear dynamcs leads to dephasng f gradent s left on for longer tmes! (2D lattce) Δφ =π M. Grener et al. PRL 87, (2001)

9 Bloch oscllaton of fermonc atoms Large contrast for mor than 100 oscllaton perodes Comparson: fermons vs. bosons (a) Momentum dstrbuton of fermons n the lattce: 1 ms (contnuous lne) and 252 ms (dashed lne). (b) Momentum dstrbuton of bosons at 0.6 ms (contnuous lne) and 3.8 ms (dashed lne). The much faster broadenng for bosons s due to the presence of nteractons. T B =1.2 s G. Roat et al., Phys. Rev. Lett. 92, (2004)

10 Ferm surfaces

11 Bose gas n a 3D lattce potental Resultng potental conssts of a smple cubc lattce where the BEC coherently populates about 100,000 lattce stes. M. Grener et al. PRL 87, (2001)

12 Interference of a superflude bose gas from a 3D lattce M. Grener et al. PRL 87, (2001)

13 Wave functon of sngle partcles n a perodc potental Bloch states: Ψ (r) = k u (r)e k kr Plane waves modulated by a lattce perodc functon Fourer transform Wanner states: j w(r R ) = e Ψ (r) j 1 L BZ k kr k Localzed wave functons. Ths pctures allows tunnelng as well as localzed states. Inverse Fourer transform lattce stes 1 kr Ψ k = L j (r) e w(r R ) j j The Hamltonan can be wrtten n Wanner bass (TONS). Second quantzaton results n the Bose-Hubbard Hamltonan. (Necessary condton: only the lowest band s populated exctaton energes are larger than the energy gap.)

14 Bose-Hubbard Hamltonan Expandng the feld operator n the Wanner bass of localzed wave functons on each lattce ste, yelds : ψ ˆ ( x) = aw ˆ ( x x ) Bose-Hubbard Hamltonan 1 H = J a a n + U nˆ ( ˆ n 1) 2, j ˆ ˆ ˆ j + ε Bosonc operators annhlaton creaton number â..., N,... = N..., N 1,... â..., N,... = N , N + 1,... + ( ˆ ) * a a ˆ + Sngle partcle energy n the trappng potental aa ˆ ˆ a,a ˆ ˆ + =δ = j j = n + j j j Tunnel matrx element (hoppng element) Onste nteracton matrx element J = d xw( x x ) + V ( ) w( ) x 2m lat x x j U 2 4π a = m 3 d x w ( x) 4 M.P.A. Fsher et al, PRB 40, 546 (1989); D. Jaksch et al., PRL 81, 3108 (1998)

15 Superflud Lmt The knetc energy domnates (weakly nteractng bosonc system). 1 H = J aˆ ˆ ˆ ( ˆ aj + U n n 1) 2, j Atoms are delocalzed over the entre lattce, macroscopc wave functon descrbes ths state. M N Ψ ˆ SF a 0 = 1 a 0 Possonan atom number dstrbuton per lattce ste n=1 n=2

16 Mott-nsulator Interacton energy domnates (strongly correlated bosonc system). 1 H = J aˆ ˆ ˆ ( ˆ aj + U n n 1) 2, j Atoms are completely localzed to lattce stes. M = 1 ( a ) n Ψ 0 a = 0 Mott Fock states wth a vanshng atom number fluctuaton are formed, e.g. n=1. n=1

17 Momentum dstrbuton for dfferent potental depths (E recol ) 0 E recol 22 E recol M. Grener et al., Nature 415, 39 (2002)

18 Restorng coherence a) Ramp up the potental for generatng the Mott-nsuator state and subsequent ramp down. b) Wdth of the zero momentum central peak The coherence s restored wthn the tunnelng tme to the negborng lattce ste. Measurement on a phase ncoherent cloud. (Dephasng was appled before reachng the Mott-nsulator state) Measurement on a phase ncoherent cloud. Before rampng down the potental M. Grener et al., Nature 415, 39 (2002)

19 Quantum phase transton from a superflud to a Mott-nsulator At the crtcal pont g c the system wll undergo a phase transton from a superflud to an nsulator. Ths phase transton occurs even at T=0 and s drven by quantum fluctuatons! Characterstc of the quantum phase transton Exctaton spectrum s dramatcally modfed at the crtcal pont. U/J < g c (Superflud regme) Exctaton spectrum s gapless U/J > g c (Mott-Insulator regme) Exctaton spectrum s gapped Crtcal rato: U/J = z * 5.8 number of next neghbors (for a cubc lattce 6)

20 Creatng exctatons n the MI phase Mott-nsulator wth n =1 atom per lattce ste Wthout gradent potental Wth gradent potental Specal case: ΔE j = U ω U U J Energy Scales: v 20

21 Measurng the exctaton gap n the MI phase (probablty vs. gradent) 10 E recol t perturb = 2 ms 13 E recol t perturb = 4 ms 16 E recol t perturb = 9 ms 20 E recol t perturb = 20 ms M. Grener et al., Nature 415, 39 (2002)

22 Lterature Bose-Ensten condensates n 1D- and 2D optcal lattces M. Grener et al. Appl. Phys. B. 73, 769 (2001) Exporng the phase coherence n a 2D lattce of Bose-Ensten condensates M. Grener et al. PRL 87, (2001) Fermonc atoms n a three dmensonal optcal lattce: observng Ferm surfaces, dynamcs and nteractons, M. Köhl et al., PRL94, (2005) Quantum phase transton from a superflud to a Mott nsulator n a gas of ultracold atoms M. Grener et al., Nature 415, 39 (2002)