The Yrast Line of a Rapidly Rotating Bose Gas: Gross-Pitaevskii Regime

The Y rast Line of a Rapidly Rot ating Bose Gas: Gross-Pitaevskii Regime Elliott H. Lieb Dep artments of Mathematics and Physics, Princ et on Univers i ty, Pri nc eton, NJ 08544, USA ∗ Rob ert Seiringer Dep artment of Physics, Princ eton University, Princ eton, NJ 08544, USA † Jakob Y ng v ason F ak ult¨ at f¨ ur Physik, Universit¨ at W ien, and Erwin Schr¨ odinger Institute for M athematic al Physics, 1090 V ienna, Austria ‡ W e consider an ultracold rotating Bose gas in a harmonic trap close to th e critical angular v e- locity so that the system can b e considered to b e confined to the low est Landau level . With this assumption we prov e t hat th e Gross-Pitaevskii en ergy functional accurately d escrib es the ground state energy of the corresp onding N -b o dy Hamiltonian with contact in t eraction provided the total angular momentum L is m u ch less than N 2 . While the Gross-Pitaevskii en ergy is alw ays an obvious v ariational upp er b oun d to the ground state energy , a more refin ed analysis is needed to establish it as an exact low er b ound. W e also discuss the q uestion of Bose-Einstein condensation in the pa- rameter range considered. Coherent states together with inequ alities in spaces of analytic functions are the main technical tools. I. INTRO D UCTION A B ose ga s rotating in a harmo nic trap has a cr iti- cal ro tation sp eed ab ove which the trap ca nnot confine it aga inst centrifugal for ces. If the tra pping p otential equals m 2 ( ω 2 ⊥ r 2 + ω 3 x 2 3 ), with m the par ticle mass and r = p x 2 1 + x 2 2 the distance fro m the a xis o f r otation, then the cr itical angular velo city is ω ⊥ . In a reference frame rotating with a ngular velocity Ω the Hamiltonian for one particle is p 2 2 m + m 2  ω 2 ⊥ r 2 + ω 2 3 x 2 3  − Ω L (1) with L the compo nent of the angular momen tum along the rotation axis. It is conv enient and instructive to co m- plete the sq uare and write (1) as 1 2 m ( p − A ) 2 + mω 2 3 2 x 2 3 + ( ω ⊥ − Ω) L (2) where A = mω ⊥ ( x 2 , − x 1 , 0). In the rapidly ro tating case, where 0 < ω ⊥ − Ω ≪ min { ω ⊥ , ω z } , it is natura l to restrict the allowed wa ve functions to the g round sta te space of the firs t tw o terms in (2), which we denote b y H , and this restr iction will b e made in this pap er. The space H consists o f functions in the lo west Landau level (LLL) for motion in the plane p erp endicular to the axis of r otation, m ultiplied by a fixed Gaus sian in the x 3 -direction. Apar t from the irrelev ant additive constants 2 ~ ω ⊥ and ~ ω 3 , the kinetic energ y in H is simply ω L (3) ∗ Electronic address: lieb@princeton.edu † Electronic address: rseiring@princeton.edu ‡ Electronic address: jako b. yngv ason@univie.ac.at with ω = ω ⊥ − Ω > 0. Note that L is no n-negative for functions in H . Its eigenv alues are 0 , ~ , 2 ~ , . . . . T o characterize the functions in the s pace H , it is nat- ural to intro duce complex notation, z = x 1 + ix 2 . F unc- tions in H are of the form f ( z ) exp  − mω ⊥ 2 ~ | z | 2 − mω 3 2 ~ | x 3 | 2  (4) with f a n a na lytic function. All the freedom is in f since the Gaussian is fixed. If the tra pping p otential in the x 3 - direction is not quadratic, the Gaussian in the x 3 -v ariable has to be r eplaced by the a ppropriate gro und state wav e function. A fancy wa y of saying this is that our Hilb ert space H co nsists of a nalytic functions on the complex pla ne C with inner pro duct given by h φ | ψ i = Z C φ ( z ) ∗ ψ ( z ) e −| z | 2 dz , (5) where dz is sho rt for dx 1 dx 2 . F or simplicit y we choos e units such that m = ~ = ω ⊥ = 1. The eigenfunction of the angular mo men tum L corresp o nding to the eigen- v a lue n is simply z n . In other w ords, L = z ∂ z . W e remark that the exp ectatio n v a lue of r 2 = | z | 2 in this state is n + 1. F or a system of N b osons, the appropria te wa ve func- tions are analy tic and symmetric functions of the bo sons co ordinates z 1 , . . . , z N . The Hilb ert space is thus H N = ⊗ N symm H . The kinetic energy is simply ω times the to ta l angular momentum. In a ddition to the k inetic energy , there is pairwise in- teraction a mong the bo sons. It is assumed to b e short range compar ed to a ny other characteristic length in the system, and can b e mo deled by g P 1 ≤ i 0. Physically , this g is prop ortiona l to aω 1 / 2 3 where a is the scatter ing leng th of the thre e -dimensional interaction p otential. O n the full, 2 original, Hilb ert space ⊗ N symm L 2 ( R 2 ), a δ -function as a repulsive interaction p otential is meaningless. On the subspace H N the matrix elements of δ ( z i − z j ) mak e p e r- fect sense, how ever, and define a b ounded op erator δ ij . Using the analyticity of the w av e functions this op erato r is easily shown to act as ( δ 12 ψ ) ( z 1 , z 2 ) = 1 2 π ψ (( z 1 + z 2 ) / 2 , ( z 1 + z 2 ) / 2) , (6) which tak es analytic functions in to analytic functions. Its matrix elements in a tw o- particle function ψ ( z 1 , z 2 ) a re h ψ | δ 12 | ψ i = Z C | ψ ( z , z ) | 2 e − 2 | z | 2 dz . (7) The dimensio nal reduction from three to tw o dimen- sions for the N -b o dy problem and the restriction to the LLL is, of course, only r easonable if the in ter action en- ergy p er particle is mu ch less tha n the ener g y gap 2 ~ ω ⊥ betw een Landau levels and the gap ~ ω 3 for the motion in the x 3 -direction. F or a dilute gas the in tera ction en- ergy p er par ticle is of the o r der aρ where ρ is the av- erage three-dimensional density [1]. Provided N g / ω is not small we ca n e stimate ρ by noting that the effec- tive radius R of the system ca n b e o btained by eq uating aρ ∼ N a/ ( R 2 ω − 1 / 2 3 ) ∼ N g / R 2 with the k inetic energy ω L ∼ ω R 2 . This gives R ∼ ( N g /ω ) 1 / 4 and the condition for the res triction to the LLL b eco mes ( N g ω ) 1 / 2 ≪ min { ω ⊥ , ω 3 } . (8) The physics of r apidly rota ting ultr a cold Bos e gases close to the LLL regime has b een the sub ject of many the- oretical and exp erimental in vestigations in recent years, starting with the pap ers [2, 3, 4 , 5, 6]. The recent re- views [7, 8, 9, 1 0] contain extensive lists of references on this sub ject. On the exp erimental side we mention in particular the pap ers [1 1, 12, 1 3] that rep ort on e xp e r - imen ts with ro tational fre q uencies exceeding 99% of the trap frequency . II. MODEL AND MAIN RESUL T The discussio n in the Intro duction leads us to the fol- lowing well-kno wn model (see, e.g., [2, 7, 10]) for N bo sons with r epulsive short-ra nge pa ir wise interactions: H = ω N X i =1 L i + g X 1 ≤ i 0 there exists a C < ∞ such that (1 1) holds if g N /ω ≥ c . In par ticular, this implies that the ratio of E 0 and E GP is clos e to one if g ≪ N ω a nd g & N − 1 ω . Note that, by simple scaling, E GP ( N , ω , g ) = N ω E GP (1 , 1 , g N / ω ). Hence the contrary cas e of s mall g N /ω is not particularly in teresting; a s g N /ω → 0 one obtains a non-interacting gas. On the other hand, if g N /ω is not small the kinetic and in tera ction energy are of the same order o f magnitude, which, acc ording to the pr evious back-of-the-env e lop e analysis, implies E GP ( N , ω , g ) ∼ N √ N g ω . Hence the total angular mo- men tum, which is obtained by ta king the deriv a tive of 3 the energy with resp ect to ω , is of the or der N p N g /ω , which is muc h less than N 2 if and only if g ≪ N ω . Hence our b ound (11) co vers the whole par a meter regime L tot ≪ N 2 . W e note also that in ter ms of the fil ling factor ν = N 2 / (2 L tot ) [6] the pa rameter regime g / ( N ω ) ≪ 1 co rre- sp onds to ν ≫ 1 . In contrast, the Laughlin w ave function has filling factor ν = 1 / 2 for N larg e. Betw een these tw o extremes rich physics r elated to the FQHE is ex p ected [10], but this reg ime is apparently still o ut of exp erimen- tal rea ch. Before giving the pro o f o f (11), w e shall discuss some of its implications for the yrast line. Recall that the yrast energ y I 0 ( L ) is defined as the gro und s tate e nergy of P i 0 and consider the mo dified Hamiltonian H ′ = H + η J 3 / 2 ω N X i =1 χ J ( L i ) . By (18) its gr ound state energy is b ounded from ab ov e by E 0 (1 + b η ), with b = 2 3 / 2 (1 + 4 √ π ). In other words, E 0 is b ounded from b elow b y the ground s tate ener g y of H ′ divided by (1 + b η ). W e choose η = ( g / N ω ) 1 / 10 . B. Step 2 In order to der ive a low er b ound on H ′ , we sha ll first extend it to F o ck s pace in the usual way . W e do this in order to b e able to utilize coherent states a nd the lo wer and uppe r symbols of the ex tended o pe rator. Let ϕ j ( z ) = z j ( π j !) − 1 / 2 denote the nor malized eigen- functions o f the ang ular momentum op erator L with eigenv alue j ∈ N . Let a † i and a i denote the co r resp ond- ing cre a tion and annihila tio n op era to rs. On F o ck space, consider the op era tor H = ω X i ≥ 0 h i + η J 3 / 2 χ J ( i ) i a † i a i + g 2 X ij k h ϕ i ⊗ ϕ j | δ | ϕ k ⊗ ϕ i + j − k i a † i a † j a k a i + j − k + µ  X i ≥ 0 a † i a i − N  2 (19) for some µ > 0 . F or simplicit y w e deno te δ 12 simply by δ . Note that beca use of conserv ation of angular momentum all other matrix elements of δ v anish. The Hamiltonian H ′ can be view ed as the re s triction o f H to the sub- space containing exactly N particle s . The v alue of µ is irrelev ant, s ince the term m ultiplying µ v anishes in this subspace. In pa rticular, the gr ound state ener g y of H ′ is b o unded from b elow by the gro und state ener gy o f H , for any v alue of µ . W e introduce coherent states for all ang ular mo men- tum states up to J . That is, in Eq. (19) w e normal- o rder all the a † i and a i in the usual wa y (which is o nly rele- v a n t for the term multiplying µ since all other terms are already no rmal or dered) and then re place all a i by com- plex num b ers ζ i and all a † i by the conjugate num b er s ζ ∗ i for 0 ≤ i ≤ J . The re sulting op era tor on the F o ck s pa ce generated by the mo des > J is c a lled the lower sym- b ol of H a nd will be denoted by h ℓ ( ~ ζ ), where ~ ζ stands for ( ζ 0 , . . . , ζ J ). Note that h ℓ ( ~ ζ ) do es not cons e rve the particle num b er, which expla ins why it was necessary to embed our N -particle Hilb ert spa ce into F o ck spa ce. The ground state energy of H is b ounded from below b y the gro und s tate energy of the upp er symb ol h u ( ~ ζ ) when minimized ov er a ll pa rameters ~ ζ . The upp er symbol is given in terms of the low er symbo l as [27] h u ( ~ ζ ) = e − P J i =0 ∂ ζ i ∂ ζ ∗ i h ℓ ( ~ ζ ) = h ℓ ( ~ ζ ) + U 1 ( ~ ζ ) + U 2 ( ~ ζ ) , (20) where U 1 ( ~ ζ ) = − P J i =0 ∂ ζ i ∂ ζ ∗ i h ℓ ( ~ ζ ) eq uals − ω X i ≤ J i − 2 g X 0 ≤ i ≤ J h φ ζ ⊗ ϕ i | δ | φ ζ ⊗ ϕ i i − 2 g X 0 J h ϕ k ⊗ ϕ i | δ | ϕ k ⊗ ϕ i i a † k a k − µ  2 k φ ζ k 2 − 2 N + 2 N > + 1  ( J + 1 ) + 2 k φ ζ k 2  and U 2 ( ~ ζ ) = 1 2 J X i =0 ∂ ζ i ∂ ζ ∗ i ! 2 h ℓ ( ~ ζ ) = g X 0 ≤ i,j ≤ J h ϕ i ⊗ ϕ j | δ | ϕ i ⊗ ϕ j i + µ ( J + 1)( J + 2) . Here, φ ζ = P J j =0 ζ j ϕ j , k φ ζ k 2 = h φ ζ | φ ζ i = P J i =0 | ζ j | 2 and N > = P k>J a † j a j . Note that in a T aylor expan- sion of the exp onential in (20) only the first three ter ms contribute since h ℓ ( ~ ζ ) is a p olynomia l of degree four. 5 F or a low er bound, we can us e the fact that U 2 ( ~ ζ ) ≥ 0. Moreov er, to b o und the v ar ious ter ms in U 1 ( ~ ζ ) we can use the fact that P i ≤ J | ϕ i ( z ) | 2 ≤ π − 1 e | z | 2 . It is then easy to see that U 1 ( ~ ζ ) is b o unded from below as U 1 ( ~ ζ ) ≥ − ω 2 J ( J + 1) −  2 g π + 2 µ ( J + 2)  k φ ζ k 2 − 2 N > h µ ( J + 1 ) + g π i . (21) In order for the first term to b e muc h smaller than the GP energ y E GP ( N , g , ω ) ∼ N √ N g ω it is cle ar that we can no t re pla ce a ll mo des by a c- num be r but must rather require that J ≪ N ( g / N ω ) 1 / 4 . C. Step 3 As we hav e explained in the previous step, H ′ is bo unded from b elow b y the minimum ov er ~ ζ of the ground state energy of h u ( ~ ζ ). Supp ose the minimum is attained a t some ~ ζ 0 , and let h · i 0 denote the corre sp o nd- ing g round sta te exp ectatio n v alue in the F o ck space o f the modes > J . The next step is to use a simple low er bo und on h u ( ~ ζ 0 ) to b ound the η -dep endent term in H inv olving the mo des > J . Let us choos e the par ameters µ and J in following wa y . Recall from s tep 1 that J 0 ≤ J < 2 J 0 , with J 0 an ar bitrary integer that we a re free to choos e. W e take J 0 = ⌊ N ( g / N ω ) 3 / 10 ⌋ , where ⌊ t ⌋ denotes the integer part of a num b er t > 0, and µ = ω J / (4 N ). The pr op or- tionality constants here are chosen mo re fo r conv enience than for optimalit y . Note tha t J 0 is a big num b er if g / ( N ω ) ≪ 1 and g N /ω & 1. Recall that η = ( g / N ω ) 1 / 10 . W e cla im that, for small g / N ω , η * X i>J χ J ( i ) a † i a i + 0 ≤ 2 E GP ( N , ω , g ) ω J 3 / 2 . (22) This b ound is similar to (18) exce pt for the additional factor η on the left side and E GP replacing E 0 on the rig ht side. In or der to prove (22), we can use the p ositivity of the interaction to o btain the following low er b ound o n h u ( ~ ζ ). The low er sy m b o l h ℓ ( ζ ) is b ounded from b elow as h ℓ ( ~ ζ ) ≥ η X i>J χ J ( i ) a † i a i + ω ( J + 1) N > + µ   k φ ζ k 2 − N  2 + 2 N > ( k φ ζ k 2 − N ) + ( N > ) 2 + k φ ζ k 2  . (23) In particula r , using (2 1), h u ( ~ ζ ) ≥ ηω J 3 / 2 X i>J χ J ( i ) a † i a i + µ  k φ ζ k 2 − N  2 + N >  ( J + 1 )( ω − 2 µ ) − 2 g π − 2 µN  − N  2 g π + 2 µ ( J + 2)  − ω 2 J ( J + 1) −  2 g π + 2 µ ( J + 2)  ( k φ ζ k 2 − N ) . With the aid of Sch warz’s inequality , we o btain h u ( ~ ζ ) ≥ ηω J 3 / 2 X i>J χ J ( i ) a † i a i + N >  ( J + 1 )( ω − 2 µ ) − 2 g π − 2 µN  − N  2 g π + 2 µ ( J + 2)  − ω 2 J ( J + 1) −  2 g π + 2 µ ( J + 2)  2 4 µ . (24) F or our choice of µ and J , µ ≪ ω and g ≪ ω J for small g / N ω . Mor e ov er, µN = ω J / 4. Hence the se cond term on the right side of (24) is p ositive for small g / N ω . Moreov er, since E GP ( N , ω , g ) ∼ N √ N ω g for N g & ω , as r emarked ea r lier, all the terms in the last line of (24) are muc h smaller than E GP ( N , ω , g ) if g ≪ N ω and, in particular, are b ounded from b elow b y − E GP / 2. Since the g r ound state ener gy of h u ( ~ ζ 0 ) is b ounded ab ove by E 0 ( N , ω , g )(1 + b η ) ≤ 1 . 5 E GP ( N , ω , g ), Inequality (22) follows. D. Step 4 W e now giv e a mo re refined low er b ound on h ℓ ( ~ ζ ). In- stead of dro pping all the interaction terms, as we did in the previous step in order to obtain the b o und (23), we use the fact tha t δ ≥ P J ⊗ P J δ P J ⊗ P J + P J ⊗ P J δ ( P J ⊗ Q J + Q J ⊗ P J ) + ( P J ⊗ Q J + Q J ⊗ P J ) δ P J ⊗ P J , (25) where Q J = I − P J and P J denotes the pro jectio n on to the J + 1 dimens io nal subspace of H spa nned by eig enfunc- tions of the angular momentum with L ≤ J . Inequality (25) is a simple Sch warz inequa lit y and follows from posi- tivit y o f δ and the fact tha t P J ⊗ P J δ Q J ⊗ Q J = 0 . Using 6 (25), we s e e that h ℓ ( ~ ζ ) is b ounded fro m below by h ℓ ( ~ ζ ) ≥ E GP ( φ ζ ) + ω X j >J j a † j a j + µ   k φ ζ k 2 − N  2 + 2 N > ( k φ ζ k 2 − N ) + ( N > ) 2 + k φ ζ k 2  + g X k>J h φ ζ ⊗ φ ζ | δ | φ ζ ⊗ ϕ k i a k + h.c. . F or the r emaining terms in the upper symbol h u ( ~ ζ ) we pro ceed as in the pr e vious step. W e r etain µ ( k φ ζ k 2 − N ) 2 / 2 for later use, how ever, and a rrive at h u ( ~ ζ ) ≥ E GP ( φ ζ ) + µ 2  k φ ζ k 2 − N  2 + g X k>J h φ ζ ⊗ φ ζ | δ | φ ζ ⊗ ϕ k i a k + h.c. − N  2 g π + 2 µ ( J + 2)  − ω 2 J ( J + 1) −  2 g π + 2 µ ( J + 2)  2 2 µ . (26) The rig h t side of (26) cont ains the des ir ed quantit y E GP ( φ ζ ). The s econd term guar antees that k φ ζ k 2 is close to N . All the terms in the last t wo lines a re sma ll co m- pared to E GP ( N , ω , g ) for o ur c ho ic e of µ and J . Hence we a re left with giving a b ound on the third term in (26), which is linear in cr eation and annihila tion op er a tors. It is this b ound tha t requires most effo r t; it will be com- pleted in the remaining three s teps . E. Step 5 A simple Sch warz inequality s hows that, for any se - quence of co mplex n um b er s c k and p ositive num b ers e k , X k  c k a k + c ∗ k a † k  ≤ X k | c k | 2 e k + X k e k a † k a k . (27) W e apply this to our case, where c k = −h φ ζ ⊗ φ ζ | δ | φ ζ ⊗ ϕ k i for k > J , a nd c k = 0 o therwise. W e pic k so me κ > 0 and cho ose e k = κ − 1 χ J ( k ) for k > J , with χ J ( k ) given in (17). Then X k | c k | 2 e k = κ h φ ζ | ρ ζ χ − 1 J ( L ) ρ ζ | φ ζ i , (28) where ρ ζ denotes the mult iplication op era tor ρ ζ ( z ) = | φ ζ ( z ) | 2 e −| z | 2 and χ − 1 J ( L ) ≡ ∞ X k = J +1 1 χ J ( k ) | ϕ k ih ϕ k | . (F or s implicit y , we a buse the no ta tion slightly , s ince ρ ζ do es not leave H inv ar iant; h ϕ k | ρ ζ | φ ζ i makes p er fectly go o d sense, how ever.) In Step 7 b elow we shall show that h φ ζ | ρ ζ χ − 1 J ( L ) ρ ζ | φ ζ i ≤ 6  Z C | φ ζ ( z ) | 4 e − 2 | z | 2 dz  3 / 2 . (29) This inequality ma y seem surprising at fir st sight. The function ρ ζ | φ ζ i con ta ins angular mo menta up to L ≤ 2 J , and χ − 1 J (2 J ) grows exp onentially with J . As will b e- come clear b elow, ho wever, |h ϕ k | ρ ζ | φ ζ i| 2 is expo nent ially small unless k is sma lle r than ro ughly J + √ J . Since χ − 1 J ( J + √ J ) is b ounded indep endently of J , this makes (29) plausible. Altogether, (27)–(29) imply that X k  c k a k + c ∗ k a † k  ≤ 6 κ  Z C | φ ζ ( z ) | 4 e − 2 | z | 2 dz  3 / 2 + X k e k a † k a k . W e hav e shown in Step 3 ab ov e that * X k e k a † k a k + 0 ≤ 2 E GP ( N , ω , g ) κη ω J 3 / 2 in the gro und s ta te of h u ( ~ ζ 0 ). After optimizing ov er κ we get the following low er b ound on − P k ( c k a k + c ∗ k a † k ) (v alid as an exp ecta tio n v alue in the ground state): − 4 √ 3  Z C | φ ζ ( z ) | 4 e − 2 | z | 2 dz  3 / 4  E GP ( N , ω , g ) η ω J 3 / 2  1 / 2 . Finally , using H¨ o lder’s inequalit y this expre s sion is bo unded from b elow by − β 2 Z C | φ ζ ( z ) | 4 e − 2 | z | 2 dz − 6 5 β 3  E GP ( N , ω , g ) η ω J 3 / 2  2 for arbitra ry β > 0. F. Step 6 The analysis in the preceding steps has shown that h u ( ~ ζ 0 ) ≥ (1 − β ) E GP ( φ ζ 0 ) + µ 2  k φ ζ 0 k 2 − N  2 − N  2 g π + 2 µ ( J + 2)  −  2 g π + 2 µ ( J + 2)  2 2 µ − ω 2 J ( J + 1) − g 6 5 β 3  E GP ( N , ω , g ) η ω J 3 / 2  2 . (30) 7 As mentioned above, we choose µ = ω J / (4 N ) and J ∼ N ( g / N ω ) 3 / 10 . Moreov er, we ma ke the choice η = β = ( g / N ω ) 1 / 10 . Then all the er ror terms are small compared to E GP ( N , ω , g ), namely they are of the o rder ( g / N ω ) 1 / 10 E GP . Since the g r ound state ener gy of h ( ~ ζ 0 ) is b ounded ab ov e b y E GP ( N , ω , g )(1 + b η ) ≤ 1 . 5 E GP ( N , ω , g ) and all the er ror terms in (30) are b ounded by E GP / 2 for small g / ( N ω ), we see that   k φ ζ k 2 − N   ≤  2 E GP ( N , ω , g ) µ  1 / 2 ∼ N  g N ω  1 / 10 for ~ ζ = ~ ζ 0 and hence (1 − β ) E GP ( φ ζ ) + µ 2  k φ ζ k 2 − N  2 ≥ E GP ( N , ω , g )  1 − C  g N ω  1 / 10  for some constant C > 0. This yields the low er bo und in (11). G. Step 7 The only thing left to do is to prov e Ineq ua lity (2 9). F or conv enience, we s hall introduce the no tation k φ k p =  Z C | φ ( z ) | p e − ( p/ 2) | z | 2 dz  1 /p for 1 ≤ p < ∞ . W e can b ound h φ ζ | ρ ζ χ − 1 J ( L ) ρ ζ | φ ζ i ≤ k φ ζ k 2 4 k √ ρ ζ χ − 1 J ( L ) √ ρ ζ k . The latter norm is an op era tor norm on L 2 ( C , e −| z | 2 dz ) (defined a s the max im um exp ectation v alue), a nd equals the op erato r no rm of q χ − 1 J ( L ) ρ ζ q χ − 1 J ( L ) on H . Next, w e derive a p o int wise bo und on ρ ζ ( z ) = | φ ζ ( z ) | 2 e −| z | 2 . Let P J ( z , z ′ ) = P J j =0 ϕ j ( z ) ϕ j ( z ′ ) ∗ de- note the integral kernel of the pro jection o n to the sub- space of H with angular momentum L ≤ J . With the aid of H¨ o lder’s inequality , | φ ζ ( z ) | =     Z C P J ( z , z ′ ) φ ζ ( z ′ ) e −| z ′ | 2 dz ′     ≤ k φ ζ k 4 k P J ( z , · ) k 4 / 3 . Another application of H¨ older’s inequality yields k P J ( z , · ) k 4 / 3 ≤ k P J ( z , · ) k (4 − 3 p ) / (4 − 2 p ) 2 k P J ( z , · ) k p/ (4 − 2 p ) p for any 1 ≤ p ≤ 4 / 3. The L 2 norm of P J ( z , · ) is given by k P j ( z , · ) k 2 2 = J X j =0 | ϕ j ( z ) | 2 . The L p norms with p 6 = 2 a re not given by simple expres- sions, how ever. In the following, w e will show that they can b e b ounded ab ove by a quantit y indep endent of J . More precisely , for any 1 < p < ∞ , we sha ll s how that k P J ( z , · ) k p ≤ (2 π /p ) 1 /p π sin( π/p ) e | z | 2 / 2 independently o f J . In o rder to s ee this, note that, for fixed z and | z ′ | , the function θ 7→ P J ( z , | z ′ | e iθ ) is o b- tained from P ∞ ( z , | z ′ | e iθ ) by restricting the F ourier c om- po nent s to 0 ≤ j ≤ J . This restr iction is a b ounded op eration (uniformly in J ) on L p ( T ) for 1 < p < ∞ [30, Ch. I I, Sec. 1], and hence k P J ( z , · ) k p is b o unded by a constant times k P ∞ ( z , · ) k p . In fact, an upp er bound on the optimal constant in this inequality is given by the norm of the Riesz pr o jection (viewed as an op era to r from L p ( T ) to itself ), whic h is known to equal 1 / sin( π/ p ) [31]. Hence we have the bo und k P J ( z , · ) k p ≤ 1 sin( π /p ) k P ∞ ( z , · ) k p =  (2 π /p ) 1 /p π sin( π /p )  e | z | 2 / 2 . (31) The last equality follows from the fa ct that P ∞ ( z , z ′ ) = π − 1 e − z z ′∗ . Let c p denote the co nstant in square br ack ets in (31) taken to the p ow er 2 p/ (4 − 2 p ). W e hav e thus s hown that, for 1 < p ≤ 4 / 3 a nd α ( p ) = (4 − 3 p ) / (4 − 2 p ) ρ ζ ( z ) ≤ c p k φ ζ k 2 4  X J j =0 | ϕ j ( z ) | 2 e −| z | 2  α ( p ) . A p ossible choice is p = 6 / 5, in which ca s e c p = (10 / 3) √ 2(5 / 3 π ) 1 / 4 ≈ 4 . 02 and α ( p ) = 1 / 4. Hence, for any function | ψ i = P k>J d k | ϕ k i , we hav e that  ψ     q χ − 1 J ( L ) ρ ζ p χ J ( L )     ψ  ≤ 4 . 02 k φ ζ k 2 4 X k ≥ J +1 | d k | 2 exp h 1 8  √ k − √ J + 1  2 i × Z C | ϕ k ( z ) | 2  X J j =0 | ϕ j ( z ) | 2 e −| z | 2  1 / 4 e −| z | 2 dz . (32) Here we have used the fact that our upp er bo und on ρ ζ ( z ) is r adial, i.e., dep ends only on | z | , a nd hence there a r e no off-diagona l ter ms on the r ight side. An applicatio n of J ensen’s inequalit y for the c o ncav e function t 7→ t 1 / 4 implies that Z C | ϕ k ( z ) | 2  X J j =0 | ϕ j ( z ) | 2 e −| z | 2  1 / 4 e −| z | 2 dz ≤  Z C | ϕ k ( z ) | 2 e −| z | 2  X J j =0 | ϕ j ( z ) | 2 e −| z | 2  dz  1 / 4 . 8 F ro m Stirling’s formula [32] it follows that j ! ≥ ( j / e) j √ 1 + j for a ny j ≥ 0. Using this b ound it is eas y to see that | ϕ j ( z ) | 2 e −| z | 2 ≤ 1 π √ 1 + j e − ( | z |− √ j ) 2 . Similarly , | ϕ k ( z ) | 2 e −| z | 2 ≤ 3 3 / 2 π √ 8 e 1 | z | e − ( | z |− √ k ) 2 for k ≥ 1. Hence Z C | ϕ j ( z ) | 2 | ϕ k ( z ) | 2 e − 2 | z | 2 dz ≤ 3 3 / 2 π √ 2 e 1 √ 1 + j Z ∞ 0 e − ( x − √ j ) 2 e − ( x − √ k ) 2 dx ≤ 3 3 / 2 √ 4 π e 1 √ 1 + j e − ( √ j − √ k ) 2 / 2 . T o obtain the last inequalit y , w e simply extended the int e gral to the whole rea l axis. The sum over j can b e bo unded from ab ove b y the integral J X j =0 1 √ 1 + j e − ( √ j − √ k ) 2 / 2 ≤ Z J +1 0 e − ( √ s − √ k ) 2 / 2 ds √ s ≤ 2 Z 0 −∞ e − ( t + √ J +1 − √ k ) 2 / 2 dt ≤ 2 Z ∞ −∞ e − ( t + √ J +1 − √ k ) 2 / 2 e − t ( √ k − √ J +1) dt = √ 8 π e − ( √ k − √ J +1) 2 / 2 . The one-fourth ro ot of this expres sion cancels ex actly the exp onential factor in (32), and we arrive at (29). IV. BOSE-EINSTEIN CONDENSA TION The technique employ ed fo r the energy b ounds can b e used to show that the mo del (9) exhibits Bo se-Einstein condensation in the gr ound state if N → ∞ with g N /ω fixed, in complete analo gy to the pro of of the corres p o nd- ing r esult in [29 ]. The nonuniqueness o f the minimizers of the GP functional (10) due to bre a king of ro ta tional sym- metry mak es the precise statement a little complicated, but in brie f the res ult is as follo ws : If Ψ N is a se quence of ground states of (9) for N = 1 , 2 , . . . and γ N are the cor - resp onding normalized 1-par ticle density matrices, then any limit γ o f the sequence γ N as N → ∞ with the coupling constant g N / ω fixed is a conv ex com bination of pro jector s onto nor malized minimizers of the GP func- tional (1 0) for the g iven coupling co nstant. A pr ecise formulation is given in Theore m 2 in [2 9]. A key s tep in the pro of is to extend the bo unds on the g round state energy of H to pe rturb ed op erato r s H ( S ) = H + P N i =1 S i where S is a b ounded op era tor on the 1-particle space H and S i the corresp onding op erator on the i - th factor in ⊗ N symm H . These b ounds lead to the ineq ualities min φ h φ | S | φ i ≤ T r S γ ≤ max φ h φ | S | φ i (33) for any limit γ of the sequence of density matrices γ N , where the max resp. min is taken over all normalized min- imizers φ o f (10) with coupling para meter g N /ω . With the a id of some arg umen ts from co nv ex analysis one can then conclude a s in [29] that γ has a repres ent ation in terms of pro jectors | φ ih φ | on to norma liz ed minimizers φ of the GP functional. The a ppr opriate extensio n o f this result to the Thomas-F ermi limit is s till an op en problem. V. CONCLUSION W e a nalyzed the g round state ener g y of a r apidly rotat- ing Bose gas in a ha rmonic trap, under the usual assump- tion that the par ticles are r estricted to the lo west Landau level, that the tw o-b o dy interaction is a δ - function, a nd that the num b er of particle s , N , is very lar ge. F or the limiting situa tio n we prov e , rig orously , tha t the Gross- Pitaevskii energy , appropria tely mo dified to the low est Landau level, is exact provided g ≪ N ω . Here, g is the strength of the δ - function interaction, and ω is the dif- ference b etw een the trapping and ro tation fre quencies. Ackno wledgme nt s W e thank Rup ert F ra nk fo r illuminating discussions. This work w as pa rtially suppo rted by U.S. National Sci- ence F oundatio n gr ants P HY-06528 54 (E .L .) and PHY- 06523 56 (R.S.), a nd the ESF netw o rk INST ANS (J.Y.). [1] E.H. Lieb, J. Yngv ason, Phys. Rev . Lett. 80 , 2504 (1998). See also: E.H. Lieb, R. S eiringer, J.P . Solov ej, J. Yn gv a- son, The Mathematics of the Bose gas and its Condensa- tion , Birkh¨ auser, Basel ( 2005). [2] N . K. Wilkin, J. M. F. Gunn, and R. A. Smith, Phys. Rev. Lett. 80 , 2265 (1998) [3] A.D. Jackson, G.M. Kav oulakis, Ph y s. R ev. Lett. 85 , 2854 (2000) [4] R.A. S mith and N.K. Wilkin, Phys. Rev. A 62 , 061602 (2000) [5] T.-L. Ho, Phys. Rev. Lett. 87 , 060403 (2001) [6] N.R. Coop er, N.K. Wilkin, J.M.F. Gunn , Phys. Rev. 9 Lett. 87 , 120405 (2001) [7] I . Blo ch, J. Dalibard, W. Zwerger, R ev. Mo d. Phys. 80 , 885 (2008) [8] K . Kasamatsu, M. Tsub ota, Prog. Low T emp . Phys. 16 , 349 (2008) [9] A .L. F etter, R otating tr app e d Bose-Einstein c ondensates , preprint arXiv:0801 .2952 [10] N.R . Coop er, R apid ly R otating Ato mic Gases , preprint arXiv 0810.439 8 [11] P . Engels, I. Codd ington, P .C. Haljan, V. S c hw eikhardt, E.A. Cornell, Phys. Rev. Lett. 90 , 170405 (2003) [12] V. Sch weikhardt, I. Coddington, P . Engels, V.P . Mogen- dorff, E.A. Cornell, Phys. Rev. Lett. 92 , 040404 (2004) [13] I. Cod d ington, P . C. Haljan, P . Engels, V. Sch weikhard, S. T ung, and E. A. Cornell, Phys. R ev . A 70 , 063607 (2004) [14] M. Lewin and R. Seiringer, Str ongly c orr elate d phases in r apid l y r otating Bose gases , preprint arXiv:0906 .0741 [15] B. Mottelson, Phys. Rev. Lett. 83 , 2695 (1999) [16] S. Viefers, T.H. H ansson, S .M. Reimann, Phys. Rev. A 62 053604 (2000) [17] G.F. Bertsch, T. Papenbro ck, Phys. Rev. Lett. 83 , 5412 (1999) [18] T. Pa p enbrock, G.F. Bertsc h, Phys. R ev. A 63 , 02361 6 (2001) [19] M.S. Hussein, O.K. V oro v, Phys. Rev. A 65 , 035603 (2002) [20] T. Nak a jima, M. Ued a, Ph y s. Rev . Lett . 91 , 140 401 (2003) [21] C. Y an n ouleas, U. Landmann , T rial wave functions, mole cular states, and r o-vibr ational sp e ctr a in the low- est landau l evel: A uni versal description for b osons and fermions , preprint arXiv:0812.314 2 [22] G. W atanabe, G. Ba ym, C.J. Pethic k, Phys. R ev. Lett. 93 , 190401 (2004) [23] G. Ba y m, J. Low T emp. Phys. 138 , 601 (2005) [24] A. A ftalion, X. Blanc, J. D alibard, Phys. Rev. A 71 , 023611 (2005) [25] A. Aftalion, X. Blanc, F. Nier, J. F u nct. An al. 241 , 661 (2006) [26] A. Aftalion, X. Blanc, S IAM J. Math. Anal. 38 , 874 (2006) [27] J.R. Klauder, B.-S. S k agerstam, Coher ent States , p. 14, W orld Scientific (1985) [28] E.H. Lieb, R. Seiringer, and J. Y ngv ason, Phys. Rev. Lett. 94 , 080401 (2005). [29] E.H. Lieb, R. S eiringer, Comm u n. Math. Ph ys. 264 , 505 (2006) [30] Y. Katznelson, An intr o duction to harmonic analysis , Wiley (1968) [31] B. Hollenbeck, I.E. V erbitsky , J. F unct. Anal. 175 , 370 (2000) [32] M. Abramow itz, I.A. St egun , Handb o ok of mathematic al functions , page 257, D o ver (1972)