Exponential concentration of fluctuations in mean-field boson dynamics

Exp onen tial concen tration of fluctuations in mean-field b oson dynamics Matias Gabriel Ginzburg ∗ , Simone Rademac her † and Giacomo De P alma ‡ F ebruary 19, 2026 Abstract W e study the mean-field dynamics of a system of N interacting bosons starting from an initially condensated state. F or a broad class of mean-field Hamiltonians, including models with arbitrary b ounded in teractions and models with un b ounded interaction potentials, we pro ve that the probability of having n particles outside the condensate deca ys exp onen tially in n for any finite evolution time. Our results strengthen previously known b ounds that pro vide only p olynomial control on the probability of having n excitations. 1 In tro duction In this pap er, we study the dynamics of a large num b er N of b osons in the mean-field regime, which arises in v arious areas of ph ysics, including spin systems [ 1 , 2 ], and Bose-Einstein condensates [ 2 , 3 ]. In t ypical exp erimen tal set-ups, for instance in exp eriments on Bose-Einstein condensation [ 4 , 5 ], the initial state is prepared in a condensed phase, meaning that a macroscopic fraction of the N particles occupies the same one-particle quantum state, referred to as the condensate. It is well established, b oth mathematically and ph ysically , that the mean-field dynamics preserves condensation [ 1 – 3 ]. W e con tribute to the analysis of condensation along the mean-field dynamics by proving that the proba- bilit y of having n particles outside the condensate decays exp onen tially with n for any finite evolution time. This result strengthens previously known b ounds, which typically prov e only a p olynomial decay of suc h a probabilit y . T o be more precise, let Sym N h be the N -fold symmetric tensor product of the single-particle Hilbert space h . W e describ e the dynamics of the N b osons by a wa v efunction Ψ N ( t ) ∈ Sym N h solving the Sc hr¨ odinger equation i∂ t Ψ N ( t ) = H N Ψ N ( t ) , (1) where the mean-field Hamiltonian H N is giv en by H N = N X i =1 T i + 1 N − 1 X 1 ≤ i 0. Mean-field mo dels of type (i) naturally arise, for example, in the description of spin systems [ 1 , 2 ], and Lipkin-Meshk ov-Glic k-type mo dels [ 6 ]. In the con tinuum, b ounded interactions arise naturally as soft-core or regularized p oten tials of finite heigh t, whic h are used to mo del effective interactions in ultracold atomic gases, for instance in Rydb erg-dressed systems and related prop osals for sup ersolid phases [ 7 ]. The Kac mo del and soft spheres can also b e describ ed by type (i) mo dels [ 8 ]. Mean-field mo dels with unbounded interaction potentials of type (ii) constitute a standard mathematical framew ork for the description of Bose–Einstein condensation, and the condition V 2 ≤ D (1 − ∆) is c hosen in order to include the Coulomb p oten tial; see for example [ 3 ]. In typical exp erimen tal realizations [ 4 , 5 ], dilute gases of b osonic atoms are confined in external traps and co oled to ultra-low temp eratures. Below a critical temp erature, a macroscopic fraction of the particles o ccupies the same one-particle quantum state, giving rise to Bose-Einstein condensation. Motiv ated in this exp erimen tal realization, we study the many b ody dynamics ( 1 ) for initial data given b y a condensate state Ψ N (0) = N X n =0 ϕ (0) ⊗ ( N − n ) ⊗ s ξ n (3) with ϕ (0) ∈ h the initial condensate wa vefunction, and ξ n are n -particle wa vefunctions describing the exci- tations. If the initial num b er of excitations is exp onen tially controlled, in a sense that we will make formal later, we will sho w that the prop ert y of exp onen tial condensation is preserv ed for p ositive times. More precisely , the condensate wa v efunction ϕ ( t ) , satisfying ∥ ϕ ( t ) ∥ 2 = 1, is exp ected to evolv e according to the Hartree equation i∂ t ϕ ( t ) = H ϕ ( t ) H ϕ ( t ) , H ϕ ( t ) H = T + w ϕ ( t ) , (4) where the effective one-particle op erator w ϕ ( t ) is defined through ⟨ φ 1 , w ϕ ( t ) φ 2 ⟩ = ⟨ φ 1 ⊗ ϕ ( t ) , w φ 2 ⊗ ϕ ( t ) ⟩ , ∀ φ 1 , φ 2 ∈ h . (5) If w is b ounded, the Hartree equation ( 4 ) alwa ys has a unique mild solution defined for any t ∈ R from [ 9 , Chapter 6, Theorem 1.4]. P articles outside the condensate are counted b y the excitation num b er op erator N + ( t ) = N X i =1 Q ( t ) i , where Q ( t ) i denotes the operator acting as the orthogonal pro jector Q ( t ) = 1 − | ϕ ( t ) ⟩⟨ ϕ ( t ) | on the i -th particle and as the identit y on all others. Mathematically , condensation at p ositiv e times is expressed by 1 N ⟨ Ψ N ( t ) , N + ( t )Ψ N ( t ) ⟩ → 0 as N → ∞ . (6) 1 If T or w are unbounded, their domain is a dense subspace of h or h ⊗ 2 rather than the full h or h ⊗ 2 , resp ectiv ely 2 This prop erty is well understo o d for mo dels of type (ii) under suitable assumptions on the interaction p oten tial and the initial data [ 1 – 3 , 10 – 18 , 18 – 21 ]. More recently , these results hav e b een refined by proving uniform b ounds on higher moments of the excitation num b er, ⟨ Ψ N ( t ) , ( N + ( t )) k Ψ N ( t ) ⟩ = O (1) , (7) for fixed k ∈ N in the limit N → ∞ ; see, for example, [ 22 – 30 ]. These bounds imply that the probability of finding more than n particles outside the condensate decays at least p olynomially in n . In this work, w e strengthen these results by proving exp onen tial deca y of this probabilit y; see Theorem 2.3 b elow for a precise formulation. 2 Exp onen tial condensation In this section we presen t our results on exp onen tial condensation, more precisely w e consider an initial condensate state ( 3 ) satisfying ⟨ Ψ N (0) , exp( β N + (0))Ψ N (0) ⟩ ≤ C β (8) for some C β > 0 indep enden t of N . Then in the large N limit ⟨ Ψ N ( t ) , exp  β N + ( t )  Ψ N ( t ) ⟩ = O (1) (9) for β ≤ β c ( t ) and suitable β c ( t ) > 0 discussed b elo w for the tw o different types of mo dels (i), (ii). The result for mean-field mo dels of t yp e (i) is given in Theorem 2.1 in subsection 2.1 , and the result for mean-field mo dels of type (ii) in Theorem 2.4 in subsection 2.2 . 2.1 Mean-field mo dels with b ounded p oten tials W e recall that in this section, we study mean-field mo dels of type (i) arising for example in the con text of quan tum spin systems. Theorem 2.1 (Exp onen tial Condensation for b ounded p oten tials) . L et w b e an arbitr ary b ounde d two-b o dy inter action, let Ψ N ( t ) ∈ Sym N h denote the solution to the Schr¨ odinger e quation ( 1 ) with initial data ( 3 ) satisfying the c ondensate c ondition ( 8 ) . F urthermor e, let ϕ ( t ) denote the solution to the Hartr e e e quation ( 4 ) with initial data ϕ (0) ∈ h . Then, the numb er of excitations N + ( t ) define d by ( 28 ) , satisfies ⟨ Ψ N ( t ) , exp( β N + ( t )) Ψ N ( t ) ⟩ ≤ C β f ( t, β ) (10) for al l t ≥ 0 and 0 ≤ β < β c ( t ) = − ln tanh(3 ∥ w ∥ t ) , wher e f ( t, β ) =  1 − tanh (3 ∥ w ∥ t ) e − β 1 − tanh (3 ∥ w ∥ t ) e β  1 / 3 . (11) Remark 2.2. W e collect a few remarks on the constan ts f ( t, β ) and β c ( t ) app earing in Theorem 2.1 . (i) Since the b ound f ( t, β ) on the moment generating function of N + ( t ) does not dep end on the total n umber of particles N , Theorem 2.1 indeed establishes exp onen tial condensation in the sense of ( 9 ). Moreo ver, f (0 , β ) = 1 for all β > 0 and f ( t, 0) = 1 for all t ≥ 0. Hence, for the pairs (0 , β ) and ( t, 0), the upper b ound pro vided by the theorem coincides with the exact v alue of the corresponding quantit y . (ii) W e note that β c ( t ) > 0 for all finite times, and therefore exp onential condensation holds for an y t ≥ 0. The critical parameter β c ( t ) scales as O (log( t − 1 )) for small times and as O ( e − t ) for large times. In particular, β c ( t ) b ecomes exp onen tially small as t → ∞ . 3 Remark 2.3 (Probabilisitc picture) . As mentioned abov e, the n umber of excitations in the state Ψ N ( t ) can b e interpreted as a random v ariable with distribution P Ψ N ( t )  N + ( t ) = n  = ⟨ Ψ N ( t ) , 1 {N + ( t )= n } Ψ N ( t ) ⟩ . (12) In this probabilistic picture, Theorem 2.1 provides an upp er b ound on the moment generating function of N + ( t ). As a consequence of Marko v’s inequality , w e obtain P Ψ N ( t )  N + ( t ) > n  ≤ C β f ( t, β ) e − β n (13) for all β < β c ( t ). Hence, the probabilit y of finding more than n particles outside the condensate decays exp onen tially in n . 2.2 Mean-field mo dels with unbounded p oten tials W e recall that in this section, we study mean-field mo dels of type (ii) used as a mathematical framework for the analysis of Bose-Einstein condensates. Thus, in this section w e study the mean-field Hamiltonian H N = N X i =1 − ∆ x i + 1 N − 1 X 1 ≤ i 0, i.e. Z x ∈ R 3 V ( x ) 2 | ψ ( x ) | 2 dx ≤ D Z x ∈ R 3  | ψ ( x ) | 2 + |∇ ψ ( x ) | 2  dx . (16) for an y ψ ∈ H 1 ( R 3 ). The asso ciated Hartree dynamics ( 4 ) has a unique mild solution, and, moreov er V := sup x Z | V ( x − y ) ϕ ( t, y ) | 2 dy < ∞ , (17) see for example [ 3 ]. In this setting, we pro ve exp onen tial condensation of the form ( 9 ), to o. Theorem 2.4 (Exp onen tial condensation for unbounded interaction p oten tials) . L et h = L 2 ( R 3 ) , let V satisfy ( 15 ) , let Ψ N ( t ) ∈ Sym N h denote the solution to the Schr¨ odinger e quation ( 1 ) with initial datum ( 3 ) satisfying the c ondensate c ondition ( 8 ) . F urthermor e, let ϕ ( t ) denote the solution to the Hartr e e e quation ( 4 ) with initial datum ϕ (0) ∈ h . Then, the numb er of excitations N + define d by ( 28 ) satisfies ⟨ Ψ N ( t ) , exp( β N + ( t )) Ψ N ( t ) ⟩ ≤ C β f ( t, β ) (18) for al l t ≥ 0 and for al l 0 ≤ β < β c ( t ) = − ln tanh(6 V t ) , wher e f ( t, β ) =  1 − tanh (6 V t ) e − β 1 − tanh (6 V t ) e β  1 / 3 . (19) 2.2.1 Comparison to the literature As discussed in the in tro duction, our results fit into a broad b ody of literature on the analysis of Bose–Einstein condensation for mean-field mo dels [ 1 – 3 , 10 – 18 , 18 – 21 ], as well as on the study of quantum fluctuations, i.e. particles outside the condensate, in such systems [ 22 – 30 ]; see also the review article [ 31 ]. Our results, ho wev er, complemen t the existing literature on mean-field dynamics by providing a stronger, exp onen tial control of the n umber of excited particles. W e remark that, for large times, the critical v alue β c deca ys exp onen tially in time, and our b ounds are consistent with earlier results on momen t estimates for the num b er of excitations, whic h t ypically gro w exp onen tially in time; see, for example, [ 3 ] under similar assumptions on the initial data and interaction p oten tial. 4 2.2.2 Probabilistic picture T o the b est of our knowledge, our w ork establishes the first b ound on the moment generating function of the n umber of excitations along the man y-b ody time ev olution; see Theorem 2.3 . F or mean-field mo dels in the ground state, b ounds on the momen t generating function hav e b een obtained earlier [ 32 – 34 ], and its asymptotic b eha vior has b een analyzed more recently in [ 35 ]. F rom this p erspective, our results provides a first step tow ards exp onential control of excitations in the time-dep enden t setting. 2.2.3 Singular s caling regimes F rom a physical p oin t of view, the most relev ant but mathematically most challenging setting is the analysis of singular in teraction p otentials in the so-called Gross-Pitaevskii regime. In this regime, the mean-field in teraction p oten tial V is replaced by an N -dep enden t p oten tial V N = N 3 V ( N · ), which scales with the total n umber of particles and conv erges, in the large-particle limit, to a δ -interaction. Remark ably , b ounds [ 33 – 35 ] and asymptotic form ulas for the momen t generating function of the n umber of excitations in the ground state ha ve b een sho wn to remain v alid even in this mathematically challenging regime. F or the corresp onding man y-b o dy dynamics, how ever, only b ounds on the first moment of the num b er of excitations are currently kno wn [ 36 ], while b ounds on higher moments remain a challenging op en problem. 2.3 Idea of the pro ofs W e prov e b oth theorems in section 4 based on the same idea: The goal is to derive a b ound for the function g N : R + × R + → R b y g N ( t, β ) = ⟨ Ψ N ( t ) , exp( β N + ( t ))Ψ N ( t ) ⟩ (20) b y a Gronw all-type argument. W e will first compute the time deriv ative of ∂ t g N and sho w in a second step that w e can b ound it from ab ov e in terms of g N and ∂ β g N . In the last step we then perform a Gronw all t yp e argumen t and arrive at the desired b ound. Before presen ting the pro of in section 4 , in the following Section we first give a brief in tro duction to the F o c k space formalism and the so-called excitation map, in tro duced b y [ 37 ], that our analysis relies on. 3 Excitation map In this section we introduce the excitation map, introduced in [ 37 ], defined for any N -particle w av efunction to factor out any contributions of the Hartree evolution ϕ ( t ). The excitation map is based on the idea that while the one-particle Hilb ert space h has a decomp osition in to h = span( { ϕ ( t ) } ) ⊕ span( { ϕ ( t ) } ) ⊥ , the N -particle Hilb ert space Sym N h has a decomp osition into pro ducts of span( { ϕ ( t ) } ) and F ≤ N ⊥ ϕ ( t ) := N M n =0 Sym n h + ( t ) , (21) with h + ( t ) = span( { ϕ ( t ) } ) ⊥ called the orthogonal or excitation F o c k space. In con trast to the standard b osonic F o c k space F := ∞ M n =0 Sym n h , (22) the orthogonal F o c k space is defined ov er h + ( t ) (instead of h ) and contains w av efunctions of at most N particles. On the F o c k space F , we hav e the usual creation and annihilation op erators giv en by a ∗ i = a ( e i ) , and a i = a ( e i ) (23) 5 where we set e 0 := ϕ ( t ) and { e i } i ∈ N + as an orthonormal basis of span( { ϕ ( t ) } ) ⊥ . W e remark that to simplify notation, we neglect the time dep endence of the basis { e i } i ∈ N + and e 0 in their notation. The creation and annihilation op erators satisfy the standard canonical commutation relations [ a ∗ i , a ∗ j ] = [ a i , a j ] = 0 , and [ a i , a ∗ j ] = δ ij . (24) F or the definition of the excitation map, we embed an y N -particle wa v efunction into the b osonic F o c k space through W N : Sym N h → F given by W N (Ψ N ) := N − 1 M n =0 0 ! ⊕ Ψ N ⊕ N − 1 M n =0 0 ! , and W ∗ N ∞ M n =0 ψ n ! = ψ N (25) where W ∗ N denotes the adjoin t of W N . In particular, note that while W ∗ N W N acts as the iden tity on Sym N h , the adjoin t W N W ∗ N acts on F as a pro jector onto the subspace of N particles. Moreov er, w e define Q N ,t : F → F as the orthogonal pro jector from the full b osonic F o c k space F on the orthogonal F o c k space F ≤ N ⊥ ϕ ( t ) b y Q N ,t := N M n =0 (1 − | ϕ ( t ) ⟩⟨ ϕ ( t ) | ) ⊗ n ! ⊕ ∞ M n = N +1 0 ! . (26) Then, the excitation map U N ,t : Sym N h → F [ 37 ], is given by U N ,t = N X n =0 Q N ,t a ( ϕ ( t )) N − n p ( N − n )! W N . (27) Note that in this picture, the num b er of excitations defined in ( 1 ) reads W N N + ( t ) W ∗ N := N − a ∗ ( ϕ ( t )) a ( ϕ ( t )) = X i ∈ N + a ∗ i a i , (28) where N = P i ∈ N a ∗ i a i . By construction, the excitation map U N ,t maps any N -particle wa v efunction into the orthogonal F o c k space by destroying all particles in the Hartree wa vefunction e 0 = ϕ ( t ). In particular note that the image of U N ,t is the orthogonal F o c k space ℑ ( U N ,t ) = ℑ ( Q N ,t ) = F ≤ N ⊥ ϕ ( t ) , resp ecting the heuristics that an N -particle w av efunction can hav e at most N excitations outside the Hartree evolution. W e can then define the initial excitations state Ξ N := U N , 0 Ψ N (0) = N M n =0 ξ n . (29) F urthermore, the conjugate of the excitation map is giv en by U ∗ N ,t = N X n =0 W ∗ N ( a ( ϕ ( t )) ∗ ) N − n p ( N − n )! Q N ,t . (30) Since U ∗ N ,t U N ,t = 1 and U N ,t U ∗ N ,t = Q N ,t , the excitation map is a partial isometry . The excitation map comes with nice prop erties (see also [ 37 ]): It transforms the pro jected creation op erator to U N ,t W ∗ N a ∗ n W N − 1 U ∗ N − 1 ,t = ( Q N ,t a ∗ n Q N − 1 ,t ∀ n ≥ 1 Q N ,t √ N − N Q N − 1 ,t n = 0 . (31) Th us, in the limit of infinitely many particles, where we exp ect that N = O (1), a particle created through a 0 = a ( ϕ ( t )) along the Hartree evolution is effectiv ely replaced by an O ( √ N ) quan tity , while excitations 6 orthogonal to the Hartree evolution, created through a ∗ n with n ≥ 1, are replaced by O (1) quan tities. In ph ysics this formal transformation is referred to as c-num b er substitution. Note that U N ,t furthermore maps the pro jected num b er of excitations ( 28 ) to the num b er of particles op erator on the orthogonal F o c k space, i.e. U N ,t N + ( t ) U ∗ N ,t = X n ≥ 1 U N ,t W ∗ N a ∗ n W N − 1 U ∗ N − 1 ,t U N − 1 ,t W ∗ N − 1 a n W N U ∗ N ,t = N Q N , (32) where we used that W N − 1 U ∗ N − 1 ,t U N − 1 ,t W ∗ N − 1 = 1 on the image of a n W N U ∗ N ,t , similarly Q N − 1 ,t = 1 on the image of a n Q N ,t and Q N ,t comm utes with N . The pro jected particle-num b er op erator N = P ∞ i =0 a ∗ i a i (that is actually time indep enden t) is mapp ed by the excitation map to U N ,t W ∗ N N W N U ∗ N ,t = N Q N ,t + ( N − N ) Q N ,t = N Q N ,t . (33) F or the pro of of the theorems, we em b ed the N -particle wa vefunction through W N in the b osonic F o c k space. F or the analysis, we therefore also need to transform the Hamiltonian H N orignially defined in ( 2 ) on the N -particle Hilb ert space Sym N h to an op erator on the F o c k space: W e recall that H N is defined as a sum of one-particle op erators T i and t wo-particle op erators w ij whose second quantization is defined as d Γ( T ) = ∞ M n =1 n X i =1 T i , d Γ( w ) = ∞ M n =2 X i 0 dep ends on the choice of the tw o-particle op erator w in Theorem 2.4 resp. Theorem 2.1 . In fact, we shall prov e that for the c hoice of b ounded interactions, in Theorem 2.1 , we hav e K = ∥ w ∥ , while for unbounded interations, in Theorem 2.4 , we hav e K = 2 V . 7 F or this we split the pro of in three steps: In the first step in subsection 4.1 , we compute the deriv ative of ∂ t g N ( t, β ) that w orks the same for b oth Theorems. In the second step, in subsection 4.2 , we then deriv e an estimate of the form ( 38 ). As discussed before, the final b ound dep ends on the c hoice of the t wo-particle interaction w and is therefore prov en differently for b oth Theorems in subsubsection 4.2.1 and subsubsection 4.2.2 . In the final third step in subsection 4.3 w e then p erform a Gronw all type argument to deriv e the final results in Theorem 2.1 and Theorem 2.4 from ( 38 ). 4.1 Step 1 In this section, we derive an explicit expression for the time deriv ative of g N ( t, β ) defined in ( 37 ). Note that in fact, in the definition of g N ( t, β ) there are tw o quan tities that dep end on time: the man y-b o dy w av efunction Ψ N ( t ) and (in abuse of notation) the orthogonal n um b er operator N + ( t ). Ho wev er, introducing the fluctuation dynamics U N ( t, 0) := U N ,t e − iH N t U ∗ N , 0 (39) and using that by our choice of initial data ( 29 ), we hav e from ( 1 ) and the identit y ( 32 ) g N ( t, β ) = ⟨U N ( t, 0)Ξ N , exp( β N ) U N ( t, 0)Ξ N ⟩ . (40) Th us, we passed the time dep endency of the exp ectation v alue to the fluctuations dynamics U N ( t ; 0) only . The fluctuation dynamics satisfies i∂ t U N ( t, 0) = L N ( t ) U N ( t, 0) , (41) with the evolution generator L N ( t ) = ( i∂ t U N ,t ) U ∗ N ,t + U N ,t H N U ∗ N ,t . (42) In the following, we use the short-hand notation ⟨·⟩ for the exp ectation v alue of any op erator in the state U N ( t, 0)Ξ N . With this notation, the time deriv ative of g N reads ∂ t g N ( t, β ) = − i ⟨ [exp( β N ) , L N ( t )] ⟩ . (43) T o calculate the commutator, we first deriv e an explicit expression for the generator L N ( t ) of the fluctuation dynamics ( 42 ) in terms of creation and annihilation op erators. W e start with the time deriv ative of the excitation map. W e recall the definition of U N ,t in ( 27 ). Since i∂ t a ( ϕ ( t )) = [ d Γ( H ϕ ( t ) H ) , a ( ϕ ( t ))] , and i∂ t Q N ,t = [ d Γ( H ϕ ( t ) H ) , Q N ,t ] , (44) w e find i∂ t U N ,t = " d Γ( H ϕ ( t ) H ) , X n ∈ N Q N ,t a ( ϕ ( t )) N n p ( N − n )! # W N = d Γ( H ϕ ( t ) H ) U N ,t − X n ∈ N Q N ,t a ( ϕ ( t )) N n p ( N − n )! ! d Γ( H ϕ ( t ) H ) W N = d Γ( H ϕ ( t ) H ) U N ,t − U N ,t W ∗ N d Γ( H ϕ ( t ) H ) W N . (45) In the last step, w e used that W N W ∗ N acts as the identit y in the image of W N , and d Γ( H H ) commutes with b oth, the num b er of particles and W N . Multiplying ( 45 ) by U ∗ N ,t w e get the first term of ( 42 ). F or the second term of ( 42 ), we use W ∗ N d Γ( H N ) W N = H N , and we thus arrive at L ( t ) = d Γ( H H ) Q N ,t + U N ,t W ∗ N  d Γ( H N ) − d Γ( H ϕ ( t ) H )  W N U ∗ N ,t . (46) 8 Since the first term comm utes with the particle-num b er op erator, it do es not con tribute to the deriv ative of g N ( t ; β ) (see ( 43 )). W e compute the second term, based on the representations ( 35 ) of d Γ( H N ) and d Γ( H ϕ ( t ) H ) and the prop erties of the excitation map ( 31 ). Using that U N ,t W ∗ N a ∗ m a ∗ n a q a p W N U ∗ N ,t = U N ,t W ∗ N a ∗ m W N − 1 U ∗ N − 1 ,t U N − 1 ,t W ∗ N − 1 a ∗ n W N − 2 U ∗ N − 2 ,t U N − 2 ,t W ∗ N − 2 a q W N − 1 U ∗ N − 1 ,t U N − 1 ,t W ∗ N − 1 a p W N U ∗ N ,t (47) and the notation P m = P m ∈ N and P m ′ = P m ∈ N + , w e arrive after a straigh tforward computation at U N ,t W ∗ N  d Γ( H N ) − d Γ( H ϕ ( t ) H )  W N U ∗ N ,t = 1 2( N − 1) Q N w 0000 ( N − N )( N − N − 1) + X mnpq ′ w mnpq a ∗ m a ∗ n a q a p + X npq ′ w 0 npq √ N − N a ∗ n a q a p + X mpq ′ w m 0 pq a ∗ m √ N − N − 1 a q a p + X mnq ′ w mn 0 q a ∗ m a ∗ n a q √ N − N + X mnp 0 ′ w mnp 0 a ∗ m a ∗ n √ N − N − 1 a p + X pq ′ w 00 pq √ N − N √ N − N − 1 a q a p + X nq ′ w 0 n 0 q √ N − N a ∗ n a q √ N − N + X np ′ w 0 np 0 √ N − N a ∗ n √ N − N − 1 a p + X mq ′ w m 00 q a ∗ m √ N − N − 1 a q √ N − N + X mp ′ w m 0 p 0 a ∗ m ( N − N − 1) a p + X mn ′ w mn 00 a ∗ m a ∗ n √ N − N − 1 √ N − N + X m ′ w m 000 a ∗ m ( N − N − 1) √ N − N + X n ′ w 0 n 00 √ N − N a ∗ n √ N − N − 1 √ N − N + X p ′ w 00 p 0 √ N − N ( N − N − 1) a p + X q ′ w 000 q √ N − N √ N − N − 1 a q √ N − N ! Q N − Q N X mp ′ w m 0 p 0 a ∗ m a p + w 0000 ( N − N ) + X m ′ w m 000 a ∗ m √ N − N + X p ′ w 00 p 0 √ N − N a p ! Q N . (48) W e note that we set e 0 = ϕ ( t ), and thus the co efficien ts w k 1 ,k 2 ,k 3 ,k 4 for k i ∈ N dep end on time. W e can write the r.h.s. as U N ,t W ∗ N ( d Γ( H N ) − d Γ( H H )) W N U ∗ N ,t = Q N ,t 2 X δ = − 2 A δ Q N ,t , (49) where A δ con tains operators that destro y δ particles and A − δ = A ∗ δ con tains operators that create δ particles. The terms further simplify by the symmetries under particle p ermutation of w . In fact, since for the deriv ative of g N ( t, β ) the op erator A 0 , comm uting with the n umber of particle op erator, do es not contribute, the only relev ant terms are A ∗ 1 = 1 ( N − 1) X mnq ′ w mn 0 q a ∗ m a ∗ n a q √ N − N − 1 ( N − 1) X m ′ w m 000 a ∗ m N √ N − N , A ∗ 2 = 1 2( N − 1) X mn ′ w mn 00 a ∗ m a ∗ n √ N − N − 1 √ N − N N − 1 , (50) and their adjoins op erators. 9 In this step the Hartree Hamiltonian plays a crucial role: the second term on A ∗ 1 has a contribution of order N 3 / 2 coming from the many-bo dy Hamiltonian which cancels when we subtract the contribution from the Hartree Hamiltonian and the final results has the correct scaling √ N . Th us, we arrive from ( 43 ) at ∂ t g N ( t, β ) = − i ⟨ [exp( β N ) , L ( t )] ⟩ = 2 X δ =1 2 Im ⟨ [exp( β N ) , A ∗ δ ] ⟩ . (51) As an easy consequence of the commutation relations, we hav e N A ∗ δ = A ∗ δ ( N + δ ) and therefore [exp ( β N ) , A ∗ δ ] = 2 sinh  β 2  exp  β N 2  A ∗ δ exp  β N 2  (52) that enables us to write ( 43 ) explicitly in terms of creation and annihilation op erators as ∂ t g ( t, β ) = 4 sinh  β 2  Im * X mnq ′ w mn 0 q N − 1 exp  β N 2  a ∗ m a ∗ n a q √ N − N e xp  β N 2  + − 4 sinh  β 2  Im * X m ′ w m 000 N − 1 exp  β N 2  a ∗ m N √ N − N exp  β N 2  + + 2 sinh ( β ) Im * X mn ′ w mn 00 N − 1 exp  β N 2  a ∗ m a ∗ n √ N − N − 1 √ N − N e xp  β N 2  + . (53) Based on this formula, we derive explicit b ounds of the form ( 38 ) in the next step. 4.2 Step 2 While the calculations of the previous section hold for any choice of the interaction p oten tial w , the bounds w e pro ve for all terms of ( 53 ) will dep end on the p eculiar assumptions on w in Theorem 2.1 resp. Theorem 2.4 . W e prov e the tw o cases in separate subsections b elo w. 4.2.1 Mean-field mo dels of t yp e (i) W e start with proving that the estimate ( 38 ) holds true for any b ounded w and with the choice K := ∥ w ∥ as in Theorem 2.1 . W e estimate all terms of ( 53 ) separately and start with the first. F or this w e define the follo wing op erators b y their matrix elements W mnpq = (1 − δ m 0 ) w mnpq δ p 0 , ∀ m, p ∈ N , n, q ∈ N + , (54) C pq mn = 1 N − 1  exp  β N 2  U N ,t W ∗ N a ∗ m a ∗ n a q a p W N U ∗ N ,t exp  β N 2  , ∀ m, p ∈ N , n, q ∈ N + . (55) W e note that the op erator W is b ounded, and in particular, its norm is b ounded b y ∥ W ∥ ≤ ∥ w ∥ . The op erator C is p ositiv e semi-definite. Indeed, for an y vector ( v mn ) m ∈ N ,n ∈ N + , w e hav e ⟨ v , C v ⟩ = * exp  β N 2  U N ,t W ∗ N X m X n ′ a ∗ m a ∗ n X p X q ′ a q a p W N U ∗ N ,t exp  β N 2  + =      X p X q ′ a q a p W N U ∗ N ,t exp  β N 2  U N ( t, 0)Ξ N      2 > 0 . (56) 10 Since w e will use the inequality | T r( W C ) | ≤ ∥ W ∥ T r( C ), first we compute the trace of C using ( 32 ) and ( 33 ) T r( C ) = X m X n ′ 1 N − 1  exp  β N 2  U N ,t W ∗ N a ∗ m a ∗ n a n a m W N U ∗ N ,t exp  β N 2  = 1 N − 1 * U N ,t W ∗ N X m a ∗ m N a m W N U ∗ N ,t exp ( β N ) + = 1 N − 1 * U N ,t W ∗ N X m ′ a ∗ m a m ( N + − 1) + a ∗ 0 a 0 N + ! W N U ∗ N ,t exp ( β N ) + = 1 N − 1  U N ,t W ∗ N N + ( N − 1) W N U ∗ N ,t exp ( β N )  = 1 N − 1 ⟨N ( N − 1) exp( β N ) ⟩ = ∂ β g N ( t, β ) , (57) where in the last step we iden tified ∂ β g N ( t, β ) = ⟨N exp( β N ) ⟩ . Using the op erators W and C defined in ( 54 ) resp. ( 55 ), the first term in equation ( 53 ) can b e b ounded b y 4 sinh  β 2     * X mnq ′ w mn 0 q N − 1 exp  β N 2  a ∗ m a ∗ n a q p N − N + exp  β N 2  +    = 4 sinh  β 2     * X mp X nq ′ (1 − δ m 0 ) w mnpq N − 1 δ p 0 exp  β N 2  a ∗ m a ∗ n a q p N − N + exp  β N 2  +    = 4 sinh  β 2  | T r ( W C ) | ≤ 4 sinh  β 2  ∥ w ∥ ∂ β g N ( t, β ) . (58) F or the second term of equation ( 53 ), we define the follo wing op erators by their matrix elements ˜ W mp = (1 − δ m 0 ) w m 0 p 0 δ p 0 , ∀ m, p ∈ N , (59) ˜ C pm = 1 N − 1  exp  β N 2  U N W ∗ N a ∗ m N + a p W N U ∗ N exp  β N 2  , ∀ m, p ∈ N . (60) W e note that ˜ W is b ounded with norm ∥ ˜ W ∥ ≤ ∥ w ∥ and the op erator ˜ C is p ositiv e with trace T r( ˜ C ) = 1 N − 1 X m  exp  β N 2  U N W ∗ N a ∗ m N + a m W N U ∗ N exp  β N 2  = 1 N − 1 * U N W ∗ N X m ′ a ∗ m a m ( N + − 1) + a ∗ 0 a 0 N + ! W N U ∗ N exp ( β N ) + = 1 N − 1 ⟨ U N W ∗ N ( N − 1) N + W N U ∗ N exp ( β N ) ⟩ = 1 N − 1 ⟨ ( N − 1) N exp ( β N ) ⟩ = ∂ β g N ( t, β ) . (61) 11 With these definitions, we estimate the second term of ( 53 ) by 4 sinh  β 2     * X m ′ w m 000 N − 1 exp  β N 2  a ∗ m N √ N − N e xp  β N 2  +    = 4 sinh  β 2     * X mp (1 − δ m 0 ) w m 0 p 0 N − 1 δ p 0 exp  β N 2  U N W ∗ N a ∗ m N + a p W N U ∗ N exp  β N 2  +    = 4 sinh  β 2  | T r( ˜ W ˜ C ) | ≤ 4 sinh  β 2  ∥ w ∥ ∂ β g N ( t, β ) . (62) F or the third term of equation ( 53 ), we will use a different b ound: we define the following vectors with en tries v mn = w mn 00 ∀ m, n ∈ N + , (63) u mn = 1 N − 1  exp  β N 2  a ∗ m a ∗ n √ N − N − 1 √ N − N e xp  β N 2  ∀ m, n ∈ N + . (64) Since w is b ounded, we hav e s X mn | w mnpq | 2 ≤ ∥ w ∥ ∀ p, q ∈ N , (65) and furthermore ∥ v ∥ 2 = X mn ′ | w mn 00 | 2 ≤ ∥ w ∥ 2 . (66) T o compute the norm of the vector u , we artificially add 1 = ( N + 1) − 1 / 2 ( N + 1) 1 / 2 and estimate with the Cauc hy–Sc hw arz inequality ∥ u ∥ 2 = 1 ( N − 1) 2 X mn ′      exp  β N 2  a ∗ m a ∗ n ( N + 1) − 1 / 2 ( N + 1) 1 / 2 √ N − N − 1 √ N − N exp  β N 2      2 ≤ 1 ( N − 1) 2 X mn ′  exp  β N 2  a ∗ m a ∗ n ( N + 1) − 1 a n a m exp  β N 2  × ⟨ ( N + 1)( N − N − 1)( N − N ) exp ( β N ) ⟩ ≤ * X mn ′ a ∗ m a ∗ n a n a m ( N − 1) − 1 exp ( β N ) +  ( N + 1) ( N − N − 1)( N − N ) ( N − 1) 2 exp ( β N )  ≤ ⟨N exp ( β N ) ⟩ ⟨ ( N + 1) exp ( β N ) ⟩ ≤ ⟨ ( N + 1) exp( β N ) ⟩ 2 . (67) With these notations and b ounds, we get for the third term of ( 53 ) 2 sinh ( β )      * X mn ′ w mn 00 N − 1 exp  β N 2  a ∗ m a ∗ n √ N − N − 1 √ N − N e xp  β N 2  +      = 2 sinh( β ) | ⟨ v , u ⟩ | ≤ 2 sinh( β ) ∥ v ∥∥ u ∥ ≤ 2 sinh( β ) ∥ w ∥ ( g N ( t, β ) + ∂ β g N ( t, β )) . (68) 12 Summing up the three b ounds ( 58 ), ( 62 ) and ( 68 ), we get as a final b ound for ( 53 ) ∂ t g N ( t, β ) ≤  8 ∥ w ∥ sinh  β 2  + 2 ∥ w ∥ sinh( β )  ∂ β g N ( t, β ) + 2 sinh( β ) ∥ w ∥ g N ( t, β ) , (69) that pro ves the claim. 4.2.2 Mean-field mo dels of t yp e (ii) No w we will prov e the claim ( 38 ) under the assumptions of Theorem 2.4 . W e start by rewriting ( 53 ) using that in this case w is a multiplication op erator. Since the interaction is given by a multiplication op erator of the form w ( x ; y ) = V ( x − y ), we hav e w mnpq = Z dxdy ¯ e m ( x ) ¯ e n ( y ) V ( x − y ) e p ( x ) e q ( y ) . (70) W e observ e that, formally , since the expectation v alue in ( 53 ) is ev aluated in w av efunctions on the orthogonal F o c k space F ≤ N ⊥ ϕ ( t ) , we can replace P ′ b y P in all three terms of ( 53 ). Therefore, using the identit y P n ¯ e n ( x ) e n ( y ) = δ ( x − y ), the linearity P n α n a ∗ n = a ∗ ( P n α n e n ), and X n ¯ e n ( x ) a ∗ n = a ∗ ( X n e n ( x ) e n ) = a ∗ ( δ ( x − · )) =: a ∗ x . (71) w e can write ( 53 ) as ∂ t g ( t, β ) = 4 sinh  β 2  Im  Z dxdy V ( x − y ) N − 1 ϕ ( t, x ) exp  β N 2  a ∗ x a ∗ y a y √ N − N exp  β N 2  − 4 sinh  β 2  Im  Z dxdy V ( x − y ) N − 1 ¯ ϕ ( t, y ) ϕ ( t, x ) ϕ ( t, y ) exp  β N 2  a ∗ x N √ N − N exp  β N 2  + 2 sinh ( β ) Im  Z dxdy V ( x − y ) N − 1 ϕ ( t, x ) ϕ ( t, y ) exp  β N 2  a ∗ x a ∗ y √ N − N − 1 √ N − N exp  β N 2  . (72) T o b ound each of these terms we use that the creation and annihilation op erators are b ounded in terms of the n umber of particles op erator by ∥ a ∗ ( f )Ψ ∥ ≤ ∥ f ∥ L 2 ∥ √ N + 1Ψ ∥ , ∥ a ( f )Ψ ∥ ≤ ∥ f ∥ L 2 ∥ √ N Ψ ∥ ∀ Ψ ∈ F , (73) v alid for all Ψ ∈ F . On the orthogonal F o ck space F ≤ N ⊥ ϕ ( t ) w e furthermore hav e ∥N Ψ ∥ ≤ N ∥ Ψ ∥ ∀ Ψ ∈ F ≤ N . (74) Moreo ver, we use that ∥ ϕ ( t ) ∥ L 2 = 1 ∀ t and the constant V defined in ( 17 ) can b e written as V = sup x ∥ V ( x − · ) ϕ ( t ) ∥ L 2 . (75) No w we b ound each term in ( 72 ) separately using that R f ( x ) a ∗ x dx = a ∗ ( f ), the Cauch y-Sc hw arz inequality and the inequalities ( 73 ), ( 74 ). 13 W e start with the first term that we estimate with     4 sinh  β 2  Im  Z dxdy V ( x − y ) N − 1 ϕ ( t, x ) exp  β N 2  a ∗ x a ∗ y a y √ N − N exp  β N 2      ≤ 4 N − 1 sinh  β 2  Z dy      exp  β N 2  a ∗ y a ∗ ( V ( y − · ) ϕ ( t )) √ N − N − 1 a y exp  β N 2      ≤ 4 N − 1 sinh  β 2  Z dy ∥ a ( V ( y − · ) ϕ ( t )) a y exp  β N 2  U N ( t, 0)Ξ N ∥ × ∥ √ N − N − 1 a y exp  β N 2  U N ( t, 0)Ξ N ∥ ≤ 4 N − 1 sinh  β 2  Z dy V ( N − 1) ∥ a y exp  β N 2  U N ( t, 0)Ξ N ∥ 2 ≤ 4 V sinh  β 2  ⟨N exp ( β N ) ⟩ = 4 V sinh  β 2  ∂ β g N ( t, β ) . (76) With similar ideas, we b ound the second term by     4 sinh  β 2  Im  Z dxdy V ( x − y ) N − 1 ¯ ϕ ( t, y ) ϕ ( t, x ) ϕ ( t, y ) exp  β N 2  a ∗ x N √ N − N e xp  β N 2      ≤ 4 N − 1 sinh  β 2  Z dy | ϕ ( t, y ) | 2      exp  β N 2  a ∗ ( V ( y − · ) ϕ ( t )) N √ N − N exp  β N 2      ≤ 4 N − 1 sinh  β 2  Z dy | ϕ ( t, y ) | 2 ∥ √ N a ( V ( y − · ) ϕ ( t )) exp  β N 2  U N ( t, 0)Ξ N ∥ × ∥ √ N √ N − N exp  β N 2  U N ( t, 0)Ξ N ∥ ≤ 4 N − 1 sinh  β 2  V N ∥ √ N exp  β N 2  U N ( t, 0)Ξ N ∥ 2 ≤ 4 V N N − 1 sinh  β 2  ⟨N exp ( β N ) ⟩ ) ≤ 4 V (1 + ϵ ) sinh  β 2  ∂ β g N ( t, β ) . (77) W e remark that the last step holds true for an y ϵ > 1 / ( N − 1). Lastly , we bound the third term by     2 sinh ( β ) Im  Z dxdy V ( x − y ) N − 1 ϕ ( t, x ) ϕ ( t, y ) exp  β N 2  a ∗ x a ∗ y √ N − N − 1 √ N − N exp  β N 2      ≤ 2 N − 1 sinh ( β ) Z dx     ϕ ( t, x )  exp  β N 2  a ∗ x √ N − N a ∗ ( V ( x − · ) ϕ ( t )) √ N − N exp  β N 2      ≤ 2 N − 1 sinh ( β ) Z dx | ϕ ( t, x ) |∥ √ N − N a x exp  β N 2  U N ( t, 0)Ξ N ∥ × ∥ a ∗ ( V ( x − · ) ϕ ( t )) √ N − N e xp  β N 2  U N ( t, 0)Ξ N ∥ ≤ 2 N − 1 sinh ( β ) ∥ √ N − N a ( ϕ ( t )) exp  β N 2  U N ( t, 0)Ξ N ∥ × V ∥ √ N + 1 √ N − N exp  β N 2  U N ( t, 0)Ξ N ∥ ≤ 2 V N N − 1 sinh ( β ) ∥ √ N + 1 exp  β N 2  U N ( t, 0)Ξ N ∥ 2 ≤ 2 V (1 + ϵ ) sinh( β ) ( ∂ β g N ( t, β ) + g N ( t, β )) . (78) 14 T o simplify the notation we multiply the b ound on ( 76 ) by 1 + ϵ . F urthermore, since N ≥ 2, we fix ϵ = 1. Summing up the three b ounds ( 76 ), ( 77 ) and ( 78 ), we arriv e at ∂ t g N ( t, β ) ≤  16 V sinh β 2 + 4 V sinh β  ∂ β g N ( t, β ) + 4 sinh β V g N ( t, β ) , (79) that pro ves the desired claim. 4.3 Step 3 In the last step, w e prov e the final estimate of Theorem 2.4 resp. Theorem 2.1 from ( 38 ) based on a Gron w all t yp e argument. Since this argumen t do es not dep end on the c hoice of K in the estimate ( 38 ), this part of the pro of is again the same for b oth Theorems. T o simplify the calculations, we first b ound 8 K sinh β 2 + 2 K sinh β ≤ 6 K sinh β . (80) and write ( 38 ) as 1 2 K sinh β ∂ t g N ( t, β ) + 3 ∂ β g N ( t, β ) ≤ g N ( t, β ) . (81) In particular note that all co efficien ts in fron t of (deriv atives of ) g N ( t, β ) are independent of N . How ever, b efore we can apply Gronw all’s lemma, we need to perform a suitable change of v ariables x = X ( t, β ) and y = Y ( t, β ) such that the inequality takes the form ∂ y g N ≤ g N . (82) In fact, from ( 82 ), we then arrive for fixed x from Gronw all’s inequality at g N ( x, y ) ≤ g N ( x, y 0 ) e y − y 0 (83) for arbitrary y 0 ∈ R . W e define y 0 through the inv erse change of v ariables t = T ( x, y ) and β = B ( x, y ) suc h that it realizes the initial conditions g N ( t = 0 , β ) ≤ C β , i.e. we define y 0 through T ( X ( t, β ) , y 0 ) = 0 , (84) suc h that g N ( x, y 0 ) ≤ C β . Thus, the change of v ariables ( X , Y ), solv es 1 2 K sinh β ∂ t X − 3 ∂ β X = 0 , and 1 2 K sinh β ∂ t Y − 3 ∂ β Y = 1 . (85) The functions X ( t, β ) = 6 K t − 2artanh( e − β ) , and Y ( t, β ) = 6 K t − 2artanh( e − β ) − β 3 (86) solv e the equations and any other solution is a reparameterization of this v ariables and will therefore not c hange the result. The in verse change of v ariables is given by T ( x, y ) = 1 6 K  x + 2artanh  e − 3( x − y )  , and B ( x, y ) = 3( x − y ) . (87) Since β ≥ 0 b y assumption, we find x − y ≥ 0. F rom ( 84 ) we finally get y 0 ( t, β ) = 6 K t − 2artanh( e − β ) + 1 3 ln  tanh  artanh( e − β ) − 3 K t  (88) for all 0 ≤ β < β c ( t ) := − ln tanh(3 K t ) , (89) that ensures the argument of the logarithm is p ositiv e. Thus the initial condition g N ( x, y 0 ) ≤ C β is realized only for β < β c ( t ). With the choices ( 86 ) and ( 88 ) in ( 83 ) w e finally arrive at g N ( t, β ) ≤ C β  1 − tanh(3 K t ) e − β 1 − tanh(3 K t ) e β  1 3 . (90) 15 5 Conclusions In this work w e hav e analyzed the p ersistence of Bose–Einstein condensation for b osonic man y-b ody systems ev olving under mean-field dynamics, with a particular fo cus on the quantitativ e control of the num b er of excitations ab o ve the condensate. Starting from initial data exhibiting an exp onen tial concentration of the n umber of excitations, we pro ved that this property is preserv ed for all finite times: the probability of finding n particles outside the condensate decays exp onen tially in n , uniformly in the total particle num b er N . Our results apply to t wo broad and ph ysically relev ant classes of mo dels: systems with arbitrary bounded t wo-bo dy interactions, encompassing a wide range of finite-dimensional and effective mean-field mo dels, and con tinuum systems with unbounded interaction potentials satisfying the op erator inequality V 2 ≤ D (1 − ∆), including Coulom b-type interactions. In b oth settings, the exp onen tial control of the num b er of excitations significan tly sharpens previously av ailable b ounds, whic h w ere limited to p olynomial decay deriv ed from uniform estimates on finitely many moments of the excitation num b er op erator. F rom a ph ysical p ersp ectiv e, the exp onen tial concentration of the num b er of excitations provides a refined description of the many-bo dy state b ey ond the leading-order Hartree dynamics. This type of control is particularly relev ant in situations where rare but large fluctuations may play a role, for instance in the study of correlation functions or in the justification of effective theories for excitations. On the mathematical side, our results contribute to the gro wing b ody of work aimed at a detailed understanding of fluctuations in b osonic mean-field limits. They suggest that, at leas t for finite times, the excitation structure of the many-bo dy wa vefunction is considerably more rigid than what moment b ounds alone can capture. It remains an open question to understand if the exp onen tial condensation holds true also in the Gross-Pitaevskii scaling regime. Ac kno wledgemen ts W e thanks Niels Benedikter for helpful discussions. GDP has b een supp orted by the HPC Italian National Cen tre for HPC, Big Data and Quantum Computing – Prop osal code CN00000013 – CUP J33C22001170001 and by the Italian Extended Partnership PE01 – F AIR F uture Artificial In telligence Research – Prop osal co de PE00000013 – CUP J33C22002830006 under the MUR National Recov ery and Resilience Plan funded b y the Europ ean Union – NextGenerationEU. F unded by the Europ ean Union – NextGenerationEU under the National Recov ery and Resilience Plan (PNRR) – Mission 4 Education and research – Component 2 F rom research to business – Inv estment 1.1 Notice Prin 2022 – DD N. 104 del 2/2/2022, from title “understanding the LEarning pro cess of QUantum Neural net w orks (LeQun)”, prop osal co de 2022WHZ5XH – CUP J53D23003890006. GDP and DP are members of the “Grupp o Nazionale p er la Fisica Matematica (GNFM)” of the “Istituto Nazionale di Alta Matematica “F rancesco Severi” (INdAM)”. 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