We study the mean-field dynamics of a system of $N$ interacting bosons starting from an initially condensated state. For a broad class of mean-field Hamiltonians, including models with arbitrary bounded interactions and models with unbounded interaction potentials, we prove that the probability of having $n$ particles outside the condensate decays exponentially in $n$ for any finite evolution time. Our results strengthen previously known bounds that provide only polynomial control on the probability of having $n$ excitations.
In this paper, we study the dynamics of a large number N of bosons in the mean-field regime, which arises in various areas of physics, including spin systems [1,2], and Bose-Einstein condensates [2,3]. In typical experimental set-ups, for instance in experiments 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, both mathematically and physically, that the mean-field dynamics preserves condensation [1][2][3].
We contribute to the analysis of condensation along the mean-field dynamics by proving that the probability of having n particles outside the condensate decays exponentially with n for any finite evolution time. This result strengthens previously known bounds, which typically prove only a polynomial decay of such a probability.
To be more precise, let Sym N h be the N -fold symmetric tensor product of the single-particle Hilbert space h. We describe the dynamics of the N bosons by a wavefunction Ψ N (t) ∈ Sym N h solving the Schrödinger equation
where the mean-field Hamiltonian H N is given by
Here, T i denotes the self-adjoint N -particle operator acting as the self-adjoint single-particle operator T : h → h on the i-th particle and as the identity on all the other particles. Similarly, w ij denotes the self-adjoint two-body interaction acting as the self-adjoint operator w : h ⊗2 → h ⊗2 on the particles i and j and as the
with ϕ(0) ∈ h the initial condensate wavefunction, and ξ n are n-particle wavefunctions describing the excitations. If the initial number of excitations is exponentially controlled, in a sense that we will make formal later, we will show that the property of exponential condensation is preserved for positive times. More precisely, the condensate wavefunction ϕ(t) , satisfying ∥ϕ(t)∥ 2 = 1, is expected to evolve according to the Hartree equation
where the effective one-particle operator w ϕ(t) is defined through
If w is bounded, the Hartree equation ( 4) always has a unique mild solution defined for any t ∈ R from [9, Chapter 6, Theorem 1.4].
Particles outside the condensate are counted by the excitation number operator
where Q(t) i denotes the operator acting as the orthogonal projector Q(t) = 1 -|ϕ(t)⟩⟨ϕ(t)| on the i-th particle and as the identity on all others.
Mathematically, condensation at positive times is expressed by
This property is well understood for models of type (ii) under suitable assumptions on the interaction potential and the initial data [1-3, 10-18, 18-21]. More recently, these results have been refined by proving uniform bounds on higher moments of the excitation number,
for fixed k ∈ N in the limit N → ∞; see, for example, [22][23][24][25][26][27][28][29][30]. These bounds imply that the probability of finding more than n particles outside the condensate decays at least polynomially in n. In this work, we strengthen these results by proving exponential decay of this probability; see Theorem 2.3 below for a precise formulation.
In this section we present our results on exponential condensation, more precisely we consider an initial condensate state (3) satisfying ⟨Ψ N (0), exp(βN
for some C β > 0 independent of N . Then in the large N limit
for β ≤ β c (t) and suitable β c (t) > 0 discussed below for the two different types of models (i), (ii). The result for mean-field models of type (i) is given in Theorem 2.1 in subsection 2.1, and the result for mean-field models of type (ii) in Theorem 2.4 in subsection 2.2.
We recall that in this section, we study mean-field models of type (i) arising for example in the context of quantum spin systems.
Theorem 2.1 (Exponential Condensation for bounded potentials). Let w be an arbitrary bounded two-body interaction, let Ψ N (t) ∈ Sym N h denote the solution to the Schrödinger equation (1) with initial data (3) satisfying the condensate condition (8). Furthermore, let ϕ(t) denote the solution to the Hartree equation (4) with initial data ϕ(0) ∈ h.
Then, the number of excitations N + (t) defined by (28), satisfies
for all t ≥ 0 and 0 ≤ β < β c (t) = -ln tanh(3∥w∥ t), where
Remark 2.2. We collect a few remarks on the constants f (t, β) and β c (t) appearing in Theorem 2.1.
(i) Since the bound f (t, β) on the moment generating function of N + (t) does not depend on the total number of particles N , Theorem 2.1 indeed establishes exponential condensation in the sense of (9). Moreover, 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 bound provided by the theorem coincides with the exact value of the corresponding quantity.
(ii) We note that β c (t) > 0 for all finite times, and therefore exponential condensation holds for any 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)
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