Mean field dynamics of some open quantum systems

Abstract

We consider a large number N of quantum particles coupled via a mean field interaction to another quantum system (reservoir). Our main result is an expansion for the averages of observables, both of the particles and of the reservoir, in inverse powers of $\sqrt{N}$. The analysis is based directly on the Dyson series expansion of the propagator. We analyse the dynamics, in the limit $N\to \mathrm{\infty }$, of observables of a fixed number n of particles, of extensive particle observables and their fluctuations, as well as of reservoir observables. We illustrate our results on the infinite mode Dicke model and on various energy-conserving models.

1. Introduction and main results

We consider a system of N (possibly distinct) quantum ‘particles’ interacting with a ‘reservoir’ quantum system. The Hilbert space associated with particle j is ${\mathcal{H}}_j$ and ${\mathcal{H}}_{\mathrm{R}}$ is that of the reservoir. The Hamiltonian acts on the total Hilbert space

$${\mathcal{H}}^{(N)}={\mathcal{H}}_1\otimes \cdots \otimes {\mathcal{H}}_N\otimes {\mathcal{H}}_{\mathrm{R}}$$ 1.1

and is given by

$$H_N=\sum _{j=1}^N{h}_j+H_{\mathrm{R}}+\frac{\lambda }{\sqrt{N}}\sum _{j=1}^N{G}_j\otimes B_j.$$ 1.2

Here, h_j is the Hamiltonian of the jth particle (a short way of writing ${\mathrm{𝟙}}_1\otimes \cdots {\mathrm{𝟙}}_{j-1}\otimes h_j\otimes {\mathrm{𝟙}}_{j+1}\cdots \otimes {\mathrm{𝟙}}_N\otimes {\mathrm{𝟙}}_{\mathrm{R}}$ acting non-trivially on ${\mathcal{H}}_j$) and H_R is the reservoir Hamiltonian acting on ${\mathcal{H}}_{\mathrm{R}}$. The interaction is characterized by self-adjoint operators $G_j\in {\mathcal{M}}_j$, where

$${\mathcal{M}}_j=\mathcal{B}({\mathcal{H}}_j),j=1,\dots ,N$$ 1.3

is the algebra of bounded operators on ${\mathcal{H}}_j$. The assumption that G_j is bounded is not necessary for our approach but simplifies the exposition and is relevant in applications. We include reservoirs consisting of free Bose particles and thus we do not wish to restrict our focus on bounded interaction operators B_j. Rather, we assume only that $B_j\in {\mathcal{M}}_{\mathrm{R}}$ is a self-adjoint (possibly unbounded) operator on ${\mathcal{H}}_{\mathrm{R}}$, where

$${\mathcal{M}}_{\mathrm{R}}=\mathcal{L}({\mathcal{H}}_{\mathrm{R}})$$ 1.4

is the set of linear operators on ${\mathcal{H}}_{\mathrm{R}}$. Of course, it is assumed that H_N is self-adjoint. For 1≤n≤N, we set

$${\mathcal{M}}_{\le n}={\mathcal{M}}_1\otimes \cdots \otimes {\mathcal{M}}_n.$$ 1.5

The Heisenberg dynamics is defined by

$${\tau }_{\lambda ,N}^t(A)={\mathrm{e}}^{\mathrm{i}\hspace{0.17em}t{H}_N}A{\mathrm{e}}^{-\mathrm{i}\hspace{0.17em}t{H}_N},A\in {\mathcal{M}}_{\le N}\otimes \mathcal{B}({\mathcal{H}}_{\mathrm{R}})$$ 1.6

and we denote the free dynamics (λ=0) by

$$A(t)\equiv {\tau }_{0,n}^t(A),A\in {\mathcal{M}}_{\le n}\otimes \mathcal{B}({\mathcal{H}}_{\mathrm{R}}).$$ 1.7

The initial state is taken of the form

$${\omega }_N={\mu }_1\otimes {\mu }_2\otimes \cdots \otimes {\mu }_N\otimes {\mu }_{\mathrm{R}},$$ 1.8

where μ_j is a state on ${\mathcal{M}}_j$ and μ_R is a state on $\mathcal{B}({\mathcal{H}}_{\mathrm{R}})$. We view a state, say ω, as a normalized linear functional on observables (as is usual in the ‘algebraic’ formulation of quantum theory). Equivalently, one might think of a state as a density matrix, say ρ. The two notions are linked by ω(A)=Tr(ρA). We are often interested in the dynamics of unbounded reservoir observables (such as the number of excitations in a Bose field) and hence we extend the definitions (1.6) and (1.7) to $A\in {\mathcal{M}}_{\le N}\otimes {\mathcal{M}}_{\mathrm{R}}$. Generally, ${\tau }_{\lambda ,N}^t(A)$ is then an unbounded operator on ${\mathcal{H}}^{(N)}$ and we assume throughout that ω_N is sufficiently ‘regular’ so that ${\omega }_N({\tau }_{\lambda ,N}^t(A))$ is well defined, for all N and all t.

Definitions —

• (1) We call the system symmetric if ${\mathcal{H}}_j={\mathcal{H}}_{\mathrm{S}}$, h_j=h, G_j=G, B_j=B and μ_j=μ_S for all j=1,…,N, where h and G are a single-particle Hamiltonian and an interaction operator, respectively, B is a reservoir interaction operator and μ_S is a state on ${\mathcal{M}}_{\mathrm{S}}=\mathcal{B}({\mathcal{H}}_{\mathrm{S}})$.

• (2) We call the system energy conserving if G_j(t)=G_j for all t and all j.

Our goal is to find an expansion of the dynamics ${\omega }_N({\tau }_{\lambda ,N}^t(A))$ in powers of N^−1/2. To do so, we proceed as follows.

• — We define coefficients X_ν,N,Y _ν,N accompanying N^−ν and N^−ν−1/2 in such an expansion. These are functionals on observables which still depend on N and are analytic in λ at λ=0. Their Taylor expansions are given in (1.10) and (1.11).

• — We introduce two conditions (A0) and (A1) and show in theorem 1.1 that the functionals X_ν,N,Y _ν,N are well defined (expressed by convergent Taylor series in λ) and that they are uniformly bounded in N.

• — We give in theorem 1.2 the expansion of ${\omega }_N({\tau }_{\lambda ,N}^t(A))$ in terms of the X_ν,N and Y _ν,N.

• — We show in theorem 1.3 that, for symmetric systems, X_ν,N and Y _ν,N have limits as $N\to \mathrm{\infty }$, which are again analytic in λ at λ=0 (cf. (3.19), (3.20)).

The coefficients X_ν,N, Y _ν,N are constructed from the Dyson series expansion of the dynamics (in which the unperturbed part is generated by H_N with λ=0). They are given by integrals over multi-commutators (see §3a for the mechanism). We define them now. Let $A(t)\in {\mathcal{M}}_{\le n}\otimes {\mathcal{M}}_{\mathrm{R}}$, r≥1 and t≥t₁≥t₂≥⋯≥t_r be fixed. Let (p₁,…,p_N) be an N-tuple of integers p_j∈{0,1,…}. We associate with (p₁,…,p_N) a set of operators, denoted by ${\mathcal{C}}_r(p_1,\dots ,p_N)$, consisting of the collection of all r-fold multi-commutators

$$T_t=[G_{j_r}(t_r)\otimes B_{j_r}(t_r),[\dots ,[G_{j_1}(t_1)\otimes B_{j_1}(t_1),A(t)]\dots ]],$$ 1.9

in which each index j₁,…,j_r varies over the values {1,…,N}, under the constraint that p₁ among the indices equal 1, p₂ among them equal 2, and so on, and p_N among them equal N. For each integer ν≥0, set

$$\begin{array}{r}{X}_{\nu ,N}(A(t))=N^{\nu }\sum _{\genfrac{}{}{0ex}{}{r\ge 2\nu ,r\ne 0}{\mathrm{even}}}{(\mathrm{i}\lambda )}^r{N}^{-r/2}\sum _{p_1+\cdots +p_n=2\nu }\hspace{0.17em}\sum _{\genfrac{}{}{0ex}{}{p_{n+1}+\cdots +p_N=r-2\nu }{\mathrm{even}}}{\int }_0^t\hspace{0.17em}\mathrm{d}{t}_1\cdots {\int }_0^{t_{r-1}}\mathrm{d}{t}_r\hspace{0.17em}\sum _{T_t\in {\mathcal{C}}_r(p_1,\dots ,p_N)}{\omega }_N(T_t).\end{array}$$ 1.10

The first three sums are over integers r,p₁,…,p_N∈{0,1,2,…} and the indication ‘even’ means that the respective sums are taken only over even summation indices (even r in the first one, even p_n+1,…,p_N in the third one). We also define, for ν≥0,

$$\begin{array}{rl}{Y}_{\nu ,N}(A(t))& =N^{\nu +1/2}\sum _{\genfrac{}{}{0ex}{}{r\ge 2\nu +1}{\mathrm{odd}}}{(\mathrm{i}\lambda )}^r{N}^{-r/2}\sum _{p_1+\cdots +p_n=2\nu +1}\hspace{0.17em}\sum _{\genfrac{}{}{0ex}{}{p_{n+1}+\cdots +p_N=r-2\nu -1}{\mathrm{even}}}\\ & \times {\int }_0^t\mathrm{d}{t}_1\cdots {\int }_0^{t_{r-1}}\hspace{0.17em}\mathrm{d}{t}_r\sum _{T_t\in {\mathcal{C}}_r(p_1,\dots ,p_N)}{\omega }_N(T_t).\end{array}$$ 1.11

We point out that the terms $\sum _{p_1+\cdots +p_n=2\nu }\cdots {\omega }_N(T_t)$ in (1.10), (1.11) still depend on N, but not on λ. It is then apparent that (1.10) and (1.11) define functions of λ analytic at λ=0. X_0,N(A(t)) has a zero of order 2 at λ=0 and, for ν≥1, X_ν,N(A(t)) has a zero of order 2ν at λ=0. Y _ν,N(A(t)) has a zero of order 2ν+1 at λ=0, for ν≥0.

The conditions (A0) and (A1) below serve to control the convergence of the series (1.10) and (1.11). We present models satisfying them in §1b.

• (A0) Vanishing odd moment condition. We assume that, for every j=1,…,N, every integer k≥0 and for all times t₁,…,t_2k+1∈[0,t],

  $${\mu }_j(G_j(t_1)\cdots G_j(t_{2k+1}))=0.$$ 1.12

Given $A_{\mathrm{R}}\in {\mathcal{M}}_{\mathrm{R}}$, t≥0 and an integer r≥1, we define

$${\beta }_r(A_{\mathrm{R}},t)\equiv \underset{t_1,\dots ,t_r\in [0,t]}{sup}\underset{\sigma \in {\mathfrak{S}}_r}{max}\underset{1\le j\le r+1}{max}|{\mu }_{\mathrm{R}}(B_{\sigma (1)}(t_1)\cdots [\hspace{0.17em}{}_j{A}_{\mathrm{R}}(t)]\cdots B_{\sigma (r)}(t_r))|$$ 1.13

and

$$b(A_{\mathrm{R}},t)\equiv \underset{r\to \mathrm{\infty }}{lim sup}\frac{{[{\beta }_r(A_{\mathrm{R}},t)]}^{1/r}}{\sqrt{r}},$$ 1.14

where ${\mathfrak{S}}_r$ is the group of permutations and the symbol [_jA_R(t)] means that A_R(t) is the jth factor inside the product. We make the following assumption.

• (A1) Let $g=\underset{j\ge 1}{sup}\parallel G_j\parallel$. We have

  $$2\sqrt{2\mathrm{e}}|\lambda |gtb(A_{\mathrm{R}},t)<1.$$ 1.15

The bound (1.15) gives the time scale for which our results hold. We show in §1b that for some reservoirs the bound is satisfied for all λ, $t\in \mathbb{R}$ (e.g. when B is a bounded operator). For others the bound imposes the constraint |λ|t≤C for some finite constant C>0 (e.g. when B is a Bose field operator and μ_R is quasi-free).

The following result shows that the functionals X_ν,N and Y _ν,N are well defined.

Theorem 1.1 —

The series (1.10) and (1.11) converge for any n (<N) and any $A=A_{\mathrm{S}}\otimes A_{\mathrm{R}}\in {\mathcal{M}}_{\le n}\otimes {\mathcal{M}}_{\mathrm{R}}$. Moreover,

• (A) The maps λ↦X_ν,N(A(t)) and λ↦Y _ν,N(A(t)) are analytic in a disc centred at λ=0, of radius R having the N- and ν-independent lower bound

  $$R\ge \frac{1}{2\sqrt{2\mathrm{e}}\hspace{0.17em}gtb(A_{\mathrm{R}},t)}.$$ 1.16
• (B) For ν≥0, we have

  $$|X_{\nu ,N}(A(t))|\le \parallel A_{\mathrm{S}}\parallel \frac{{(2n|\lambda |gt)}^{2\nu }}{(2\nu )!}{S}_{\nu }(A_{\mathrm{R}},t)$$ 1.17

  and

  $$|Y_{\nu ,N}(A(t))|\le \parallel A_{\mathrm{S}}\parallel \frac{{(2n|\lambda |gt)}^{2\nu +1}}{(2\nu +1)!}{S}_{\nu +1/2}(A_{\mathrm{R}},t),$$ 1.18

  where

  $$S_{\nu }(A_{\mathrm{R}},t)=\sum _{s\ge {\delta }_{0,\nu }}\frac{{(2|\lambda |gt)}^{2s}}{s!}{\beta }_{2(\nu +s)}(A_{\mathrm{R}},t).$$ 1.19

  The summation in (1.19) starts at s=0 if ν>0 and at s=1 for ν=0. The series (1.19) converges.

We point out that the upper bounds in (1.17) and (1.18) are independent of N. Denoting for a moment the general term in the series (1.19) by ${\varkappa }_s$, we use (1.15) to obtain $\underset{s}{lim sup}|{\varkappa }_s{|}^{1/s}\le 2\hspace{0.17em}\mathrm{e}\hspace{0.17em}{b}^2(A_{\mathrm{R}},t){(2|\lambda |gt)}^2$. This shows that the series (1.19) converges due to (1.15). The linear functionals X_ν,N and Y _ν,N are used to express the dynamics as follows.

Theorem 1.2 —

Let $A=A_{\mathrm{S}}\otimes A_{\mathrm{R}}\in {\mathcal{M}}_{\le n}\otimes {\mathcal{M}}_{\mathrm{R}}$, where n<N, and suppose that

$${(n\mathrm{e}|\lambda |gt)}^2\underset{\nu \to \mathrm{\infty }}{lim sup}\frac{{[S_{\nu }(A_{\mathrm{R}},t)]}^{1/\nu }}{{\nu }^2}<1.$$ 1.20

Then we have the expansion

$${\omega }_N({\tau }_{\lambda ,N}^t(A))={\omega }_n(A(t))+\sum _{\nu =0}^{\mathrm{\infty }}{N}^{-\nu }{X}_{\nu ,N}(A(t))+N^{-\nu -1/2}{Y}_{\nu ,N}(A(t)),$$ 1.21

where the series converges absolutely and uniformly in N≥1, and uniformly in t for t varying in compact sets.

We show in lemma 1.4 that $X_{0,N}{\upharpoonright }_{{\mathcal{M}}_{\le n}}=0$ for any n, while X_0,N does not vanish on ${\mathcal{M}}_{\mathrm{R}}$ (see §1a(ii)). In the symmetric situation (see the definition after (1.8)) the functionals X_ν,N and Y _ν,N have limits as $N\to \mathrm{\infty }$.

Theorem 1.3 (symmetric case) —

Suppose that the system is symmetric and that the conditions of theorem 1.2 hold. Then the limits

$$\underset{N\to \mathrm{\infty }}{lim}{X}_{\nu ,N}(A(t))=X_{\nu }(A(t))\mathrm{and}\underset{N\to \mathrm{\infty }}{lim}{Y}_{\nu ,N}(A(t))=Y_{\nu }(A(t))$$ 1.22

exist and are uniform in t for t varying in compact sets. The limiting X_ν, Y _ν are functionals analytic in λ at λ=0. The observable A_S does not have to be symmetric with respect to the permutation of particles for this result to hold.

The Taylor expansions of X_ν and Y _ν are given in (3.19) and (3.20). Combining (1.22) with theorem 1.2 yields the expansion

$${\omega }_N({\tau }_{\lambda ,N}^t(A))={\omega }_n(A(t))+\sum _{\nu =0}^{\mathrm{\infty }}{N}^{-\nu }\{X_{\nu }(A(t))+o_N\}+N^{-\nu -1/2}\{Y_{\nu }(A(t))+o_N\},$$ 1.23

with $o_N\to 0$ uniformly in t for t in compacta. We have $X_0{\upharpoonright }_{{\mathcal{M}}_{\le n}}=0$ for any n (see lemma 1.4).

(a). Consequences of theorems 1.2 and 1.3

In this section, we present results that follow from our main theorems above. The proofs of all the lemmas given below are presented in §3e.

(i). The n-particle dynamics

Lemma 1.4 —

Let n<N. We have $X_{0,N}{\upharpoonright }_{{\mathcal{M}}_{\le n}}=0$, meaning that, in the limit $N\to \mathrm{\infty }$, the dynamics on ${\mathcal{M}}_{\le n}$ is just the non-interacting one,

$$\underset{N\to \mathrm{\infty }}{lim}{\omega }_N({\tau }_{\lambda ,N}^t(A))={\omega }_n(A(t)),\mathrm{\forall }A\in {\mathcal{M}}_{\le n}.$$

Remarks —

• (1) The fact that $X_{0,N}{\upharpoonright }_{{\mathcal{M}}_{\le n}}=0$ follows directly from (1.10). Indeed, for ν=0, all the p₁=⋯=p_n=0. This means that, in the commutator T_t in (1.10), all the operators G_j(t_j) act on particles j with j≥n+1, so they commute with any $A\in {\mathcal{M}}_{\le n}$ (see (1.9)). Hence T_t=0. In the special case of the Dicke model (see §2a), the result of lemma 1.4 was obtained in [1].

• (2) Lemma 1.4 shows that the influence of the reservoir on the dynamics of any n-particle observable is negligible, as $N\to \mathrm{\infty }$. However, $X_{0,N}{\upharpoonright }_{{\mathcal{M}}_{\mathrm{R}}}\ne 0$ and the reservoir experiences a non-trivial dynamics due to the presence of the particles (see §1a(ii)).

(ii). The reservoir dynamics

Consider the symmetric case. According to theorem 1.3, we have

$${\omega }_N({\tau }_{\lambda ,N}^t(A_{\mathrm{R}}))={\mu }_{\mathrm{R}}(A_{\mathrm{R}}(t))+X_0(A_{\mathrm{R}}(t))+o_N$$ 1.24

with $o_N\to 0$ as $N\to \mathrm{\infty }$. The reservoir dynamics is not the free one in the large N limit. The influence of the ‘particles’ is given by the term X₀. The particles may themselves be seen as a ‘mean field reservoir’, acting on the ‘system’ R.

The interaction term X₀ is analytic in λ and we know its Taylor series; cf. (3.19). In the energy-conserving situation, we can re-sum the Taylor series to obtain the following result.

Lemma 1.5 —

Suppose that the system is symmetric and the dynamics is energy conserving. Let μ_HO be the vacuum state of a quantum harmonic oscillator with creation and annihilation operators a,a*, satisfying $[a,a^{\ast }]=\mathrm{𝟙},$ and set $\phi =(1/\sqrt{2})(a+a^{\ast })$. We have

$$\underset{N\to \mathrm{\infty }}{lim}{\omega }_N({\tau }_{\lambda ,N}^t(A_{\mathrm{R}}))={\mu }_{\mathrm{HO}}\otimes {\mu }_{\mathrm{R}}({\mathrm{e}}^{\mathrm{i}\hspace{0.17em}t(H_{\mathrm{R}}+2\sqrt{\varkappa }\phi \otimes B)}({\mathrm{𝟙}}_{\mathrm{HO}}\otimes A_{\mathrm{R}}){\mathrm{e}}^{-\mathrm{i}\hspace{0.17em}t(H_{\mathrm{R}}+2\sqrt{\varkappa }\phi \otimes B)}),$$ 1.25

where $\varkappa ={\lambda }^2{\mu }_{\mathrm{S}}(G^2)$.

The result shows that the net effect of the $N\to \mathrm{\infty }$ single particles on the reservoir is described by a single, zero-frequency (H_HO=0) quantum harmonic oscillator interacting linearly with the reservoir. The only trace of the original single-particle system, in the limit $N\to \mathrm{\infty }$, is the variance μ_S(G²), determining the interaction strength between the oscillator and the reservoir R.

(iii). The leading orders

We examine the terms with ν=0 in (1.21) in more detail. The following result holds for systems which do not have to be symmetric.

Lemma 1.6 —

Assume the conditions of theorem 1.2 and set μ_≤n=μ₁⊗μ₂⊗⋯⊗μ_n. We have

$$\begin{array}{rl}& {\omega }_N({\tau }_{\lambda ,N}^t(A))={\mu }_{\le n}(A_{\mathrm{S}}(t)){\mu }_{\mathrm{R}}(A_{\mathrm{R}}(t))\\ & -2\frac{{\lambda }^2}{N}{\mu }_{\le n}(A_{\mathrm{S}}(t))\sum _{j=n+1}^N{\int }_0^t\mathrm{d}s{\int }_0^s\mathrm{d}{s}^{\prime }\hspace{0.17em}\mathrm{Re}\{{\mu }_j(G_j(s^{\prime })G_j(s)){\mu }_{\mathrm{R}}(B_j(s^{\prime })[B_j(s),A_{\mathrm{R}}(t)])\}\\ & -2\frac{\lambda }{\sqrt{N}}\sum _{j=1}^n{\int }_0^t\mathrm{d}s\hspace{0.17em}\mathrm{Im}\{{\mu }_{\le n}(G_j(s)A_{\mathrm{S}}(t)){\mu }_{\mathrm{R}}(B_j(s)A_{\mathrm{R}}(t))\}+O\left({\lambda }^4+\frac{{\lambda }^3}{\sqrt{N}}+\frac{1}{N}\right).\end{array}$$ 1.26

The remainder is uniform in t varying in compact subsets of $|t|<{(2\sqrt{2\mathrm{e}}|\lambda |gb(A_{\mathrm{R}},t))}^{-1}$ and it is uniform in $t\in \mathbb{R}$ if b(A_R,t)=0.

The second term on the right-hand side of (1.26) has a (finite) limit as $N\to \mathrm{\infty }$ if, for instance, the model is asymptotically constant, i.e. if ${\mu }_j\to {\mu }_{\mathrm{S}}$, $G_j\to G$ and $B_j\to B$ as $j\to \mathrm{\infty }$. Indeed,

$$\begin{array}{rl}& -2\frac{{\lambda }^2}{N}\sum _{j=n+1}^N{\int }_0^t\mathrm{d}s{\int }_0^s\mathrm{d}{s}^{\prime }\hspace{0.17em}\mathrm{Re}\{{\mu }_j(G_j(s^{\prime })G_j(s)){\mu }_{\mathrm{R}}(B_j(s^{\prime })[B_j(s),A_{\mathrm{R}}(t)])\}\\ & =-2{\lambda }^2{\int }_0^t\mathrm{d}s{\int }_0^s\mathrm{d}{s}^{\prime }\hspace{0.17em}\mathrm{Re}\{{\mu }_{\mathrm{S}}(G(s^{\prime })G(s)){\mu }_{\mathrm{R}}(B(s^{\prime })[B(s),A_{\mathrm{R}}(t)])\}+o\left(\frac{1}{N}\right),\end{array}$$ 1.27

where o(1/N) is a quantity that converges to zero as $N\to \mathrm{\infty }$. One can generalize (1.27) to the case where the quantities μ_j,G_j,B_j only converge in the sense of ergodic averages.

(iv). The dynamics of intensive and fluctuation observables

We show here that the average of intensive system observables evolves according to the free dynamics, but the reservoir induces system fluctuations. Consider the symmetric case (see after (1.8)) and let $A\in {\mathcal{M}}_{\mathrm{S}}$ be a single-particle observable. The associated intensive observable is

$$A_{(N)}\equiv N^{-1}\sum _{n=1}^N{A}_n,A_n\equiv {\mathrm{𝟙}}_{\mathrm{S}}\otimes \cdots A\cdots \otimes {\mathrm{𝟙}}_{\mathrm{S}}\in {\mathcal{M}}_{\le N},$$

where A acts on the nth factor. According to lemma 1.4, we have

$$\underset{N\to \mathrm{\infty }}{lim}{\omega }_N({\tau }_{\lambda ,N}^t(A_{(N)}))={\mu }_{\mathrm{S}}(A(t)),$$ 1.28

meaning that the expectation of intensive observables evolves according to the free dynamics, as $N\to \mathrm{\infty }$. One may interpret (1.28), which holds for all states μ_S, as a manifestation of the law of large numbers (for each fixed t). Namely the sample average of the ‘random variables’ ${\tau }_{\lambda ,N}^t(A_n)$, n=1,…,N, converges to the limit A(t). One defines the fluctuation observable [2]

$$F_N(A,t)=\sqrt{N}({\tau }_{\lambda ,N}^t(A_{(N)})-{\mu }_{\mathrm{S}}(A(t))).$$

The following result gives the limiting dynamics of fluctuation observables.

Lemma 1.7 —

Consider the symmetric situation. We have

$$\begin{array}{rl}& \underset{N\to \mathrm{\infty }}{lim}{\omega }_N(F_N(A,t))=\mathrm{i}\lambda {\int }_0^t{\mu }_{\mathrm{S}}([G(t_1),A(t)]){\mu }_{\mathrm{R}}(B(t_1))\hspace{0.17em}\mathrm{d}{t}_1\\ & +\sum _{r\ge 3,\mathrm{odd}}\frac{{(\mathrm{i}\lambda )}^r}{((r-1)/2)!}{\int }_0^t\mathrm{d}{t}_1\cdots {\int }_0^{t_{r-1}}\mathrm{d}{t}_r\hspace{0.17em}{\mu }_{\mathrm{S}}([G(t_1),A(t)])\\ & \times {\sum _{j_2,\dots ,j_r}}^{\ast }{\omega }_{(r-1)/2}([G_{j_r}(t_r)\otimes B(t_r),[\cdots ,[G_{j_2}(t_2)\otimes B(t_2),B(t_1)]]]),\end{array}$$ 1.29

where the starred sum is over all indices j₂,…,j_r with the constraint that two among them equal 1, two among them equal 2, and so on, and two among them equal (r−1)/2.

Lemma 1.7 shows that the fluctuations vanish if μ_R(B(t₁)⋯B(t_2k+1))=0 for all k≥1. This is, in particular, the case for a free Bose reservoir in a gauge-invariant state μ_R (e.g. the equilibrium state, or the vacuum, or a state with a definite number of excitations), coupled to the system via a field operator B=φ(f).

We now discuss an example of a system with non-vanishing fluctuations. Consider each system particle to be a spin $\frac{1}{2}$, with $h_j=\frac{1}{2}{\omega }_0{\sigma }_z$, and the reservoir to be a harmonic oscillator with H_R=ω_Ra*a in a coherent state μ_R=〈α|⋅|α〉, $\alpha \in \mathbb{C}$. Let the interaction operator be G_j=σ_x and $B=\phi =(1/\sqrt{2})(a^{\ast }+a)$.¹ (This is the single-mode Dicke maser model with initial field in a coherent state; cf. §2a.) Then a|α〉=α|α〉 and μ_R(B(s))=2 Re(e^iω_Rsα). The lowest order of the fluctuation is then explicitly

$$\underset{N\to \mathrm{\infty }}{lim}{\omega }_N(F_N(A,t))=-4\lambda {\int }_0^t\mathrm{Im}({\mu }_{\mathrm{S}}([{\mathrm{e}}^{\mathrm{i}\hspace{0.17em}s{\omega }_0}{\sigma }_{+},A(t)]))\hspace{0.17em}\mathrm{Re}({\mathrm{e}}^{\mathrm{i}\hspace{0.17em}s{\omega }_{\mathrm{R}}}\alpha )\hspace{0.17em}\mathrm{d}s+O({\lambda }^3).$$ 1.30

Consider the spins to be in equilibrium at temperature T and assume that the initial field coherent state is given by $\alpha \in \mathbb{R}$, and that the off-diagonal of the observable A is also real, $A_{\uparrow \downarrow }=⟨\uparrow |A|\downarrow ⟩\in \mathbb{R}$. Then we calculate (1.30) further to be

$$\underset{N\to \mathrm{\infty }}{lim}{\omega }_N(F_N(A,t))=\frac{-4\lambda \alpha A_{\uparrow \downarrow }}{1+{\mathrm{e}}^{-{\omega }_0/T}}\sin ({\omega }_{0}t)\left(\frac{\sin ({\omega }_0+{\omega }_{\mathrm{R}})t}{{\omega }_0+{\omega }_{\mathrm{R}}}+\frac{\sin ({\omega }_0-{\omega }_{\mathrm{R}})t}{{\omega }_0-{\omega }_{\mathrm{R}}}\right)+O({\lambda }^3).$$ 1.31

This shows that only the coherences (off-diagonals in the energy basis) of the observable A contribute to the fluctuations and for diagonal A they vanish. Moreover, the fluctuations are oscillating in time, their onset is quadratic in time and the magnitude of the fluctuations is a decreasing function of the temperature T≥0, but still has a non-zero value for $T\to \mathrm{\infty }$.

(b). Satisfying conditions (A0), (A1) and (1.20)

(i). Condition (A0)

• (1) Let the particle j be described by a d_j-level system with Hamiltonian

  $$h_j=\mathrm{diag}(e_j^{(1)},\dots ,e_j^{(d_j)}).$$

  Take for G_j an operator inducing single-step excitations and de-excitations, written in the diagonal basis of h as

  $$G_j=\left(\begin{array}{cccc}0& {\gamma }_{j,1}& 0& \\ {\overline{\gamma }}_{j,1}& 0& {\gamma }_{j,2}& \\ 0& {\overline{\gamma }}_{j,2}& 0& \ddots & 0\\ & & \ddots & \ddots & {\gamma }_{j,d_j-1}\\ & & 0& {\overline{\gamma }}_{j,d_j-1}& 0\end{array}\right),$$ 1.32

  where the upper and lower diagonal consists of possibly non-zero numbers and all other entries vanish. Denote by V _{ℓ1,ℓ2,…} the span of the vectors ${\varphi }_j^{({\ell }_1)},{\varphi }_j^{({\ell }_2)},\dots$, where ${\varphi }_j^{(\ell )}$ is the ℓth excited state, i.e. $h_j{\varphi }_j^{(\ell )}=e_j^{(\ell )}{\varphi }_j^{(\ell )}$. Applying successively G_j(t₁), G_j(t₂), G_j(t₃),… to the state ${\varphi }_j^{(1)}$ gives vectors belonging to the subspaces

  $$V_1\mapsto V_2\mapsto V_{1,3}\mapsto V_{2,4}\mapsto \cdots$$

  and it is manifest that $G_j(t_1)\cdots G_j(t_{2k+1}){\varphi }_j^{(1)}\perp {\varphi }_j^{(1)}$. The same holds for ${\varphi }_j^{(1)}$ replaced by ${\varphi }_j^{(\ell )}$ for any ℓ. Therefore, the relation (1.12) is satisfied for ${\mu }_j(\cdot )=⟨{\varphi }_j^{(\ell )},\hspace{0.17em}\cdot \hspace{0.17em}{\varphi }_j^{(\ell )}⟩$, any ℓ.

• (2) A special case of the above example is d_j=2 and G_j=σ_x, the Pauli (spin flip) matrix.

• (3) If (1.12) holds for states ${\mu }_j^{(1)},{\mu }_j^{(2)},\dots ,{\mu }_j^{(m)}$, then it holds for any of their mixture, $p_1{\mu }_j^{(1)}+p_2{\mu }_j^{(2)}+\cdots +p_m{\mu }_j^{(m)}$. In particular, in the examples (1), (2) above, condition (1.12) is satisfied for the equilibrium states given by the density matrix ρ_j∝e^−βh_j, and more generally for any density matrix which is a function ρ_j=f(h_j).

(ii). Condition (A1)

We introduce two classes, (E1) and (E2), of examples, which we will refer to below repeatedly.

Example (E1) —

Each B_j is a bounded operator, with $g_{\mathrm{R}}\equiv \underset{j}{sup}\parallel B_j\parallel <\mathrm{\infty }$. Then we have β_r(t)≤ ∥A_R∥(g_R)^r and, consequently, b(λ,t)=0. It follows that (1.15) is satisfied for all $t,\lambda \in \mathbb{R}$.

Example (E2) —

The reservoir describes a bosonic quantum field and the B_j are field operators, i.e. B_j=φ(f_j), $f_j\in L^2({\mathbb{R}}^3,d^{3}x)$ with $\phi (f)=(1/\sqrt{2})[a(f)+a^{\ast }(f)]$ and the dynamics is given by B_j(t)= φ(e^i th_Rf), for some self-adjoint one-particle Hamiltonian h_R. The field state μ_R is a Gaussian (gauge-invariant, quasi-free) state with two-point function μ_R(B(s)B(s′)). We prove the following result in §3e.

Lemma 1.8 —

Consider the example (E1).

(1) Let $A_{\mathrm{R}}\in {\mathcal{M}}_{\mathrm{R}}$. Then

$${\beta }_r(A_{\mathrm{R}},t)\le \frac{\mathrm{e}}{\sqrt{\pi }}\parallel A_{\mathrm{R}}\parallel {(C(t)/\mathrm{e})}^{r/2}{r}^{r/2}\text{and}b(A_{\mathrm{R}},t)\le \sqrt{\frac{C(t)}{\mathrm{e}}},$$

where $C(t)\equiv \underset{1\le i,j\le N}{max}\underset{0\le s\le s^{\prime }\le t}{max}|{\mu }_{\mathrm{R}}(\phi ({\mathrm{e}}^{\mathrm{i}\hspace{0.17em}s{h}_{\mathrm{R}}}{f}_i)\phi ({\mathrm{e}}^{\mathrm{i}\hspace{0.17em}{s}^{\prime }{h}_{\mathrm{R}}}{f}_j))|$.

(2) Let A_R be a possibly unbounded reservoir operator, denote by $\hat{N}$ the number operator and suppose that the state μ_R carries at most n₀ particles. (More precisely, ${\mu }_{\mathrm{R}}(P(\hat{N}>n_0)B_{\mathrm{R}})={\mu }_{\mathrm{R}}(B_{\mathrm{R}}P(\hat{N}>n_0))=0$ for any $B_{\mathrm{R}}\in {\mathcal{M}}_{\mathrm{R}}$, where $P(\hat{N}>n_0)$ is the spectral projection of $\hat{N}$.) Then the upper bound for β_r given in point (1) above applies with ∥A_R∥ replaced by $\parallel A_{\mathrm{R}}P(\hat{N}\le n_0+r/2)\parallel$ and the upper bound for b is the same as the one given in point (1).

(3) Suppose A_R=φ(g₁)⋯φ(g_k) is the product of k field operators. Then

$${\beta }_r(A_{\mathrm{R}},t)\le \frac{\mathrm{e}}{\sqrt{\pi }}{\left(\frac{C(A_{\mathrm{R}},t)}{\mathrm{e}}\right)}^{(r+k)/2}{(r+k)}^{(r+k)/2}\text{and}b(A_{\mathrm{R}},t)\le \sqrt{\frac{C(A_{\mathrm{R}},t)}{\mathrm{e}}},$$

where $C(A_{\mathrm{R}},t)=\underset{f,f^{\prime }\in \{f_1,\dots ,f_N,g_{,}\dots ,g_k\}}{max}\underset{0\le s,s^{\prime }\le t}{max}|{\mu }_{\mathrm{R}}(\phi ({\mathrm{e}}^{\mathrm{i}\hspace{0.17em}s{h}_{\mathrm{R}}}f)\phi ({\mathrm{e}}^{\mathrm{i}\hspace{0.17em}{s}^{\prime }{h}_{\mathrm{R}}}{f}^{\prime }))|$.

A sufficient parameter constraint for (1.15) to hold is 8λ²g²t²C<1, where C is the appropriate (time-dependent) constant given in lemma 1.8 (1), (2) or (3).

(iii). Condition (1.20)

In example (E1), we have $S_{\nu }(A_{\mathrm{R}},t)\le \parallel A_{\mathrm{R}}\parallel {(g_{\mathrm{R}})}^{2\nu }{\mathrm{e}}^{{(2|\lambda |g{g}_{\mathrm{R}}t)}^2}$ and $\underset{\nu }{lim sup}{[S_{\nu }(A_{\mathrm{R}},t)]}^{1/\nu }/{\nu }^2=0$. It follows that (1.20) holds for arbitrary $t,\lambda \in \mathbb{R}$.

For example (E2), we have the following result (proved in §3e).

Lemma 1.9 —

Suppose that, in the situations (1)–(3) of lemma 1.8, we have 16λ²g²t²C(t)<1 or 16λ²g²t²C(A_R,t)<1. Then $\underset{\nu }{lim sup}{[S_{\nu }(A_{\mathrm{R}},t)]}^{1/\nu }$ grows at most linearly in ν, so $\underset{\nu }{lim sup}{[S_{\nu }(A_{\mathrm{R}},t)]}^{1/\nu }/{\nu }^2=0$. Hence (1.20) does not introduce any additional bounds on t,λ.

(c). Embedding in previous work

The literature on the mean field limits of closed systems of interacting particles is huge. The topic has been a very active research field in physics and mathematical physics for many decades. We find Spohn’s paper [3] particularly useful to understand, with little cumbersome technicality, the essence of the phenomena emerging in the mean field limit and how to show them using the ‘BBGKY’ hierarchy method. An excellent, more detailed account and overview of the literature is presented in [4]. The literature on mean field limits of open quantum systems is less rich, to our knowledge, with the exception of the Dicke model in various variations, whose thermodynamics and dynamics have been studied in great detail by many authors, for instance [1,5–8]. The methods allowing us to treat the Dicke model use its symmetries in an essential way and are very specific to the model at hand. A more general approach is taken in [9,10], where nonlinear evolutions for density matrices are examined. However, a rigorous derivation of Markovian nonlinear dynamics emerging from the mean field limit has not been derived (except in explicitly solvable models). The problem is that one should control the two limits of small coupling (λ small) and high complexity (N large) simultaneously. Our present work is an attempt at this, in that we derive a controlled expansion of the dynamics in both parameters λ and N. We are not yet able to derive the limiting ($N\to \mathrm{\infty }$) evolution equation in a ‘closed’ form, say as a Hartree equation. Instead, here we only derive an expansion of the limiting dynamics in λ, which is generally not ‘resummable’, except for energy-conserving systems, as exemplified in our lemma 1.5 and in the previous work [11]. On the other hand, our approach is a very direct analysis of the Dyson series expansion of the propagator and does not rely on many of the models’ specifics.

2. Illustrations

(a). The Dicke model

The Dicke (maser) model describes the interaction of (idealized, two-level) atoms with the quantized electromagnetic field. Its Hamiltonian is given by Hepp & Lieb [1] and Hioe [6]

$$H_N^{\prime }=\frac{{\omega }_0}{2}\sum _{j=1}^N{\sigma }_j^z+\nu a^{\ast }a+\frac{\lambda }{\sqrt{N}}\sum _{j=1}^N{\sigma }_j^x\otimes (a^{\ast }+a),$$ 2.1

where each atom has a transition (Bohr) frequency ω₀, and where ${\sigma }_j^{x,z}$ are the Pauli matrices of the jth atom. In (2.1), the radiation is described by a single bosonic mode of frequency ν, with associated creation and annihilation operators a* and a, satisfying $[a,a^{\ast }]=\mathrm{𝟙}$. The factor $1/\sqrt{N}$ is actually a factor V ^−1/2, where V is the volume of the cavity containing the atoms. Considering V/N fixed leads to the prefactor in the interaction in (2.1). Often the rotating wave approximation is considered, where ${\sigma }_j^x\otimes (a^{\ast }+a)$ is replaced by ${\sigma }_j^{+}\otimes a+{\sigma }_j^{-}\otimes a^{\ast }$, and then one can exploit the conservation of the total number of particles. We do not use this approximation here. A multi-mode model is given by Fannes et al. [7]

$$H_N^{″}=\frac{{\omega }_0}{2}\sum _{j=1}^N{\sigma }_j^z+\sum _{j=1}^N{\nu }_j{a}^{\ast }a+\frac{\lambda }{N}\sum _{i,j=1}^N{\sigma }_j^x\otimes (a_i^{\ast }+a_i).$$ 2.2

In the limit of continuous modes the a_i are replaced by a(k), where $k\in {\mathbb{R}}^3$ is a continuous (momentum) parameter, satisfying [a(k),a*(l)]=δ(k−l) (Dirac delta); one obtains a Hamiltonian of the form

$$\begin{array}{r}{H}_N=\frac{{\omega }_0}{2}\sum _{j=1}^N{\sigma }_j^z+H_{\mathrm{R}}+\frac{\lambda }{\sqrt{N}}\sum _{j=1}^N{\sigma }_j^x\otimes \phi (g),\end{array}$$ 2.3

$$\begin{array}{r}{H}_{\mathrm{R}}={\int }_{{\mathbb{R}}^3}|k|a^{\ast }(k)a(k)\hspace{0.17em}{\mathrm{d}}^{3}k\end{array}$$ 2.4

$$\begin{array}{r}\text{and}B\equiv \phi (g)=\frac{1}{\sqrt{2}}(a^{\ast }(g)+a(g)),\end{array}$$ 2.5

acting on the Hilbert space ${\mathcal{H}}_N={\otimes }_{j=1}^N{\mathbb{C}}^2\otimes \mathcal{F}$, where $\mathcal{F}$ is the bosonic Fock space over the one-particle Hilbert space $L^2({\mathbb{R}}^3,{\mathrm{d}}^{3}k)$ (Fourier representation). The value g(k) of the form factor $g\in L^2({\mathbb{R}}^3,{\mathrm{d}}^{3}k)$ governs the strength of the coupling of mode k to the collection of atoms. One factor $1/\sqrt{N}$ in (2.2) is absorbed into the continuous creation and annihilation operators. The Hamiltonian (2.3) with $\lambda /\sqrt{N}$ replaced by λ/N (and in the rotating wave approximation) was considered by Davies [5]. Davies’ model is thus a weak coupling limit of (2.3), as was also remarked in [7]. We analyse the average field excitation

$$\mathcal{N}(t)=\underset{N\to \mathrm{\infty }}{lim}{\omega }_N({\tau }_{\lambda ,N}^t({\hat{N}}_{\mathrm{R}}))={\mu }_{\mathrm{R}}({\hat{N}}_{\mathrm{R}})+X_0({\hat{N}}_{\mathrm{R}}),$$ 2.6

where ${\hat{N}}_{\mathrm{R}}={\int }_{{\mathbb{R}}^3}{a}^{\ast }(k)a(k)\hspace{0.17em}{\mathrm{d}}^{3}k$ is the number operator (see also (1.24)). We take as the initial state the form (1.8) in which the field is in the vacuum μ_R=|Ω〉〈Ω| (we can deal with excited states in just the same manner) and, for some p∈[0,1], we take

$${\mu }_j=p|\uparrow ⟩⟨\uparrow |+(1-p)|\downarrow ⟩⟨\downarrow |.$$ 2.7

We thus have (see (3.19))

$$\mathcal{N}(t)=X_0({\hat{N}}_{\mathrm{R}})=\sum _{q\ge 1}\frac{{(-{\lambda }^2)}^q}{q!}{\int }_0^t\mathrm{d}{t}_1\cdots {\int }_0^{t_{2q-1}}d{t}_{2q}\sum _{T_t\in {\mathcal{D}}_q}{\omega }_q(T_t).$$ 2.8

Here, ${\mathcal{D}}_q$ is the class of all (r=2q)-fold multi-commutators of the form (1.9) for which exactly two among the indices j₁,…,j_2q take each one of the values 1,…,q. The lowest order in λ can be calculated directly either from (2.8) or using lemma 1.6, (1.26) and a few easy calculations. We obtain

$$\mathcal{N}(t)={\lambda }^2{\int }_{{\mathbb{R}}^3}|g(k){|}^2\left[p\frac{1-\cos ({\omega }_0-\omega )t}{{({\omega }_0-\omega )}^2}+(1-p)\frac{1-\cos ({\omega }_0+\omega )t}{{({\omega }_0+\omega )}^2}\right]{\mathrm{d}}^{3}k+O({\lambda }^4),$$ 2.9

with a remainder uniform in t for |t|<(2|λ| ∥g∥)⁻¹. (Use lemma 1.8 (2), to see that $b(\hat{N},t)\le \parallel g\parallel /\sqrt{2\mathrm{e}}$ .) In the parameter regime considered, the average number of field excitations is oscillating in time. This has also been observed in [1] for λ smaller than a critical value λ_c. Beyond this regime an exponential increase in time of $\mathcal{N}(t)$ is expected (superradiance). Uncovering this behaviour might be difficult in the present set-up, as one should include all orders of λ and take λ not too small. Furthermore, n-body observables show non-trivial dynamics in the superradiant phase (see for instance [1], theorem 4.2), and hence, in view of lemma 1.4, one might have to relax condition (A0) to capture non-trivial effects in this regime (maybe even in the photon field).

The thermodynamic properties and dynamics of the Dicke maser model have been studied in great detail in the references mentioned above (and many others). The analysis is based tightly on the specifics of the model (in particular, certain reductions due to conservation laws). The goal here was to illustrate how our general approach applies to the Dicke maser model, reproducing some of the previous results.

(b). Energy-conserving models

(i). Single-particle dynamics

The system dynamics for the energy-conserving and symmetric situation has been solved explicitly [11] and without imposing the vanishing odd moment condition. In that paper, each particle is given by a d-level system, the reservoir is an infinitely extended thermal, free bosonic quantum field and the interaction operator B is the field operator $\phi (g)=(1/\sqrt{2})(a^{\ast }(g)+a(g))$. One of the motivations for this model is the analysis of coherence and entanglement of qubits subject to a collective noise. It is shown in [11] that, for each fixed time t, the reduced-density matrix of n particles (obtained by tracing out all other particles as well as the reservoir) converges as $N\to \mathrm{\infty }$ to the n-fold tensor product of single-particle density matrices, ${\rho }_{\mathrm{\infty }}(t)\otimes \cdots \otimes {\rho }_{\mathrm{\infty }}(t)$. The limiting single-particle density matrix obeys the quadratic evolution equation

$$i\hspace{0.17em}\frac{\mathrm{d}}{\mathrm{d}t}{\rho }_{\mathrm{\infty }}(t)=[h,{\rho }_{\mathrm{\infty }}(t)]+{\lambda }^{2}F(t)\hspace{0.17em}{\mathrm{Tr}}_2[G\otimes G,{\rho }_{\mathrm{\infty }}(t)\otimes {\rho }_{\mathrm{\infty }}(t)],$$ 2.10

where h and G are the single-particle Hamiltonian and the interaction operator (cf. (1.2)) and F(t) is an explicit complex-valued function. Tr₂ denotes the partial trace over the second factor. We now explain how the findings of [11] relate to those obtained here.

Writing the partial trace in (2.10) as

$${\mathrm{Tr}}_2[G\otimes G,{\rho }_{\mathrm{\infty }}(t)\otimes {\rho }_{\mathrm{\infty }}(t)]=[G,{\rho }_{\mathrm{\infty }}(t)]\hspace{0.17em}\mathrm{Tr}({\rho }_{\mathrm{\infty }}(t)G)$$ 2.11

and using the fact that h and G commute, it follows easily that $(\mathrm{d}/\mathrm{d}t)\hspace{0.17em}\mathrm{Tr}({\rho }_{\mathrm{\infty }}(t)G)=0$ for all t. This conservation law, when used in (2.11) and (2.10), shows that, in fact, the evolution (2.10) takes the form

$$i\hspace{0.17em}\frac{\mathrm{d}}{\mathrm{d}t}{\rho }_{\mathrm{\infty }}(t)=[h_{\mathrm{eff}}(t),{\rho }_{\mathrm{\infty }}(t)]\text{with}\hspace{0.17em}{h}_{\mathrm{eff}}(t)=h+{\lambda }^2{⟨G⟩}_{0}F(t)G,$$ 2.12

where ${⟨G⟩}_0=\mathrm{Tr}({\rho }_{\mathrm{\infty }}(0)G)$ is the average of G in the initial single-particle state. Note that not only is the effective Hamiltonian in (2.12) time dependent, but it depends also on the initial condition (this is the hidden non-linearity). In the setting considered in this paper, due to the vanishing odd moment condition (A0), we have 〈G〉₀=0 and hence h_eff=h. That is, the particles evolve according to the free dynamics, as predicted by lemma 1.4.

(ii). Reservoir dynamics

We are not aware that the reservoir dynamics has been considered before in the literature. Recall from lemma 1.5 that, in the large N limit, any energy-conserving model is equivalent to a single harmonic oscillator interacting with the reservoir. The only quantity of the particles playing a role is μ_S(G²) and we do not have to specify the particles system further. Here, we will solve explicitly the reservoir dynamics given in lemma 1.5, (1.25) for a reservoir of free bosons, where H_R and B are given by (2.4), (2.5). Denote by dE(x), $x\in \mathbb{R}$, the projection-valued spectral measure of φ, so that $\phi ={\int }_{\mathbb{R}}x\hspace{0.17em}\mathrm{d}E(x)$. We have

$${\mathrm{e}}^{\mathrm{i}\hspace{0.17em}t(H_{\mathrm{R}}+2\sqrt{\varkappa }\phi \otimes B)}({\mathrm{𝟙}}_{\mathrm{HO}}\otimes A_{\mathrm{R}})\hspace{0.17em}{\mathrm{e}}^{-\mathrm{i}\hspace{0.17em}t(H_{\mathrm{R}}+2\sqrt{\varkappa }\phi \otimes B)}={\int }_{\mathbb{R}}\mathrm{d}E(x)\otimes {\mathrm{e}}^{\mathrm{i}\hspace{0.17em}t(H_{\mathrm{R}}+\phi (\xi g))}{A}_{\mathrm{R}}{\mathrm{e}}^{-\mathrm{i}\hspace{0.17em}t(H_{\mathrm{R}}+\phi (\xi g))},$$ 2.13

where $\xi =2\sqrt{\varkappa }x$ and, on the right-hand side, φ(ξg)=ξφ(g) is the free Bose field operator smeared out with the form factor ξg(k), $k\in {\mathbb{R}}^3$. For a general form factor $f\in L^2({\mathbb{R}}^3)$ let W(f)=e^iφ(f) be the Weyl operator. Using the standard relations $W(f)H_{\mathrm{R}}W(-f)=H_{\mathrm{R}}-\phi (\mathrm{i}\omega f)+\frac{1}{2}\parallel \sqrt{\omega }f{\parallel }_2^2$ (with ω=|k|) and W(f)φ(g)W(−f)=φ(g)−Im〈f,g〉, we readily verify the following relation, choosing f(k)=ξg(k)/iω:

$$W(f){\mathrm{e}}^{\mathrm{i}\hspace{0.17em}t(H_{\mathrm{R}}+\phi (\xi g))}W(-f)={\mathrm{e}}^{\mathrm{i}\hspace{0.17em}t{H}_{\mathrm{R}}}\hspace{0.17em}{\mathrm{e}}^{\frac{\mathrm{i}}{2}t{\xi }^2\parallel g/\sqrt{\omega }{\parallel }_2^2}.$$ 2.14

(This is the ‘polaron transformation’.) Combining (2.13) and (2.14) yields

$$\begin{array}{rl}& e^{\mathrm{i}\hspace{0.17em}t(H_{\mathrm{R}}+2\sqrt{\varkappa }\phi \otimes B)}({\mathrm{𝟙}}_{\mathrm{HO}}\otimes A_{\mathrm{R}}){\mathrm{e}}^{-\mathrm{i}\hspace{0.17em}t(H_{\mathrm{R}}+2\sqrt{\varkappa }\phi \otimes B)}\\ & ={\int }_{\mathbb{R}}\mathrm{d}E(x)\otimes W(-f){\mathrm{e}}^{\mathrm{i}\hspace{0.17em}t{H}_{\mathrm{R}}}W(f)A_{\mathrm{R}}W(-f){\mathrm{e}}^{-\mathrm{i}\hspace{0.17em}t{H}_{\mathrm{R}}}W(f)\\ & ={\int }_{\mathbb{R}}\mathrm{d}E(x)\otimes W(({\mathrm{e}}^{\mathrm{i}\omega t}-1)f)A_{\mathrm{R}}(t)W(-({\mathrm{e}}^{\mathrm{i}\omega t}-1)f).\end{array}$$ 2.15

In the last step, we have used ${\mathrm{e}}^{\mathrm{i}\hspace{0.17em}t{H}_{\mathrm{R}}}W(f){\mathrm{e}}^{-\mathrm{i}\hspace{0.17em}t{H}_{\mathrm{R}}}=W({\mathrm{e}}^{\mathrm{i}\hspace{0.17em}t\omega }f)$ and the Weyl canonical commutation relations W(f)W(h)=e^{−(i/2) Im〈f,h〉}W(f+h). Taking for A_R=W(h) a general Weyl operator ($h\in L^2({\mathbb{R}}^3)$), the second tensor factor on the right-hand side of (2.15) is ${\mathrm{e}}^{-\mathrm{i}\hspace{0.17em}\mathrm{Im}⟨f,(1-{\mathrm{e}}^{\mathrm{i}\omega t})h⟩}W({\mathrm{e}}^{\mathrm{i}\omega t}h)={\mathrm{e}}^{-2\mathrm{i}\sqrt{\varkappa }x\mathrm{Re}⟨g,((1-{\mathrm{e}}^{\mathrm{i}\omega t})/\omega )h⟩}W({\mathrm{e}}^{\mathrm{i}\omega t}h)$. Using this information in (2.15), we find

$${\mathrm{e}}^{\mathrm{i}\hspace{0.17em}t(H_{\mathrm{R}}+2\sqrt{\varkappa }\phi \otimes B)}({\mathrm{𝟙}}_{\mathrm{HO}}\otimes W(h)){\mathrm{e}}^{-\mathrm{i}\hspace{0.17em}t(H_{\mathrm{R}}+2\sqrt{\varkappa }\phi \otimes B)}={\mathrm{e}}^{-2\mathrm{i}\sqrt{\varkappa }\hspace{0.17em}\mathrm{Re}⟨g,\frac{1-{\mathrm{e}}^{\mathrm{i}\omega t}}{\omega }h⟩\phi }\otimes W({\mathrm{e}}^{\mathrm{i}\omega t}h).$$ 2.16

The average in the harmonic oscillator vacuum state is

$${\mu }_{\mathrm{HO}}\left({\mathrm{e}}^{-2\mathrm{i}\sqrt{\varkappa }\hspace{0.17em}\mathrm{Re}⟨g,((1-{\mathrm{e}}^{\mathrm{i}\omega t})/\omega )h⟩\phi }\right)={\mathrm{e}}^{-\varkappa {(\mathrm{Re}⟨g,\frac{1-{\mathrm{e}}^{\mathrm{i}\omega t}}{\omega }h⟩)}^2}$$

and hence combining (2.16) with (1.25) gives the explicit limiting dynamics for the reservoir,

$$\underset{N\to \mathrm{\infty }}{lim}{\omega }_N({\tau }_{\lambda ,N}^t(W(h)))={\mathrm{e}}^{-{\lambda }^2{\mu }_{\mathrm{S}}(G^2){(\mathrm{Re}⟨g,((1-{\mathrm{e}}^{\mathrm{i}\omega t})/\omega )h⟩)}^2}{\mu }_{\mathrm{R}}(W({\mathrm{e}}^{\mathrm{i}\omega t}h)).$$ 2.17

One can readily carry out this analysis for $A_{\mathrm{R}}={\hat{N}}_{\mathrm{R}}$, the number operator of the reservoir Bose field. Let $\mathcal{N}(t)=\underset{N\to \mathrm{\infty }}{lim}{\omega }_N({\tau }_{\lambda ,N}^t({\hat{N}}_{\mathrm{R}}))$ be the average number of particles, as in §2a. Then one obtains

$$\mathcal{N}(t)=\mathcal{N}(0)+{\lambda }^2{\mu }_{\mathrm{S}}(G^2)\parallel ({\mathrm{e}}^{\mathrm{i}\omega t}-1)g/\omega {\parallel }_2^2.$$ 2.18

This is an exact formula, in that there are no higher than quadratic-order terms in λ in the quantity $\mathcal{N}(t)$. The number of particles is again oscillatory in time, for all values of λ. This indicates that, for an energy-conserving Dicke maser model, there is no superradiant phase transition. This is in contrast with the true (energy-exchanging) Dicke model (see §2a), for which $\mathcal{N}(t)$ is oscillatory in time for λ smaller than a critical value λ_c, while for λ>λ_c, $\mathcal{N}(t)$ increases exponentially in time [1].

3. Proofs

(a). The mechanism

For $A\in {\mathcal{M}}_{\le n}\otimes {\mathcal{M}}_{\mathrm{R}}$ we set $A(t)={\tau }_{0,n}^t(A)$ and we expand in a Dyson series,

$${\omega }_N({\tau }_{\lambda ,N}^t(A))={\omega }_n(A(t))+\sum _{r=1}^{\mathrm{\infty }}{(\mathrm{i}\lambda )}^r{\int }_0^t\mathrm{d}{t}_1\cdots {\int }_0^{t_{r-1}}\mathrm{d}{t}_r{\omega }_N(P_{r,N}(A(t))),$$ 3.1

where, for r≥1,

$$P_{r,N}(A(t))=N^{-r/2}\sum _{j_1,j_2,\dots ,j_r=1}^N[G_{j_r}(t_r)\otimes B_{j_r}(t_r),[\dots ,[G_{j_1}(t_1)\otimes B_{j_1}(t_1),A(t)]\dots ]].$$ 3.2

It is convenient to make a change of variables in the sum (3.2). Given a fixed r-tuple (j₁,…,j_r)∈{1,…,N}^r and a number 1≤k≤N, define 0≤p_k≤r to be the number of js in the tuple which are equal to k. We have p₁+⋯+p_N=r. To a given (j₁,…,j_r) there corresponds a unique (p₁,…,p_N)∈{0,…,r}^N, while a given (p₁,…,p_N) is associated with exactly

$$\left(\begin{array}{c}r\\ p_1,\dots ,p_N\end{array}\right)\equiv \frac{r!}{(p_1)!\cdots (p_N)!}$$ 3.3

distinct (j₁,…,j_r)∈{1,…,N}^r. We denote by ${\mathcal{C}}_r(p_1,\dots ,p_N)$ the set of all $\left(\genfrac{}{}{0em}{}{r}{p_1,\dots ,p_N}\right)$ summands in (3.2) with different values of (j₁,…,j_r) associated with the same (p₁,…,p_N). Using this change of variables in (3.2) results in the expression

$$P_{r,N}(A(t))=N^{-r/2}\sum _{p_1+\cdots +p_N=r}\hspace{0.17em}\sum _{T_t\in \mathcal{C}(p_1,\dots ,p_N)}{T}_t,$$ 3.4

which defines the terms T_t. We split the sum over the p₁,…,p_N to obtain

$$P_{r,N}(A(t))=N^{-r/2}\sum _{p=0}^r\hspace{0.17em}\sum _{p_1+\cdots +p_n=r-p}\hspace{0.17em}\sum _{p_{n+1}+\cdots +p_N=p}\hspace{0.17em}\sum _{T_t\in \mathcal{C}(p_1,\dots ,p_N)}{T}_t.$$ 3.5

Here, p=p_n+1+⋯+p_N∈{0,…r} is the sum of the powers of all factors G in T_t acting on particles with index larger than n. By the vanishing odd moment condition, all terms with odd values of any of the p_n+1,…,p_N give a vanishing contribution to (3.5). Set p_j=2q_j for j=n+1,…,N. Then

$$P_{r,N}(A(t))=N^{-r/2}\sum _{p=0}^r{}^{\prime }\hspace{0.17em}\sum _{p_1+\cdots +p_n=r-p}\hspace{0.17em}\sum _{q_{n+1}+\cdots +q_N=p/2}\hspace{0.17em}\sum _{T_t\in \mathcal{C}(p_1,\dots ,p_n,2q_{n+1},\dots ,2q_N)}{T}_t,$$ 3.6

where the prime ′ indicates that we only sum over even numbers.

To get an idea of the N-dependence of (3.6), we estimate the number of terms,

$$\sum _{T_t\in \mathcal{C}(p_1,\dots ,2q_N)}1=\left(\begin{array}{c}r\\ p_1,\dots ,2q_N\end{array}\right)<\frac{r!}{(p_1)!\cdots (p_n)!}\frac{1}{(p/2)!}\left(\begin{array}{c}\frac{p}{2}\\ q_{n+1},\dots ,q_N\end{array}\right).$$ 3.7

Then, as

$$\sum _{q_{n+1}+\cdots +q_N=p/2}\left(\begin{array}{c}\frac{p}{2}\\ q_{n+1},\dots ,q_N\end{array}\right)={(N-n)}^{p/2}\sim N^{p/2},$$ 3.8

we expect an upper bound

$$|\omega (P_{r,N}(A(t)))|\hspace{0.17em}\lesssim \hspace{0.17em}\sum _{p=0}^r{}^{\prime }\hspace{0.17em}\hspace{0.17em}{N}^{(p-r)/2}\sum _{p_1+\cdots +p_n=r-p}f(p_1,\dots ,p_n,r,p,t),$$ 3.9

where f does not depend on N. For r even, the terms generated in (3.9) are of the orders N^−r/2,N^−r/2+1,…,N⁰, all powers being integers. For r odd, the terms generated in (3.9) are of the orders N^−r/2,N^−r/2+1,…,N^−1/2, all powers being half-integers. Having in mind an expansion of (3.1) in negative powers of N, we can ask which terms will give a contribution to N^−ν. For an integer ν≥0, such a term will be associated with even r only. From (3.9), (p−r)/2=−ν, or p=r−2ν. Similarly, terms with N^−ν−1/2, ν≥0, are associated with odd r in (3.9), such that $(p-r)/2=-\nu -\frac{1}{2}$, i.e. p=r−2ν−1. It follows that the quantities X_ν,N and Y _ν,N, defined in (1.10) and (1.11), give the contributions to the series on the right-hand side of (3.1) of order N^−ν and N^−ν−1/2, respectively.

(b). Proof of theorem 1.1

We first prove (B). The bound |ω_N(T_t)|≤∥A_S∥(2g)^rβ_r(A_R,t) follows directly from the form of T_t as an r-fold multi-commutator (cf. (3.2), (3.4)). Using the equality in (3.7) gives

$$\left|{\int }_0^t\mathrm{d}{t}_1\cdots {\int }_0^{t_{r-1}}\mathrm{d}{t}_r\sum _{T_t\in {\mathcal{C}}_r(p_1,\dots ,p_N)}{\omega }_N(T_t)\right|\le \parallel A_{\mathrm{S}}\parallel \frac{{(2gt)}^r}{r!}{\beta }_r(A_{\mathrm{R}},t)\left(\begin{array}{c}r\\ p_1,\dots ,p_N\end{array}\right).$$ 3.10

Setting p_j=2q_j, for n+1≤j≤N, we get

$$\left(\begin{array}{c}r\\ p_1,\dots ,p_N\end{array}\right)=\frac{r!}{(p_1)!\cdots (p_n)!}\frac{1}{(r/2-\nu )!}\left(\begin{array}{c}\frac{r}{2}-\nu \\ 2q_{n+1},\dots ,2q_N\end{array}\right)$$ 3.11

and the latter multinomial is bounded above by $\left(\genfrac{}{}{0em}{}{r/2-\nu }{q_{n+1},\dots ,q_N}\right)$. We have

$$\sum _{q_{n+1}+\cdots +q_N=r/2-\nu }\left(\begin{array}{c}\frac{r}{2}-\nu \\ q_{n+1},\dots ,q_N\end{array}\right)={(N-n)}^{r/2-\nu }.$$ 3.12

Using (3.10)–(3.12) in (1.10) yields

$$|X_{\nu ,N}(A(t))|\le \parallel A_{\mathrm{S}}\parallel \sum _{r\ge 2\nu \hspace{0.17em}\mathrm{even},r\ne 0}\frac{{(2|\lambda |gt)}^r}{(r/2-\nu )!(2\nu )!}{\beta }_r(A_{\mathrm{R}},t)\sum _{p_1+\cdots +p_n=2\nu }\left(\begin{array}{c}2\nu \\ p_1,\dots ,p_n\end{array}\right).$$ 3.13

The last sum on the right-hand side of (3.13) equals n^2ν (cf. (3.8)). Finally, we make a change of variable s=r/2−ν in (3.13) to arrive at the bound (1.17). The bound (1.18) is obtained in the same way.

Now we prove (A). We write (1.10) in the form $X_{\nu ,N}(A(t))=\sum _{r\ge 2\nu \hspace{0.17em}\mathrm{even},\hspace{0.17em}r\ne 0}{\lambda }^r\hspace{0.17em}{a}_r$ with the obvious identification of the Taylor coefficients a_r. The radius of convergence is given by $R_{\nu }=1/\underset{r}{lim sup}|a_r{|}^{1/r}$. Using the same bounds as in the proof of (A) above, we readily get |a_r|≤∥A_S∥(n^2ν/(2ν)!)((2gt)^r/(r/2−ν)!)β_r(A_R,t), so that

$$\underset{r}{lim sup}|a_r{|}^{1/r}\le 2gt\underset{r}{lim sup}{\left[\frac{{\beta }_r(A_{\mathrm{R}},t)}{(r/2-\nu )!}\right]}^{1/r}=2gt\underset{r}{lim sup}{\left[\frac{{\beta }_r(A_{\mathrm{R}},t)}{(r/2)!}\right]}^{1/r}.$$

Using Stirling’s bound $(r/2)!\ge \sqrt{2\pi }{(r/2)}^{r/2+1/2}{\mathrm{e}}^{-r/2}$ then implies that

$$\underset{r}{lim sup}|a_r{|}^{1/r}\le 2\sqrt{2\mathrm{e}}\hspace{0.17em}gt\underset{r}{lim sup}\frac{{[{\beta }_r(A_{\mathrm{R}},t)]}^{1/r}}{\sqrt{r}}.$$

This shows (1.16). The bound on the radius of convergence for Y _ν,N(A(t)) is obtained in the same way. ▪

(c). Proof of theorem 1.2

The expression for the left-hand side of (1.21) is given by (3.1) and (3.5). We split the sum (3.1) into two, one sum for r even and one for r odd. As indicated before (1.10), for the even part, we make a change of variables, passing from (r,p) to (ν,r), where ν=(r−p)/2. For the sum with r odd, we pass from (r,p) to (ν,r), with ν=(r−p−1)/2. This leads (formally) to the expansion (1.21). If the series $\sum _{\nu \ge 0}\{|X_{\nu ,N}(A(t))|+|Y_{\nu ,N}(A(t))|\}$ converges uniformly in N≥1, then this rearrangement does not affect the value of the series and (1.21) holds. We now show that $\sum _{\nu \ge 0}\{|X_{\nu ,N}(A(t))|+|Y_{\nu ,N}(A(t))|\}$ converges uniformly in N≥1. Consider the bound (1.17). We have

$$\sum _{\nu \ge 0}|X_{\nu ,N}(A(t))|\le \parallel A_{\mathrm{S}}\parallel \sum _{\nu \ge 0}\frac{{(2n|\lambda |gt)}^{2\nu }}{(2\nu )!}{S}_{\nu }(A_{\mathrm{R}},t)$$

and the right-hand side converges if ${(2n|\lambda |gt)}^2\underset{\nu }{lim sup}{[S_{\nu }(A_{\mathrm{R}},t)/(2\nu )!]}^{1/\nu }<1$. From Stirling’s estimate, we have $(2\nu )!\ge \sqrt{2\pi }{(2\nu )}^{2\nu +1/2}{\mathrm{e}}^{-2\nu }$ and hence $\underset{\nu }{lim sup}{[S_{\nu }(A_{\mathrm{R}},t)/(2\nu )!]}^{1/\nu }\le ({\mathrm{e}}^2/4)\underset{\nu }{lim sup}{S}_{\nu }{(A_{\mathrm{R}},t)}^{1/\nu }/{\nu }^2$. Similarly, one sees that $\sum _{\nu \ge 0}|Y_{\nu ,N}(A(t))|$ converges. This shows (1.21). ▪

(d). Proof of theorem 1.3

For fixed r and ν, consider the sum over the p_n+1+⋯+p_N=r−2ν in the general term of the series (1.10) defining X_ν,N(A(t)). We split this sum as

$$\sum _{\genfrac{}{}{0ex}{}{p_{n+1}+\cdots +p_N=r-2\nu }{\mathrm{even}}}=\sum _{\genfrac{}{}{0ex}{}{p_{n+1}+\cdots +p_N=r-2\nu }{\mathrm{even}}}^{\ast }+\sum _{\genfrac{}{}{0ex}{}{p_{n+1}+\cdots +p_N=r-2\nu }{\mathrm{even}}}^{\ast \ast },$$ 3.14

where, in the first sum (*) on the right-hand side, all p_j∈{0,2}, and, in the second one (**), some p_j≥4 (recall that the p_j are even). We now show that only the sum with the single star in (3.14) contributes to the expression (1.10) in the limit $N\to \mathrm{\infty }$. From the relations²

$$\sum _{q_1+\cdots +q_L=T}1=\left(\begin{array}{c}L+T-1\\ T\end{array}\right)\text{and}\sum _{q_1+\cdots +q_L=T}^{\ast }1=\left(\begin{array}{c}L\\ T\end{array}\right),$$ 3.15

where the q_j=0,1,… and the star in (3.15) means that we sum only over values q_j∈{0,1}, we deduce that (L=N−n, T=r/2−ν)

$$\begin{array}{rl}& \sum _{\genfrac{}{}{0ex}{}{p_{n+1}+\cdots +p_N=r-2\nu }{\mathrm{even}}}^{\ast \ast }1=\left(\begin{array}{c}L+T-1\\ T\end{array}\right)-\left(\begin{array}{c}L\\ T\end{array}\right)\\ & =\frac{L^T}{T!}\left\{\left(1+\frac{T-1}{L}\right)\left(1+\frac{T-2}{L}\right)\cdots \left(1+\frac{1}{L}\right)-\left(1-\frac{1}{L}\right)\left(1-\frac{2}{L}\right)\cdots \left(1-\frac{T-1}{L}\right)\right\}\\ & =N^{r/2-\nu }{o}_N,\end{array}$$ 3.16

where $o_N\to 0$ as $N\to \mathrm{\infty }$. The general term in the series (1.10) carries a factor N^ν−r/2 and therefore, for each fixed r, the part of the summand in (1.10) associated with the doubly starred sum (cf. (3.14)) converges to zero as $N\to \mathrm{\infty }$. It follows (from the dominated convergence theorem for the series (1.10)) that

$$\begin{array}{rl}\underset{N\to \mathrm{\infty }}{lim}{X}_{\nu ,N}(A(t))& =\underset{N\to \mathrm{\infty }}{lim}\hspace{0.17em}\sum _{\genfrac{}{}{0ex}{}{r\ge 2\nu ,r\ne 0}{\mathrm{even}}}{(\mathrm{i}\lambda )}^r{N}^{\nu -r/2}\sum _{p_1+\cdots +p_n=2\nu }\hspace{0.17em}\sum _{\genfrac{}{}{0ex}{}{p_{n+1}+\cdots +p_N=r-2\nu }{\mathrm{even}}}^{\ast }\\ & \times {\int }_0^t\mathrm{d}{t}_1\cdots {\int }_0^{t_{r-1}}\mathrm{d}{t}_r\sum _{T_t\in {\mathcal{C}}_r(p_1,\dots ,p_N)}{\omega }_N(T_t).\end{array}$$ 3.17

Owing to the invariance of ω_N(T_t) with respect to permutation of any of the particle indices j=n+1,…,N, the value of ω_N(T_t) is the same for every one of the configurations (p_n+1,…,p_N) in the starred sum of (3.17). (This does not hold for j≤n because the observable A_S acts on the first n particles.) We may thus set p_n+1=⋯=p_n+r/2−ν=2 and p_j=0 for j=n+r/2−ν+1,…,N, and take this term with the multiplicity (N−n r/2−ν) , which is the number of terms in the sum according to (3.15). In other words,

$$\sum _{\genfrac{}{}{0ex}{}{p_{n+1}+\cdots +p_N=r-2\nu }{\mathrm{even}}}^{\ast }\hspace{0.17em}\sum _{T_t\in {\mathcal{C}}_r(p_1,\dots ,p_N)}{\omega }_N(T_t)=\left(\begin{array}{c}N-n\\ \frac{r}{2}-\nu \end{array}\right)\sum _{T_t\in {\mathcal{D}}_r(p_1,\dots ,p_n)}{\omega }_{n+r/2-\nu }(T_t),$$ 3.18

where ${\mathcal{D}}_r(p_1,\dots ,p_n)$ is the set of all r-fold multi-commutators (1.9), where p_j among the indices j₁,…,j_r equal j, for j=1,…,n (under the additional constraint that p₁+⋯+p_n=2ν), and two among the indices j₁,…,j_r equal each one of the values n+1,…,n+r/2−ν. Combining (3.17) and (3.18) gives

$$\begin{array}{rl}& \underset{N\to \mathrm{\infty }}{lim}{X}_{\nu ,N}(A(t))\\ & =\sum _{\genfrac{}{}{0ex}{}{r\ge 2\nu ,r\ne 0}{\mathrm{even}}}\frac{{(\mathrm{i}\lambda )}^r}{(r/2-\nu )!}\sum _{p_1+\cdots +p_n=2\nu }{\int }_0^t\mathrm{d}{t}_1\cdots {\int }_0^{t_{r-1}}\mathrm{d}{t}_r\sum _{T_t\in {\mathcal{D}}_r(p_1,\dots ,p_n)}{\omega }_{n+r/2-\nu }(T_t)\\ & \equiv X_{\nu }(A(t)).\end{array}$$ 3.19

The same argument applies to Y _ν,N(A(t)), (1.11), and yields

$$\begin{array}{rl}& \underset{N\to \mathrm{\infty }}{lim}{Y}_{\nu ,N}(A(t))=\sum _{\genfrac{}{}{0ex}{}{r\ge 2\nu +1}{\mathrm{odd}}}\frac{{(\mathrm{i}\lambda )}^r}{(r/2-\nu -1/2)!}\\ & \times \sum _{p_1+\cdots +p_n=2\nu +1}{\int }_0^t\mathrm{d}{t}_1\cdots {\int }_0^{t_{r-1}}\mathrm{d}{t}_r\sum _{T_t\in {\mathcal{E}}_r(p_1,\dots ,p_n)}{\omega }_{n+r/2-\nu -1/2}(T_t)\\ & \equiv Y_{\nu }(A(t)),\end{array}$$ 3.20

where ${\mathcal{E}}_r(p_1,\dots ,p_n)$ denotes the set of all multi-commutators as in the sum of (3.2) with the following constraint: p_j among the indices j₁,…,j_r equal j, for j=1,…,n (with the constraint p₁+⋯+p_n=2ν+1) and two among the indices j₁,…,j_r equal each of the values $n+1,\dots ,n+r/2-\nu -\frac{1}{2}$.

Using the bound (1.17), one readily sees that $\sum _{\nu =0}^{\mathrm{\infty }}{N}^{-\nu }[X_{\nu ,N}(A(t))-X_{\nu }(A(t))]\to 0$ in the limit $N\to \mathrm{\infty }$. The analogous result holds for Y _ν,N, and (1.23) follows. ▪

(e). Proofs of lemmas

(i). Proof of lemma 1.8

According to Wick’s theorem,

$$|{\mu }_{\mathrm{R}}(B(t_{\sigma (1)})\cdots B(t_{\sigma (r)}))|\le \sum _{\mathrm{pairings}\hspace{0.17em}\sigma }\prod _{j=1}^{r/2}|{\mu }_{\mathrm{R}}(B(t_j)B(t_{\sigma (j)}))|\le \frac{r!}{2^{r/2}(r/2)!}C{(t)}^{r/2},$$ 3.21

where the sum is over r!/2^r/2(r/2)! pairings. For r odd, μ_R(B(t_σ(1))⋯B(t_σ(r)))=0.

We now prove (1). We use the Cauchy–Schwarz inequality for states, $|\mu (XAY)|\le \parallel A\parallel \sqrt{\mu (X{X}^{\ast })\mu (Y^{\ast }Y)}$, to estimate

$$\begin{array}{rl}& |{\mu }_{\mathrm{R}}(B_{{\sigma }^{\prime }(1)}(t_{\sigma (1)})\cdots [\hspace{0.17em}{}_j{A}_{\mathrm{R}}(t)]\cdots B_{{\sigma }^{\prime }(r)}(t_{\sigma (r)}))|\\ & \le \parallel A_{\mathrm{R}}\parallel {[{\mu }_{\mathrm{R}}(B_{{\sigma }^{\prime }(1)}(t_{\sigma (1)})\cdots B_{{\sigma }^{\prime }(j-1)}(t_{\sigma (j-1)})B_{{\sigma }^{\prime }(j-1)}^{\ast }(t_{\sigma (j-1)})\cdots B_{{\sigma }^{\prime }(1)}^{\ast }(t_{\sigma (1)}))]}^{1/2}\\ & \times {[{\mu }_{\mathrm{R}}(B_{{\sigma }^{\prime }(r)}^{\ast }(t_{\sigma (r)})\cdots B_{{\sigma }^{\prime }(j)}^{\ast }(t_{\sigma (j)})B_{{\sigma }^{\prime }(j)}(t_{\sigma (j)})\cdots B_{{\sigma }^{\prime }(r)}(t_{\sigma (r)}))]}^{1/2}.\end{array}$$ 3.22

The right-hand side of (3.22) is estimated using Wick’s theorem (see (3.21)), yielding

$$\begin{array}{rl}& |{\mu }_{\mathrm{R}}(B_{{\sigma }^{\prime }(1)}(t_{\sigma (1)})\cdots [\hspace{0.17em}{}_j{A}_{\mathrm{R}}(t)]\cdots B_{{\sigma }^{\prime }(r)}(t_{\sigma (r)}))|\\ & \le \parallel A_{\mathrm{R}}\parallel {(C(t)/2)}^{r/2}{\left\{\frac{(2(j-1))!}{(j-1)!}\frac{(2(r-j+1))!}{(r-j+1)!}\right\}}^{1/2}.\end{array}$$ 3.23

With the usual Stirling approximations,

$$\sqrt{2\pi }{n}^{n+1/2}{\mathrm{e}}^{-n}\le n!\le \mathrm{e}{n}^{n+1/2}{\mathrm{e}}^{-n},$$ 3.24

valid for all integers n≥1, we obtain

$$\frac{(2n)!}{n!}\le \frac{e}{\sqrt{\pi }}{\left(\frac{4}{e}\right)}^n{n}^n$$ 3.25

and hence

$$\frac{(2(j-1))!}{(j-1)!}\frac{(2(r-j+1))!}{(r-j+1)!}\le \frac{{\mathrm{e}}^2}{\pi }{\left(\frac{4}{\mathrm{e}}\right)}^r{\ell }^{\ell }{(r-\ell )}^{r-\ell },$$ 3.26

where ℓ=j−1. The function ℓ↦ℓ^ℓ(r−ℓ)^r−ℓ=(ℓ/(r−ℓ))^ℓ(r−ℓ)^r is readily seen to be maximal at ℓ=r/2 (use simple calculus), where this function takes the value (r/2)^r. Combining the bound ℓ^ℓ(r−ℓ)^r−ℓ≤(r/2)^r with (3.26) and (3.23) yields the upper bound on β_r(A_R,t) given in (1) of lemma 1.8. The upper bound on b(A_R,t) is then immediate from the definition (1.15).

The proof of (2) is obtained in the same way as (1). Indeed, as μ_R has at most n₀ particles, and each B can produce at most one particle, we may use the bound (3.22) with ∥A∥ replaced by $\parallel A\hspace{0.17em}P(\hat{N}\le n_0+r/2)\parallel$.

Next we prove statement (3). We simply have to estimate μ_R applied to a product of r+k field operators. Wick’s theorem gives the bound (3.21),

$${\beta }_r(A_{\mathrm{R}},t)\le \frac{1}{2^{(r+k)/2}}\frac{(r+k)!}{((r+k)/2)!}C{(A_{\mathrm{R}},t)}^{(r+k)/2}.$$ 3.27

Using the bound (3.25) with n=r+k yields the upper bound on β_r(A_S,t) in (2) of the lemma. The upper bound on b(A_R,t) is then immediate from the definition (1.15). ▪

(ii). Proof of lemma 1.9

As the bounds on β_r(A_S,t) in (1), (2) and (3) of lemma 1.8 have the same form, it suffices to give the proof in the case (3), for k even. We obtain (recall (1.19))

$$\begin{array}{rl}{S}_{\nu }(A_{\mathrm{R}},t)& \le \frac{\mathrm{e}}{\sqrt{\pi }}{\left(\frac{C}{\mathrm{e}}\right)}^{\nu +k/2}{(2\nu +k)}^{\nu +k/2}+\frac{\mathrm{e}}{\sqrt{\pi }}{\left(\frac{C}{\mathrm{e}}\right)}^{\nu +k/2}\sum _{s\ge 1}\frac{{(C{\varkappa }^2/\mathrm{e})}^s}{s!}{(2(\nu +s)+k)}^{\nu +s+k/2}\\ & \le \frac{\mathrm{e}}{\sqrt{\pi }}{\left(\frac{C}{\mathrm{e}}\right)}^{\nu +k/2}{(2\nu +k)}^{\nu +k/2}+\frac{\mathrm{e}}{\sqrt{2}\pi }{\left(\frac{C}{\mathrm{e}}\right)}^{\nu +k/2}\sum _{s\ge 1}{(C{\varkappa }^2)}^s{\left(\frac{2+(2\nu +k)}{s}\right)}^s\\ & \times {(2(\nu +s)+k)}^{\nu +k/2},\end{array}$$ 3.28

where $\varkappa =2|\lambda |gt$, C=C(A_R,t), and, in the second step, we used Stirling’s bound (3.24) for s!, s≥1. Using (2+(2ν+k)/s)^s=2^s(1+(ν+k/2)/s)^s≤2^se^ν+k/2 in (3.28) yields

$$\begin{array}{rl}{S}_{\nu }(A_{\mathrm{R}},t)& \le \frac{\mathrm{e}}{\sqrt{\pi }}{\left(\frac{C}{\mathrm{e}}\right)}^{\nu +k/2}{(2\nu +k)}^{\nu +k/2}+\frac{\mathrm{e}}{\sqrt{2}\pi }{(2C)}^{\nu +k/2}\sum _{s\ge 1}{(2C{\varkappa }^2)}^s{\left(s+\nu +\frac{k}{2}\right)}^{\nu +k/2}\\ & \le \frac{\mathrm{e}}{\sqrt{\pi }}{\left(\frac{C}{\mathrm{e}}\right)}^{\nu +k/2}{(2\nu +k)}^{\nu +k/2}+\frac{\mathrm{e}}{\sqrt{2}\hspace{0.17em}\pi }{(2C)}^{\nu +k/2}{(2C{\varkappa }^2)}^{-\nu -k/2}\sum _{s\ge \nu +k/2}{(2C{\varkappa }^2)}^s{s}^{\nu +k/2}.\end{array}$$ 3.29

To estimate the last series, we use the equality $s^{\nu +k/2}={\mathrm{\partial }}_{\alpha }^{\nu +k/2}{|}_{\alpha =0}\hspace{0.17em}{\mathrm{e}}^{\alpha s}$ to obtain

$$\sum _{s\ge \nu +k/2}{(2C{\varkappa }^2)}^s{s}^{\nu +k/2}\le \sum _{s\ge 0}{(2C{\varkappa }^2)}^s{s}^{\nu +k/2}={\mathrm{\partial }}_{\alpha }^{\nu +k/2}{|}_{\alpha =0}{(1-2e^{\alpha }C{\varkappa }^2)}^{-1},$$ 3.30

which holds provided $2C{\varkappa }^2<1$. Combining the bound

$$|{\mathrm{\partial }}_{\alpha }^m{|}_{\alpha =0}{(1-2e^{\alpha }C{\varkappa }^2)}^{-1}|\le 2^{m}m!\underset{1\le j\le m}{max}{(1-2C{\varkappa }^2)}^{-j}={\left(\frac{2}{1-2C{\varkappa }^2}\right)}^{m}m!$$ 3.31

(for m=ν+k/2) with (3.29) and (3.30), we arrive at

$$S_{\nu }(A_{\mathrm{R}},t)\le \frac{\mathrm{e}}{\sqrt{\pi }}{\left(\frac{C}{\mathrm{e}}\right)}^{\nu +k/2}{(2\nu +k)}^{\nu +k/2}+\frac{\mathrm{e}}{\sqrt{2}\pi }{(2C)}^{\nu +k/2}{(2C{\varkappa }^2[1-2C{\varkappa }^2])}^{-\nu -k/2}\left(\nu +\frac{k}{2}\right)!.$$

Therefore, (S_ν(A_R,t))^1/ν grows at most as ((ν+k/2)!)^1/ν∼ν for large values of ν. The result of lemma 1.9 follows. ▪

(iii). Proof of lemma 1.5

As G_j(t)=G_j for all times, the commutator terms T_t in (3.19) simply equal $T_t=G_{n+1}^2\cdots G_{n+r/2}^2[B(t_r),[\cdots ,[B(t_1),A_{\mathrm{R}}(t)]\cdots ]]$, and hence

$$\begin{array}{rl}{X}_0(A_{\mathrm{R}})& =\sum _{q\ge 1}\frac{{(-\varkappa )}^q}{q!}{\int }_0^t\mathrm{d}{t}_1\cdots {\int }_0^{t_{2q-1}}\mathrm{d}{t}_{2q}\hspace{0.17em}{\mu }_{\mathrm{R}}([B(t_{2q}),[\cdots ,[B(t_1),A_{\mathrm{R}}]\cdots ]])\\ & =\sum _{r\ge 2\hspace{0.17em}\mathrm{even}}\frac{{(-\varkappa )}^{r/2}}{(r/2)!}{\int }_0^t\mathrm{d}{t}_1\cdots {\int }_0^{t_{r-1}}\mathrm{d}{t}_r{\mu }_{\mathrm{R}}([B(t_r),[\cdots ,[B(t_1),A_{\mathrm{R}}]\cdots ]]).\end{array}$$ 3.32

Next we introduce an ‘ancilla’ harmonic oscillator with creation and annihilation operator a* and a, satisfying $[a,a^{\ast }]=\mathrm{𝟙}$, acting on the Hilbert space ${\mathcal{H}}_{\mathrm{HO}}$. Let $\phi =(1/\sqrt{2})(a^{\ast }+a)$ be the harmonic oscillator field operator and consider the multi-commutator acting on the Hilbert space ${\mathcal{H}}_{\mathrm{HO}}\otimes {\mathcal{H}}_{\mathrm{R}}$, r≥1,

$$[\phi \otimes B(t_r),[\cdots ,[\phi \otimes B(t_1),{\mathrm{𝟙}}_{\mathrm{HO}}\otimes A_{\mathrm{R}}]\cdots ]]={\phi }^r\otimes [B(t_r),[\cdots ,[B(t_1),A_{\mathrm{R}}]\cdots ]].$$ 3.33

The vacuum state μ_HO=〈Ω_HO,⋅ Ω_HO〉 satisfies ${\mu }_{\mathrm{HO}}({\phi }^2)=\frac{1}{2}$ and, by Wick’s theorem,

$${\mu }_{\mathrm{HO}}({\phi }^r)=\left\{\begin{array}{ll}\frac{r!}{2^r(r/2)!}& r\text{ even}\\ 0& r\text{ odd}.\end{array}\right.$$ 3.34

As the odd moments of φ in μ_HO vanish, and 1/(r/2)!=(2^r/r!)μ_HO(φ^r) for even r, we obtain from (3.32) and (3.33)

$$\begin{array}{rl}{X}_0(A_{\mathrm{R}})& =\sum _{r\ge 1}\frac{{(2i\sqrt{\varkappa })}^r}{r!}{\int }_0^t\mathrm{d}{t}_1\cdots {\int }_0^{t_{r-1}}\mathrm{d}{t}_r\\ & {\mu }_{\mathrm{HO}}\otimes {\mu }_{\mathrm{R}}([\phi \otimes B(t_r),[\cdots ,[\phi \otimes B(t_1),{\mathrm{𝟙}}_{\mathrm{HO}}\otimes A_{\mathrm{R}}]\cdots ]]).\end{array}$$ 3.35

The series on the right-hand side (3.35) is readily identified as a Dyson series, namely

$$X_0(A_{\mathrm{R}})={\mu }_{\mathrm{HO}}\otimes {\mu }_{\mathrm{R}}({\mathrm{e}}^{\mathrm{i}\hspace{0.17em}t(H_{\mathrm{R}}+2\sqrt{\varkappa }\phi \otimes B)}{\mathrm{e}}^{-\mathrm{i}\hspace{0.17em}t{H}_{\mathrm{R}}}({\mathrm{𝟙}}_{\mathrm{HO}}\otimes A_{\mathrm{R}}){\mathrm{e}}^{\mathrm{i}\hspace{0.17em}t{H}_{\mathrm{R}}}{\mathrm{e}}^{-\mathrm{i}\hspace{0.17em}t(H_{\mathrm{R}}+2\sqrt{\varkappa }\phi \otimes B)})-{\mu }_{\mathrm{R}}(A_{\mathrm{R}}).$$ 3.36

Combining this with (1.24) yields

$${\omega }_N({\tau }_{\lambda ,N}^t(A_{\mathrm{R}}))={\mu }_{\mathrm{HO}}\otimes {\mu }_{\mathrm{R}}({\mathrm{e}}^{\mathrm{i}\hspace{0.17em}t(H_{\mathrm{R}}+2\sqrt{\varkappa }\phi \otimes B)}({\mathrm{𝟙}}_{\mathrm{HO}}\otimes A_{\mathrm{R}}){\mathrm{e}}^{-\mathrm{i}\hspace{0.17em}t(H_{\mathrm{R}}+2\sqrt{\varkappa }\phi \otimes B)})+o_N.$$ 3.37

This proves lemma 1.5. ▪

(iv). Proof of lemma 1.7

An application of theorem 1.3 yields

$${\omega }_N(F_N(A,t))=\frac{1}{N}\sum _{n=1}^N{Y}_0(A_n(t))+o(N),$$ 3.38

where $o(N)\to 0$ as $N\to \mathrm{\infty }$ and Y ₀ is given by (3.20). Consider Y ₀(A_n(t)) for a fixed n. For ν=0, the sum over p₁,…,p_n in (3.20) has only terms where exactly one of the p_j equals 1 and all others vanish. As the observable is $A_n(t)\in {\mathcal{M}}_n$, only p_n=1 contributes (all other terms vanish, as for them we have T_t=0). This forces p₁=⋯=p_n−1=0, p_n=1 in (3.20). The relation (1.29) then follows from (3.20) by using that the system is symmetric. ▪

(v). Proof of lemma 1.6

Consider the lowest order of X_0,N in λ given by r=2 in (1.10). For this term, we have p₁=⋯=p_n=0 and p_n+1+⋯+p_N=2. Owing to the vanishing odd moment condition, the last constraint implies that exactly one of the p_j, for a single j∈{n+1,…,N}, equals 2 and all other p_j are zero. Therefore, the term with r=0 in (1.10) equals

$$-{\lambda }^2\sum _{j=n+1}^N{\int }_0^t\mathrm{d}{t}_1{\int }_0^{t_1}\mathrm{d}{t}_2{\omega }_N([G_j(t_2)\otimes B_j(t_2),[G_j(t_1)\otimes B_j(t_1),A(t)]]).$$ 3.39

Taking into account that A(t)=A_S(t)⊗A_R(t) with $A_{\mathrm{S}}(t)\in {\mathcal{M}}_{\le n}$ commuting with G_j for j≥n+1, we expand the double commutator in (3.39) to obtain

$$\begin{array}{rl}& {\omega }_N([G_j(t_2)\otimes B_j(t_2),[G_j(t_1)\otimes B_j(t_1),A(t)]])\\ & =2{\mu }_{\le n}(A_{\mathrm{S}}(t))\hspace{0.17em}\mathrm{Re}\hspace{0.17em}{\mu }_j(G_j(t_2)G_j(t_1))\hspace{0.17em}{\mu }_{\mathrm{R}}(B_j(t_2)[B_j(t_1),A_{\mathrm{R}}(t)]).\end{array}$$ 3.40

This yields the term with the double integral on the right-hand side of (3.41). The terms with r>2 in (1.10) give an error O(λ⁴), uniformly in N.

Next take the term r=1 in Y _0,N(A(t)), (1.11). The constraint on the p₁,…,p_n is that exactly one of them equals 1; all others vanish. Moreover, p_n+1=⋯=p_N=0. Thus the term with r=1 is

$$\mathrm{i}\lambda \sum _{j=1}^n{\int }_0^t\mathrm{d}s{\omega }_N([G_j(s)\otimes B_j(s),A(t)]).$$ 3.41

The integrand, μ_≤n(G_j(s)A_S(t))μ_R([B_j(s),A_R(t)])+μ_≤n([G_j(s),A_S(t)])μ_R(A_R(t)B_j(s)), is readily seen to become that of the integral $\propto \lambda /\sqrt{N}$ on the right-hand side of (1.26). The remaining terms in the series (1.11), for r>1, give an error O(λ³).

Finally, the remaining series of all terms with ν>0 in (1.21) adds up to an error O(1/N). ▪

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Authors' contributions

M.M. conceived the mathematical model and both M.M. and A.R. worked on obtaining the results and crafting the proofs. Both authors were involved in the writing of the paper and both authors gave their final approval for publication.

Competing interests

We have no competing interests.

Funding

Both authors were supported by a Discovery Grant (PI MM) from the Natural Sciences and Engineering Research Council of Canada (NSERC).