Minimal Photon Velocity Bounds in Non-relativistic Quantum Electrodynamics

Abstract

We consider non-relativistic quantum particle systems, such as atoms and molecules, coupled to the quantized electromagnetic field. We prove several photon velocity bounds for total energies below the ionization threshold. We also consider phonons coupled to such particle systems and prove velocity bounds for them as well.

Appendices

Appendix A: Photon # and Low Momentum Estimate

For simplicity, consider hamiltonians of the form (1.4)–(1.5), with the coupling operators g(k) satisfying (1.6) and (1.7) with μ>−1/2. The extension to hamiltonians of the form (1.23)–(1.24) is done along the lines of Sect. 4. Recall the notations 〈A〉 _ψ =〈ψ,Aψ〉, N _ρ =dΓ(ω ^−ρ) and $\varUpsilon_{\rho}= \{\psi_{0}\in f (H) D( N_{\rho}^{1/2}), \mbox{ for some } f \in\mathrm {C}_{0}^{\infty}( (-\infty, \varSigma) )\}$. The idea of the proof of the following estimate follows [30] and [9].

Proposition A.1

Let ρ∈[−1,1]. For any ψ ₀∈ϒ _ρ ,

$$\begin{aligned} \langle N_\rho\rangle_{\psi_t} \lesssim t^{\nu_\rho} \| \psi_0 \|_\rho ^2,\quad \nu_\rho= \frac{1+\rho}{2+\mu}. \end{aligned}$$

(A.1)

Proof

Decompose N _ρ =K ₁+K ₂, where

$$\begin{aligned} K_1 := \mathrm{d}\varGamma\bigl(\omega^{-\rho} \chi_{t^{\alpha}\omega\leq1}\bigr) \quad \mbox{and} \quad K_2 := \mathrm{d}\varGamma\bigl(\omega^{-\rho}\chi_{t^{\alpha}\omega\geq1}\bigr). \end{aligned}$$

Then, by (1.19),

$$\begin{aligned} \langle K_2\rangle_\psi\leq\bigl\langle \mathrm{d}\varGamma\bigl(t^{\alpha(1+\rho)} \omega \chi_{t^{\alpha}\omega\geq1}\bigr) \bigr\rangle _{\psi_t} \le t^{\alpha(1+\rho )} \langle H_f \rangle_{\psi_t}\lesssim t^{\alpha(1+\rho)} \|\psi_0\|. \end{aligned}$$

(A.2)

On the other hand, we have by (B.10),

$$\begin{aligned} D K_1 = \mathrm{d}\varGamma\bigl(\alpha \omega^{1+\rho} t^{\alpha-1} \chi'_{t^{\alpha }\omega\leq1}\bigr) - I \bigl( i \omega^{-\rho}\chi_{t^{\alpha}\omega\leq1} g\bigr). \end{aligned}$$

(A.3)

Since $\| \eta_{1} g(k) \|_{\mathcal {H}_{p}} \lesssim|k|^{\mu}\langle k \rangle ^{-2-\mu}$ (see (1.6)), we obtain

$$\begin{aligned} \int d k \, \omega(k)^{-2\rho} \chi_{t^\alpha\omega\leq1} \bigl\| g(k) \bigr\| _{\mathcal {L}(\mathcal {H}_{p})}^2 \bigl(\omega(k)^{-1}+1\bigr) \lesssim t^{-2 (1+\mu-\rho) \alpha}. \end{aligned}$$

(A.4)

This together with (B.11) and (1.19) gives

$$\begin{aligned} \bigl|\bigl\langle I \bigl( i \omega^{-\rho}\chi_{t^{\alpha} \omega\leq1} g \bigr)\bigr\rangle _{\psi_t}\bigr| \lesssim t^{- (1+\mu-\rho) \alpha}\| \psi_0 \|^2. \end{aligned}$$

(A.5)

Hence, by (A.3), since $\partial_{t} \langle K_{1}\rangle_{\psi_{t}} = \langle DK_{1}\rangle_{\psi_{t}}$, $\chi_{t^{\alpha}\omega\leq1}' \leq 0$, we obtain

$$\begin{aligned} \partial_t \langle K_1\rangle_{\psi_t} \lesssim t^{-(1+\mu-\rho) \alpha } \| \psi_0 \|^2, \end{aligned}$$

and therefore

$$\begin{aligned} \langle K_1\rangle_{\psi_t} \leq C t^{\nu'} \| \psi_0 \|^2 + \langle N_\rho\rangle_{\psi_0}, \end{aligned}$$

(A.6)

where ν′=1−(1+μ−ρ)α, if (1+μ−ρ)α<1 and ν′=0, if (1+μ−ρ)α>1. Estimates (A.6) and (A.2) with $\alpha= \frac{1}{2+\mu}$, if ρ>−1, give (A.1). The case ρ=−1 follows from (1.19). □

Remark

A minor modification of the proof above give the following bound for ρ>0 and $\nu_{\rho}':={\frac{\rho}{\frac{3}{2}+\mu}}$,

$$\begin{aligned} \langle N_\rho\rangle_{\psi_t} \lesssim t^{ \nu_\rho'} \bigl( \| \psi _t \|_N^2 + \| \psi_0 \|^2 \bigr) + \langle N_\rho \rangle_{\psi_0} . \end{aligned}$$

(A.7)

Corollary A.2

For any ψ ₀∈ϒ _ρ , γ≥0 and c>0,

$$\begin{aligned} \| \chi_{N_\rho\geq c t^\gamma} \psi_t \|^2 \lesssim t^{-\frac{\gamma }{2} + \frac{1+\rho}{2(2+\mu)}}\| \psi_0 \|^2 +t^{-\frac{\gamma }{2}}\langle N_\rho\rangle_{\psi_0}. \end{aligned}$$

(A.8)

Proof

We have

$$\begin{aligned} \| \chi_{N_\rho\geq c t^\gamma} \psi_t \|\ \leq c^{-\frac{\gamma}{2}} t^{-\frac{\gamma}{2}} \bigl\| \chi_{N_\rho\geq c t^\gamma} K_\rho^{\frac {1}{2}} \psi_t \bigr\| \leq c^{-\frac{\gamma}{2}} t^{-\frac{\gamma}{2}} \bigl\| N_\rho^{\frac{1}{2}} \psi_t \bigr\| \end{aligned}$$

Now applying (A.1) we arrive at (A.8). □

Corollary A.3

Let ψ ₀∈ϒ ₁. Then ψ ₀∈D(N) and

$$\begin{aligned} \bigl\langle N^2 \bigr\rangle _{\psi_t} \lesssim t^{ \frac{2}{2+\mu} } \| \psi_0 \|_{1}^2. \end{aligned}$$

(A.9)

Proof

By the Cauchy-Schwarz inequality, we have N ²≤dΓ(ω)dΓ(ω ⁻¹)=H _f N ₁, and hence

$$\begin{aligned} \bigl\langle N^2 \bigr\rangle _{\psi_t} \le&\bigl\langle N_1^{\frac{1}{2}} H_f N_1^{\frac{1}{2}} \bigr\rangle _{\psi_t} \\ =& \bigl\langle N_1^{\frac{1}{2}} H_f (H - E_{\mathrm{gs}} + 1 )^{-1} N_1^{\frac{1}{2}} ( H - E_{\mathrm{gs}} + 1 ) \bigr\rangle _{\psi_t} \\ &{}+ \bigl\langle N_1^{\frac{1}{2}} H_f \bigl[ N_1^{\frac{1}{2}} , (H- E_{\mathrm{gs}} + 1)^{-1} \bigr] ( H- E_{\mathrm{gs}} + 1 ) \bigr\rangle _{\psi_t} . \end{aligned}$$

Under the assumption (1.6) with μ>0, one verifies that $H_{f} [ N_{1}^{\frac{1}{2}} , (H- E_{\mathrm{gs}} + 1)^{-1} ]$ is bounded. Since H _f (H−E _gs+1)⁻¹ is also bounded, we obtain

$$\begin{aligned} \bigl\langle N^2 \bigr\rangle _{\psi_t} \lesssim\bigl\| N_1^{\frac{1}{2}} \psi_t \bigr\| \bigl( \bigl\| N_1^{\frac{1}{2}} ( H- E_{\mathrm{gs}} + 1) \psi_t \bigr\| + \bigl\| ( H- E_{\mathrm{gs}} + 1) \psi_t \bigr\| \bigr). \end{aligned}$$

(A.10)

Applying Proposition A.1 gives

$$\begin{aligned} \bigl\| N_1^{\frac{1}{2}} \psi_t \bigr\| \lesssim t^{\frac{1}{2+\mu}} \| \psi_0 \| + \bigl\| N_1^{\frac{1}{2}} \psi_0 \bigr\| , \end{aligned}$$

(A.11)

and

$$\begin{aligned} \bigl\| N_1^{\frac{1}{2}} ( H- E_{\mathrm{gs}} + 1) \psi_t \bigr\| \lesssim& t^{\frac {1}{2+\mu}} \| \psi_0 \| + \bigl\| N_1^{\frac{1}{2}} ( H- E_{\mathrm{gs}} + 1) \psi_0 \bigr\| \\ \lesssim& t^{\frac{1}{2+\mu}} \| \psi_0 \| + \bigl\| N_1^{\frac{1}{2}} \psi_0 \bigr\| , \end{aligned}$$

(A.12)

where we used in the last inequality that $N_{1}^{\frac{1}{2}} \tilde{f}(H) (N_{1}+\mathbf {1})^{-\frac{1}{2}}$ is bounded for any $\tilde{f} \in\mathrm {C}_{0}^{\infty}( \mathbb{R}^{3} )$. Combining (A.10), (A.11) and (A.12), we obtain

$$\begin{aligned} \bigl\langle N^2 \bigr\rangle _{\psi_t} \lesssim t^{\frac{2}{2+\mu}} \bigl( \bigl\| N_1^{\frac{1}{2}} \psi_0 \bigr\| ^2 + \| \psi_0 \|^2 \bigr). \end{aligned}$$

(A.13)

Hence (A.9) is proven. □

Appendix B: Method of Propagation Observables

Many steps of our proof the minimal velocity estimates use the method of propagation observables which we formalize in what follows. Let ψ _t =e ^−itH ψ ₀, where H is a hamiltonian of the form (1.4)–(1.5), with the coupling operator g(k) satisfying (1.6). The method reduces propagation estimates for our system say of the form

$$\begin{aligned} \int_0^\infty dt \, \bigl\| G_t^{\frac{1}{2}}\psi_t \bigr\| ^2 \lesssim\|\psi _0\|^2_\#, \end{aligned}$$

(B.1)

for some norm ∥⋅∥_#≥∥⋅∥, to differential inequalities for certain families ϕ _t of positive, one-photon operators on the one-photon space $L^{2}(\mathbb {R}^{3})$. Let

$$\begin{aligned} d \phi_t:= \partial_t \phi_t + i [ \omega , \phi_t ] , \end{aligned}$$

and let ν _ρ ≥0 be determined by the estimate (1.17). We isolate the following useful class of families of positive, one-photon operators:

Definition B.1

A family of positive operators ϕ _t on $L^{2}(\mathbb {R}^{3})$ will be called a one-photon weak propagation observable, if it has the following properties

• there are δ≥0 and a family p _t of non-negative operators, such that

  $$\begin{aligned} \bigl\| \omega^{\delta/2} \phi_t \omega^{ \delta/2}\bigr\| \lesssim\langle t \rangle^{-\nu_\delta} \quad \mbox{and} \quad d \phi_t \ge p_t + \sum _{\textrm{finite}}\mathrm{rem}_i , \end{aligned}$$

  (B.2)

  where rem _i are one-photon operators satisfying

  $$\begin{aligned} \bigl\| \omega^{ \rho_i/2} \, \mathrm{rem}_i \, \omega^{ \rho_i/2}\bigr\| \lesssim\langle t \rangle^{-{\lambda }_i} , \end{aligned}$$

  (B.3)

  for some ρ _i and λ _i , s.t. ${\lambda }_{i} > 1+\nu_{\rho_{i}}$,

• for some λ′>1+ν _δ and with η ₁, η ₂ satisfying (1.7),

  $$\begin{aligned} \biggl( \int\bigl\| \eta_1 \eta_2^2 ( \phi_t g ) (k) \bigr\| _{\mathcal {L}(\mathcal {H}_{p})}^2 \omega (k)^{\delta} dk \biggr)^{\frac{1}{2}} \lesssim\langle t \rangle^{-{\lambda }'}. \end{aligned}$$

  (B.4)

  (Here ϕ _t acts on g as a function of k.)

Similarly, a family of operators ϕ _t on $L^{2}(\mathbb {R}^{3})$ will be called a one-photon strong propagation observable, if

$$\begin{aligned} d \phi_t \le-p_t + \sum _{\textrm{finite}}\mathrm{rem}_i , \end{aligned}$$

(B.5)

with p _t ≥0, rem _i are one-photon operators satisfying (B.3) for some ${\lambda }_{i} > 1+\nu_{\rho_{i}}$, and (B.4) holds for some λ′>1+ν _δ .

Recall the notations N _ρ =dΓ(ω ^−ρ) and

$$\begin{aligned} \varUpsilon_\rho= \bigl\{ \psi_0\in f (H) D\bigl( N_\rho^{\frac{1}{2}}\bigr), \mbox{ for some } f \in \mathrm{C}_0^\infty\bigl( (-\infty, \varSigma) \bigr) \bigr\} . \end{aligned}$$

(B.6)

Notice that, since N ₋₁ f(H)=H _f f(H) is bounded, one easily verifies that ϒ _ρ ⊂ϒ _ρ′ for ρ≥ρ′≥−1. The following proposition reduces proving inequalities of the type of (B.1) to showing that ϕ _t is a one-photon weak or strong propagation observable, i.e. to one-photon estimates of dϕ _t and ϕ _t g.

Proposition B.2

If ϕ _t is a one-photon weak (resp. strong) propagation observable, then we have either the weak propagation estimate, (B.1), or the strong propagation estimate,

$$\begin{aligned} \langle \psi_t, \varPhi_{t} \psi_t\rangle +\int_0^\infty dt \, \bigl\| G_t^{\frac{1}{2}}\psi_t \bigr\| ^2 \lesssim\|\psi_0\|^2_\# , \end{aligned}$$

(B.7)

with the norm $\|\psi_{0}\|^{2}_{\#}:=\|\psi_{0}\|^{2}_{\diamondsuit}+ \|\psi_{0}\| ^{2}_{*}$, where Φ _t :=dΓ(ϕ _t ), G _t :=dΓ(p _t ), ∥ψ ₀∥_∗:=∥ψ ₀∥ _δ and $\|\psi_{0}\|_{\diamondsuit}:= \sum\|\psi_{0}\| _{\rho_{i}}$, on the subspace $\varUpsilon_{\max(\delta, \rho_{i})}$.

Before proceeding to the proof we present some useful definitions. Consider families Φ _t of operators on $\mathcal {H}$ and introduce the Heisenberg derivative

$$\begin{aligned} D \varPhi_t:= \partial_t \varPhi_t + i [ H, \varPhi_t ] , \end{aligned}$$

with the property

$$\begin{aligned} \partial_t \langle \psi_t, \varPhi_t \psi_t \rangle =\langle \psi_t, D \varPhi_t \psi_t \rangle . \end{aligned}$$

(B.8)

Definition B.3

A family of self-adjoint operators Φ _t on a subspace $\mathcal {H}_{1}\subset {\mathcal {H}}$ will be called a (second quantized) weak propagation observable, if for all $\psi_{0}\in \mathcal {H}_{1}$, it has the following properties

• $\sup_{t} \langle \psi_{t}, \varPhi_{t} \psi_{t}\rangle \lesssim\|\psi_{0}\|_{*}^{2}$;

• DΦ _t ≥G _t +Rem, where G _t ≥0 and $\int_{0}^{\infty}dt \, | \langle\psi_{t} , \mathrm{Rem} \, \psi_{t} \rangle| \lesssim\|\psi_{0}\|^{2}_{\diamondsuit}$,

for some norms ∥ψ ₀∥_∗, ∥⋅∥_♢≥∥⋅∥. Similarly, a family of self-adjoint operators Φ _t will be called a strong propagation observable, if it has the following properties

• Φ _t is a family of non-negative operators;

• DΦ _t ≤−G _t +Rem, where G _t ≥0 and $\int_{0}^{\infty}dt \, | \langle\psi_{t} , \mathrm{Rem} \, \psi_{t} \rangle | \lesssim\|\psi_{0}\|^{2}_{\#}$,

for some norm ∥⋅∥_#≥∥⋅∥.

If Φ _t is a weak propagation observable, then integrating the corresponding differential inequality sandwiched by ψ _t ’s and using the estimate on 〈ψ _t ,Φ _t ψ _t 〉 and on the remainder Rem, we obtain the (weak propagation) estimate (B.1), with $\|\psi_{0}\| ^{2}_{\#}:=\|\psi_{0}\|^{2}_{\diamondsuit}+ \|\psi_{0}\|^{2}_{*}$. If Φ _t is a strong propagation observable, then the same procedure leads to the (strong propagation) estimate (B.7).

Proof

Proof of Proposition B.2. Let Φ _t :=dΓ(ϕ _t ). To prove the above statement we use the relations (see Supplement I)

$$\begin{aligned} D_0 \mathrm{d}{\varGamma }(\phi_t)= \mathrm{d} {\varGamma }(d \phi_t) ,\qquad i\bigl[I(g), \mathrm{d}{\varGamma }(\phi_t) \bigr]= - I( i \phi_t g), \end{aligned}$$

(B.9)

where D ₀ is the free Heisenberg derivative,

$$\begin{aligned} D_0 \varPhi_t:= \partial_t \varPhi_t + i [ H_0, \varPhi_t ], \end{aligned}$$

valid for any family of one-particle operators ϕ _t , to compute

$$\begin{aligned} D\varPhi_{t} = \mathrm{d} {\varGamma} (d \phi_t) - I ( i \phi_t g). \end{aligned}$$

(B.10)

Denote 〈A〉 _ψ :=〈ψ,Aψ〉. Applying the Cauchy-Schwarz inequality, we find the following version of a standard estimate

$$\begin{aligned} \bigl|\bigl\langle I (g)\bigr\rangle _{\psi} \bigr| \leq2 \biggl( \int\bigl\| \eta_1 \eta_2^2 g(k) \bigr\| _{\mathcal {L}(\mathcal {H}_{p})}^2 \omega(k)^{\delta} d^3 k \biggr)^{\frac{1}{2}}\bigl\| \eta_1^{-1} \eta_2^{-2} \psi\bigr\| \| \psi\|_\delta. \end{aligned}$$

(B.11)

Using that ψ _t =f ₁(H)ψ _t , with $f_{1} \in\mathrm{C}_{0}^{\infty}( (-\infty,\varSigma)),\ f_{1} f=f$, and using (1.7), we find $\| \eta_{1}^{-1} \eta_{2}^{-2} \psi_{t} \|$ ≲∥ψ _t ∥. Taking this into account, we see that the equations (B.11), (B.4) and (1.19) yield

$$\begin{aligned} \bigl|\bigl\langle I (i \phi_t g)\bigr\rangle _{\psi_t}\bigr| \lesssim\langle t \rangle ^{-{\lambda }' + \nu_\delta}\| \psi_0 \|_\delta^2. \end{aligned}$$

(B.12)

Next, using (B.3), we find $\pm\mathrm{rem}_{i} \le\| \omega ^{ \rho_{i}/2} \, \mathrm{rem}_{i} \, \omega^{ \rho_{i}/2} \| \omega^{\rho_{i}} \lesssim\langle t \rangle^{-{\lambda }_{i} } \omega^{-\rho_{i}}$. This gives $\pm\mathrm{d}{\varGamma }(\mathrm{rem_{i}}) \lesssim\langle t \rangle^{-{\lambda }_{i}} \mathrm{d}\varGamma ( \omega^{ -\rho_{i}} )$, which, due to the bound (1.17), leads to the estimate

$$\begin{aligned} \bigl| \bigl\langle \mathrm{d}{\varGamma }(\mathrm{rem_i}) \bigr\rangle _{\psi_t} \bigr| \lesssim\langle t \rangle^{-{\lambda }_i+\nu_{\rho_i}}\| \psi_0\|^2_{\rho_i}. \end{aligned}$$

(B.13)

Let G _t :=dΓ(p _t ) and $\mathrm{Rem} := \sum_{\textrm {finite}} \mathrm{d}{\varGamma} (\mathrm{rem_{i}}) - I ( i \phi_{t} g)$. We have G _t ≥0, and, by (B.12) and (B.13),

$$\begin{aligned} \int_0^\infty dt \, \bigl| \langle \psi_t , \mathrm{Rem} \, \psi_t \rangle \bigr| \lesssim\| \psi_0\|^2_\diamondsuit, \end{aligned}$$

(B.14)

with $\|\psi_{0}\|^{2}_{\#}:=\|\psi_{0}\|^{2}_{\diamondsuit}+ \|\psi_{0}\|^{2}_{*}$, ∥ψ ₀∥_∗:=∥ψ ₀∥ _δ , $\|\psi_{0}\|_{\diamondsuit}:= \sum\|\psi _{0}\|_{\rho_{i}}$.

In the strong case, (B.5) and (B.10) imply

$$\begin{aligned} D\varPhi_{t} \le- G_t + \mathrm{Rem}, \end{aligned}$$

(B.15)

and hence by (B.14), Φ _t is a strong propagation observable.

In the weak case, (B.2) and (B.10) imply

$$\begin{aligned} D\varPhi_{t} \ge G_t + \mathrm{Rem}. \end{aligned}$$

(B.16)

Since $\phi_{t} \le\| \omega^{ \delta/2} \phi_{t} \omega^{ \delta/2} \| \omega^{ - \delta} \lesssim\langle t \rangle^{-\nu_{\delta}} \omega ^{-\delta}$, we have $\mathrm{d}{\varGamma }(\phi_{t}) \lesssim\langle t \rangle^{-\nu _{\delta}} \mathrm{d}\varGamma( \omega^{ -\delta} )$. Using this estimate and using again the bound (1.17), we obtain

$$\begin{aligned} \langle \psi_t, \varPhi_{t} \psi_t\rangle \lesssim\langle t \rangle^{-\nu_\delta }\bigl\langle \mathrm{d}{\varGamma }\bigl(\omega^{-\delta}\bigr) \bigr\rangle _{\psi_t} \lesssim\|\psi_0\| ^2_\delta. \end{aligned}$$

(B.17)

Estimates (B.14) and (B.17) show that Φ _t is a weak propagation observable. □

To prove Theorem 1.1, in Sect. 2, we also used the following proposition.

Proposition B.4

Let ϕ _t be a one-photon family satisfying

• either, for some δ≥0 ,

  $$\begin{aligned} \bigl\| \omega^{ \delta/2} \phi_t \omega^{ \delta/2}\bigr\| \lesssim\langle t \rangle^{-\nu_\delta}\quad \mathit{and} \quad d \phi_t \ge p_t - d\tilde{\phi}_t + \mathrm{rem} , \end{aligned}$$

  (B.18)

  or

  $$\begin{aligned} d \phi_t \le-p_t+ d \tilde{\phi}_t + \sum_{\mathrm{finite}} \mathrm {rem_i} , \end{aligned}$$

  (B.19)

  where p _t ≥0, rem _i are one-photon operators satisfying (B.3), and $\tilde{\phi}_{t}$ is a weak propagation observable,

• (B.4) holds.

Then, depending on whether (B.18) or (B.19) is satisfied, Φ _t :=dΓ(ϕ _t ) is a weak, or strong, propagation observable, on the subspace $\varUpsilon_{\max(\delta, \rho _{i})}$, and therefore we have either the weak or strong propagation estimates, (B.1) or (B.7), on this subspace.

Proof

Given Proposition B.2 and its proof, the only term we have to control is $\mathrm{d}{\varGamma }( d\tilde{\phi}_{t})$. Using that $\tilde{\phi}_{t}$ is a weak propagation observable and using (B.8), (B.10) and (B.12) for $\tilde{\varPhi}_{t} :=\mathrm{d}{\varGamma }( \tilde{\phi}_{t})$, we obtain

$$\begin{aligned} \biggl| \int_0^{\infty} dt \, \bigl\langle \mathrm{d}\varGamma( d \tilde{\phi}_t ) \bigr\rangle _{\psi_t} \biggr| \lesssim\|\psi_0\|^2_\# , \end{aligned}$$

(B.20)

with $\|\psi_{0}\|^{2}_{\#}:=\|\psi_{0}\|^{2}_{\diamondsuit}+ \|\psi_{0}\|^{2}_{*}$, ∥ψ ₀∥_∗:=∥ψ ₀∥ _δ , $\|\psi_{0}\|_{\diamondsuit}:= \sum\|\psi _{0}\|_{\rho_{i}}$, which leads to the desired estimates. □

Remarks

• (1) Proposition B.2 reduces a proof of propagation estimates for the dynamics (1.9) to estimates involving the one-photon datum (ω,g) (an ‘effective one-photon system’), parameterizing the hamiltonian (1.4). (The remaining datum H _p does not enter our analysis explicitly, but through the bound states of H _p which lead to the localization in the particle variables, (1.7)).

• (2) The condition on the remainder in (B.2) can be weakened to rem=rem′+rem″, with rem′ and rem″ satisfying (B.3) and

  $$\begin{aligned} | \mathrm{rem}'' | \lesssim\chi_{|y|\ge c' t}, \end{aligned}$$

  (B.21)

  for c′ as in (1.13), respectively. Moreover, (B.3) can be further weakened to

  $$\begin{aligned} \int_0^\infty dt \, \bigl|\bigl\langle \psi_t , \mathrm{d}{\varGamma }( \mathrm{rem}_i ) \psi_t \bigr\rangle \bigr| <\infty. \end{aligned}$$

  (B.22)

• (3) An iterated form of Proposition B.4 is used to prove Theorem 1.1.

Appendix C: One-Particle Commutator Estimates

In this appendix, we estimate some localization terms and commutators appearing in Sect. 2. We begin with recalling the Helffer-Sjöstrand formula that will be used several times. Let f be a smooth function satisfying the estimates $\vert\partial_{s}^{n} f(s) \vert\le C_{n} \langle s \rangle ^{\rho- n}$ for all n≥0, with ρ<0. We consider an almost analytic extension $\tilde{f}$ of f, which means that $\tilde{f}$ is a C^∞ function on $\mathbb {C}$ such that $\tilde{f} \vert_{\mathbb{R}} = f$,

$$\begin{aligned} \operatorname {supp}\tilde{f} \subset \bigl\{ z \in \mathbb {C}, \ \vert \operatorname {Im}z \vert\le C \langle \operatorname {Re}z \rangle \bigr\} , \end{aligned}$$

$\vert\tilde{f} (z) \vert\le C \langle \operatorname {Re}z \rangle^{\rho}$ and, for all $n \in {\mathbb{N}}$,

$$\begin{aligned} \biggl\vert \frac{\partial\tilde{f} }{\partial\bar{z}} (z) \biggr\vert \leq C_n \langle \operatorname {Re}z \rangle ^{\rho- 1 - n} \vert \operatorname {Im}z \vert^n . \end{aligned}$$

Moreover, if f is compactly supported, we can assume that this is also the case for $\tilde{f}$. Given a self-adjoint operator A, the Helffer–Sjöstrand formula (see e.g. [16, 38]) allows one to express f(A) as

$$\begin{aligned} f ( A ) = \frac{1}{\pi} \int\frac{\partial\tilde{f} (z)}{\partial\bar{z}} ( A - z )^{-1} \operatorname {d\:\!Re}z \operatorname {d\:\!Im}z . \end{aligned}$$

(C.1)

Now recall that $b_{{\epsilon }}:=\frac{1}{2} (\theta_{{\epsilon }}\nabla\omega\cdot y + {\hbox{ h.c.}})$, where $\theta_{{\epsilon }}=\frac{\omega}{\omega_{{\epsilon }}},\ \omega _{{\epsilon }}:=\omega+ {\epsilon }$, ϵ=t ^−κ, with κ≥0. We have the relations

$$\begin{aligned} i [ \omega, b_{\epsilon }] = \theta_{\epsilon }, \qquad i \bigl[ \omega, y^2 \bigr] = \frac{1}{2} ( \nabla\omega\cdot y + y \cdot\nabla\omega) , \end{aligned}$$

(C.2)

and, using in particular Hardy’s inequality, one can verify the estimate

$$\begin{aligned} \bigl\| \bigl[ y^2 , b_{\epsilon }\bigr] \langle y \rangle^{-2} \bigr\| = \mathcal{O} \bigl( t^\kappa\bigr). \end{aligned}$$

(C.3)

The following lemma is a straightforward consequence of the Helffer-Sjöstrand formula together with (C.2) and (C.3). We do not detail the proof.

Lemma C.1

Let $h,\tilde{h}$ be smooth function satisfying the estimates $\vert \partial_{s}^{n} h(s) \vert\leq\mathrm{C}_{n} \langle s \rangle ^{-n}$ for n≥0 and likewise for $\tilde{h}$. Let w _α =(|y|/c ₁ t ^α)², v _β =b _ϵ /(c ₂ t ^β), with 0<α,β≤1. The following estimates hold

$$\begin{gathered} \bigl[ h ( w_\alpha) , \omega\bigr] = {\mathcal{O}}\bigl(t^{-\alpha}\bigr) , \qquad\bigl[ \tilde{h} ( v_\beta) , \omega\bigr] = {\mathcal{O}}\bigl(t^{-\beta}\bigr) , \\ \bigl[ h ( w_\alpha) , \theta_{\epsilon }^{ \frac{1}{2}} \bigr] = \mathcal{O} \bigl( t^{ \frac{1}{2} \kappa- \frac{1}{2} \alpha} \bigr) , \qquad\langle y \rangle\bigl[ h ( w_\alpha) , \theta_{\epsilon }^{ \frac{1}{2}} \bigr] = \mathcal{O} \bigl( t^{ \frac{1}{2} \kappa+ \frac{1}{2} \alpha} \bigr), \\ \bigl[ \tilde{h} ( v_\beta) , \omega_{\epsilon }^{-\frac{1}{2}} \bigr] = \mathcal{O} \bigl( t^{\frac{3}{2} \kappa- \beta} \bigr) , \qquad b_{\epsilon }\bigl[ \tilde{h} ( v_\beta) , \omega_{\epsilon }^{-\frac{1}{2}} \bigr] = \mathcal{O} \bigl( t^{\frac{3}{2} \kappa} \bigr), \qquad\bigl[ \tilde{h} ( v_\beta) , \theta_{\epsilon }^{\frac{1}{2}} \bigr] = \mathcal{O} \bigl( t^{\kappa - \beta} \bigr) , \\ \bigl[ h ( w_\alpha) , b_{\epsilon }\bigr] = {\mathcal{O}}\bigl(t^{\kappa}\bigr), \qquad\bigl[ h( w_\alpha) , \tilde{h} ( v_\beta) \bigr] = {\mathcal{O}}\bigl( t^{-\beta+\kappa} \bigr) , \qquad b_{\epsilon }\bigl[ h( w_\alpha) , \tilde{h} ( v_\beta) \bigr] = {\mathcal{O}}\bigl( t^{\kappa} \bigr) . \end{gathered}$$

Now we prove the following abstract result.

Lemma C.2

Let h be a smooth function satisfying the estimates $\vert \partial_{s}^{n} h(s) \vert\leq\mathrm{C}_{n} \langle s \rangle ^{-n}$ for n≥0. Assume an operator v is s.t. the commutators [v,ω] and [v,[v,ω]] are bounded, and for some z in $\mathbb{C} \setminus\mathbb{R}$, (v−z)⁻¹ preserves D(ω). Then the operator r:=[h(v),ω]−[v,ω]h′(v) is bounded as

$$\begin{aligned} \|r\|\lesssim \bigl\Vert \bigl[ v , [ v , \omega] \bigr]\bigr\Vert . \end{aligned}$$

(C.4)

Proof

We would like to use the Helffer–Sjöstrand formula (C.1) for h. Since h might not decay at infinity, we cannot directly express h(v) by this formula. Therefore, we approximate h(v) as follows. Consider $\varphi\in\mathrm{C}_{0}^{\infty} ( \mathbb {R}; [ 0 , 1 ] )$ equal to 1 near 0 and φ _R (⋅)=φ(⋅/R) for R>0. Let $\widetilde{h}$ be an almost analytic extensions of h such that $\widetilde{h} \vert_{\mathbb{R}} = h$,

$$\begin{aligned} \operatorname {supp}\widetilde{h} \subset \bigl\{ z \in \mathbb {C}; \ \vert \operatorname {Im}z \vert\leq \mathrm{C} \langle \operatorname {Re}z \rangle \bigr\} , \end{aligned}$$

(C.5)

$\vert\widetilde{h} (z) \vert\leq\mathrm{C}$ and, for all $n \in {\mathbb{N}}$,

$$\begin{aligned} \bigl\vert \partial_{\bar{z}} \widetilde{h} (z) \bigr\vert \leq\mathrm {C}_n \langle \operatorname {Re}z \rangle ^{\rho- 1 - n} \vert \operatorname {Im}z \vert^n . \end{aligned}$$

(C.6)

Similarly let $\widetilde{\varphi} \in\mathrm{C}_{0}^{\infty} ( \mathbb {C})$ be an almost analytic extension of φ satisfying these estimates. As a quadratic form on D(ω), we have

$$\begin{aligned} \bigl[ h ( v ) , \omega \bigr] = \mathop {\mathrm {s}\mbox {-}\mathrm {lim}}_{R \to\infty} \bigl[ ( \varphi_{R} h ) ( v ) , \omega \bigr] . \end{aligned}$$

(C.7)

Since (v−z)⁻¹ preserves D(ω) for some z in the resolvent set of v (and hence for any such z, see [2, Lemma 6.2.1]), we can compute, using the Helffer–Sjöstrand representation (see (C.1)) for (φ _R h)(v),

$$\begin{aligned} \bigl[ ( \varphi_{R} h ) ( v ) , \omega \bigr] =& \frac{1}{\pi} \int \partial_{\bar{z}} ( \widetilde{\varphi}_{R} \widetilde{h} ) (z) \bigl[ ( v - z )^{-1} , \omega \bigr] \operatorname {d\:\!Re}z \operatorname {d\:\!Im}z \\ =& - \frac{1}{\pi} \int\partial_{\bar{z}} ( \widetilde{ \varphi}_{R} \widetilde{h} ) (z) ( v - z )^{-1} [ v , \omega] ( v - z )^{-1} \operatorname {d\:\!Re}z \operatorname {d\:\!Im}z \\ =& [ v , \omega] ( \varphi_{R} h )^{\prime} ( v ) + r_{R} , \end{aligned}$$

(C.8)

as a quadratic form on D(ω), where

$$\begin{aligned} r_{R} =& - \frac{1}{\pi } \int\partial_{\bar{z}} ( \widetilde{\varphi }_{R} \widetilde{h} ) (z) \bigl[ ( v - z )^{-1} , [ v , \omega] \bigr] ( v - z )^{-1} \operatorname {d\:\!Re}z \operatorname {d\:\!Im}z \\ =& \frac{1}{\pi } \int\partial_{\bar{z}} ( \widetilde{ \varphi}_{R} \widetilde{h} ) (z) ( v - z )^{-1} \bigl[ v , [ v , \omega] \bigr] ( v - z )^{-2} \operatorname {d\:\!Re}z \operatorname {d\:\!Im}z . \end{aligned}$$

(C.9)

Now, using $(v - z )^{-1}= {\mathcal{O}}( \vert \operatorname {Im}z \vert^{- 1} )$, we obtain that

$$\begin{aligned} \bigl\Vert ( v - z )^{-1} \bigl[ v , [ v , \omega] \bigr] ( v - z )^{-2} \bigr\Vert \lesssim\vert \operatorname {Im}z \vert^{- 3} \bigl\Vert \bigl[ v , [ v , \omega] \bigr] \bigr\Vert . \end{aligned}$$

(C.10)

Besides, for all $n \in {\mathbb{N}}$,

$$\begin{aligned} \bigl\vert \partial_{\bar{z}} ( \widetilde{\varphi}_{R} \widetilde{h} ) (z) \bigr\vert \leq\mathrm{C}_n \langle \operatorname {Re}z \rangle ^{\rho- 1 - n} \vert \operatorname {Im}z \vert^n , \end{aligned}$$

(C.11)

where C _n >0 is independent of R≥1. Using (C.9) together with (C.10), we see that there exists C>0 such that ∥r _R ∥≤C∥[v,[v,ω]]∥, for all R≥1. Finally, since (φ _R h)′(v) converges strongly to h′(v), the lemma follows from (C.8) and the previous estimate. □

We want apply the lemma above to the time-dependent self-adjoint operator $v = \frac{b_{{\epsilon }}}{c t^{\alpha}}$.

Corollary C.3

Let h be a smooth function satisfying the estimates $\vert \partial_{s}^{n} h(s) \vert\leq\mathrm{C}_{n} \langle s \rangle ^{-n}$ for n≥0 and let $v:=\frac{b_{{\epsilon }}}{ct^{\alpha}}$, where c>0, ϵ=t ^−κ, with 0≤κ≤β≤1. Then the operator r:=dh(v)−(dv)h′(v) is bounded as

$$\begin{aligned} \|r\|\lesssim t^{-{\lambda }},\ {\lambda }:= 2\alpha-\kappa. \end{aligned}$$

(C.12)

Proof

Observe that

$$\begin{aligned} d h(v) - (d v) h'(v) = \bigl[ h(v) , i \omega\bigr] - [ v , i \omega] h'( v ) + {\partial }_t h(v) - ( {\partial }_t v ) h'(v). \end{aligned}$$

It is not difficult to verify that (v−z)⁻¹ preserves D(ω) for any $z \in\mathbb{C} \setminus\mathbb{R}$. Hence it follows from the computations

$$\begin{aligned}{} [ v , i \omega] = t^{-\alpha} \theta_{\epsilon }, \qquad \bigl[ v , [ v , i \omega] \bigr] = t^{-2\alpha} \theta_{\epsilon }\omega_{\epsilon }^{-2} {\epsilon }, \end{aligned}$$

(C.13)

that we can apply Lemma C.2. The estimate

$$\begin{aligned} \bigl[ v , [ v , \omega] \bigr]={\mathcal{O}}\bigl( \omega _{\epsilon }^{- 1} t^{- 2\alpha} \bigr)= {\mathcal{O}}\bigl( t^{- 2\alpha+\kappa} \bigr) \end{aligned}$$

(C.14)

then gives

$$\begin{aligned} \bigl\| \bigl[h(v) , i \omega\bigr] - [ v , i \omega] h'( v ) \bigr\| \lesssim t^{-2\alpha +\kappa}. \end{aligned}$$

It remains to estimate ∥∂ _t h(v)−(∂ _t v)h′(v)∥. It is not difficult to verify that D(b _ϵ ) is independent of t. Using the notations of the proof of Lemma C.2 and the fact that ∂ _t h(v)=s-lim _R→∞ ∂ _t (φ _R h)(v), we compute

$$\begin{aligned} {\partial }_t ( \varphi_{R} h ) ( v ) =& \frac{1}{\pi} \int\partial_{\bar{z}} ( \widetilde{\varphi}_{R} \widetilde{h} ) (z) {\partial }_t ( v - z )^{-1} \operatorname {d\:\!Re}z \operatorname {d\:\!Im}z \\ =& - \frac{1}{\pi} \int\partial_{\bar{z}} ( \widetilde{ \varphi}_{R} \widetilde{h} ) (z) ( v - z )^{-1} ( {\partial }_t v) ( v - z )^{-1} \operatorname {d\:\!Re}z \operatorname {d\:\!Im}z \\ = &({\partial }_t v) ( \varphi_{R} h )^{\prime} ( v ) + r'_{R} , \end{aligned}$$

where

$$\begin{aligned} r'_{R} =& - \frac{1}{\pi } \int \partial_{\bar{z}} ( \widetilde{\varphi }_{R} \widetilde{h} ) (z) \bigl[ ( v - z )^{-1} , {\partial }_t v \bigr] ( v - z )^{-1} \operatorname {d\:\!Re}z \operatorname {d\:\!Im}z \\ =& \frac{1}{\pi } \int\partial_{\bar{z}} ( \widetilde{ \varphi}_{R} \widetilde{h} ) (z) ( v - z )^{-1} [ v , {\partial }_t v ] ( v - z )^{-2} \operatorname {d\:\!Re}z \operatorname {d\:\!Im}z . \end{aligned}$$

(C.15)

Now using ${\partial }_{t} v = - \frac{\alpha b_{{\epsilon }}}{c t^{\alpha+1}}+\frac{1}{c t^{\alpha}} {\partial }_{t} b_{{\epsilon }}$ together with (2.8), we estimate

$$\begin{aligned}{} [ v , {\partial }_t v ] = {\mathcal{O}}\bigl( t^{-1-2\alpha+ \kappa} \bigr) b_{\epsilon }+ {\mathcal{O}}\bigl( t^{-1-2\alpha+2\kappa} \bigr). \end{aligned}$$

From this, the properties of $\tilde{\varphi}$, $\tilde{h}$, and κ≤β, we deduce that $\| r'_{R} \| \lesssim t^{ - 1 - \alpha+ \kappa} \lesssim t^{-2\alpha+ \kappa}$ uniformly in R≥1. This concludes the proof of the corollary. □

The following lemma is taken from [9]. Its proof is similar to the proof of Lemma C.2

Lemma C.4

Let h be a smooth function satisfying the estimates $\vert \partial_{s}^{n} h(s) \vert\leq\mathrm{C}_{n} \langle s \rangle ^{-n}$ for n≥0 and 0≤δ≤1. Let w _α =(|y|/ct ^α)² with 0<α≤1. We have

$$\begin{aligned} \bigl[ h ( w_\alpha ) , i \omega \bigr] = \frac{1}{ c t^\alpha} h^{\prime} ( w_\alpha) \biggl( \frac{ y }{ ct^\alpha} \cdot\nabla \omega+ \nabla\omega\cdot\frac{ y }{ ct^\alpha} \biggr) + \mathrm {rem} , \end{aligned}$$

with

$$\begin{aligned} \bigl\Vert \omega^{\frac{\delta}{2}} \, \mathrm{rem} \, \omega^{\frac {\delta}{2}} \bigr\Vert \lesssim t^{-\alpha(1+ \delta) } . \end{aligned}$$

Now we prove a localization lemma. Let $v_{\alpha}:=\frac{b_{{\epsilon }}}{c' t^{{\alpha }}}$, w _α :=(|y|/ct ^α)².

Lemma C.5

Let κ<α. We have, for c<c′/2,

$$\begin{aligned} \chi_{v_\alpha\geq1} \chi_{w_\alpha\leq1} = {\mathcal{O}}\bigl(t^{-({\alpha }-\kappa)}\bigr). \end{aligned}$$

(C.16)

Proof

We omit the subindex α in w _α and v _α write w≡w _α and v≡v _α . Observe that by the definition of χ (see Introduction) and the condition c<c′/2, we have $\chi_{ |y| \ge c' t^{{\alpha }}} \chi_{ |y| \le c t^{{\alpha }}} = 0$. Let $c<\bar{c} < c'/2$ and let $\tilde{\chi}_{ |y| \le \bar{c} t }$ be such that $\chi_{ |y| \le c t } \tilde{\chi}_{ |y| \le \bar{c} t } = \chi_{ |y| \le c t }$ and $\chi_{ |y| \ge c' t } \tilde{\chi}_{ |y| \le\bar{c} t } = 0$. Define $\bar{b}_{{\epsilon }}:= \tilde{\chi}_{\mid y \mid\leq\bar{c} t^{\alpha}} b_{{\epsilon }}\tilde{\chi}_{\mid y \mid\leq\bar{c} t^{\alpha}}$. It follows from the expression of b _ϵ that |〈u,b _ϵ u〉|≤∥u∥∥|y|u∥, and hence we deduce that $| \langle u , \bar{b}_{{\epsilon }}u \rangle| \le\bar{c} t^{\alpha} \| u \|^{2}$. This gives $\chi_{\bar{b}_{{\epsilon }}\geq c' t^{\alpha}}=0$. Using this, we write

$$\begin{aligned} \chi_{b_{\epsilon }\geq c't^\alpha}\chi_{\mid y \mid\leq ct^{\alpha}}=(\chi _{b_{\epsilon }\geq c't^\alpha}-\chi_{\bar{b}_{\epsilon }\geq c't^\alpha})\chi_{\mid y \mid\leq ct^{\alpha}}. \end{aligned}$$

(C.17)

Let $\bar{v}:=\frac{\bar{b}_{{\epsilon }}}{c' t^{\alpha}}$. Denote g(v):=χ _v≥1 and $g(\bar{v}):=\chi_{\bar{v} \geq 1}$. We will use the construction and notations of the proof of Lemma C.2. Using the Helffer-Sjöstrand formula for (φ _R g)(c), we write

$$ \begin{aligned}[b] ( \varphi_{R} g ) (v)-( \varphi_{R} g ) (\bar{v}) &= \frac{1}{\pi} \int \partial_{\bar{z}}(\widetilde{\varphi_{R} g}) (z) \bigl[ ( v - z )^{-1} -( \bar{v} - z )^{-1} \bigr] \operatorname {d\:\!Re}z \operatorname {d\:\!Im}z \\ &= - \frac{1}{\pi} \int\partial_{\bar{z}} ( \widetilde{ \varphi}_{R} \widetilde{g} ) (z) ( v - z )^{-1} ( v -\bar{v} ) (\bar{v} - z )^{-1} \operatorname {d\:\!Re}z \operatorname {d\:\!Im}z . \end{aligned} $$

(C.18)

Now we show that $( v -\bar{v} )(\bar{v} - z )^{-1}\chi_{\mid y \mid \leq c t^{\alpha}}= {\mathcal{O}}(t^{-({\alpha }-\kappa)} |\operatorname {Im}z|^{-2})$. We have

$$\begin{aligned} v - \bar{v} = ( 1 - \tilde{\chi}_{\mid y \mid\leq\bar{c} t^{\alpha}} ) \frac{ b_{\epsilon }}{ c' t^\alpha} +\tilde{\chi}_{\mid y \mid\leq\bar{c} t^{\alpha}} \frac{ b_{\epsilon }}{ c' t^\alpha} ( 1 - \tilde{\chi}_{\mid y \mid \leq\bar{c} t^{\alpha}} ), \end{aligned}$$

and we observe that, by Lemma C.1,

$$\begin{aligned} \bigl[ ( 1 - \tilde{\chi}_{\mid y \mid\leq\bar{c} t^{\alpha}} ) , b_{\epsilon }\bigr] = {\mathcal{O}}\bigl(t^{\kappa}\bigr). \end{aligned}$$

(C.19)

Thus

$$\begin{aligned} v - \bar{v} = (1 + \tilde{\chi}_{\mid y \mid\leq\bar{c} t^{\alpha}} ) \frac{ b_{\epsilon }}{ c' t^\alpha} ( 1 - \tilde{\chi}_{\mid y \mid\leq\bar{c} t^{\alpha}} ) + {\mathcal{O}}\bigl(t^{- ( \alpha-\kappa)}\bigr), \end{aligned}$$

Moreover, we can write

$$\begin{aligned} ( 1 - \tilde{\chi}_{\mid y \mid\leq\bar{c} t^{\alpha}} ) (\bar{v} - z )^{-1} \chi_{\mid y \mid\leq c t^{\alpha}} =& \bigl[ ( 1 - \tilde{\chi}_{\mid y \mid\leq\bar{c} t^{\alpha}} ) , (\bar{v} - z )^{-1} \bigr] \chi_{\mid y \mid\leq c t^{\alpha}} \\ =& - (\bar{v} - z )^{-1} \biggl[ ( 1 - \tilde{\chi}_{\mid y \mid\leq\bar{c} t^{\alpha}} ) , \frac{ b_{\epsilon }}{ c t^\alpha} \biggr] (\bar{v} - z )^{-1} \chi_{\mid y \mid\leq c t^{\alpha}} \\ =& {\mathcal{O}}\bigl( t^{-(\alpha-\kappa)} |\operatorname {Im}z|^{-2} \bigr) , \end{aligned}$$

where we used (C.19) to obtain the last estimate. This implies the statement of the lemma. □

Remark

The estimate (C.16) can be improved to $\chi_{v_{{\alpha }}\geq1} \chi_{w_{{\alpha }}\leq1} = {\mathcal{O}}(t^{-m({\alpha }-\kappa)})$, for any m>0, if we replace ω _ϵ :=ω+ϵ in the definition of b _ϵ by the smooth function $\omega _{{\epsilon }}:=\sqrt{\omega ^{2}+{\epsilon }^{2}}$.

Supplement I. Creation and Annihilation Operators on Fock Spaces

Recall that the propagation speed of the light and the Planck constant divided by 2π are set equal to 1. Recall also that the one-particle space is $\mathfrak {h}:= \mathrm{L}^{2} ( \mathbb {R}^{3} ; \mathbb {C})$, for phonons, and $\mathfrak {h}:= \mathrm{L}^{2} ( \mathbb {R}^{3} ; \mathbb {C}^{2})$, for photons. In both cases we use the momentum representation and write functions from this space as u(k) and u(k,λ), respectively, where $k\in \mathbb {R}^{3}$ is the wave vector or momentum of the photon and λ∈{−1,+1} is its polarization.

With each function $f \in \mathfrak {h}$, one associates creation and annihilation operators a(f) and a ^∗(f) defined, $\mbox {for}\ u\in\bigotimes_{s}^{n}\mathfrak {h}$, as

$$\begin{aligned} a^*(f) : u\rightarrow \sqrt{n+1} f\bigotimes _s u \quad \mbox{and} \quad a(f) : u\rightarrow \sqrt{n} \langle f, u\rangle _\mathfrak {h}, \end{aligned}$$

(I.1)

with $\langle f, u\rangle _{\mathfrak {h}}:=\int\overline{f(k)} u(k, k_{1}, \ldots, k_{n-1}) \, \mathrm{d} k$, for phonons, and $\langle f, u\rangle _{\mathfrak {h}}:= \sum_{\lambda = 1 , 2} \int \, \mathrm{d} k \overline{f(k, {\lambda })} u_{n}(k, {\lambda }, k_{1}, \lambda _{1}, \ldots, k_{n-1}, \lambda_{n-1})$, for photons. They are unbounded, densely defined operators of ${\varGamma }(\mathfrak {h})$, adjoint of each other (with respect to the natural scalar product in $\mathcal {F}$) and satisfy the canonical commutation relations (CCR):

$$\begin{aligned} \bigl[ a^{\#}(f) , a^{\#}(g) \bigr] = 0 , \qquad \bigl[ a(f) , a^*(g) \bigr] = \langle f, g\rangle , \end{aligned}$$

where a ^#=a or a ^∗. Since a(f) is anti-linear and a ^∗(f) is linear in f, we write formally

$$\begin{aligned} a(f) = \int\overline{f(k)} a(k) \, d k , \qquad a^*(f) = \int f(k) a^*(k) \, d k , \end{aligned}$$

for phonons, and

$$\begin{aligned} a(f) = \sum_{\lambda= 1 , 2} \int\overline{f(k, {\lambda })} a_\lambda(k) \, d k , \qquad a^*(f) = \sum_{\lambda= 1 , 2} \int f(k, {\lambda }) a_\lambda^*(k) d k , \end{aligned}$$

for photons. Here a(k) and a ^∗(k) and a _λ (k) and $a_{\lambda}^{*}(k)$ are unbounded, operator-valued distributions, which obey (again formally) the canonical commutation relations (CCR):

$$\begin{aligned} \bigl[ a^{\#}(k) , a^{\#}\bigl(k'\bigr) \bigr] = 0 , \qquad \bigl[ a(k) , a^*\bigl(k'\bigr) \bigr] = \delta \bigl(k-k'\bigr) , \\ \bigl[ a_{\lambda}^{\#}(k) , a_{\lambda'}^{\#} \bigl(k'\bigr) \bigr] = 0 , \qquad \bigl[ a_{\lambda}(k) , a_{\lambda'}^*\bigl(k'\bigr) \bigr] = \delta_{\lambda, \lambda'} \delta\bigl(k-k'\bigr) , \end{aligned}$$

where a ^#=a or a ^∗ and $a_{\lambda}^{\#}= a_{\lambda}$ or $a_{\lambda}^{*}$.

Given an operator τ acting on the one-particle space $\mathfrak {h}$, the operator dΓ(τ) (the second quantization of τ) defined on the Fock space $\mathcal {F}$ by (1.3), can be written (formally) as dΓ(τ):=∫dk  a ^∗(k)τa(k), for phonons, and $\mathrm{d}\varGamma( \tau) : = \sum_{\lambda= 1 , 2} \int d k \, a_{\lambda}^{*} ( k ) \tau a_{\lambda}( k )$, for photons. Here the operator τ acts on the k-variable. The precise meaning of the latter expression is (1.3). In particular, one can rewrite the quantum Hamiltonian H _f in terms of the creation and annihilation operators, a and a ^∗, as

$$\begin{aligned} H_f = \sum_{\lambda= 1 , 2} \int d k \, a_\lambda^*(k) \omega(k) a_\lambda(k) \end{aligned}$$

(I.2)

for photons, and similarly for phonons.

The relations below are valid for both phonon and photon operators. Commutators of two dΓ operators reduces to commutators of the one-particle operators:

$$\begin{aligned} \bigl[\mathrm{d}\varGamma( \tau) , \mathrm{d}\varGamma\bigl( \tau' \bigr)\bigr] = \mathrm{d}\varGamma\bigl( \bigl[ \tau, \tau' \bigr] \bigr). \end{aligned}$$

(I.3)

Let τ be a one-photon self-adjoint operator. The following commutation relations involving the field operator $\varPhi( f ) = \frac {1}{\sqrt{2}} ( a^{*} ( f ) + a (f ))$ can be readily derived from the definitions of the operators involved:

$$\begin{aligned} \bigl[ \varPhi(f) , \varPhi(g) \bigr] = i \operatorname {Im}\langle f , g \rangle _{\mathfrak{h}} , \end{aligned}$$

(I.4)

$$\begin{aligned} \bigl[ \varPhi(f) , \mathrm{d}\varGamma( \tau) \bigr] = i \varPhi( i \tau f ) , \end{aligned}$$

(I.5)

$$\begin{aligned} \bigl[{\varGamma}(\tau), \varPhi(f)\bigr]=\varGamma( \tau) a \bigl((1-\tau) f \bigr) - a^*\bigl((1-\tau ) f\bigr)\varGamma( \tau) . \end{aligned}$$

(I.6)

Exponentiating these relations, we obtain

$$\begin{aligned} e^{i \varPhi(f)} \varPhi(g) e^{- i \varPhi(f)} =& \varPhi(g) - \operatorname {Im}\langle f , g \rangle _{\mathfrak{h}} , \end{aligned}$$

(I.7)

$$\begin{aligned} e^{i \varPhi(f)} \mathrm{d}\varGamma( \tau) e^{- i \varPhi(f)} =& \mathrm{d} \varGamma( \tau ) - \varPhi( i \tau f ) + \frac{1}{2} \operatorname {Re}\langle \omega f , f \rangle _{\mathfrak {h}} \end{aligned}$$

(I.8)

$$\begin{aligned} e^{i \varPhi(f)} \varGamma( \tau) e^{- i \varPhi(f)} =& \varGamma( \tau) + \int _0^1 ds \, e^{i s\varPhi(f)} \bigl( \varGamma( \tau) a \bigl((1-\tau) f\bigr) \\ &{} - a^*\bigl((1-\tau) f\bigr)\varGamma( \tau) \bigr) e^{-s i \varPhi(f)} . \end{aligned}$$

(I.9)

Finally, we have the following standard estimates for annihilation and creation operators a(f) and a ^∗(f), whose proof can be found, for instance, in [6], [29, Sect. 3], [34]:

Lemma I.1

For any $f \in\mathfrak{h}$ such that $\omega^{-\rho/2} f \in\mathfrak{h}$, the operators a ^#(f)(dΓ(ω ^ρ)+1)^−1/2, where a ^#(f) stands for a ^∗(f) or a(f), extend to bounded operators on $\mathcal{H}$ satisfying

$$\begin{gathered} \bigl\Vert a(f) \bigl( \mathrm{d}\varGamma\bigl( \omega^\rho\bigr) +1 \bigr)^{- \frac{1}{2} } \bigr\Vert \leq\bigl\Vert \omega^{-\rho/2} f \bigr\Vert _{ \mathfrak{h}} , \\ \bigl\Vert a^{*} (f) \bigl( \mathrm{d}\varGamma\bigl( \omega^\rho\bigr) +1 \bigr)^{- \frac{1}{2} } \bigr\Vert \leq\bigl\Vert \omega^{-\rho/2} f \bigr\Vert _{ \mathfrak{h} } + \Vert f \Vert_{ \mathfrak{h} }. \end{gathered}$$

If, in addition, $g \in\mathfrak{h}$ is such that $\omega^{-\rho/2} g \in\mathfrak{h}$, the operators a ^#(f)a ^#(g)(dΓ(ω ^ρ)+1)⁻¹ extend to bounded operators on $\mathcal{H}$ satisfying

$$\begin{gathered} \bigl\Vert a(f) a(g) \bigl( \mathrm{d}\varGamma\bigl( \omega^\rho\bigr) +1 \bigr)^{ -1 } \bigr\Vert \leq\bigl\Vert \omega^{-\rho/2} f \bigr\Vert _{ \mathfrak{h}} \bigl\Vert \omega^{-\rho /2} g \bigr\Vert _{ \mathfrak{h}} , \\ \bigl\Vert a^*(f) a(g) \bigl( \mathrm{d}\varGamma\bigl( \omega^\rho \bigr) +1 \bigr)^{ -1 } \bigr\Vert \leq \bigl( \bigl\Vert \omega^{-\rho/2} f \bigr\Vert _{ \mathfrak{h} } + \Vert f \Vert_{ \mathfrak{h} } \bigr) \bigl\Vert \omega^{-\rho/2} g \bigr\Vert _{ \mathfrak{h}} , \\ \bigl\Vert a^{*} (f) a^*(g) \bigl( \mathrm{d}\varGamma\bigl( \omega^\rho\bigr) +1 \bigr)^{ - 1 } \bigr\Vert \leq \bigl( \bigl\Vert \omega^{-\rho/2} f \bigr\Vert _{ \mathfrak{h} } + \Vert f \Vert_{ \mathfrak{h} } \bigr) \bigl( \bigl\Vert \omega^{-\rho/2} g \bigr\Vert _{ \mathfrak{h} } + \Vert g \Vert_{ \mathfrak{h} } \bigr). \end{gathered}$$