We slightly extend prior results about the derivation of the Maxwell–Schrödinger equations from the bosonic Pauli–Fierz Hamiltonian. More concretely, we show that the findings from Leopold and Pickl [SIAM J. Math. Anal. 52(5), 4900–4936 (2020)] about the coherence of the quantized electromagnetic field also hold for soft photons with small energies. This is achieved with the help of an estimate from Ammari et al. [arXiv:2202.05015 (2022)], which proves that the domain of the number of photon operator is invariant during the time evolution generated by the Pauli–Fierz Hamiltonian.
In this short paper, we derive the Maxwell–Schrödinger system of equations as an effective model describing a Bose–Einstein condensate of charged particles immersed in a coherent electromagnetic field. More precisely, we prove quantitatively that the Maxwell–Schrödinger system approximates well the many-body quantum evolution generated by the Pauli–Fierz Hamiltonian, provided that the total number of particles N is large, the particles are initially in a Bose–Einstein condensate, and the quantum nature of the field—quantified by the semiclassical parameter ℏ—is negligible. In particular, we focus on the combined regime . Equivalently, the same effective dynamics approximates well N → ∞ bosons weakly interacting with a quantized electromagnetic field; see the discussion below.
This problem has already been studied by one of the authors, together with Pickl, in Ref. 1. The main focus here is to build on the results and techniques introduced there and to strengthen them by studying the convergence for the photons’ reduced density matrix. In the previous work, the quantum fluctuations around the coherent state of photons have been classified only by means of their energy. The extension to the reduced density matrix is physically relevant and mathematically nontrivial because the coherence of photons with small frequencies cannot be shown by the energy of the electromagnetic field due to its massless nature. We often refer to Ref. 1 throughout this paper, hopefully striking a good balance between being concise and being self-contained.
A. The Maxwell–Schrödinger system of equations
The Maxwell–Schrödinger system of equations describes the wave function φ of a quantum particle (with a nontrivial charge distribution κ) interacting with the classical electromagnetic field, described by the vector potential A and the electric field . We choose the Coulomb gauge
and this makes, indeed, A and E the only dynamical degrees of freedom of the field. Let us also preliminarily define the current
The Maxwell–Schrödinger system thus takes the form
where is an interaction term for the quantum particle and is the Helmholtz projection onto the subspace of divergence free vector fields. The choice of gauge, Coulomb’s in this case, can be seen as a constraint, for it is preserved by the Maxwell–Schrödinger flow. A typical example for the particle interaction could be
where W is an external potential and ( |φ|2)φ is a nonlinear term, usually originating from a microscopic pair interaction. With this choice, (1.3) can used as a system of mean-field equations to model the dynamics of identical particles interacting via the potential v and the classical electromagnetic field which they generate. The Cauchy problem associated with (1.3) is obtained by fixing an initial datum (φ0, A0, E0), subjected to the constraint ∇ · A0 = 0. In order to do so, it is convenient to introduce the complex scalar fields by defining
where are the polarization vectors satisfying
which implement the Coulomb gauge. In fact, there is such a unique decomposition for any time, i.e.,
which respects both the Coulomb gauge and . This makes it possible to rewrite Maxwell’s equations (same as in Ref. 2, Chap. 1.C) in terms of αt and to consider the equivalent system
with initial datum (φ0, α0(·, 1), α0(·, 2)). Throughout this article, we use to denote the Fourier transform of f. As it will be clarified shortly, φt and αt appear naturally as the effective counterparts of the microscopic dynamical variables. Note that the energy functional of the Maxwell–Schrödinger system is given by
with A being defined in analogy to (1.5). Global well-posedness for the Maxwell–Schrödinger system with , κ(x) = e δ(x) (where e is the electric charge of the Schrödinger particle), and has been proven in Refs. 3 and 4. We will also consider only the case , but we may require the charge distribution κ to be extended, in order to well-define the microscopic system, as discussed below. Typical examples of charge distributions that we will consider are of the form
representing a charged particle with total charge , distributed in a Gaussian fashion (“smoothed” spherical distribution of “diameter” σ), or
representing a sharp cutoff in momentum space with total charge . Let us remark that globally neutral particles can be considered as long as they have a nontrivial charge distribution: for example,
yields null total charge but nontrivial dipole and quadrupole interactions with the electromagnetic field.
If the charge distribution is not concentrated in a single point and the potential v represents an electrostatic mean-field self-interaction, then the form of the latter changes as well: a physically sensible choice would be . We will allow some liberty in the choices of κ and v; the specific requirements on the two will be made precise in Assumption 2.1. Concerning global well-posedness, let us remark that compared to the literature,3,4 our choices for κ and v will be at most “better” (i.e., more regular) and therefore do not affect the proof in any way since both κ and v act by convolution in the equation.
- (rough solutions) For everythere exists a unique global solutionof the Cauchy problem associated with (1.3), being the unique strong limit of a sequence of regular solutions in (1), whose initial data approximate the rough initial datum in H1 × H1 × L2.
(continuous dependence on initial data) The solutions (φ, A) in (2) continuously depend on the initial datum (φ0, A0, E0) ∈ H1 × H1 × L2.
For , let denote the weighed -space with norm
Throughout this work, we will rely on the following statement, which results almost immediately from Proposition 1.1 (see the Appendix).
Let . For every initial datum such that , the Maxwell–Schrödinger system (1.10) has a unique global solution in .
B. The microscopic model: Pauli–Fierz Hamiltonian
The microscopic model corresponding to the Maxwell–Schrödinger system with mean-field self-interaction |φ|2φ consists of many identical nonrelativistic particles—obeying Bose–Einstein condensation—interacting among themselves by means of a weak pair potential and with a quantized electromagnetic field in Coulomb’s gauge. Contrarily to the “classical” case, the microscopic model is known to be well-defined only for extended charges. Let us start by defining a Hilbert space depending on two parameters as follows:
where is the natural Hilbert space of N identical bosons (the subscript s indicates symmetry under the interchange of variables) and is the second quantization functor associating with any (pre-)Hilbert space the corresponding Fock representation of the canonical commutation relations with ℏ being a semiclassical parameter measuring the degree of noncommutativity of the quantum field. The Fock representation is the natural one to describe noninteracting or regularized quantum field theories, the latter being the case here with . With this interpretation, the limits N → ∞ and ℏ → 0 describe, respectively, the regimes in which the bosons are many and the quantum effects of the field are negligible. The time evolution is dictated by the Schrödinger equation,
where the Hamiltonian , called Pauli–Fierz Hamiltonian, is given by
where describes the coupling strength between the particles and the field, gN is the coupling strength between the particles, κ and v are the charge distribution and the pair potential introduced previously,
is the field’s kinetic energy, with being the polarized creation and annihilation operators satisfying the canonical commutation relations (CCR),
is the smeared quantized electromagnetic vector potential in Coulomb’s gauge. Let us remark that both and Hf depend on ℏ through the creation and annihilation operators, which have ℏ-dependent CCRs. The Hamiltonian is self-adjoint on , where whenever and v is Kato-infinitesimal with respect to −Δ.5–7
C. Scaling regime
Our aim is to prove that the Maxwell–Schrödinger system emerges in some limit N → ∞ and/or ℏ → 0 as an effective model of the microscopic Pauli–Fierz dynamics. This is true only if we couple the parameters N, ℏ suitably and choose the coupling constants , gN accordingly. A possible choice is given by N → ∞, , , and . In this regime, the electromagnetic field becomes classical inverse proportionally to the increasing number of bosons. At the same time, the coupling between the particles and field is of order one, while the coupling between pairs of particles becomes weak (of order ). Defining and as new creation and annihilation operators leads to the mathematically equivalent but physically different choice N → ∞, ℏ = 1, , and . Here, the physical interpretation is of many bosons that interact weakly both with the quantized electromagnetic field (coupling of order ) and among themselves (pair coupling of order ). Our result reads as follows.
Provided that we choose an initial microscopic state that is “close enough” to a non-interacting state representing a complete condensate and a coherent field of minimal uncertainty, then at any time t ≥ 0, the evolution keeps the state “close” (in the same sense as above) to an analogous configuration in which the one-particle wave function and the argument of the coherent field have been evolved by the coupled Maxwell–Schrödinger equations.
The described scaling regime has been considered in earlier works for the Nelson model with an ultraviolet cutoff,8–11 the renormalized Nelson model,12 and the Fröhlich model.13,14 In Ref. 15, the Nelson model with an ultraviolet cutoff has been studied in a limit of many weakly interacting fermions. The classical behavior of quantum fields has also been proven in different scaling regimes.16–31 We also would like to mention Ref. 32, which derives the Maxwell–Schrödinger equations in a nonrigorous manner by neglecting certain terms in the Pauli–Fierz Hamiltonian.
II. MAIN RESULT
From now on, we will keep N as the single parameter and choose ℏ = 1, , and . We will use the notations , with vacuum Ω, HN = HN,1, and ΨN = ΨN,1. Concerning the interaction potential and charge distribution, we will make the assumptions.
In order to state our result, we define for the one-particle reduced density matrix of the charged particles by
where Tr2,…,N denotes the partial trace over the coordinates x2, …, xN and is the trace over Fock space. In addition, we introduce the number of photon operators,
and the unitary Weyl operator ,
Our result is as follows.
III. PROOF OF THE RESULT
The rest of the article outlines the Proof of Theorem 2.2. We will proceed as follows:
We define a functional β[ΨN, φ, α], which measures if the charges of the many-body state ΨN form a Bose–Einstein condensate with condensate wave function φ and if the photons are in a coherent state with mean photon number .
Afterward, we compute the change of β[ΨN,t, φt, αt] in time.
Finally, we control the growth of β[ΨN,t, φt, αt] with the help of Grönwall’s inequality. This concludes the proof.
A. Definition of the functional
We define a functional that consists of three parts.
The functional βa measures if the charges of the many-body state are in a Bose–Einstein condensate (we refer to Refs. 33 and 34 for a comprehensive introduction). Its relation to the trace norm distance of the one-particle reduced density matrix is given by (see, e.g., Ref. 1, Lemma 5.3)
The functional βb quantifies the fluctuations of ΨN around the coherent state . Using property (2.6) of the Weyl operators, it can be written as
showing that it is the same quantity as usually considered in the coherent state approach.35 While βa and βb measure the deviation of ΨN from the product state , the functional βc is introduced for technical reasons. It quantifies the fluctuations of the many-body energy per particle around the energy of the Maxwell–Schrödinger system.
In the original proof of Ref. 1, the functional β was considered with being replaced by
This definition has the advantage that it can be defined for many-body states ΨN in the domain , which is invariant under the time evolution . The additional difficulties with respect to Ref. 1 actually originate from the fact that [in contrast to ] is not contained in the domain of the Pauli–Fierz Hamiltonian. On the contrary, does not allow us to investigate the coherence of photons with small frequencies because the factor in the integral on the right-hand side of (3.5) suppresses contributions from photons with small energies.
B. Invariance of the domain
Throughout the rest of this article, (φt, αt) and ΨN,t denote the solutions of (1.10) and (1.17) from Theorem 2.2. In this section, we show that for all t ≥ 0. The condition on the Maxwell–Schrödinger solutions is satisfied because of Corollary 1.2. While is invariant under the evolution of the Pauli–Fierz Hamiltonian, due to Stone’s theorem, the invariance of is less clear because the photon number is not conserved during the time evolution. The next statement, however, displays that the number of photons can be controlled by the energy of the system.38
C. Computing the change of β in time
In order to estimate the emerging correlations between the particles and the photons during the evolution of the system, we compute the change of in time.
From Ref. 1, Secs. 6.2 and 6.4 and Duhamel’s formula, we immediately obtain the following.
D. Controlling the growth of β in time
In this section, we classify the growth of in time. The first two inequalities of Theorem 2.2 then follow from (3.3) and (3.4) and the statement below. By similar estimates as in Ref. 1, Chap. 7, one obtains (2.12) and (2.13).
N.L. would like to thank Peter Pickl for many fruitful discussions within the project.1 M.F. acknowledges the support from “Istituto Nazionale di Alta Matematica (INdAM)” through the “Progetto Giovani GNFM 2020: Emergent Features in Quantum Bosonic Theories and Semiclassical Analysis.” N.L. acknowledges the support from the Swiss National Science Foundation through the NCCR SwissMap and funding from the European Union’s Horizon 2020 Research and Innovation Programme under Marie Skłodowska-Curie Grant Agreement No. 101024712.
Conflict of Interest
The authors have no conflicts to disclose.
Marco Falconi: Writing – original draft (equal); Writing – review & editing (equal). Nikolai Leopold: Writing – original draft (equal); Writing – review & editing (equal).
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
APPENDIX: PROPERTIES OF (1.10)
To simplify the notation we assume for all . Theorem 2.2 equally applies if is complex valued. In this case, .
Inequality (3.6) was originally proven by Hiroshima and appeared in a slightly different form (for the second instead of the first moment of the number operator) in Ref. 16, Proposition 3.11. We would like to thank Hiroshima for sharing his notes with us. The proof is presented again for the convenience of the reader.