Project A11 • Electromagnetic fields interacting with quantum matter
Principal investigators
Dr. Ioannis Anapolitanos
(7/2019 - )
Prof. Dr. Dirk Hundertmark
(7/2019 - )
Project summary
We study quantum particles interacting with classical and quantized electromagnetic fields. A system of $N$ classical particles interacting with a classical electromagnetic field is described by the inhomogeneous Maxwell-Lorenz equations \begin{equation}\label{eq:Maxwell1} c^{-1} \partial_t \mathbf{B}= - \text{curl } \mathbf{E}, \quad c^{-1} \partial_t {\bf E}= \text{curl } \mathbf{B} - c^{-1}\mathbf{J}, \quad \text{div} {\bf E}= \rho, \quad \text{div} {\bf B}=0, \end{equation} where $\bf E$ and $\bf B$ are the electric and magnetic fields and $c$ is the speed of light. If $q_j$, $v_j$, $e_j$ are respectively the position, velocity, and charge of the $j$-th particle, then the charge and current density are given by \begin{equation}\label{eq:Maxwell2} \rho(x,t)= \sum_{j=1}^N e_j \varphi(x-q_j(t)), \quad \mathbf{J}(x,t)= \sum_{j=1}^N e_j \varphi(x-q_j(t)) v_j(t). \end{equation} Here, $\varphi$ is the normalized charge density distribution of each particle. The equations \eqref{eq:Maxwell1}–\eqref{eq:Maxwell2} can be written as a classical Hamiltonian system. In reality, particles and electromagnetic fields are not classical but quantum. In order to take into account quantum effects we study a quantized version of the classical Hamiltonian system, given by a Hamiltonian of the form \begin{equation}\label{eq:genHam} H=\sum_{j=1}^N \left(\frac{1}{2m_j}\left(p_j-c^{-1} e_j {\bf A}_\varphi(q_j)-c^{-1} e_j {\bf A}_{\text{ex}}(q_j)\right)^2 + e_j V_{\text{ex}}(q_j) \right) + V_{\varphi,\text{coul}} + H_f. \end{equation} Here $p_j$ are the momentum operators of the particles, $V_{\text{ex}}$ and ${\bf A}_{\text{ex}}$ the external electric and magnetic potentials, $V_{\varphi,\text{coul}}$ are interaction terms between the particles, which are typically Coulomb potentials, $H_f$ is the energy operator for the photon field, which comes from the quantization of the electromagnetic filed, and ${\bf A}_\varphi$ describes the interaction of the photon field with the particles. The operator $H$ acts on a subspace of $L^2(\mathbb{R}^{3N}) \otimes F$, where $F$ is the bosonic Fock space over $L^2(\mathbb{R}^3)$, which allows for a variable number of photons.
In this general formulation, the dynamics of the quantum system is extremely complicated. Depending on the physical situation certain terms in the Hamiltonian simplify. A first step towards the study of the full dynamics is often the investigation of static properties of the quantum system.
Among others, the questions we study are dynamical localization/delocalization, interatomic and intermolecular forces such as the van der Waals force, a special type of chemical reactions called isomerizations, and fine spectral properties of quantum systems. This includes cases where the minimum energy, the ground state energy, is at the edge of the continuous spectrum.
Dynamical localization
Our work focuses on Hamiltonians of two dimensional systems with external magnetic field ${\bf B}_{\text{ex}} = \text{curl }{\bf A}_{\text{ex}}$ which are often used to describe graphene. Studying localization of particles in a magnetic field is especially difficult if the electric or magnetic field is not radially symmetric. In this case transitions from one eigenmode of the angular momentum to another one is possible, which may lead to delocalization. This is why all prior works required that both the magnetic and electric fields are radially symmetric. In [CHSV21] we obtained for $N=1, {\bf A}_\varphi=0, H_f=0$ the first results for the case that we do not have radial symmetry of $V_{\text{ex}}$. One way to quantify the dynamical properties of a quantum system is to look at moments of the time evolution of the position. When particles are localized, classical intuition tells us that in the quantum situation moments of the position $\langle \psi(t), |x|^n \psi(t)\rangle $ should be bounded or grow slowly. Here $\psi(t)$ is the time evolution of the state of the quantum system. We proved that moments of the position grow at most logarithmically which is far from the ballistic threshold, where particles move freely with a constant velocity through the system.
Interatomic/Intermolecular forces and isomerizations
As discovered by van der Waals [Waa88], when he derived the now famous van der Waals equation which described a large class of real gases – as opposed to the ideal gas law, there is a long range attractive force between neutral atoms. This phenomenon, the now called van der Waals attraction, cannot be explained classically and is important in physics, chemistry and biology.
In this case the Hamiltonian arises from \eqref{eq:genHam} for ${\bf A}_\varphi=0, {\bf A}_{\text{ex}}=0, H_f=0$ and we write it as $H_D$ to indicate its dependence on a scaling factor $D$ of the distances between the nuclei. The question is how the ground state energy $E_D$ of $H_D$ behaves in the limit $D\to\infty$, where the distances of the nuclei grow to infinity. The limit $E_\infty=\lim_{D\to\infty}E_D$ is simply the energy of the separated system. Physicists predicted that \begin{equation}\label{eq:axilrod et al} E_D-E_\infty = -\frac{a_1}{D^6} - \frac{a_2}{D^8} + \frac{a_{ATM}}{D^9} + O(D^{-10})\, ,\quad D\to\infty, \end{equation} where the $a_1, a_2>0$ so there is attraction in the long range behaviour. The term $a_{ATM}/D^9$ was derived by Axilrod, Teller [AT43] and, independently, by Muto [Mut43] in the 1940s, using non-rigorous perturbation theory. It is a genuine non-additive three body effect which plays an important role in atomic physics [LT10, DGT14]. The coefficient $a_{ATM}$ has, unlike $a_1$, $a_2$, not a fixed sign but whether it is attractive or repulsive depends highly on the geometry of the three atoms.
In our previous work [AS17] we proved, under some conditions, that the van der Waals attraction $-a_1/D^6$ is the leading order of $E_D − E_\infty$. In [BHHV22] we went further and proved the higher order corrections \eqref{eq:axilrod et al} to the van der Waals attraction. This is the first proof of the Axilrod–Teller–Muto correction. In addition, we took relativistic effects into account, which become relevant for heavy atoms. We stress that in the relativistic case the nonlocal kinetic energy poses several additional challenges, compared to the non-relativistic case.
Figure 1. A hydrogen atom (left).
interacting with its mirror image
In [ABH20] we investigated the interaction of a molecule with a (half-infinite) metallic plate. In this case the interaction is due to the deformation of the electric field caused by the metal. It can be described by mirror charges, see Figure 1 on the right for the case of a hydrogen atom. The fact that the molecule is described by a Schrödinger operator acting on a half space poses some additional challenges.
In [Oli22] the interaction of a hydrogen atom with a metallic plate was studied taking into account the finiteness of the speed of propagation of the electromagnetic field namely for ${\bf A}_\varphi\neq 0$ and $H_f\neq 0$ in \eqref{eq:genHam}. It is a model, where the field energy is quantized, i.e., the electromagnetic interaction is due to the exchange of photons. In this case the minimum energy of the system is at the edge of the continuous spectrum, which makes the analysis considerably harder.
The above works described static properties of the systems. The results [AS17] and [BHHV22] helped us to go further and obtain new information concerning quasistatical chemical reactions, where the atoms and molecules, or individual parts of the molecules, move slowly.
A special case of such chemical reactions are isomerizations, where a single molecule changes its configuration, e.g., $HCN \to CNH$. Under the so-called adiabatic, or quasistatic approximation, it can be described with the help of a path $t \mapsto y(t)$, $t \in [0,1]$, in the configuration space from the reactant to the product, see Figure 2 i). Let $E(y)$ be the energy as function of the shape of the molecule, see Figure 2 ii).
Figure 2. (i) Hypothetical paths for HCN $\to$ CNH. (ii) hypothetical graph of $E(y).$
Quantum chemists often assume that the reaction happens along a path which is energetically optimal, see red path in Figure 2. The existence of such a path is mathematically open. In [AL20, AOZ22] we proved, in some cases, that an isomerization can happen so that the molecule does not split infinitely apart. Figure 3 gives a hypothetical case, which we excluded.
Figure 3. Hypothetical graph of $E(y)$ where a minimizing sequence of paths has to be unbounded.
Other directions
Quantum systems at criticality
Spectral moment inequalities
Absence of (positive) eigenvalues of magnetic Schrödinger/Dirac/Pauli Operators
Imagine a quantum system, where a parameter is tuned in such a way that the ground states energy approaches the continuum threshold (the lower edge of the essential spectrum) when the parameter reaches a critical value.
Does the ground state grow in diameter until it spreads out to infinity and dissolves when the parameter is at the critical value, or does it stay bounded even at criticality and then suddenly blow up. In the first case, the ground state eigenvalues ceases to exist at critical coupling, while in the second case, the ground state is a true eigenvalue at the critical coupling, which is embedded at the edge of the essential spectrum. The latter case provides a quantum system, which is extremely sensitive to external perturbations. This can be useful for the development of sensors.
In [HJL22a] we studied a restricted two–particle quantum system, i.e., helium-type type atoms, at criticality. Here the parameter is the nuclear charge. Criticality means here that the nuclear charge is such that ground state energy of the two particle systems equals the ground state energy of hydrogen. Thus there is no energy penalty which prevents breakup, i.e., there is no energy barrier which prevents one particle to move to infinity while the other one stays bound to the nucleus. This problem might sound artificial, since the nuclear charge cannot be continuously varied. Nevertheless, since the work of Stillinger [Sti66], see also [StS74], these helium–type atoms at critical coupling are an intensely studied benchmark problem in quantum chemistry. For a review see, e.g., [GB15] and the references therein.
For these helium-type systems we prove the first sharp non-isotropic upper and lower bounds for the ground state of this two electron system at and near criticality. Our bounds clearly show how the decay of the ground state of this quantum system changes from exponential decay below criticality, to a stretched exponential decay at critical coupling. This is the first proof of the precise asymptotic behavior of the ground state for this benchmark problem in quantum chemistry.
Our a-priori bounds also yield a proof of existence of ground states with the help of standard variational methods. The existence of such a ground state at critical coupling had been known previously by PDE methods [HOS83].
In addition, we show for the first time the existence of a ground state of this quantum critical system without the infinite nuclear mass approximation, i.e., for a finite nuclear mass, which is a true three particle system.
In [HJL22b] we identify a sequence of potentials which mark the boundary when a Schrödinger operator can have a zero energy eigenvalue at the edge of the essential spectrum. This sequence of potentials is similar in spirit to the famous Hardy potential and its logarithmic corrections [Lun09], [PCM86], which mark the boundary of the Schrödinger operator $P^2+V$ to have finitely versus infinitely many negative eigenvalues.
Unlike the known Hardy potential, our newly discovered sequence of potentials changes its sign depending on the dimension: In dimensions $d\le 4$ the potential it is positive, whereas if $d\ge 5$, then it is negative for all large enough $|x|$. We emphasize that for dimension $d=4$ one needs the second order correction term to see this, since the first order term vanished in four dimensions.
This sign change gives a straightforward explanation for the folklore that there is a spectral phase transition with critical dimension four, concerning the existence/non-existence of zero energy ground states.
Spectral moment inequalities
Since the famous work of Lieb and Thirring, bounds on the moments of negative eigenvalues have played a central role in the rigorous study of many particle quantum systems. These bounds are of the form \begin{align}\label{eq:LTh bound2} \sum_{j}|E_j|^\gamma= \text{tr}(P^2+V)_-^\gamma \le C_{\gamma,d} \iint_{\mathbb{R}^\times\mathbb{R}^d}(\eta^2+V(x))_-^\gamma \frac{d\eta d x}{(2\pi)^d} = C_{\gamma,d} L_{\gamma,d}^{cl} \int V_-(x)^{d/2+\gamma} \, dx, \end{align} for $\gamma$-moments (Riesz' moments of order $\gamma\ge 0$) of the negative eigenvalues $E_j$ of a Schrödinger operator $P^2+V$, where $C_{\gamma,d}>0$ is a constant. The right hand side of \eqref{eq:LTh bound2} is the semiclassical prediction. Special cases are the celebrated Cwikel–Lieb–Rozenblum (CLR) bound for the limiting case $\gamma=0$, \begin{align}\label{eq:CLR bound} N(P^2+V) \le C_{0,d} \iint_{\eta^2+V(x)<0} \frac{d\eta d x}{(2\pi)^d} = C_{0,d} L_{0,d}^{cl} \int V_-(x)^{d/2} \, dx, \end{align} which gives an upper bound for the number of negative energy bound states of the quantum system in terms of the volume of the phase space where the classical energy is negative.
Another special case of \eqref{eq:LTh bound2} is given by \begin{align}\label{eq:LTh bound1} \text{tr}(P^2+V)_- \le C_{1,d} L_{1,d}^{cl} \int V_-(x)^{d/2+1} \, dx. \end{align} for the sum of the negative eigenvalues of $P^2+V$. It is well known from the early work of Lieb and Thirring that the bound \eqref{eq:LTh bound1} is dual to a lower bound for the kinetic energy of Fermions, which is important for the stability of matter problem for large Coulomb systems.
In [HKRV23] we provided first improvements on the constants $C_{0,d}$ since the original work of Cwikel, Lieb, and Rozenblum at the beginning of the 1970s. More importantly, we significantly improved Cwikel's approach by discovering a deep connection between his proof and bounds for maximal Fourier multipliers. These maximal Fourier multipliers are a well known tool in harmonic analysis.
In [FHJN21] we drastically generalized Rumin's approach for bounds of the form \eqref{eq:LTh bound1}. This improved existing bounds on the sum of the negative eigenvalues of Schrödinger operators.
Absence of (positive) eigenvalues of magnetic Schrödinger/Dirac/Pauli Operators
Absence of positive eigenvalues of magnetic Schrödinger operators is important for the derivation of Strichartz estimates for magnetic quantum systems. In [FKV18] the authors derived a form- subordinate smallness condition on the magnetic field and the electric potential for the absence of all eigenvalues of a magnetic Schrödinger operator. The method of multipliers, which they developed, also works for magnetic Schrödinger operators with complex-valued electric potentials. Our work [CFK20] further built upon this method to derive sufficient conditions on the magnetic field and the complex matrix-valued electric potential which guarantee that the corresponding system of magnetic Dirac and Pauli operators have empty point spectrum.
In [AHK22] we proved absence of positive eigenvalues of magnetic Schrödinger operators under physically reasonable assumptions solely on the magnetic field and the electric potential, which involve mild decay conditions and are based on virial identites. The easiest, or most straightforward approach to virial identities is via dilations, which are unitary operators, corresponding to a change of scale. For a real parameter $s\in\mathbb{R}$ a dilation $U_s:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$ is given by \begin{equation}\label{eq:dilation} (U_s \varphi)(x) ≔ e^{ds/2} \varphi(e^s x)\,.\end{equation} One of the key insights in [AHK22] is that the Poincare gauge is extraordinarily useful when working with dilations. In the Poincare gauge the magnetic vector potential is given by \begin{equation}\label{eq:magnetic} A(x) = \int_0^1 B(tx)\wedge x t\, dt = \int_0^1 \widetilde{B}(tx)\, dt \end{equation} with $\widetilde{B}(x)= B(x)\wedge x$, where $B$ is the magnetic field. In arbitrary dimensions $d\ge 2$, the vector field $\widetilde{B}$ is given by $\widetilde{B}(x)= B(x)[x]$, where one interprets the magnetic field at the point $x$ as an antisymmetic matrix.
Using the commutation rule $VU_s= U_s V(e^{-s}\cdot) = U_sV_{-s}$ for (vector-valued) multiplication operators $V$ and change of variables $t=e^{-s}$ in \eqref{eq:magnetic} one sees that \begin{equation} A= \int_0^\infty \widetilde{B}_{-t} \, ds = \int_0^\infty U_s^*\widetilde{B} U_s\, e^{-s} \, ds \, , \end{equation} expressing the magnetic vector potential $A$ in terms of the magnetic field as a weighted average over dilations. This representation clearly shows that there is a deep connection between dilations and the Poincare gauge. It is the key for a proof of invariance of the domain of the magnetic momentum $P-A$ under dilations and the basis for deriving the virial identities under mild physically reasonable conditions solely on the magnetic field and the electric potential.
In addition, the famous Miller–Simon examples [MS80], with a calculation error corrected in [AHK22], show that our decay conditions on the magnetic field in [AHK22] are sharp.
The Efimov effect
The methods and ideas developed in [HJL22b], [HJL22a] turned out to be useful for the study of decay of resonances of many particle systems [BBV21], [BBV22].
In turn, the bounds for the asymptotic decay of resonances of many particle quantum systems turned out to be extremely useful for the study of the Efimov effect. This effect was first discovered by Vitali Efimov in the seventies. It underwent a remarkable transition from purely academic curiosity to a hot topic when experimental evidence of this effect was discovered only recently in ultracold gas of caesium atoms. It is a purely quantum mechanical effect, where a system of three particles interacting with short range forces has an infinite number of bound states with rather unusual properties. This effect cannot be explained classically. It is caused by an effective long range interaction, which emerges thanks to the slowly decaying zero energy resonances of two particle subsystems.
By generalizing the approach of [HJL22b], Barth, Bitter, and Vugalter in [BBV21], [BBV22] showed that in many cases of $N \geq4$ particles, zero energy resonances for the $N-1$ subsystems cannot exist at the bottom of the essential spectrum. More precisely, any zero energy resonance already decays so fast at infinity, such that it becomes square integrable, i.e., it is a true ground state at the bottom of the essential spectrum of the $N-1$ particle subsystem.
This fast decay of the resonances of the $N-1$ particle subsystems prevents the emergence of a long-range attractive interaction for the $N$ particle system. In particular, this provides a proof that the Efimov effect does not exist for $N \geq 4$ three dimensional particles, as predicted by physicists early on. Similar results on the absence of the Efimov effect were proven for systems of particles in dimension $d \leq 2$, confirming the predictions made by physicists, except for dimension $d=2$ with $N=4$ particles.
J. D. van der Waals. On the continuity of the gaseous and liquid states. Studies in Statistical Mechanics, XIV. North-Holland Publishing Co., Amsterdam, 1988. Translated from the Dutch, Edited and with an introduction by J. S. Rowlinson.[bibtex]
