Abstract Most theoretical studies for correlated light–matter systems are performed within the long-wavelength limit, i.e., the electromagnetic field is assumed to be spatially uniform. In this limit the so-called length-gauge transformation for a fully quantized light–matter system gives rise to a dipole self-energy term in the Hamiltonian, i.e., a harmonic potential of the total dipole matter moment. In practice this term is often discarded as it is assumed to be subsumed in the kinetic energy term. In this work we show the necessity of the dipole self-energy term. First and foremost, without it the light–matter system in the long-wavelength limit does not have a ground-state, i.e., the combined light–matter system is unstable. Further, the mixing of matter and photon degrees of freedom due to the length-gauge transformation, which also changes the representation of the translation operator for matter, gives rise to the Maxwell equations in matter and the omittance of the dipole self-energy leads to a violation of these equations. Specifically we show that without the dipole self-energy the so-called ‘depolarization shift’ is not properly described. Finally we show that this term also arises if we perform the semi-classical limit after the length-gauge transformation. In contrast to the standard approach where the semi-classical limit is performed before the length-gauge transformation, the resulting Hamiltonian is bounded from below and thus supports ground-states. This is very important for practical calculations and for density-functional variational implementations of the non-relativistic QED formalism. For example, the existence of a combined light–matter ground-state allows one to calculate the Stark shift non-perturbatively.
Abstract Most theoretical studies for correlated light–matter systems are performed within the long-wavelength limit, i.e., the electromagnetic field is assumed to be spatially uniform. In this limit the so-called length-gauge transformation for a fully quantized light–matter system gives rise to a dipole self-energy term in the Hamiltonian, i.e., a harmonic potential of the total dipole matter moment. In practice this term is often discarded as it is assumed to be subsumed in the kinetic energy term. In this work we show the necessity of the dipole self-energy term. First and foremost, without it the light–matter system in the long-wavelength limit does not have a ground-state, i.e., the combined light–matter system is unstable. Further, the mixing of matter and photon degrees of freedom due to the length-gauge transformation, which also changes the representation of the translation operator for matter, gives rise to the Maxwell equations in matter and the omittance of the dipole self-energy leads to a violation of these equations. Specifically we show that without the dipole self-energy the so-called ‘depolarization shift’ is not properly described. Finally we show that this term also arises if we perform the semi-classical limit after the length-gauge transformation. In contrast to the standard approach where the semi-classical limit is performed before the length-gauge transformation, the resulting Hamiltonian is bounded from below and thus supports ground-states. This is very important for practical calculations and for density-functional variational implementations of the non-relativistic QED formalism. For example, the existence of a combined light–matter ground-state allows one to calculate the Stark shift non-perturbatively.
We present a general theoretical framework to capture light-matter interactions beyond the electric-dipole approximation (EDA), applicable to extended nano- and microscale materials interacting with spatially structured electric fields without truncation at finite multipolar order. The approach is based on the Power-Zienau-Woolley (PZW) Hamiltonian for light-matter interactions and a representation of the material's Hamiltonian in a basis of maximally localized Wannier functions (MLWFs), obtainable from first-principles calculations. We utilize this approach to clarify the limitations of the ubiquitous dipole approximation. We consider electric fields with both uniform and non-uniform intensities and a range of ratios of system size to the wavelength of light. Through this analysis, we identify the conditions under which the EDA breaks down, leading to significant errors in the light-induced dynamics. Contrary to conventional belief, we find that the EDA is remarkably robust for uniformly illuminated 1-D or 2-D materials when light propagates perpendicular to the material. For 3-D materials or non-perpendicular illumination of lower-dimensional materials, conventional wisdom holds and the EDA begins to break down when the wavelength becomes comparable to the system size. Furthermore, the EDA fails when the material is illuminated partially or non-uniformly. For slowly varying field intensities this failure can be corrected by finite-order multipolar corrections. However, for fields that vary substantially, correcting via multipolar terms becomes computationally impractical. In contrast, our approach captures beyond-dipole light-matter interactions at the computational cost of a standard dipole calculation. This efficiency enables accurate first-principles simulations of spatially structured light-matter dynamics in nanoscale devices, quantum materials, and interfaces.
The loss of gauge invariance in models of light-matter interaction which arises from material and photonic space truncation can pose significant challenges to conventional quantum optical models when matter and light strongly hybridize. In structured photonic environments, necessary in practice to achieve strong light-matter coupling, a rigorous model of field quantization within the medium is also needed. Here we use the framework of macroscopic QED by quantizing the fields in an arbitrary material system, with a spatially dependent dispersive and absorptive dielectric, starting from a fundamental light-matter action. We truncate the material and mode degrees of freedom while respecting the gauge principle by imposing a partial gauge-fixing constraint during canonical quantization, which admits a large number of gauges including the Coulomb and multipolar gauges commonly used in quantum optics. We also consider gauge conditions with explicit time dependence, enabling us to unambiguously introduce additional phenomenologically time-dependent light-matter interactions in any gauge. Our results allow one to derive rigorous nonrelativistic models of ultrastrong light-matter interactions in structured photonic environments with no gauge ambiguity. Results for two-level systems and the dipole approximation are discussed, as well as how to go beyond the dipole approximation for effective single-particle models. By comparing with the limiting case of an inhomogeneous dielectric, where dispersion and absorption can be neglected and the fields can be expanded in terms of the generalized transverse eigenfunctions of the dielectric, we show how lossy systems can introduce an additional gauge ambiguity, which we resolve and predict to have fundamental implications for open quantum system models. Finally, we show how observables in mode-truncated systems can be calculated without ambiguity by using a simple gauge-invariant model of photodetection.
Classical, semi-classical, and quantum-field descriptions for the interaction of light with matter are systematically discussed. Applications of interest include precise determinations of the linear and the non-linear electromagnetic response relevant to resonant pump-probe optical phenomena, such as electromagnetically induced transparency. In the quantum-mechanical description of matter systems, we introduce a general reduced-density-matrix framework. Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations are developed in a unified and self-consistent manner, using a Liouville-space operator representation. A preliminary semi-classical perturbation treatment of the electromagnetic interaction is adopted, in which the electromagnetic field is described as a classical field satisfying the Maxwell equations. Compact Liouville-space operator expressions are derived for the linear and the general (nth order) non-linear electromagnetic-response tensors describing moving many-electron systems. The tetradic matrix elements of the Liouville-space self-energy operators, which are introduced in the time-domain and frequency-domain formulations, are evaluated for environmental collisional and radiative interactions, in order to provide explicit forms for the quantum kinetic equations and the spectral-line shape formulas. It is emphasized that a quantized-field approach is essential for a fully self-consistent quantum-mechanical description of the interacting light-matter system.
Devoting some thought to the interactions between light and matter quickly conjure a myriad of different possibilities; different models for light and matter, different possible interaction Hamiltonians, different simplifying approximations and different initial conditions. This of course means that light-matter interactions has just as many uses and surprises yet to be explored. In this thesis we will approach light-matter interactions from three different perspectives: 1) How can light-matter interactions be used to manipulate quantum fields into exotic energy distributions, 2) How well does a classical light-matter approximation carry through to quantum light-matter interactions and 3) In a relativistic theory, what are the causal consequences of approximations commonly used to reduce the analytical and numerical complexity of quantum fields. \n \nMy first aim in this thesis is to develop an operationally feasible procedure for generating exotic energy distributions. Current proposals for generating negative energy densities include using the dynamical Casimir effects or by squeezed vacua, which require relativistically accelerating mirrors or non-linear crystals. Instead we seek an optimal protocol that does not require relativistically accelerating elements and exploits the coherent control of a detector. The quantum energy teleportation (QET) protocol is ideal for this task, and we present a QET protocol acting on a relativistic scalar field optimised for the generation of negative energy densities. We show that QET can be used to generate local negative energy densities using stationary qubits. In addition we discuss the consequences of detector smearings on the (QET generated) energy distribution, providing simple and intuitive guidelines for sculpting negative energy distributions. We also show that this protocol is capable of generating regions with an arbitrary amount of negative energy, with the total amount of negative energy ∆E increasing as the negative energy well’s width ∆r is decreased by ∆E ∼ ∆r⁻³ . However, this is accompanied by increasingly large (∼ ∆r⁻³ ) positive energy peaks on either side of the well. We further find this energy-distance scaling saturates the quantum interest conjecture, suggesting the near optimality of QET for generating negative energy densities. \n \nMy second aim is to determine the validity of the dipole model of light-matter interaction with a quantum EM field. This model, classically derived from the more fundamental minimal model, is dependent on the existence of a dominant mode, whose wavelength is required to be much larger than the size of the atom it is interacting with (dipole approximation criterion). Quantisation of the EM field result in vacuum fluctuations without a dominant wavelength. Past works have attempted to overcome this by using point-like atoms, however this introduces UV divergences in the response of the atoms and is inadequate for describing light-matter interactions. The full influence of gauge considerations and quantum behaviour in light matter interaction is quite involved and complex, particularly when the atomic nucleus’ mass is considered finite and the centre of mass degrees of freedom become relevant. We consider effective models (infinite nucleus mass), which neglect the centre of mass degrees of freedom and their additional complexity, whilst allowing accurate investigations of the electronic orbital behaviour. Here we will attempt to determine under what circumstances the dipole and minimal models agree; and hope to clarify and extend the dipole approximation validity criteria for general use in quantum fields. \n \nUsing the 'dressed state' formalism to remove gauge issues, we compare the transition probabilities of Hydrogen-like atoms under both models to determine the validity of the dipole model. We show that for atomic transitions with an initial EM vacuum state, both models noticeably disagree for short interaction times (i.e. shorter than the light-crossing time of the atom). We find the transition rates predicted by both models begin to converge for longer interaction times, particularly when considering vacuum excitations. Vacuum emissions have the additional requirement that the atomic energy gap must satisfy the dipole approximation criterion due to single mode approximation effects. In addition, we find that shrinking the atom (by increasing the atom’s proton number) does not improve the accuracy of the dipole model, a result of the infinite number of UV modes in the vacuum fluctuations. We determine that the dipole model can be used with a quantum EM field, provided any intrinsically dominant modes, e.g. atomic energy gap (for vacuum emissions) or excited EM modes, satisfy the dipole approximation criterion and the interaction time is longer than the light crossing time of the atom. \n \nOur final aim is to determine the causal consequences of the commonly used rotating wave approximation (RWA) and determine in what regimes it may be accurately used. The RWA removes terms from the (detector-field) interaction Hamiltonian that do not conserve excitation number, e.g. field creation operators tensored with SU(2) raising operators. This is justified by noting that these terms oscillate quickly in time and therefore, for long interaction times (T), will integrate to zero, where long interaction time is defined by the RWA criterion ΩT≫1, where Ω is the detector’s energy gap. We wish to determine the extent of the non-locality and causality violation introduced into this originally local and causal theory; as well as determine in what regime the RWA model can be used as a faithful substitute for the unapproximated UDW model. We quantify these violations by studying the non-locality of the RWA interaction Hamiltonian, inspecting the acausal expectation values of field amplitude squared (ϕ²) and the local energy density (T₀₀); and quantifying the superluminal communication in a 2 qubit communication protocol. \n \nWe verify and extend the previous results by showing the RWA interaction Hamiltonian has a 1/r² polynomial non-locality, proportional to the vacuum Wightman function of the field. We also show that this non-locality translates into causality violations of 1/d⁴ and 1/d⁶ for the expectation values ϕ² and T₀₀ respectively; where d is measured from the surface of the detector’s lightcone and this polynomial causality violation is independent of time. We also find similar 1/d² non-localities in the channel capacity of the 2 qubit communication protocol, with non-local influence in the space-like and time-like regions of the first detector’s interaction. We show that in setups that are not causally sensitive, e.g. measuring field observables within the bulk of a source detector’s light-cone, then the RWA criterion ΩT≫1 is sufficient for the RWA model with interaction strength λ to converge to the unapproximated UDW model with interaction strength λ/2. We establish that this factor of 1/2 comes from a mathematically unsound commutation of limits and integrals. \n \nIn the case of cavity fields, we introduced a numerical trick to make infinite mode sums computationally possible without requiring single or few mode approximations. Using this trick we found the RWA non-local behaviour to be generally similar to the free space case. We find the RWA model introduces some severe polynomial non-localities and causality violations, but if used to model light-matter interactions well within the bulk of the detector’s light-cone and ΩT≫1 then it is a good approximation for the half strength UDW model.
Floquet engineering and cavity quantum electrodynamics represent two complementary descriptions of light-matter interaction, yet their precise connection beyond the small-coupling regime remains subtle. Here, by representing the cavity field in a large-photon-number phase basis, we show that the time-dependent Schrödinger equation of a classically driven system emerges directly from the fully quantized problem, without replacing the field by a classical trajectory or assuming a mean-field decoupling. This establishes a one-to-one correspondence between quasienergies and photon-sector-shifted quantum energies, while providing a direct quantum interpretation of the Floquet mean energy and Anandan phase. Because gauge invariance is exactly preserved at the level of the quantum Hamiltonian, the quantum-to-classical correspondence remains valid beyond perturbative and small-coupling regimes. We further show that light--matter entanglement is encoded in matter--harmonic correlations of the corresponding Floquet mode, thus identifying a semiclassical limit in which Floquet physics emerges without requiring the quantum field itself to approach a coherent classical state.
The space-translation method for treating the interaction of electromagnetic radiation and matter in the electric dipole approximation is investigated from the standpoint of gauge invariance. Unless the wavefunction of the space-translation method is properly transformed, the probability of finding the system in an energy eigenstate is not in general correct. In the electric dipole approximation the probability of finding the system in an energy eigenstate is most conveniently calculated in the electric field gauge where the interaction is of the length form. The length form of the interaction is not in general equal to the velocity form or the acceleration form which comes from the space-translation method. The charged harmonic oscillator is used as an example.