Investigations into quantum light-matter interactions, their approximations and applications
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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. My 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. My 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. Using 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. Our 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. We 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. In 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.
Cite this version of the work
Nicholas Funai (2021). Investigations into quantum light-matter interactions, their approximations and applications. UWSpace. http://hdl.handle.net/10012/16741