Periodic Reporting for period 1 - PhonologiQ (Theoretical Investigation of Surface Phonon Polariton-Based Quantum Photonic Circuits)
Okres sprawozdawczy: 2022-10-01 do 2024-09-30
The key idea of my work is to produce a model of quantum electrodynamics that is robust enough to define the full dynamics of the phonon-polariton states of single or coupled finite polar crystal particles. As this class of particles may provide an avenue toward engineering more energy- and time-efficient computing architectures in both classical and quantum computing paradigms, a complete and economical theory of the quantum dynamics of such particles could have a materially beneficial effect on future research efforts in optical computing and related fields.
In particular, I built my work upon the theory of macroscopic quantum electrodynamics (MQED). This theory was invented in the early 2000s to capture the quantum observables of nanometer- to micron-scale crystals or particles made of lossy materials without the need to keep track of individual atomic states. While it has been used successfully since then to model certain quantum fluctuation phenomena, it has not yet been broadly applied to practical problems in cavity optics, i.e. the study of light “trapped” in or near small structures.
Much of the theoretical work that has unlocked progress in cavity optics experiments this millennium has been based on classical cavity electrodynamics techniques. The most successful such technique is called a mode decomposition, wherein the independent characteristic patterns of motion of the currents and fields of a particle are identified. Secondary effects, such as influence of some electromagnetic driving source or nearby resonances, can then be treated as perturbations of the underlying mode structure. These mode theories allow intuitive, economical models of such ensembles to be constructed to capture important information—how and where energy is being transferred between particles, how interactions and hybridization affect observed signals, etc.—without the overhead cost of running expensive full-field simulations.
MQED does not yet contain the tools to reproduce such a mode decomposition, even though the separation of a system’s dynamics into individual state motion is more important to do in quantum mechanics than in classical mechanics. Adding mode-decomposition functionality to MQED is therefore a key first step in recreating the success of classical cavity optics.
In particular, I built my work upon the theory of macroscopic quantum electrodynamics (MQED). This theory was invented in the early 2000s to capture the quantum observables of nanometer- to micron-scale crystals or particles made of lossy materials without the need to keep track of individual atomic states. While it has been used successfully since then to model certain quantum fluctuation phenomena, it has not yet been broadly applied to practical problems in cavity optics, i.e. the study of light “trapped” in or near small structures.
Much of the theoretical work that has unlocked progress in cavity optics experiments this millennium has been based on classical cavity electrodynamics techniques. The most successful such technique is called a mode decomposition, wherein the independent characteristic patterns of motion of the currents and fields of a particle are identified. Secondary effects, such as influence of some electromagnetic driving source or nearby resonances, can then be treated as perturbations of the underlying mode structure. These mode theories allow intuitive, economical models of such ensembles to be constructed to capture important information—how and where energy is being transferred between particles, how interactions and hybridization affect observed signals, etc.—without the overhead cost of running expensive full-field simulations.
MQED does not yet contain the tools to reproduce such a mode decomposition, even though the separation of a system’s dynamics into individual state motion is more important to do in quantum mechanics than in classical mechanics. Adding mode-decomposition functionality to MQED is therefore a key first step in recreating the success of classical cavity optics.
I have pursued this approach by building on studies of polariton formation in infinite (boundaryless) dielectrics from the early 1990s. In contrast to the current formulation of finite-particle MQED, which focuses on describing polariton fields as superpositions of the observable fields of the system, the finite-medium formulation of MQED describes the system’s observables in as excitations of polariton modes. While these two formulations of quantum mechanics are equally correct, only the second provides the intuitive physical picture that I’m seeking.
Applying the mode-expansion technique to a sphere made of a model polar crystal, the Hamiltonian of the system describes the motion of a set of discrete quantum states that describe the motion of the particle’s nuclei. One can think of these states as bosonic particle-in-a-box states, akin to the states of the photon field under a box-quantization regime, that describe the amplitudes of oscillation of the particle’s nuclei in spherical harmonic patterns. These polarization states are coupled to a series of state continua that model losses to thermal reservoirs and interactions with the photon field. Because the number of states of the continua is enormous and a state exists at every possible resonance frequency, these reservoirs allow the discrete states to transfer energy to the continua and never recover it.
As is discussed below, one of the primary challenges of constructing this model lies in the diagonalization of the coupled light-matter system, a process that allows the independent patterns of motion of the system as a whole to be discovered. With these new hybrid modes calculated, the Hamiltonian takes a simple form, describing a series of polariton modes into which quanta can be deposited by external sources or from which information can be harvested. Importantly, just as the coupling between the polarization modes and the continua is straightforward to implement mathematically, so too are the couplings between the final polariton modes and any external sources.
The primary achievement of this work has been to accomplish the full diagonalization of the Hamiltonian of a finite sphere of the kind described above, producing an analytic description of the quantum operators that describe the motion of the system’s independent polariton modes.
Applying the mode-expansion technique to a sphere made of a model polar crystal, the Hamiltonian of the system describes the motion of a set of discrete quantum states that describe the motion of the particle’s nuclei. One can think of these states as bosonic particle-in-a-box states, akin to the states of the photon field under a box-quantization regime, that describe the amplitudes of oscillation of the particle’s nuclei in spherical harmonic patterns. These polarization states are coupled to a series of state continua that model losses to thermal reservoirs and interactions with the photon field. Because the number of states of the continua is enormous and a state exists at every possible resonance frequency, these reservoirs allow the discrete states to transfer energy to the continua and never recover it.
As is discussed below, one of the primary challenges of constructing this model lies in the diagonalization of the coupled light-matter system, a process that allows the independent patterns of motion of the system as a whole to be discovered. With these new hybrid modes calculated, the Hamiltonian takes a simple form, describing a series of polariton modes into which quanta can be deposited by external sources or from which information can be harvested. Importantly, just as the coupling between the polarization modes and the continua is straightforward to implement mathematically, so too are the couplings between the final polariton modes and any external sources.
The primary achievement of this work has been to accomplish the full diagonalization of the Hamiltonian of a finite sphere of the kind described above, producing an analytic description of the quantum operators that describe the motion of the system’s independent polariton modes.
The main contributions of my work in pushing the state of the art forward are two new methods I invented for setting up the Hamiltonian described above and then diagonalizing it.
The first method stems from the fact that MQED uses a physical object called a polarization field to approximate the large number of electrons and nuclei in a particle with a simpler fluid-like description. The computational savings afforded by this simplification come with a side effect: the polarization field isn’t perfectly well-defined unless the boundary of the particle is treated with assumptions beyond the usual Maxwellian prescription. Such assumptions, often referred to as additional boundary conditions, are under active research and are avoided wherever they are not strictly needed. In many experimentally relevant systems, the polarization field’s lack of definition turns out not to be an issue when calculating observables. However, if a reframing of MQED using individual states is to be created, the polarization field needs to be treated as the result of the excitation of the states of the system, such that the system’s states cannot be well-defined unless the polarization field is also.
Therefore, after identifying this issue, I derived new boundary conditions for the particle consistent with both the requirements of quantum mechanics and the subfields of classical electrodynamics that treat boundaries with more care (for example, nonlocal dielectric research). These boundary conditions force the surface-perpendicular component of each mode of the polarization field to go to zero at the surface of the sphere, guaranteeing that the mode decomposition provides a complete reconstruction of the total field and prohibiting the oscillating nuclei within the particle from “spilling out” into the space surrounding the sphere. These new boundary conditions help to merge the disconnected mathematical frameworks of quantum mechanics and electromagnetism in a new and powerful way, introducing a “particle-in-a-box” picture of optical cavity resonances rigorously without forcing undue simplifications onto the system.
The second major challenge of my problem was mathematical in nature, rather than physical. Starting from the coupled light-matter Hamiltonian described above, one arrives at a diagonalization problem with a structure that has never been properly treated before. Specifically, working at the level of creation and annihilation operators that describe the input and output of quanta into the various states of the problem, the Hamiltonian separates into a series of independent subsystems based on angular symmetry. Each subsystem features N discrete polarization states coupled to N+1 continuous reservoirs that model thermal and radiation losses. Here, N is taken to be a large number. Existing techniques do not provide solutions except when the ratios of discrete states and continua are 1:N or N:1, such that I needed to generate a new solution by iteratively diagonalizing the Hamiltonian as a succession of coupled 1:N problems. Importantly, this solution method can be applied to a wide array of problems, such that this research can be extended to future coupled-mode problems.
The first method stems from the fact that MQED uses a physical object called a polarization field to approximate the large number of electrons and nuclei in a particle with a simpler fluid-like description. The computational savings afforded by this simplification come with a side effect: the polarization field isn’t perfectly well-defined unless the boundary of the particle is treated with assumptions beyond the usual Maxwellian prescription. Such assumptions, often referred to as additional boundary conditions, are under active research and are avoided wherever they are not strictly needed. In many experimentally relevant systems, the polarization field’s lack of definition turns out not to be an issue when calculating observables. However, if a reframing of MQED using individual states is to be created, the polarization field needs to be treated as the result of the excitation of the states of the system, such that the system’s states cannot be well-defined unless the polarization field is also.
Therefore, after identifying this issue, I derived new boundary conditions for the particle consistent with both the requirements of quantum mechanics and the subfields of classical electrodynamics that treat boundaries with more care (for example, nonlocal dielectric research). These boundary conditions force the surface-perpendicular component of each mode of the polarization field to go to zero at the surface of the sphere, guaranteeing that the mode decomposition provides a complete reconstruction of the total field and prohibiting the oscillating nuclei within the particle from “spilling out” into the space surrounding the sphere. These new boundary conditions help to merge the disconnected mathematical frameworks of quantum mechanics and electromagnetism in a new and powerful way, introducing a “particle-in-a-box” picture of optical cavity resonances rigorously without forcing undue simplifications onto the system.
The second major challenge of my problem was mathematical in nature, rather than physical. Starting from the coupled light-matter Hamiltonian described above, one arrives at a diagonalization problem with a structure that has never been properly treated before. Specifically, working at the level of creation and annihilation operators that describe the input and output of quanta into the various states of the problem, the Hamiltonian separates into a series of independent subsystems based on angular symmetry. Each subsystem features N discrete polarization states coupled to N+1 continuous reservoirs that model thermal and radiation losses. Here, N is taken to be a large number. Existing techniques do not provide solutions except when the ratios of discrete states and continua are 1:N or N:1, such that I needed to generate a new solution by iteratively diagonalizing the Hamiltonian as a succession of coupled 1:N problems. Importantly, this solution method can be applied to a wide array of problems, such that this research can be extended to future coupled-mode problems.