Overview

Motivation

Quantum information is a promising platform for solving various computation problems, simulating complex systems, and both breaking and resecuring current cybersecurity methods. However, these systems are still being developed, with superconducting circuits in waveguide QED (quantum electrodynamics) platforms currently being one of the leading hardware choices for quantum information processing [1]. One option for a qubit (quantum bit) in superconducting systems is a "giant atom," so called because it breaks the dipole approximation [2]. This is important, because atoms and most other systems that exhibit quantum behavior are so small that they can be treated essentially as single points; their interaction with other systems can be assumed to occur at just one spot. This is the dipole approximation. On the other hand, giant atoms are constructed in such a way that they can have multiple spatially separated interaction points, but otherwise, they act just as any other two (or more) level quantum system. Giant atoms are not atoms themselves; they are specially constructed superconducting circuits.

Routing Problem

An essential element of any information processing system is the ability to control the flow of information within it. At a very low level, how does the qubit get from one logic gate to another? Or, how can a qubit be sent through a network to arrive at the intended destination? One popular scheme uses photons as 'flying' qubits, meaning they transfer the information, and some other hardware (e.g. superconducting qubits) performs the computations. Waveguides are a widely studied platform that can efficiently channel these photons. 

In this study we concern ourselves with routing a single photon from one such information channel to another. At the most fundamental level, we are only guaranteed to know the probability that the photon can be found in a given waveguide moving in a particular direction. Therefore our goal is to route an incident photon to another waveguide in a particular direction with near certainty. To mediate this interaction we introduce giant atoms coupled to each waveguide at two distinct points. 

Methods

For each of the systems we studied, the same general process was used, so a qualitative description is listed here in lieu of repeating them for each system.

The first step was to write out the Hamiltonian of the system, which contains all of the information about the energy of the system. It had contributions from the giant atoms themselves, the waveguides, and the allowed interactions between the two. We used a "real-space" method for writing the contributions of the photons, as in the work of Shen and Fan [3].

Next, we wrote out the general state of our system, which takes the form of a weighted sum of all possible measurable states. Since we are modelling the system with a single photon, the only possible individual states are either finding one photon in one of the waveguides with all atoms in the ground state or a single atom excited with zero photons in the waveguides. 

In quantum mechanics, the Schrodinger equation governs all the physics, so we used the time independent Schrodinger equation, which required both the Hamiltonian and state we had found, to produce equations of motion for the system.

These equations were solved to find the routing probabilities for each of the ports. We were particularly interested in the probabilities of the photon being found on the right end of the upper waveguide. This final step of equation solving and result plotting was done using the simulation software Mathematica.

Conclusions

We looked at various numbers and connection methods for giant atoms within a single photon router. We found that the arrangement method chosen between the giant atoms and waveguides had a major impact on the routing efficiency of the system, and that increasing the number of giant atoms in the system was crucial for increasing the range of parameters for which the system would efficiently route photons (so long as the configuration allowed any efficient routing).

Next steps for this project include looking at the efficiency of a generalized braided and/or nesting configuration when expanded to more atoms, including the effects of atom-atom interactions, including photon losses to the environment, and considering what happens when multiple photons are sent into the system.

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