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Arbitrary gauge quantisation of light-matter theories with time-dependent constraints

Adam Stokes, Ahsan Nazir

TL;DR

The paper develops a general framework for quantising light-matter theories with time-dependent holonomic constraints, showing that a unique canonical description exists only when time dependence is built into the Lagrangian from the outset. It defines the irrotational gauge as the gauge in which the Hamiltonian remains correct when time dependence is introduced later at the Hamiltonian level, and demonstrates that the Coulomb gauge is not generally irrotational for time-dependent light-matter interactions. Through concrete applications to superconducting circuits with variable external flux and a moving atom, it shows how external controls impose time-dependent constraints and how gauge choices determine the correct description, including the appearance of Röntgen-current terms. The work provides a unified, gauge-aware approach to time-dependent light-matter dynamics, clarifying when naive phenomenological modulations are valid and how observable spectra depend on the chosen gauge, with implications for the design and interpretation of quantum technologies. Overall, the framework informs how to construct physically meaningful, gauge-consistent models in time-dependent quantum electrodynamics and related platforms.

Abstract

We provide a general framework for the quantisation of light-matter theories with time-dependent holonomic constraints. Unless time dependence is present from the outset at the Lagrangian level, different gauges generally produce non-equivalent canonical theories. The irrotational gauge is defined as that which also yields a correct theory when time dependence is introduced at the Hamiltonian level. Our framework unifies examples of such gauges found in existing literature. In particular, we show that for describing time-dependent light-matter interactions the Coulomb gauge is not generally irrotational, so it does not enjoy any special status.

Arbitrary gauge quantisation of light-matter theories with time-dependent constraints

TL;DR

The paper develops a general framework for quantising light-matter theories with time-dependent holonomic constraints, showing that a unique canonical description exists only when time dependence is built into the Lagrangian from the outset. It defines the irrotational gauge as the gauge in which the Hamiltonian remains correct when time dependence is introduced later at the Hamiltonian level, and demonstrates that the Coulomb gauge is not generally irrotational for time-dependent light-matter interactions. Through concrete applications to superconducting circuits with variable external flux and a moving atom, it shows how external controls impose time-dependent constraints and how gauge choices determine the correct description, including the appearance of Röntgen-current terms. The work provides a unified, gauge-aware approach to time-dependent light-matter dynamics, clarifying when naive phenomenological modulations are valid and how observable spectra depend on the chosen gauge, with implications for the design and interpretation of quantum technologies. Overall, the framework informs how to construct physically meaningful, gauge-consistent models in time-dependent quantum electrodynamics and related platforms.

Abstract

We provide a general framework for the quantisation of light-matter theories with time-dependent holonomic constraints. Unless time dependence is present from the outset at the Lagrangian level, different gauges generally produce non-equivalent canonical theories. The irrotational gauge is defined as that which also yields a correct theory when time dependence is introduced at the Hamiltonian level. Our framework unifies examples of such gauges found in existing literature. In particular, we show that for describing time-dependent light-matter interactions the Coulomb gauge is not generally irrotational, so it does not enjoy any special status.

Paper Structure

This paper contains 16 sections, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Time-dependent constraints
  3. Elimination of coordinates using constraints
  4. The canonical formalism and the irrotational gauge
  5. Hamiltonian
  6. Naive Hamiltonians and the irrotational gauge
  7. Application to light-matter systems
  8. Superconducting circuit with variable external flux
  9. A moving atom
  10. System and Constraints
  11. Hamiltonian
  12. Implications and discussion
  13. Modulated light-matter interactions
  14. Beyond phenomenological modulation
  15. Natural lineshape
  16. ...and 1 more sections

Figures (1)

  • Figure 1: Two Josephson junctions form a closed loop , $L_0$, which is threaded by an external flux $\phi(t)$. The right and left branch fluxes are $x_0$ and $x_1$, respectively. Since $x_0$ and $x_1$ both possess the same orientation as $\phi(t)$, we have $R_{00} =1= R_{01}$. The lower node (0) is designated as ground, and the non-ground node flux is $q =G{\bf x}$. The latter linear combination of the $x_n$ is determined by the choice of spanning tree (gauge).