Coherent states in projected Hilbert spaces

TL;DR

This paper develops coherent state bases and phase-space representations for projected Hilbert spaces, enabling efficient analysis of quantum systems with conserved quantities or conditional measurements, with applications in photonic quantum technologies.

Contribution

It introduces new coherent state frameworks and dynamical equations tailored for projected Hilbert spaces, facilitating analysis of complex quantum systems with restrictions.

Findings

Effective calculation of recurrences in anharmonic oscillators

Analysis of phase-noise effects on Schrödinger cat states

Application to boson sampling interferometry with many modes

Abstract

Coherent states in a projected Hilbert space have many useful properties. When there are conserved quantities, a representation of the entire Hilbert space is not necessary. The same issue arises when conditional observations are made with post-selected measurement results. In these cases, only a part of the Hilbert space needs to be represented, and one can define this restriction by way of a projection operator. Here coherent state bases and normally-ordered phase-space representations are introduced for treating such projected Hilbert spaces, including existence theorems and dynamical equations. These techniques are very useful in studying novel optical or microwave integrated photonic quantum technologies, such as boson sampling or Josephson quantum computers. In these cases states become strongly restricted due to inputs, nonlinearities or conditional measurements. This paper focuses on coherent phase states, which have especially simple properties. Practical applications are reported on calculating recurrences in anharmonic oscillators, the effects of arbitrary phase-noise on Schrodinger cat fringe visibility, and on boson sampling interferometry for large numbers of modes.