  We introduce Quantum Time-Frequency Analysis, which expands the approach of
Quantum Harmonic Analysis to include modulations of operators in addition to
translations. This is done by a projective representation of double-phase
space, and we consider the associated matrix coefficients and integrated
representation. This leads to the polarised Cohen's class, which is an
isomorphism from Hilbert-Schmidt operators to a reproducing kernel Hilbert
space, and has orthogonality relations similar to many objects in classical
time-frequency analysis. By considering a class of windows for the polarised
Cohen's class that is smaller than the class of Hilbert-Schmidt operators, then
we find spaces of modulation spaces of operators, and we consider the
properties of these spaces, including discretisation results and mapping
properties between function modulation spaces. We also compare modulation
spaces of operators to known symbol classes for pseudodifferential operators.
In many cases, using rank-one examples of operators, we recover familiar
objects and results from classical time-frequency analysis and the theory of
pseudodifferential operators.
