  The ratio of two consecutive level spacings has emerged as a very useful
metric in investigating universal features exhibited by complex spectra. It
does not require the knowledge of density of states and is therefore quite
convenient to compute in analyzing the spectrum of a general system. The
Wigner-surmise-like results for the ratio distribution are known for the
invariant classes of Gaussian random matrices. However, for the crossover
ensembles, which are useful in modeling systems with partially broken
symmetries, corresponding results have remained unavailable so far. In this
work, we derive exact results for the distribution and average of the ratio of
two consecutive level spacings in the Gaussian orthogonal to unitary crossover
ensemble using a $3\times 3$ random matrix model. This crossover is useful in
modeling time-reversal symmetry breaking in quantum chaotic systems. Although
based on a $3\times 3$ matrix model, our results can also be applied in the
study of large spectra, provided the symmetry-breaking parameter facilitating
the crossover is suitably scaled. We substantiate this claim by considering
Gaussian and Laguerre crossover ensembles comprising large matrices. Moreover,
we apply our result to investigate the violation of time-reversal invariance in
the quantum kicked rotor system.
