  We investigate correlations between orthogonally polarized cavity modes of a
bimodal micropillar laser with a single layer of self-assembled quantum dots in
the active region. While one emission mode of the microlaser demonstrates a
characteristic s-shaped input-output curve, the output intensity of the second
mode saturates and even decreases with increasing injection current above
threshold. Measuring the photon auto-correlation function g^{(2)}(\tau) of the
light emission confirms the onset of lasing in the first mode with g^{(2)}(0)
approaching unity above threshold. In contrast, strong photon bunching
associated with super-thermal values of g^{(2)}(0) is detected for the other
mode for currents above threshold. This behavior is attributed to gain
competition of the two modes induced by the common gain material, which is
confirmed by photon crosscorrelation measurements revealing a clear
anti-correlation between emission events of the two modes. The experimental
studies are in excellent qualitative agreement with theoretical studies based
on a microscopic semiconductor theory, which we extend to the case of two modes
interacting with the common gain medium. Moreover, we treat the problem by an
extended birth-death model for two interacting modes, which reveals, that the
photon probability distribution of each mode has a double peak structure,
indicating switching behavior of the modes for the pump rates around threshold.
