We conduct a theoretical study of the bistable optical response of a nanoparticle heterodimer comprised of a closely spaced semiconductor quantum dot and metal nanoparticle. The bistable nature of the response results from the interplay between the quantum dots optical nonlinearity and its self-action (feedback) originating from the presence of the metal nanoparticle. We show that the feedback is governed by a complex valued coupling parameter $G$. Both the real and imaginary parts of $G$ ($G_mathrm{R}$ and $G_mathrm{I}$) play an important role in the occurrence of bistability, which is manifested in an S-shaped dependence of the quantum dot excited state population on the intensity of the external field, and hysteresis of the population. From our calculations, we find that at $G_mathrm{R} = 0$, the critical value for bistability to occur is $G_mathrm{I} = 8Gamma$, whereas at $G_mathrm{I} = 0$, the critical value of $G_mathrm{R} = 4Gamma$, where $Gamma$ is the polarization dephasing rate. Thus, there exist two different (limiting) mechanisms of bistability, depending on whether $G_mathrm{R}$ is much larger or much smaller than $G_mathrm{I}$. We also calculate the bistability phase diagram within the systems parameter space: spanned by $G_mathrm{R}$, $G_mathrm{I}$ and $Delta$, the latter being the detuning between the driving frequency and the transition frequency of the quantum dot. Additionally, switching times from the lower stable branch to the upper one (and {it vise versa}) are calculated as a function of the intensity of the driving field. We show that the conditions for bistability to occur can be realized, for example, for a heterodimer comprised of a closely spaced CdSe (or CdSe/ZnSe) quantum dot and a gold nanosphere.
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