Fluctuation and dissipation are byproducts of coupling to the "environment." The Caldeira-Leggett model, a successful paradigm of quantum Brownian motion, views the environment as a collection of harmonic oscillators linearly coupled to the system. However, symmetry considerations may forbid a linear coupling, e.g., for a neutral particle in quantum electrodynamics. We argue that the absence of linear couplings can lead to a fundamentally different behavior. Specifically, we consider a heavy particle quadratically coupled to quantum fluctuations of the bath. In one dimension the particle undergoes anomalous diffusion, unfolding as a power-law distribution in space, reminiscent of Levy flights. We suggest condensed matter analogs where similar effects may arise.