Breaking the symmetry in a coupled wave system can result in unusual amplification behavior. In the case of difference parametric amplification the resonant pump frequency is equal to the difference, instead of the sum, frequency of the normal modes. We show that sign reversal in the symmetry relation of parametric coupling give rise to difference parametric amplification as a dual of optical parametric amplification. For optical systems, our result can potentially be used for efficient XUV amplification.PACS numbers: 42.65. Yj, 42.60.Da, 42.65.Sf, 42.50.Nn Parametric processes are essential to quantum optical applications including frequency conversion, quantum communication, and nonclassical state generation [1][2][3][4]. In particular, the application of squeezed light in precision measurement has led to enhanced sensitivity for gravitational wave detection [5]. Parametric interaction occurs when driving a nonlinear dipole with two frequency inputs. In a doubly resonant cavity, two nondegenerate target frequencies, ω e > ω g , can be parametrically coupled to a pump frequency through a nonlinear medium [6,7]. When the pump frequency ν is equal to the sum-frequency Σω ≡ ω e + ω g or the differencefrequency ∆ω ≡ ω e −ω g , resonant parametric interaction occurs. A sum-frequency will facilitate energy transfer from the pump field E p to the target fields E e,g , leading to amplification. A difference-frequency will promote energy exchange between the target fields without changing their total energy [8]. In the framework of quantum optics, the former corresponds to anti-Jaynes-Cummings interaction and the latter amounts to Jaynes-Cummings interaction [9,10].In a recent proposal by Svidzinsky et al.[11], a semiclassical approach was used to show that JaynesCummings interaction could lead to strong amplification of light in a superradiant atomic gas, if such a coupled system is driven with an external difference-frequency pump. This quickly leads to the conceptual difficulty that energy conservation is violated. In optical parametric amplification (OPA) energy transfers from the pump field to the target fields because one sum-frequency photon, having higher energy, breaks into two target-frequency photons with smaller energy [10]. In the case where the difference-frequency pump drives the amplification, such a photon picture cannot apply since the energy of one difference-frequency photon is less than the total energy of two target-frequency photons. Assuming that this effect exists, what is then the mechanism for energy transfer? To shed light on this puzzle, we turn to Maxwell equations where OPA was originally studied [12][13][14].In this Letter, we show that difference parametric amplification (DPA), i.e. amplification based on a difference-frequency drive, does not violate energy conservation at the level of classical physics. We illustrate the dualism between DPA and OPA through the symmetry relation of parametric coupling. Given that quantum mechanics is a more superior theory than classical mechanics, a...