Linear Hamiltonian

A linear Hamiltonian is a Hamiltonian when the Hamiltonian is at most first-order in terms of the creation and annihilation operators  [1].

$$\hat{H}=\sum _{j,k}{A}_{jk}{\hat{a}}_j^{†}{\hat{a}}_k$$

Examples of first-order linear optical operations include Phase Shifters and Beam Splitters.

References

[1]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Linear Optical Quantum Computing with Photonic Qubits, Reviews of Modern Physics 79, 135 (2007).