Observation of parity–time symmetry in optics

One of the fundamental axioms of quantum mechanics is associated with the Hermiticity of physical observables1. In the case of the Hamiltonian operator, this requirement not only implies real eigenenergies but also guarantees probability conservation. Interestingly, a wide class of non-Hermitian Hamiltonians can still show entirely real spectra. Among these are Hamiltonians respecting parity–time (PT) symmetry2–7. Even though the Hermiticity of quantum observables was never in doubt, such concepts have motivated discussions on several fronts in physics, including quantum field theories8, nonHermitian Anderson models9 and open quantum systems10,11, to mention a few. Although the impact of PT symmetry in these fields is still debated, it has been recently realized that optics can provide a fertile ground where PT-related notions can be implemented and experimentally investigated12–15. In this letter we report the first observation of the behaviour of a PT optical coupled system that judiciously involves a complex index potential. We observe both spontaneous PT symmetry breaking and power oscillations violating left–right symmetry. Our results may pave the way towards a new class of PT-synthetic materials with intriguing and unexpected properties that rely on non-reciprocal light propagation and tailored transverse energy flow. Before we introduce the concept of spacetime reflection in optics, we first briefly outline some of the basic aspects of this symmetry within the context of quantum mechanics. In general, a Hamiltonian Ĥ = p̂2/2m+V (x̂) (where x̂ and p̂ are position and momentumoperators respectively,m ismass andV is the potential) is considered to be PT symmetric, PTĤ = ĤPT , provided that it shares common eigenfunctions with the PT operator1,16–21. This condition corresponds to an exact or unbroken PT symmetry, as opposed to that of broken PT symmetry, where, even though PT Ĥ = ĤPT is still valid, Ĥ and PT (or any other antilinear operator) possess different eigenvectors22. For the case considered here, given that the action of the parity P and time T operators is defined as p̂→ −p̂, x̂ → −x̂ and p̂→ −p̂, x̂ → x̂, i→ −i, respectively, it then follows that a necessary (but not sufficient) condition for aHamiltonian to be PT symmetric isV (x̂)=V (−x̂). In other words, PT symmetry requires that the real part of the potential V is an even function of position x , whereas the imaginary part is odd; that is, the Hamiltonian must have the form Ĥ = p̂/2m+VR(x̂)+ iεVI(x̂), where VR,I are the symmetric and antisymmetric components of V , respectively12–14. Clearly, if ε = 0, this Hamiltonian is Hermitian. It turns out that, even if the antisymmetric imaginary component is finite, this class of potentials can still allow for both bound and radiation states, all with entirely real spectra. This is possible as long as ε is below some threshold, ε < εth. If, on the other hand, this limit is crossed