Carl M. Bender

Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry

The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition of

Complex extension of quantum mechanics

\mathrm{PT}

PT-symmetric quantum mechanics

symmetry, one obtains new infinite classes of complex Hamiltonians whose spectra are also real and positive. These

Parity–time-symmetric whispering-gallery microcavities

\mathrm{PT}

A new perturbative approach to nonlinear problems

symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space. This paper describes the unusual classical and quantum properties of these theories.

Loss-induced suppression and revival of lasing

Requiring that a Hamiltonian be Hermitian is overly restrictive. A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but satisfies the less restrictive and more physical condition of space-time reflection symmetry (PT symmetry). One might expect a non-Hermitian Hamiltonian to lead to a violation of unitarity. However, if PT symmetry is not spontaneously broken, it is possible to construct a previously unnoticed symmetry C of the Hamiltonian. Using C, an inner product whose associated norm is positive definite can be constructed. The procedure is general and works for any PT-symmetric Hamiltonian. Observables exhibit CPT symmetry, and the dynamics is governed by unitary time evolution. This work is not in conflict with conventional quantum mechanics but is rather a complex generalization of it.

Introduction to PT-symmetric quantum theory

This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition H†=H on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition. This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum. However, replacing this mathematical condition by the weaker and more physical requirement H‡=H, where ‡ represents combined parity reflection and time reversal PT, one obtains new classes of complex Hamiltonians whose spectra are still real and positive. This generalization of Hermiticity is investigated using a complex deformation H=p2+x2(ix)e of the harmonic oscillator Hamiltonian, where e is a real parameter. The system exhibits two phases: When e⩾0, the energy spectrum of H is real and positive as a consequence of PT symmetry. However, when −1<e<0, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues b...

Must a Hamiltonian be Hermitian

It is now shown that coupled optical microcavities bear all the hallmarks of parity–time symmetry; that is, the system’s dynamics are unchanged by both time-reversal and mirror transformations. The resonant nature of microcavities results in unusual effects not seen in previous photonic analogues of parity–time-symmetric systems: for example, light travelling in one direction is resonantly enhanced but there are no resonance peaks going the other way.

No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model.

A recently proposed perturbative technique for quantum field theory consists of replacing nonlinear terms in the Lagrangian such as φ4 by (φ2)1+δ and then treating δ as a small parameter. It is shown here that the same approach gives excellent results when applied to difficult nonlinear differential equations such as the Lane–Emden, Thomas–Fermi, Blasius, and Duffing equations.

Faster than Hermitian Quantum Mechanics

Controlling and reversing the effects of loss are major challenges in optical systems. For lasers, losses need to be overcome by a sufficient amount of gain to reach the lasing threshold. In this work, we show how to turn losses into gain by steering the parameters of a system to the vicinity of an exceptional point (EP), which occurs when the eigenvalues and the corresponding eigenstates of a system coalesce. In our system of coupled microresonators, EPs are manifested as the loss-induced suppression and revival of lasing. Below a critical value, adding loss annihilates an existing Raman laser. Beyond this critical threshold, lasing recovers despite the increasing loss, in stark contrast to what would be expected from conventional laser theory. Our results exemplify the counterintuitive features of EPs and present an innovative method for reversing the effect of loss. Introducing loss into a coupled optical system can result in an enhancement of the optical properties. [Also see Perspective by Schwefel] Achieving gain despite increasing loss When energy is pumped into an optically active material, the buildup (or gain) of excitations within the material can reach a critical point where the emission of coherent light, or lasing, can occur. In many systems, however, the buildup of the excitations is suppressed by losses within the material. Overturning conventional wisdom that loss is bad and should be minimized, Peng et al. show that carefully tweaking the coupling strength between the various components of a coupled optical system can actually result in an enhancement of the optical properties by adding more loss into the system (see the Perspective by Schwefel). The results may provide a clever design approach to counteract loss in optical devices. Science, this issue p. 328; see also p. 304