Carl M. Bender, Mariagiovanna Gianfreda

In 1980 Englert examined the classic problem of the electromagnetic self-force on an oscillating charged particle. His approach, which was based on an earlier idea of Bateman, was to introduce a charge-conjugate particle and to show that the two-particle system is Hamiltonian. Unfortunately, Englert’s model did not solve the problem of runaway modes, and the corresponding quantum theory had ghost states. It is shown here that Englert’s Hamiltonian is PT symmetric, and that the problems with his model arise because the PT symmetry is broken at both the classical and quantum level. However, by allowing the charged particles to interact and by adjusting the coupling parameters to put the model into an unbroken PT-symmetric region, one eliminates the classical runaway modes and obtains a corresponding quantum system that is ghost free.

http://arxiv.org/abs/1409.3828

High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Bo Peng, Sahin Kaya Ozdemir, Fuchuan Lei, Faraz Monifi, Mariagiovanna Gianfreda, Gui Lu Long, Shanhui Fan, Franco Nori, Carl M. Bender, Lan Yang

Optical systems combining balanced loss and gain profiles provide a unique platform to implement classical analogues of quantum systems described by non-Hermitian parity-time- (PT-) symmetric Hamiltonians and to originate new synthetic materials with novel properties. To date, experimental works on PT-symmetric optical systems have been limited to waveguides in which resonances do not play a role. Here we report the first demonstration of PT-symmetry breaking in optical resonator systems by using two directly coupled on-chip optical whispering-gallery-mode (WGM) microtoroid silica resonators. Gain in one of the resonators is provided by optically pumping Erbium (Er3+) ions embedded in the silica matrix; the other resonator exhibits passive loss. The coupling strength between the resonators is adjusted by using nanopositioning stages to tune their distance. We have observed reciprocal behavior of the PT-symmetric system in the linear regime, as well as a transition to nonreciprocity in the PT symmetry-breaking phase transition due to the significant enhancement of nonlinearity in the broken-symmetry phase. Our results represent a significant advance towards a new generation of synthetic optical systems enabling on-chip manipulation and control of light propagation.

http://arxiv.org/abs/1308.4564

Optics (physics.optics); Materials Science (cond-mat.mtrl-sci); Mathematical Physics (math-ph); Classical Physics (physics.class-ph); Quantum Physics (quant-ph)

Carl M. Bender, Mariagiovanna Gianfreda

The inspiration for this theoretical paper comes from recent experiments on a PT-symmetric system of two coupled optical whispering galleries (optical resonators). The optical system can be modeled as a pair of coupled linear oscillators, one with gain and the other with loss. If the coupled oscillators have a balanced loss and gain, the system is described by a Hamiltonian and the energy is conserved. This theoretical model exhibits two PT transitions depending on the size of the coupling parameter \epsilon. For small \epsilon the PT symmetry is broken and the system is not in equilibrium, but when \epsilon becomes sufficiently large, the system undergoes a transition to an equilibrium phase in which the PT symmetry is unbroken. For very large \(\epsilon\) the system undergoes a second transition and is no longer in equilibrium. The classical and the quantized versions of the system exhibit transitions at exactly the same values of \(\epsilon\).

http://arxiv.org/abs/1305.7107

High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Carl M. Bender, Mariagiovanna Gianfreda

The C operator in PT-symmetric quantum mechanics satisfies a system of three simultaneous algebraic operator equations, \(C^2=1\), \([C,PT]=0\), and \([C,H]=0\). These equations are difficult to solve exactly, so perturbative methods have been used in the past to calculate C. The usual approach has been to express the Hamiltonian as \(H=H_0+\epsilon H_1\), and to seek a solution for C in the form \(C=e^Q P\), where \(Q=Q(q,p)\) is odd in the momentum p, even in the coordinate q, and has a perturbation expansion of the form \(Q=\epsilon Q_1+\epsilon^3 Q_3+\epsilon^5 Q_5+\ldots\). [In previous work it has always been assumed that the coefficients of even powers of $\epsilon$ in this expansion would be absent because their presence would violate the condition that \(Q(p,q)\) is odd in p.] In an earlier paper it was argued that the C operator is not unique because the perturbation coefficient \(Q_1\) is nonunique. Here, the nonuniqueness of C is demonstrated at a more fundamental level: It is shown that the perturbation expansion for Q actually has the more general form \(Q=Q_0+\epsilon Q_1+\epsilon^2 Q_2+\ldots\) in which {\it all} powers and not just odd powers of \(\epsilon\) appear. For the case in which \(H_0\) is the harmonic-oscillator Hamiltonian, \(Q_0\) is calculated exactly and in closed form and it is shown explicitly to be nonunique. The results are verified by using powerful summation procedures based on analytic continuation. It is also shown how to calculate the higher coefficients in the perturbation series for Q.

http://arxiv.org/abs/1302.7047

High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)