Twofold transition inPT-symmetric coupled oscillators

Abstract

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 . For small the PT symmetry is broken and the system is not in equilibrium, but when becomes sufficiently large, the system undergoes a transition to an equilibrium phase in which the PT symmetry is unbroken. For very large 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 .

Higher-order time-symmetry-breaking phase transition due to meeting of an exceptional point and a Fano resonance

“…Therefore we call this a Time-Symmetry Breaking Phase Transition (TSBPT) where the imaginary part of the eigenvalue can be regarded as an order parameter that has a singularity at the critical point in terms of a system control parameter [19][20][21][22][23][24][25][26]. Very * stanaka@p.s.osakafu-u.ac.jp recently TSBPT has been experimentally observed in mesoscopic quantum systems [17,27], and also in many analogous optical systems where interesting collective dynamical properties have been studied, such as superradiance and lasing [28][29][30][31].…”

Systematic pathway toPT-symmetry breaking in scattering systems

“…Spontaneously breaking the PT -symmetry one observes a transition from a real to a complex eigenvalue spectrum [1,3]. Recently it has been impressively demonstrated that PT -symmetry can be realized in wave optical devices [3][4][5][6][7][8][9]. The first studies in this direction were motivated by the fact that the time-dependent Schrödinger equation maps on the paraxial approximation of the electromagnetic wave equation, describing the transverse variation of the electric field [3,10], where the variation on the z axis plays the role of time in the corresponding Schrödinger equation.…”

Algebraic Treatment of 𝓟𝓣-Symmetric Coupled Oscillators

“…PT HPT = H since it is invariant under the combined effect of parity P : {x, y, p x , p y } → {−y, −x, −p y , −p x } and time reversal T : {x, y, p x , p y } → {x, y, −p x , −p y } [1]. Note that we can also choose P : {x, y, p x , p y } → {y, x, p y , p x } for exactly the same purpose.…”

“…In a recent paper Bender et al [1] discussed the classical and quantum-mechanical versions of a system of two coupled linear oscillators one with gain and the other with loss. When the gain and loss parameters are equal the Hamiltonian derived from the classical equations of motion is PT -symmetric and exhibits two PT -transitions in terms of the coupling parameter.…”