Anti-parity–time symmetry with flying atoms

Abstract

The recently-developed notion of 'parity-time (PT) symmetry' in optical systems with a controlled gain-loss interplay has spawned an intriguing way of achieving optical behaviors that are presently unattainable with standard arrangements. In most experimental studies so far, however, the implementations rely highly on the advances of nanotechnologies and sophisticated fabrication techniques to synthesize solid-state materials. Here, we report the first experimental demonstration of optical anti-PT symmetry, a counterpart of conventional PT symmetry, in a warm atomic-vapor cell. By exploiting rapid coherence transport via flying atoms, our scheme illustrates essential features of anti-PT symmetry with an unprecedented precision on phase-transition threshold, and substantially reduces experimental complexity and cost. This result represents a significant advance in non-Hermitian optics by bridging a firm connection with the field of atomic, molecular and optical physics, where novel phenomena and applications in quantum and nonlinear optics aided by (anti-)PT symmetry could be anticipated.Canonical quantum mechanics postulates Hermitian Hamiltonians to describe closed physical systems so that the system energy is conserved with real eigenvalues and the orthogonality between eigenstates with different eigen-energy is ensured. For systems with open boundaries, non-Hermitian Hamiltonians with complex eigenvalues and non-orthogonal eigen-functions are commonly expected. However, a counterintuitive discovery by Bender and Boettcher in 1998 1 has radically challenged this cognition and evoked considerable efforts on extending canonical quantum theory to non-Hermitian Hamiltonian systems 2,3,4 . In their pioneering work 1 , they showed that below some phase-transition point, a wide class of non-Hermitian Hamiltonians (̂) display entirely real spectra if they are invariant under the anti-linear parity-time (PT) operator (̂̂), [̂,̂̂] = 0 . In non-relativistic quantum mechanics, governed by Schrӧdinger equation, a necessary but not sufficient condition 1 for PT symmetry to hold mandates the complex potential involved to satisfy ( ) = * (− ), which implies its real (imaginary) part is an even (odd) function of position. Even more exciting is the occurrence of a sharp, symmetry-breaking transition once a non-Hermitian parameter crosses an exceptional point. As such, the Hamiltonian and PT operator no longer share the same set of eigen-functions and the real eigen-spectra of the system start to become complex.Despite the ramifications of these theoretical developments 1,2,3,4 , unfortunately, quantum mechanics is by nature an Hermitian theory and thus any attempt to observe PT symmetry in such systems is out of reach. On the other hand, due to the presence of gain and loss, optics has been recognized as a fertile ground for experimental investigations of PT symmetry. Given such optical settings, they can be fully realized without introducing any conflict with the Hermiticity of standard quantum mechanics. Thanks to the mathemat...

Pseudo-Hermitian Hamiltonians generating waveguide mode evolution

“…In this case, the eigenvalues are either purely imaginary, or occur in negative-conjugate pairs [20][21][22][23]. A physical example of such a symmetry can be found in parametric amplifiers, where the P operator interchanges signal and idler waveguide channels [21].…”

“…Real eigenvalues correspond to propagating modes, and complex eigenvalues correspond to evanescent (in-gap) modes. As shown below, H supports real eigenvalues despite being nonHermitian because it satisfies a certain pair of symmetries: pseudo-Hermiticity [13][14][15][16] and anti-PT symmetry [20][21][22][23]. These symmetries are tied to the physical conditions of flux conservation and time-reversal invariance in the underlying waveguide.…”

“…We show that they form a subset of the "pseudo-Hermitian" matrices, a class of non-Hermitian matrices identified by Mostafazadeh and co-workers as a generalization of the PT symmetry concept [13][14][15][16][17][18][19]. This subset is restricted further by a generalized anti-PT symmetry [20][21][22][23], where the P operation interchanges forward-going and backward-going waveguide modes. A * Electronic address: yidong@ntu.edu.sg similar anti-PT symmetry was previously identified by Sukhorukov et al in the context of beam evolution in parametric amplifiers [21].…”

Wide-range-tunable Dirac-cone band structure in a chiral-time-symmetric non-Hermitian system

“…We say that a system with Hamiltonian H has chiral symmetry, if {H, C} = 0. The physics of C depends on the model discussed [42][43][44][45][46][47]. Here we emphasize "chiral symmetry" due to its anticommutation relation with its Hamiltonians.…”

“…However, the CT symmetry is like anti PT symmetry [46]. Actually, an anti-PT -symmetric Hamiltonian can be simply constructed from a conventional PT -symmetric Hamiltonian by multiplying i.…”