PT-symmetric Quantum Mechanics Research Articles

Anti-[formula omitted] transformations and complex non-Hermitian [formula omitted]-symmetric superpartners

We propose a new algebraic formalism for constructing complex non-Hermitian PT-symmetric superpartners by extending a conventional shape-invariant superpotential into the complex domain. The resulting potential is an unbroken super- and parity-time (PT)-symmetric shape-invariant potential with real energy eigenvalues, maintaining this property for all parameter values. In order to restore the probabilistic interpretation within a true quantum theory, a new inner product called the CPT-inner product is defined in PT-symmetric quantum mechanics, replacing the Dirac Hermitian inner product. In this work, we propose a new version of the inner product called the anti-PT (APT)-inner product, 〈A|B〉≡|A〉APT.|B〉, which replaces the previous versions without any additional considerations. This PT-supersymmetric quantum mechanics framework also allows for the unification of various areas of physics, including classical optics and quantum mechanics. To validate the theory, we present exact solutions for optical waveguides and the quantum tunneling probability, demonstrating excellent agreement with experimental data for the probability of crossing the potential barrier in the 3H(d,n)4He reaction.

Observing a changing Hilbert-space inner product

In quantum mechanics, physical states are represented by rays in Hilbert space $\mathscr H$, which is a vector space imbued by an inner product $\langle\,|\,\rangle$, whose physical meaning arises as the overlap $\langle\phi|\psi\rangle$ for $|\psi\rangle$ a pure state (description of preparation) and $\langle\phi|$ a projective measurement. However, current quantum theory does not formally address the consequences of a changing inner product during the interval between preparation and measurement. We establish a theoretical framework for such a changing inner product, which we show is consistent with standard quantum mechanics. Furthermore, we show that this change is described by a quantum channel, which is tomographically observable, and we elucidate how our result is strongly related to the exploding topic of PT-symmetric quantum mechanics. We explain how to realize experimentally a changing inner product for a qubit in terms of a qutrit protocol with a unitary channel.

Observation of PT-symmetric quantum interference

Parity-Time (PT) symmetric quantum mechanics is a complex extension of conventional Hermitian quantum mechanics in which physical observables possess a real eigenvalue spectrum. However, an experimental demonstration of the true quantum nature of PT symmetry has been elusive thus far, as only single-particle physics has been exploited to date. In our work, we demonstrate two-particle quantum interference in a PT-symmetric system. We employ integrated photonic waveguides to reveal that PT-symmetric bunching of indistinguishable photons shows strongly counterintuitive features. We substantiate our experimental results by modelling the system by a quantum master equation, which we analytically solve using Lie algebra methods. Our work paves the way for nonlocal PT-symmetric quantum mechanics as a novel building block for future quantum devices.

PT -symmetric quantum field theory in D dimensions

PT-symmetric quantum mechanics began with a study of the Hamiltonian $H=p^2+x^2(ix)^\varepsilon$. A surprising feature of this non-Hermitian Hamiltonian is that its eigenvalues are discrete, real, and positive when $\varepsilon\geq0$. This paper examines the corresponding quantum-field-theoretic Hamiltonian $H=\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}\phi^2(i\phi)^\varepsilon$ in $D$-dimensional spacetime, where $\phi$ is a pseudoscalar field. It is shown how to calculate the Green's functions as series in powers of $\varepsilon$ directly from the Euclidean partition function. Exact finite expressions for the vacuum energy density, all of the connected $n$-point Green's functions, and the renormalized mass to order $\varepsilon$ are derived for $0\leq D<2$. For $D\geq2$ the one-point Green's function and the renormalized mass are divergent, but perturbative renormalization can be performed. The remarkable spectral properties of PT-symmetric quantum mechanics appear to persist in PT-symmetric quantum field theory.

Two- and four-dimensional representations of the PT - and CPT -symmetric fermionic algebras

Fermionic systems differ from their bosonic counterparts, the main difference with regard to symmetry considerations being that $T^2=-1$ for fermionic systems. In PT-symmetric quantum mechanics an operator has both PT and CPT adjoints. Fermionic operators $\eta$, which are quadratically nilpotent ($\eta^2=0$), and algebras with PT and CPT adjoints can be constructed. These algebras obey different anticommutation relations: $\eta\eta^{PT}+\eta^{PT}\eta=-1$, where $\eta^{PT}$ is the PT adjoint of $\eta$, and $\eta\eta^{CPT}+\eta^{CPT}\eta=1$, where $\eta^{CPT}$ is the CPT adjoint of $\eta$. This paper presents matrix representations for the operator $\eta$ and its PT and CPT adjoints in two and four dimensions. A PT-symmetric second-quantized Hamiltonian modeled on quantum electrodynamics that describes a system of interacting fermions and bosons is constructed within this framework and is solved exactly.

PT -symmetric gain and loss in a rotating Bose-Einstein condensate

PT-symmetric quantum mechanics allows finding stationary states in mean-field systems with balanced gain and loss of particles. In this work we apply this method to rotating Bose-Einstein condensates with contact interaction which are known to support ground states with vortices. Due to the particle exchange with the environment transport phenomena through ultracold gases with vortices can be studied. We find that even strongly interacting rotating systems support stable PT-symmetric ground states, sustaining a current parallel and perpendicular to the vortex cores. The vortices move through the non-uniform particle density and leave or enter the condensate through its borders creating the required net current.

PT Symmetry, Non-Gaussian Path Integrals, and the Quantum Black-Scholes Equation

The Accardi-Boukas quantum Black-Scholes framework, provides a means by which one can apply the Hudson-Parthasarathy quantum stochastic calculus to problems in finance. Solutions to these equations can be modelled using nonlocal diffusion processes, via a Kramers-Moyal expansion, and this provides useful tools to understand their behaviour. In this paper we develop further links between quantum stochastic processes, and nonlocal diffusions, by inverting the question, and showing how certain nonlocal diffusions can be written as quantum stochastic processes. We then go on to show how one can use path integral formalism, and PT symmetric quantum mechanics, to build a non-Gaussian kernel function for the Accardi-Boukas quantum Black-Scholes. Behaviours observed in the real market are a natural model output, rather than something that must be deliberately included.

PT symmetric interpretation of effective potentials

Conventional systems in equilibrium should have convex effective potentials. PT symmetry applies to systems which are in between open and closed systems. A PT symmetric interpretation can allow some non-convex effective potentials to be entirely physical. The oneloop effective potentials of the Higgs field in the Standard Model and the gravitino condensate in dynamically broken supergravity are conventionally unstable at large field values. In the specially simple case of space-independent and time-independent fields, the effective potentials are governed by PT-symmetric quantum mechanics. The PT-symmetric reinterpretation of these models at a quantum-mechanical level eliminates these instabilities and involves unusual semi-classical analysis involving many Riemann sheets. This interpretation is based on a combination of numerical analysis and semi-classical asymptotics.

Light neutrino masses from a non-Hermitian Yukawa theory

Working within the context of PT-symmetric quantum mechanics, we begin by describing a non-Hermitian extension of QED that is both Lorentz invariant and consistent with unitarity. We show that the non-Hermitian Dirac mass matrix of this theory exhibits an exceptional point, corresponding to an effectively massless theory whose conserved current is either right- or left-chiral dominated. With this inspiration, we are able to construct a non- Hermitian model of light Dirac neutrino masses from Hermitian and anti-Hermitian Yukawa couplings that are both of order unity. We finish by highlighting potential phenomenological implications of this model.

Pseudo-Hermitian Systems with P T $\mathcal {P}\mathcal {T}$ -Symmetry: Degeneracy and Krein Space

We show in the present paper that pseudo-Hermitian Hamiltonian systems with even PT-symmetry admit a degeneracy structure. This kind of degeneracy is expected traditionally in the odd PT-symmetric systems which is appropriate to the fermions as shown by Jones-Smith and Mathur [1] who extended PT-symmetric quantum mechanics to the case of odd time-reversal symmetry. We establish that the pseudo-Hermitian Hamiltonians with even PT-symmetry admit a degeneracy structure if the operator PT anticommutes with the metric operator {\eta} which is necessarily indefinite. We also show that the Krein space formulation of the Hilbert space is the convenient framework for the implementation of unbroken PT-symmetry. These general results are illustrated with great details for four-level pseudo-Hermitian Hamiltonian with even PT-symmetry.

On the quantisation of complex higher derivative theories and avoiding the Ostrogradsky ghost

Generic higher derivative theories are believed to be fundamentally unphysical because they contain Ostrogradsky ghosts. We show that within complex classical mechanics it is possible to construct higher derivative theories that circumvent the Ostrogradsky theorem and have a real energy spectrum that is bounded from below. The complex theory can be canonically quantised. The resulting quantum theory does not suffer from the kinetic instability and maintains the usual probabilistic interpretation without violating the correspondence principle. As a proof of concept, we construct a class of stable interacting complex higher derivative theories and present a concrete example. This consistent and canonical framework allows us to analyse the previous attempts to avoid ghosts that use non-canonical quantisation schemes, such as the Lee–Wick theories, Dirac–Pauli quantisation or PT-symmetric quantum mechanics. The key to understand the would-be ghosts in any kinetically stable higher derivative theory is to accept the complex system behind it.

Classical irregular blocks, Hill’s equation and PT-symmetric periodic complex potentials

The Schroedinger eigenvalue problems for the Whittaker-Hill potential $Q_{2}(x)=\tfrac{1}{2} h^2\cos4x+4h\mu\cos2x$ and the periodic complex potential $Q_{1}(x)=\tfrac{1}{4}h^2{\rm e}^{-4ix}+2h^2\cos2x$ are studied using their realizations in two-dimensional conformal field theory (2dCFT). It is shown that for the weak coupling (small) $h\in\mathbb{R}$ and non-integer Floquet parameter $\nu\notin\mathbb{Z}$ spectra of hamiltonians $H_{i}\!=\!-{\rm d}^2/{\rm d}x^2 + Q_{i}(x)$, $i=1,2$ and corresponding two linearly independent eigenfunctions are given by the classical limit of the "single flavor" and "two flavors" ($N_f=1,2$) irregular conformal blocks. It is known that complex non-hermitian hamiltonians which are PT-symmetric (= invariant under simultaneous parity P and time reversal T transformations) can have real eigenvalues. The hamiltonian $H_{1}$ is PT-symmetric for $h,x\in\mathbb{R}$. It is found that $H_{1}$ has a real spectrum in the weak coupling region for $\nu\in\mathbb{R}\setminus\mathbb{Z}$. This fact in an elementary way follows from a definition of the $N_f=1$ classical irregular block. Thus, $H_{1}$ can serve as yet another new model for testing postulates of PT-symmetric quantum mechanics.

Non-self-adjoint Hamiltonians with complex eigenvalues

Motivated by what one observes dealing with PT-symmetric quantum mechanics, we discuss what happens if a physical system is driven by a diagonalizable Hamiltonian with not all real eigenvalues. In particular, we consider the functional structure related to systems living in finite-dimensional Hilbert spaces, and we show that certain intertwining relations can be deduced also in this case if we introduce suitable antilinear operators. We also analyze a simple model, computing the transition probabilities in the broken and in the unbroken regime.

Consistency of PT-symmetric quantum mechanics

In recent reports, suggestions have been put forward to the effect that parity and time-reversal (PT) symmetry in quantum mechanics is incompatible with causality. It is shown here, in contrast, that PT-symmetric quantum mechanics is fully consistent with standard quantum mechanics. This follows from the surprising fact that the much-discussed metric operator on Hilbert space is not physically observable. In particular, for closed quantum systems in finite dimensions there is no statistical test that one can perform on the outcomes of measurements to determine whether the Hamiltonian is Hermitian in the conventional sense, or PT-symmetric—the two theories are indistinguishable. Nontrivial physical effects arising as a consequence of PT symmetry are expected to be observed, nevertheless, for open quantum systems with balanced gain and loss.

Pseudospectra in non-Hermitian quantum mechanics

We propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non-Hermitian operators. We relate pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint operators, and basis properties of eigenfunctions. The abstract results are illustrated by unexpected wild properties of operators familiar from PT-symmetric quantum mechanics.

Quantum control and the challenge of non-Hermitian model-building

In a way inspired by the brief 2002 note “The challenge of nonhermitian structures in physics” by Ramirez and Mielnik (with the text most easily available via arXiv:quant- ph/0211048) the situation in the theory is briefly summarized here as it looks twelve years later. Our text has three parts. In the first one we briefly mention the pre-history (dating back to the Freeman Dyson's proposal of the non-Hermitian-Hamiltonian method in 1956 and to its subsequent successful “interacting boson model” applications in nuclear physics) and, first of all, the amazing recent progress reached, in the stationary case, using, in essence, an inversion of the Dyson's approach. The impact on the latter idea upon abstract quantum physics is sampled, first of all, by the reference to papers by Bender et al. (who made the non-Hermitian model-building popular under the nickname of parity-times-time-reflection- symmetric alias PT-symmetric quantum mechanics) and by Mostafazadeh (who reinterpreted PT-symmetry as P-pseudo-Hermiticity). In the second part of our review the emphasis is shifted to the newest, non-stationary upgrade of the formalism which we proposed in the year 2009 and which is characterized by the simultaneous participation of a triplet of Hilbert spaces H in the representation of a single quantum system. In the third part of the review we finally emphasize that the majority of applications of our three-Hilbert-space (THS) recipe is still ahead of us because the enhancement of the flexibility is necessarily accompanied by an enhancement of the technical difficulties. An escape out of the technical trap is proposed to be sought in a restriction of attention to quantum models living in finite-dimensional Hilbert spaces H. As long as the use of such spaces is so typical for the quantum-control considerations, we conclude with conjecture that the THS formalism should start searching for implementations in the field of quantum control.

Jarzynski Equality in PT-Symmetric Quantum Mechanics.

We show that the quantum Jarzynski equality generalizes to PT-symmetric quantum mechanics with unbroken PT symmetry. In the regime of broken PT symmetry, the Jarzynski equality does not hold as also the CPT norm is not preserved during the dynamics. These findings are illustrated for an experimentally relevant system-two coupled optical waveguides. It turns out that for these systems the phase transition between the regimes of unbroken and broken PT symmetry is thermodynamically inhibited as the irreversible work diverges at the critical point.

Ad Hoc Physical Hilbert Spaces in Quantum Mechanics

The overall principles of what is now widely known as PT-symmetric quantum mechanics are listed, explained and illustrated via a few examples. In particular, models based on an elementary local interaction V(x) are discussed as motivated by the naturally emergent possibility of an efficient regularization of an otherwise unacceptable presence of a strongly singular repulsive core in the origin. The emphasis is put on the constructive aspects of the models. Besides the overall outline of the formalism we show how the low-lying energies of bound states may be found in closed form in certain dynamical regimes. Finally, once these energies are found real we explain that in spite of a manifest non-Hermiticity of the Hamiltonian the time-evolution of the system becomes unitary in a properly amended physical Hilbert space.

Adiabatic approximation in PT-symmetric quantum mechanics

In this paper, we present a quantitative sufficient condition for adiabatic approximation in PT-symmetric quantum mechanics, which yields that a state of the PT-symmetric quantum system at any time will remain approximately in the m-th eigenstate up to a multiplicative phase factor whenever it is initially in the m-th eigenstate of the Hamiltonian. In addition, we estimate the approximation errors by the distance and the fidelity between the exact solution and the adiabatic approximate solution to the time evolution equation, respectively.

Non-self-adjoint model of a two-dimensional noncommutative space with an unbound metric

We demonstrate that a non self-adjoint Hamiltonian of harmonic oscillator type defined on a two-dimensional noncommutative space can be diagonalized exactly by making use of pseudo-bosonic operators. The model admits an antilinear symmetry and is of the type studied in the context of PT-symmetric quantum mechanics. Its eigenvalues are computed to be real for the entire range of the coupling constants and the biorthogonal sets of eigenstates for the Hamiltonian and its adjoint are explicitly constructed. We show that despite the fact that these sets are complete and biorthogonal, they involve an unbounded metric operator and therefore do not constitute (Riesz) bases for the Hilbert space $\Lc^2(\Bbb R^2)$, but instead only D-quasi bases. As recently proved by one of us (FB), this is sufficient to deduce several interesting consequences.