arXiv subjects
Publications and source records attributed to Carl M. Bender.
In some recent experiments the distinction between synthetic magnetic monopoles and Dirac monopoles has been blurred. A case in point is the work in a letter by Ray {\it et al.} [arXiv:1408.3133] in which a beautiful experiment is reported but claims with regard to Dirac monopoles are misleading.
Logarithmic time-like Liouville quantum field theory has a generalized PT invariance, where T is the time-reversal operator and P stands for an S-duality reflection of the Liouville field $ϕ$. In Euclidean space the Lagrangian of such a theory, $L=\frac{1}{2}(\nablaϕ)^2-igϕ\exp(iaϕ)$, is analyzed using the techniques of PT-symmetric quantum theory. It is shown that L defines an infinite number of unitarily inequivalent sectors of the theory labeled by the integer n. In one-dimensional space (quantum mechanics) the energy spectrum is calculated in the semiclassical limit and the mth energy level in the nth sector is given by $E_{m,n}\sim(m+1/2)^2a^2/(16n^2)$.
The Hamiltonian for a PT-symmetric chain of coupled oscillators is constructed. It is shown that if the loss-gain parameter $γ$ is uniform for all oscillators, then as the number of oscillators increases, the region of unbroken PT-symmetry disappears entirely. However, if $γ$ is localized in the sense that it decreases for more distant oscillators, then the unbroken-PT-symmetric region persists even as the number of oscillators approaches infinity. In the continuum limit the oscillator system is described by a PT-symmetric pair of wave equations, and a localized loss-gain impurity leads to a pseudo-bound state. It is also shown that a planar configuration of coupled oscillators can have multiple disconnected regions of unbroken PT symmetry.
Complex trajectories for Hamiltonians of the form H=p^n+V(x) are studied. For n=2 time-reversal symmetry prevents trajectories from crossing. However, for n>2 trajectories may indeed cross, and as a result, the complex trajectories for such Hamiltonians have a rich and elaborate structure. In past work on complex classical trajectories it has been observed that turning points act as attractors; they pull on complex trajectories and make them veer towards the turning point. In this paper it is shown that the poles of V(x) have the opposite effect --- they deflect and repel trajectories. Moreover, poles shield and screen the effect of turning points.
This paper proposes a very simple perturbative technique to calculate the low-lying eigenvalues and eigenstates of a parity-symmetric quantum-mechanical potential. The technique is to solve the time-independent Schroedinger eigenvalue problem as a perturbation series in which the perturbation parameter is the energy itself. Unlike nearly all perturbation series for physical problems, for the ground state this perturbation expansion is convergent and, even though the ground-state energy is in general not small compared with 1, the perturbative results are numerically accurate. The perturbation series is divergent for higher energy levels but can be easily evaluated by using methods such as Pade summation.
This paper presents a detailed asymptotic study of the nonlinear differential equation y'(x)=\cos[πxy(x)] subject to the initial condition y(0)=a. Although the differential equation is nonlinear, the solutions to this initial-value problem bear a striking resemblance to solutions to the time-independent Schroedinger eigenvalue problem. As x increases from x=0, y(x) oscillates and thus resembles a quantum wave function in a classically allowed region. At a critical value x=x_{crit}, where x_{crit} depends on a, the solution y(x) undergoes a transition; the oscillations abruptly cease and y(x) decays to 0 monotonically as x-->\infty. This transition resembles the transition in a wave function that occurs at a turning point as one enters the classically forbidden region. Furthermore, the initial condition a falls into discrete classes; in the nth class of initial conditions a_{n-1} \infty, a_n~A\sqrt{n}, where A=2^{5/6}. Numerical analysis reveals that the first Painleve transcendent has an eigenvalue structure that is quite similar to that of the equation y'(x)=\cos[πxy(x)] and that the nth eigenvalue grows with n like a constant times n^{3/5} as n-->\infty. Finally, it is noted that the constant A is numerically very close to the lower bound on the power-series constant P in the theory of complex variables, which is associated with the asymptotic behavior of zeros of partial sums of Taylor series.
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.
In an earlier paper it was argued that the conventional double-scaling limit of an O(N)-symmetric quartic quantum field theory is inconsistent because the critical coupling constant is negative and thus the integral representing the partition function of the critical theory does not exist. In this earlier paper it was shown that for an O(N)-symmetric quantum field theory in zero-dimensional spacetime one can avoid this difficulty if one replaces the original quartic theory by its PT-symmetric analog. In the current paper an O(N)-symmetric quartic quantum field theory in one-dimensional spacetime [that is, O(N)-symmetric quantum mechanics] is studied using the Schroedinger equation. It is shown that the global PT-symmetric formulation of this differential equation provides a consistent way to perform the double-scaling limit of the O(N)-symmetric anharmonic oscillator. The physical nature of the critical behavior is explained by studying the PT-symmetric quantum theory and the corresponding and equivalent Hermitian quantum theory.
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 ε. For small εthe PT symmetry is broken and the system is not in equilibrium, but when εbecomes sufficiently large, the system undergoes a transition to an equilibrium phase in which the PT symmetry is unbroken. For very large εthe 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 ε.
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 $H=H_0+εH_1$, 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 $Q=εQ_1+ε^3 Q_3+ε^5 Q_5+\ldots$. [In previous work it has always been assumed that the coefficients of even powers of $ε$ 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 $Q=Q_0+εQ_1+ε^2 Q_2+\ldots$ in which {\it all} powers and not just odd powers of $ε$ 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.
The conventional double-scaling limit of an O(N)-symmetric quartic quantum field theory is inconsistent because the critical coupling constant is negative. Thus, at the critical coupling the Lagrangian defines a quantum theory with an upside-down potential whose energy appears to be unbounded below. Worse yet, the integral representation of the partition function of the theory does not exist. It is shown that one can avoid these difficulties if one replaces the original theory by its PT-symmetric analog. For a zero-dimensional O(N)-symmetric quartic vector model the partition function of the PT-symmetric analog is calculated explicitly in the double-scaling limit.
It was shown recently that a PT-symmetric $iϕ^3$ quantum field theory in $6-ε$ dimensions possesses a nontrivial fixed point. The critical behavior of this theory around the fixed point is examined and it is shown that the corresponding phase transition is related to the existence of a nontrivial solution of the gap equation. The theory is studied first in the mean-field approximation and the critical exponents are calculated. Then, it is examined beyond the mean-field approximation by using renormalization-group techniques, and the critical exponents for $6-ε$ dimensions are calculated to order $ε$. It is shown that because of its stability the PT-symmetric $iϕ^3$ theory has a higher predictive power than the conventional $ϕ^3$ theory. A comparison of the $iϕ^3$ model with the Lee-Yang model is given.
The interplay of dilatonic effects in dilaton cosmology and stochastic quantum space-time defects within the framework of string/brane cosmologies is examined. The Boltzmann equation describes the physics of thermal dark-matter-relic abundances in the presence of rolling dilatons. These dilatons affect the coupling of stringy matter to D-particle defects, which are generic in string theory. This coupling leads to an additional source term in the Boltzmann equation. The techniques of asymptotic matching and boundary-layer theory, which were recently applied by two of the authors (CMB and SS) to a Boltzmann equation, are used here to find the detailed asymptotic relic abundances for all ranges of the expectation value of the dilaton field. The phenomenological implications for the search of supersymmetric dark matter in current colliders, such as the LHC, are discussed.
The paper by Bowen, Mancini, Fessatidis, and Murawski (2012 Phys. Scr. {\bf 85}, 065005) demonstrates in a dramatic fashion the serious difficulties that can arise when one rushes to perform numerical studies before understanding the physics and mathematics of the problem at hand and without understanding the limitations of the numerical methods used. Based on their flawed numerical work, the authors conclude that the work of Bender and Boettcher is wrong even though it has been verified at a completely rigorous level. Unfortunately, the numerical procedures performed and described in the paper by Bowen et al are incorrectly applied and wrongly interpreted.
It is shown that if the C operator for a PT-symmetric Hamiltonian with simple eigenvalues is not unique, then it is unbounded. Apart from the special cases of finite-matrix Hamiltonians and Hamiltonians generated by differential expressions with PT-symmetric point interactions, the usual situation is that the C operator is unbounded. The fact that the C operator is unbounded is significant because, while there is a formal equivalence between a PT-symmetric Hamiltonian and a conventionally Hermitian Hamiltonian in the sense that the two Hamiltonians are isospectral, the Hilbert spaces are inequivalent. This is so because the mapping from one Hilbert space to the other is unbounded. This shows that PT-symmetric quantum theories are mathematically distinct from conventional Hermitian quantum theories.
In this work we analyze PT-symmetric double-well potentials based on a two-mode picture. We reduce the problem into a PT-symmetric dimer and illustrate that the latter has effectively two fundamental bifurcations, a pitchfork (symmetry-breaking bifurcation) and a saddle-center one, which is the nonlinear analog of the PT-phase-transition. It is shown that the symmetry breaking leads to ghost states (amounting to growth or decay); although these states are not true solutions of the original continuum problem, the system's dynamics closely follows them, at least in its metastable evolution. Past the second bifurcation, there are no longer states of the original continuum system. Nevertheless, the solutions can be analytically continued to yield a new pair of branches, which is also identified and dynamically examined. Our explicit analytical results for the dimer are directly compared to the full continuum problem, yielding a good agreement.
Non-Hermitian PT-symmetric quantum-mechanical Hamiltonians generally exhibit a phase transition that separates two parametric regions, (i) a region of unbroken PT symmetry in which the eigenvalues are all real, and (ii) a region of broken PT symmetry in which some of the eigenvalues are complex. This transition has recently been observed experimentally in a variety of physical systems. Until now, theoretical studies of the PT phase transition have generally been limited to one-dimensional models. Here, four nontrivial coupled PT-symmetric Hamiltonians, $H=p^2/2+x^2/2+q^2/2+y^2/2+igx^2y$, $H=p^2/2+x^2/2+q^2/2+y^2+igx^2y$, $H=p^2/2+x^2/2+q^2/2+y^2/2+r^2/2+z^2/2+igxyz$, and $H=p^2/2+x^2/2+q^2/2+y^2+r^2/2+3z^2/2+igxyz$ are examined. Based on extensive numerical studies, this paper conjectures that all four models exhibit a phase transition. The transitions are found to occur at $g\approx 0.1$, $g\approx 0.04$, $g\approx 0.1$, and $g\approx 0.05$. These results suggest that the PT phase transition is a robust phenomenon not limited to systems having one degree of freedom.
If a Hamiltonian is PT symmetric, there are two possibilities: Either the eigenvalues are entirely real, in which case the Hamiltonian is said to be in an unbroken-PT-symmetric phase, or else the eigenvalues are partly real and partly complex, in which case the Hamiltonian is said to be in a broken-PT-symmetric phase. As one varies the parameters of the Hamiltonian, one can pass through the phase transition that separates the unbroken and broken phases. This transition has recently been observed in a variety of laboratory experiments. This paper explains the phase transition in a simple and intuitive fashion and then describes an extremely elementary experiment in which the phase transition is easily observed.