Muga G. Time in Quantum Mechanics

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336 G.G. Paulus and D. Bauer

respectively). The second half of “slit 1a” and the ﬁrst half of “slit 1b” give rise

to right-going electrons now whereas it is vice versa for the left-going electrons.

The opposite applies to slits 2a and 2b where the absolute value of the electric

ﬁeld, however, is already smaller. As a consequence, quantum trajectories leading

to positive and negative ﬁnal momentum p



are weighted differently, and the sym-

metry in the photoelectron momentum spectrum is broken. This is clearly visible

in Fig. 11.16 where the differential photoelectron energy spectra are shown for the

“right-going” electrons with p



> 0 and the “left-going” electrons with p



< 0.

The spectrum for the right-going electrons is less structured than the spectrum in

(right)

(left)

Yield (arb. units)Yield (arb. units)

/ 2 (a.u.)

Fig. 11.16 Photoelectron spectra (11.63) in laser polarization direction (upper panel, “right-going”

electrons) and in the opposite direction (lower panel, “left-going” electrons) for the pulse analyzed

in Fig. 11.15. The exact SFA spectra from the numerical integration of (11.59) (gray and bold)

are compared with the saddle-point results (11.65), taking into account only one of the dominating

quantum trajectories starting from slit 1a or 1b for a given ﬁnal momentum p



(dotted), both

trajectories from slit 1a and 1b (dashed), and all four trajectories (slits 1a, 1b, 2a, and 2b, solid).

For the right-going electrons (upper panel) slits 2a and 2b are not important so that the interference

pattern is less complex than for the left-going electrons (lower panel)

11 Double-Slit Experiments in the Time Domain 337

Fig. 11.14 for ϕ

cep

= 0: The bigger scale interference pattern (arising from slits

1a and 1b) is similar but the small-scale modulations are absent because slits 2a

and 2b are not important for the right-going electrons. However, for the left-going

electrons the situation is the opposite: The small-scale modulations are much more

pronounced than in Fig. 11.14 because quantum trajectories originating from both

slit 2a and slit 2b are contributing to the spectrum of the left-going electrons. Hence,

by varying the carrier-envelope phase between +π/2 and −π/2 the attosecond time

slits are shifted and the relative weight of their contributions is affected. As a result

the pronounced interference structure visible in Fig. 11.16, lower panel, can be con-

tinuously transferred from the left-going electrons to the right-going electrons, with

the left/right-symmetric spectrum shown in Fig. 11.16 being the intermediate case

cep

= 0.

11.4.6 Beyond the SFA

We do not want to give the wrong impression that the plain SFA (i.e., without

rescattering and Coulomb corrections) introduced above is sufﬁcient to understand

all aspects of strong-ﬁeld laser atom interaction. Replacing the ﬁnal state |Ψ

)

by a plane wave |p and neglecting further interactions with the Coulomb potential

in the transition from the (still exact) Eq. (11.55) to the Keldysh amplitude (11.57)

is not as innocent as it may seem. First of all, gauge invariance is lost, which can

even have consequences on a qualitative level, as was shown in [6]. The formal

replacement of the ﬁnal state and the omission of the second term in (11.55) may be

physically interpreted as neglecting the Coulomb interaction between the electron

and the ion once the electron is emitted. However, this Coulomb interaction after the

emission of the electron inﬂuences signiﬁcantly momentum distributions, including

the left/right asymmetry discussed in Sect. 11.4.4 [13, 14], the interference structure

of spectra, and the angular distributions of the photoelectrons in, e.g., elliptically

polarized laser ﬁelds [56]. It is good news that the quantum orbit approach to the

SFA introduced in Sect. 11.4.3 also offers the possibility to overcome these (and

other) shortcomings [56, 57].

Acknowledgments DB is grateful to Sergei Popruzhenko for illuminating discussions and proof-

reading. GGP acknowledges support by The Welch Foundation (A-1562) and the Air Force Ofﬁce

of Scientiﬁc Research (FA9550-07-1-0069).

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328

Chapter 12

Optimal Time Evolution for Hermitian

and Non-Hermitian Hamiltonians

Carl M. Bender and Dorje C. Brody

The shortest path between two truths in the real domain

passes through the complex domain.

Jacques Hadamard, The Mathematical Intelligencer 13 (1991)

12.1 Introduction

Interest in optimal time evolution dates back to the end of the seventeenth century,

when the famous brachistochrone problem was solved almost simultaneously by

Newton, Leibniz, l’H

opital, and Jacob and Johann Bernoulli. The word brachis-

tochrone is derived from Greek and means shortest time (of ﬂight). The classical

brachistochrone problem is stated as follows: A bead slides down a frictionless wire

from point A to point B in a homogeneous gravitational ﬁeld. What is the shape of

the wire that minimizes the time of ﬂight of the bead? The solution to this problem

is that the optimal (fastest) time evolution is achieved when the wire takes the shape

of a cycloid, which is the curve that is traced out by a point on a wheel that is rolling

on ﬂat ground.

In the past few years there has been much interest in the quantum brachistochrone

problem, which is formulated in a somewhat similar fashion: Consider two ﬁxed

quantum states, an initial state |ψ

 and a ﬁnal state |ψ

 in a Hilbert space. We

then consider the set of all Hamiltonians satisfying the energy constraint that the

difference between the largest and smallest eigenvalues is a ﬁxed energy ω: E

max

−

min

= ω. Some of the Hamiltonians in this set allow the initial state |ψ

 to evolve

into the ﬁnal state |ψ

 in time t:

|ψ

=e

−iHt/

|ψ

 . (12.1)

C.M. Bender (B)

Department of Physics, Washington University, St. Louis, MO 63130, USA, cmb@wustl.edu

D.C. Brody

Department of Mathematics, Imperial College London, London SW7 2AZ, UK,

d.brody@imperial.ac.uk

Bender, C.M., Brody, D.C.: Optimal Time Evolution for Hermitian and Non-Hermitian

Hamiltonians. Lect. Notes Phys. 789, 341–361 (2009)

DOI 10.1007/978-3-642-03174-8

 Springer-Verlag Berlin Heidelberg 2009

342 C.M. Bender and D.C. Brody

The quantum brachistochrone problem is to ﬁnd the optimal Hamiltonian, that is,

the Hamiltonian that accomplishes this evolution in the shortest possible time, which

we denote by τ .

In this chapter we show that for Hermitian Hamiltonians the shortest evo-

lution time τ is a nonzero quantity whose size depends on the Hilbert-space

distance between the ﬁxed initial and ﬁnal state vectors. However, for complex

non-Hermitian Hamiltonians, the value of τ can be made arbitrarily small. Thus,

non-Hermitian Hamiltonians permit arbitrarily fast time evolution.

Of course, a non-Hermitian Hamiltonian may be physically unrealistic because it

may possess complex eigenvalues and/or it may generate nonunitary time evolution,

that is, time evolution in which probability is not conserved. However, there is a

special class of non-Hermitian Hamiltonians that are PT symmetric, that is, Hamil-

tonians that are invariant under combined space and time reﬂection. Although such

Hamiltonians are not Hermitian in the Dirac sense, they do have entirely real spectra

and give rise to unitary time evolution. Thus, such Hamiltonians deﬁne consistent

and acceptable theories of quantum mechanics. We show in this chapter that if we

use Hamiltonians of this type to solve the quantum brachistochrone problem, we

can achieve arbitrarily fast time evolution without violating any principles of quan-

tum mechanics. Thus, if it were possible to implement faster-than-Hermitian time

evolution, then non-Hermitian Hamiltonians might have important applications in

quantum computing.

This chapter is organized as follows: In Sect. 12.2 we introduce and describe

PT quantum mechanics and explain how a Hamiltonian that is not Dirac Hermi-

tian can still deﬁne a consistent theory of quantum mechanics. Then in Sect. 12.3

we explain why complex classical mechanics allows for faster-than-conventional

time evolution. In Sect. 12.4 we discuss the quantum brachistochrone for Her-

mitian Hamiltonians. Then, in Sect. 12.5 we extend the discussion in Sect. 12.4

to Hamiltonians that are not Dirac Hermitian. In Sect. 12.6 we explain how it

might be possible for a complex Hamiltonian to achieve faster-than-Hermitian time

evolution.

12.2 PT Quantum Mechanics

Based on the training that one receives in a traditional quantum mechanics course,

one would expect a theory deﬁned by a non-Hermitian Hamiltonian to be physically

unacceptable for a closed system

because the energy levels would most likely be

complex and the time evolution would most likely be nonunitary (not probability

conserving). However, theories deﬁned by a special class of non-Hermitian Hamil-

tonians called PT-symmetric Hamiltonians can have positive real energy levels and

can exhibit unitary time evolution. Such theories are consistent quantum theories. It

N. of E.: For open systems non-Hermitian Hamiltonians and non unitary evolution may be per-

fectly physical, see e.g., Chap. 6 by G. Hegerfeldt or Chap. 4 by A. Ruschhaupt et al., this volume.

12 Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians 343

may be possible to distinguish these theories experimentally from theories deﬁned

by Hermitian Hamiltonians because, in principle, non-Hermitian Hamiltonians can

be used to generate arbitrarily fast time evolution.

We use the following terminology in this chapter: By Hermitian, we mean Dirac

Hermitian, where the Dirac Hermitian adjoint symbol

†

represents combined matrix

transposition and complex conjugation. The parity operator P performs spatial

reﬂection and thus in quantum mechanics it changes the sign of the position operator

x and the momentum operator p: PxP =−x and PpP =−p. Because the parity

operator P is a reﬂection operator, its square is the unit operator: P

= 1.The

time-reversal operator T performs the time reﬂection t →−t, and thus it changes

the sign of the momentum operator p, TpT =−p, but it leaves the position oper-

ator invariant: TxT = x. The square of T is the unit operator T

= 1. We require

that the operators P and T individually leave invariant the fundamental Heisenberg

algebra of quantum mechanics [x, p] = i. Thus, while P is a linear operator, we see

that T must perform complex conjugation T zT = z

∗

, and hence T is an antilinear

operator.

The ﬁrst class of PT-symmetric quantum mechanical Hamiltonians was intro-

duced in 1998 [6]. Since then there have been many papers on this subject by a wide

range of authors. There have also been three recent review articles [4, 5, 30]. In

[6] it was discovered that even if a Hamiltonian is not Hermitian, its energy levels

can be all real and positive so long as the eigenfunctions are symmetric under PT

reﬂection.

These new kinds of Hamiltonians are obtained by deforming ordinary Hermi-

tian Hamiltonians into the complex domain. The original class of PT-symmetric

Hamiltonians that was proposed in [6] has the form

H = p

+ x

(ix)

(ε>0) , (12.2)

where ε is a real deformation parameter. Two particularly interesting special cases

are obtained by setting ε = 1 to get H = p

+ ix

and by setting ε = 2 to get

H = p

− x

. Surprisingly, these Hamiltonians have real, positive, discrete energy

levels even though the potential for ε = 1 is imaginary and the potential for ε = 2

is upside down. The ﬁrst complete proof of spectral reality and positivity for H in

(12.2) was given by Dorey et al. [28, 29].

The philosophical background of PT quantum mechanics is simply this: One

of the axioms of quantum mechanics requires that the Hamiltonian H be Dirac

Hermitian. This axiom is distinct from all other quantum mechanical axioms because

it is mathematical rather than physical in character. The other axioms of quantum

mechanics are stated in physical terms; these other axioms require locality, causality,

stability and uniqueness of the vacuum state, conservation of probability, Lorentz

Another way to see that T is associated with complex conjugation is to require that the time-

dependent Schr

odinger equation be invariant under time reversal. This implies that time reﬂection

t →−t must be accompanied by complex conjugation i →−i. See [60] for a discussion of the

properties of antilinear operators.

344 C.M. Bender and D.C. Brody

invariance, and so on. The condition of Dirac Hermiticity H = H

†

is mathematical,

but the condition of PT symmetry H = H

= (PT)H(PT) (space–time reﬂec-

tion symmetry) is physical because P and T are elements of the Lorentz group.

The spectrum of H in (12.2) is real, which poses the question of whether this

Hamiltonian speciﬁes a quantum mechanical theory. That is, is the theory speciﬁed

by H associated with a Hilbert space endowed with a positive inner product and does

H specify unitary (norm-preserving) time evolution? The answer to these questions

is yes. Positivity of the inner product and unitary time evolution was established in

[11, 12] for quantum-mechanical systems having an unbroken PT symmetry (an

analogous result was obtained by Mostafazadeh [51]) and in [13] for quantum ﬁeld

theory.

To demonstrate that the theory speciﬁed by H in (12.2) is a quantum mechanical

theory, we construct a linear operator C that satisﬁes the three simultaneous alge-

braic equations [11]: C

= 1,[C, PT] = 0, and [C, H] = 0. Using C, which in

quantum ﬁeld theory is a Lorentz scalar [8], we can then construct the appropriate

inner product for a PT-symmetric Hamiltonian: a|b≡a

CPT

·b. This inner prod-

uct, which uses the CPT adjoint, has a strictly positive norm: a|a > 0. Because

H commutes with both PT and C, H is self-adjoint with respect to CPT con-

jugation. Also, the time-evolution operator e

−iHt/

is unitary with respect to CPT

conjugation. Note that the Hilbert space and the CPT inner product are dynamically

determined by the Hamiltonian itself.

We have explained why a PT-symmetric Hamiltonian gives rise to a unitary

theory, but in doing so we raise the question of whether PT-symmetric Hamilto-

nians are useful. The answer to this question is simply that PT-symmetric Hamil-

tonians have already been useful in many areas of physics. For example, in 1959

Wu showed that the ground state of a Bose system of hard spheres is described by

a non-Hermitian Hamiltonian [61]. Wu found that the ground-state energy of this

system is real and he conjectured that all of the energy levels were real. Hollowood

showed that the non-Hermitian Hamiltonian for a complex Toda lattice has real

energy levels [42]. Cubic non-Hermitian Hamiltonians of the form H = p

+ ix

(and also cubic quantum ﬁeld theories having an imaginary self-coupling term) arise

in studies of the Lee-Yang edge singularity [25, 26, 32, 62] and in various Reggeon

ﬁeld-theory models [23, 39, 40]. In all of these cases a non-Hermitian Hamiltonian

having a real spectrum appeared mysterious at the time, but now the explanation is

clear: In every case the non-Hermitian Hamiltonian is PT symmetric. Hamiltonians

having PT symmetry have also been used to describe magnetohydrodynamic sys-

tems [35, 38] and to study nondissipative time-dependent systems interacting with

electromagnetic ﬁelds [31].

An important application of PT quantum mechanics is in the revitalization of

theories that have been thought to be dead because they appear to have ghosts.

Ghosts are states having negative norm. We have explained above that in order to

construct the quantum mechanical theory deﬁned by a PT-symmetric Hamiltonian,

we must construct the appropriate adjoint from the C operator. Having constructed

the CPT adjoint, one may ﬁnd that the so-called ghost state is actually not a ghost

at all because when its norm is calculated using the appropriate deﬁnition of the

12 Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians 345

adjoint, the norm turns out to be positive. This is what happens in the case of the

Lee model.

The Lee model was proposed in 1954 as a quantum ﬁeld theory in which mass,

wave-function, and charge renormalization could be performed exactly and in closed

form [48]. However, in 1955 K

all

en and Pauli showed that when the renormalized

coupling constant is larger than a critical value, the Hamiltonian becomes non-

Hermitian (in the Dirac sense) and a ghost state appears [45]. The importance of

theworkofK

all

en and Pauli was emphasized by Salam in his review of their paper

[58] and the appearance of the ghost was assumed to be a fundamental defect of

the Lee model. However, in 2005 it was shown that the non-Hermitian Lee-model

Hamiltonian is PT symmetric and when the norms of the states of this model are

determined using the C operator, which can be calculated exactly and in closed

form, the ghost state is seen to be an ordinary physical state having positive norm

[9]. Thus, the following assertion by Barton [3] is not correct: “A non-Hermitian

Hamiltonian is unacceptable partly because it may lead to complex energy eigen-

values, but chieﬂy because it implies a nonunitary S matrix, which fails to conserve

probability and makes a hash of the physical interpretation.”

Another example of a quantum model that was thought to have ghost states,

but in fact does not, is the Pais–Uhlenbeck oscillator model [16–18]. This model

has a fourth-order ﬁeld equation, and for the past several decades it was thought

(incorrectly) that all such higher-order ﬁeld equations lead inevitably to ghosts. It is

shown in [16] that when the Pais–Uhlenbeck model is quantized using the methods

of PT quantum mechanics, it does not have any ghost states at all.

There are many potential applications for PT quantum mechanics in areas such

as particle physics, cosmology, gravitation, quantum ﬁeld theory, and solid-state

physics. These applications are discussed in detail in the recent review article in [5].

Furthermore, there are now indications that theories described by PT-symmetric

Hamiltonians can be observed in table-top experiments [49, 55, 56].

Having shown the validity and potential usefulness of PT quantum mechan-

ics, one may ask why PT quantum mechanics works. The reason is that CP is

a positive operator, and thus it can be written as the exponential of another oper-

ator Q: CP = e

. The square root of e

can then be used to construct a new

Hamiltonian

H via a similarity transformation on the PT-symmetric Hamiltonian

H ≡ e

−Q/2

Q/2

. The new Hamiltonian

H has the same energy eigenvalues

as the original Hamiltonian H because a similarity transformation is isospectral.

Moreover,

H is Dirac Hermitian [52]; PT quantum mechanics works because there

is an isospectral equivalence between a non-Hermitian PT-symmetric Hamiltonian

and a conventional Dirac Hermitian Hamiltonian.

There are a number of elementary examples of this equivalence, but a nontrivial

illustration is provided by the Hamiltonian H in (12.2) at ε = 2. This Hamiltonian

is not Hermitian because boundary conditions that violate the L

norm must be

imposed in Stokes wedges in the complex plane in order to obtain a real, positive,

discrete spectrum. The exact equivalent Hermitian Hamiltonian is

H = p

+4x

−

2 x, where  is Planck’s constant [10, 24, 44]. The term proportional to  vanishes

in the classical limit and is thus an example of a quantum anomaly.

346 C.M. Bender and D.C. Brody

Since PT symmetry is equivalent by means of a similarity transformation to

conventional Dirac Hermiticity, one may wonder whether PT quantum mechanics

is actually fundamentally different from ordinary quantum mechanics. The answer

is yes and, at least in principle, there is an experimentally observable difference

between PT-symmetric and ordinary Dirac Hermitian Hamiltonians. The quantum

brachistochrone provides a setting for examining this difference and provides a way

to discriminate between the class of PT-symmetric Hamiltonians and the class of

Dirac Hermitian Hamiltonians.

12.3 Complex Classical Motion

It is implicitly assumed in the derivation of the classical brachistochrone that the

path of shortest time of descent is real. However, it is interesting that if one allows

for the possibility of complex paths of motion, one can achieve an even shorter

time of ﬂight. In this section we consider a simple classical-mechanical system.

Our purpose is to explain heuristically how extending a dynamical system into the

complex domain can result in faster-than-real time evolution.

To demonstrate that a shorter time of ﬂight can be achieved by means of complex

paths, let us consider the classical harmonic oscillator, whose Hamiltonian is

H = p

+ x

. (12.3)

If a particle has energy E = 1, then the classical turning points of the motion of

the particle are located at x =±1. The particle undergoes simple harmonic motion

in which it oscillates sinusoidally between these two turning points. This periodic

motion is indicated in Fig. 12.1 by a solid line connecting the turning points. How-

ever, in addition to this oscillatory motion on the real-x axis, there are an inﬁnite

number of other trajectories that a particle of energy E can have [7]. These classical

trajectories, which are also shown in Fig. 12.1, are all ellipses whose foci are located

at precisely the positions of the turning points. All of the classical orbits are periodic,

and all orbits have the same period T = 2π . Thus, a classical particle travels faster

along more distant ellipses.

Now suppose that a classical particle of energy E = 1 travels along the real-x

axis from some point x =−a to x = a, where a > 1. If the potential V (x)is

everywhere zero along its path, then it will travel at a constant velocity. However,

suppose that the particle suddenly ﬁnds itself in the parabolic potential V (x) = x

when it reaches the turning point at x =−1 and that it suddenly escapes the inﬂu-

ence of this potential at x = 1. Then, the time of ﬂight from x =−a to x = a

will be changed because the particle does not travel at a constant velocity between

the turning points. Next, let us imagine that the potential V (x) = x

is suddenly

turned on before the particle reaches the turning point at x =−1. In this case,

the particle will follow one of the elliptical paths in the complex plane around the

positive real axis. Just as the particle reaches the positive real axis the potential

is turned off, so the particle proceeds onward along the real axis until it reaches

Muga G. Time in Quantum Mechanics - Vol. 2

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