Graefe, E-M., Rush, A., & Schubert, R. (2018). Propagation ofGaussian beams in the presence of gain and loss. IEEE Journal ofSelected Topics in Quantum Electronics, 22(5), [5000906].https://doi.org/10.1109/JSTQE.2016.2555800

Peer reviewed versionLicense (if available):OtherLink to published version (if available):10.1109/JSTQE.2016.2555800

Link to publication record in Explore Bristol ResearchPDF-document

This is the accepted author manuscript (AAM). The final published version (version of record) is available onlinevia [insert publisher name] at [insert hyperlink]. Please refer to any applicable terms of use of the publisher.

University of Bristol - Explore Bristol ResearchGeneral rights

This document is made available in accordance with publisher policies. Please cite only thepublished version using the reference above. Full terms of use are available:http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/

Propagation of Gaussian beams in the presence of gain and loss

Eva-Maria Graefe, Alexander Rush, and Roman Schubert ∗†

October 1, 2018

Abstract

We consider the propagation of Gaussian beamsin a waveguide with gain and loss in the paraxialapproximation governed by the Schrodinger equa-tion. We derive equations of motion for the beamin the semiclassical limit that are valid when thewaveguide profile is locally well approximated byquadratic functions. For Hermitian systems, with-out any loss or gain, these dynamics are given byHamilton’s equations for the center of the beam andits conjugate momentum. Adding gain and/or lossto the waveguide introduces a non-Hermitian com-ponent, causing the width of the Gaussian beam toplay an important role in its propagation. Here weshow how the width affects the motion of the beamand how this may be used to filter Gaussian beamslocated at the same initial position based on theirwidth.

1 Introduction

While quantum dynamics is in general very differ-ent from classical dynamics, for short times ini-tially Gaussian wave packets stay approximatelyGaussian and follow classical trajectories. This hasbeen shown by Hepp and Heller using the fact thatGaussian states stay exactly Gaussian for harmonicHamiltonians, and Taylor expanding more generalHamiltonians up to second order around the cen-tre of the wave packet [1] (see also [2] for a reviewon the properties and dynamics of Gaussian wavepackets) . As long as the wave packet stays well lo-calized compared to the scale on which the higher

∗Eva-Maria Graefe and Alexander Rush are with the De-partment of Mathematics, Imperial College London, Lon-don, SW7 2AZ, United Kingdom†Roman Schubert is with the School of Mathematics,

University of Bristol, Bristol BS8 1TW, United Kingdom

order terms in the Taylor expansion of the Hamilto-nian are negligible this is a good approximation. Itis also the main ingredient of more powerful meth-ods to approximate dynamics in the deep quantumregime such as the frozen Gaussian propagator andthe method of hybrid mechanics [3].

There are many analogies between the wave na-ture of light and the quantum mechanical wavefunction. In particular, the propagation of lightin wave guides in the paraxial approximation isdescribed by a time-dependent Schrodinger equa-tion (where the propagation direction acts as time).This has led to spectacular demonstrations of quan-tum dynamical effects, as well as to new applica-tions in optics (see, e.g. [4] and references therein).In recent years, there has been a surge of interest inoptical waveguides with gain and loss, mimickingnon-Hermitian, and in particular PT -symmetric[5], quantum systems [6]. It is an interesting ques-tion what might be the fate of initial Gaussianwave packets in the limit of large locally approx-imately harmonic potentials in the spirit of Heppand Heller. In [7] this question has been investi-gated, and a new type of classical dynamics guidingthe centers of Gaussian wave packets has been de-rived. In this dynamics the changing shape of thewave packet couples back into the dynamics of thecentre. That is, even in the limit where the beamwidth is completely negligible compared to the ex-ternal potential and where the dynamics is accu-rately described by the Gaussian approximation,there are many classical trajectories for a given setof initial position and momentum, depending onthe width of the wave packet.

Here we review the approximation and give ex-plicit equations of motion for the special case ofa Hamiltonian with complex potential, and inves-tigate the resulting dynamics in an example of aPT -symmetric multimode waveguide. The approx-

1

arX

iv:1

601.

0780

2v1

[qu

ant-

ph]

28

Jan

2016

imation accurately describes the propagation andeffects such as the typical power oscillations in PT-symmetric systems if the potential is chosen largeenough compared to the beam width associated tothe ground state. We demonstrate how the shapeof the beam affects the propagation, and how thismight be used as a filtering device.

2 Gaussian beams in theparaxial wave equation

For simplicity we consider the propagation of lightin a two-dimensional waveguide where the (com-plex) refractive index n is constant along the z-direction and varies along the x-direction. The re-sults are straight forwardly generalised to higherdimensions and refractive indices that are modu-lated in z-direction as well.

In the paraxial approximation the propagationof the electric field amplitude ψ is described by theSchrodinger equation

i~∂ψ

∂z= − ~2

2n0

∂2ψ

∂x2+ V (x)ψ, (1)

where ~ := λ2π , and λ is the wavelength, and where

the effective potential

V (x) =n20 − n2(x)

2n0≈ n0 − n(x) (2)

is approximately given by the refractive index pro-file n(x) of the waveguide, where n0 is the referencerefractive index of the substrate. Without loss ofgenerality we use rescaled units with n0 = 1 = ~ inwhat follows.

We consider the propagation of an initial Gaus-sian beam of the form

ψ(x, 0) =(

Im(B0)π

) 14

ei(B02 (x−q0)2+p0(x−q0)), (3)

where q0 ∈ R is the position of the center ofthe beam, p0, the expectation value of the quan-tum momentum operator, describes the angle ofincidence, and B0 ∈ C encodes the shape of thebeam. In particular it holds ∆q0 = 1√

2 ImB0, and

∆p0 = |B0|√2 ImB0

, and we demand Im(B0) > 0 such

that the wave packet is normalisable.

-15 -10 -5 0 5 10 15

x

-10

-5

0

5

10

V(x)

-30 -20 -10 0 10 20 30

x

-50

-25

0

25

50

V(x)

Figure 1: Real (solid blue lines) and imaginary (dashed ma-genta lines) parts of the potential (7) with γ = 1, ω = 1,and η = 5 (on the left) and η = 10 (on the right). Theimaginary part is scaled up by a factor of two, to make itmore visible. For comparison we also show the probabilitydistribution of a Gaussian beam with B = i, correspondingto the width of the approximate ground state of the real partof the potential.

In the spirit of Hepp and Heller we make a Gaus-sian ansatz for the propagated beam

ψ(x, z)=N(z)(Im(B(z))

π

) 14

ei(B(z)

2 (x−q(z))2+p(z)(x−q(z))+α(z)),

(4)with q(z), p(z) ∈ R, B(z) ∈ C. Here N(z) ∈ Rdenotes the norm, and α(z) ∈ R is an additionalphase. We Taylor expand the potential around thecenter q(z) of the wave packet and equate terms ofthe same orders of (x− q) to find the dynamicalequations for the parameters [7]

p =− V ′R(q) +Re(B)

Im(B)V ′I (q),

q =p+1

Im(B)V ′I (q),

B =−B2 − V ′′R (q)− iV ′′I (q),

N =

(1

~VI(q) +

1

4 Im(B)V ′′I (q)

)N,

α =p q − p2

2− VR(q)− 1

2Im(B).

(5)

Here VR and VI denote the real and imaginary partof the (negative) refractive index, respectively. Thedot denotes a derivative with respect to z, and theprime a derivative with respect to q. To keep thenotation compact we have dropped the explicit de-pendence on z of the parameters, and we shall con-tinue to do so in what follows.

The first two equations in (5) are the generalisa-tions of the standard Hamilton’s equations for thedynamics of the position and the momentum to sys-tems with gain and loss. In stark contrast to the

2

well-known conservative case these equations de-pend explicitly on the time-dependent parameterB that describes the shape of the Gaussian beamand whose dynamics is governed by the third equa-tion in (5). The remaining two equations describethe dynamics of the overall norm and an additionalphase, and can be trivially integrated once the firstthree equations are solved. In the present paperwe shall not be concerned with the dynamics of thephase α.

We can eliminate p to derive a second orderdifferential equation for the position of the wavepacket center

q = −V ′R(q) +Re(B)

Im(B)V ′I (q) +

1

Im(B)V ′′I (q)q, (6)

that couples to the dynamical equation for thewidth parameter B, and both in turn determinethe dynamics of the norm N . Since the approxi-mation relies on the Gaussian to be well localized,the imaginary part of B encodes the reliability ofthe approximation. In what follows we shall con-sider an example of a model for a PT -symmetricwaveguide to demonstrate the quality of the ap-proximation for initial Gaussian states, as well asthe influence of the beam shape on the propagation.

3 Gaussian propagation ina waveguide with PT -symmetric gain-loss profile

We consider the complex potential

V = −(

1− iγ

ηtanh

(x

η

))η2e−ω2x2

2η2 (7)

as a model for a multimode waveguide with PT -symmetric gain-loss profile. The parameter η de-termines the scale of the potential, which extendsfurther and is deeper for larger values of η. Thus,the larger the value of η, the more classical (i.e.described by equations (5)) we expect the propa-gation of Gaussian states to be. The strength ofthe loss-gain profile is described by the parameterγ. Close to the origin the real part of the potentialis approximately harmonic with frequency ω inde-pendently of the value of η. Thus, the fundamentalmode of the real part of the potential is approx-imately a Gaussian state with B = iω, i.e., with

-15 -10 -5 0 5 10 15

x

0

5

10

15

20

25

30

z

0

0.2

0.4

0.6

0.8

-15 -10 -5 0 5 10 15

x

0

5

10

15

20

25

30

z

0

0.2

0.4

0.6

0.8

-15 -10 -5 0 5 10 15

x

0

5

10

15

20

25

30

z

0

0.2

0.4

0.6

0.8

-15 -10 -5 0 5 10 15

x

0

5

10

15

20

25

30

z

0

0.2

0.4

0.6

0.8

-15 -10 -5 0 5 10 15

x

0

5

10

15

20

25

30

z

0

0.2

0.4

0.6

0.8

-15 -10 -5 0 5 10 15

x

0

5

10

15

20

25

30

z

0

0.2

0.4

0.6

0.8

-15 -10 -5 0 5 10 15

x

0

5

10

15

20

25

30

z

0

0.2

0.4

0.6

0.8

-15 -10 -5 0 5 10 15

x

0

5

10

15

20

25

30

z

0

0.2

0.4

0.6

0.8

Figure 2: Renormalized beam propagation in the Gaussianapproximation (left) and for a numerically exact propagationusing the split-operator method (right) for p0 = 0, B0 = iand different values of q0 and η (q0 = 1 and η = 5, q0 = 2and η = 10, q0 = 3 and η = 15, and q0 = 1 and η = 15respectively, from top to bottom).

width ∆q = 1√2ω

, which provides the natural length

scale on which the potential should be locally welldescribed by the second order Taylor expansion forthe Gaussian approximation to be valid. Figure 1depicts the real and imaginary part of the potentialfor γ = 1 and ω = 1 for two values of η. For com-parison the intensity profile for a Gaussian beamwith B = i is also depicted.

To illustrate how the Gaussian approximationimproves with increasing η we show examples ofthe propagation in the approximation as well asthe numerically exact solution of the Schrodingerequation (using a split-operator method) in Figure2 for three different values of η for B0 = i

2 , p0 = 0and initial positions slightly off the central axis.We have chosen a value of the width that is differ-

3

0 5 10 15 20 25 30

z

-2

-1

0

1

2

q

0 5 10 15 20 25 30

z

0

10

20

30

40

50

60

N2

0 5 10 15 20 25 30

z

-3

-2

-1

0

1

2

3

q

0 5 10 15 20 25 30

z

0

100

200

300

400

500

N2

0 5 10 15 20 25 30

z

-4

-2

0

2

4

q

0 10 20 30

z

0

500

1000

1500

2000

2500

3000

3500

N2

0 5 10 15 20 25 30

z

-1.5

-1

-0.5

0

0.5

1

1.5

2

q

0 5 10 15 20 25 30

z

0

20

40

60

80

100

N2

Figure 3: Gaussian approximation (solid blue lines) and ex-act quantum propagation using the split-operator method(dashed magenta lines) for the mean position (left) and thenorm (right) for the same parameters as in figure 2.

ent from that of the coherent state of the harmonicapproximation around the origin, such that thereis a stronger contribution from the width. Sincethe potential is not completely scale-invariant withrespect to η we cannot directly compare the prop-agation for different values of η and correspond-ing initial values of q0. To provide an overviewwe show three examples with different values of ηfor the same value of q0

η as well as a fourth exam-ple with a large value of η where the initial valueof q0 is chosen the same as for the example withthe smallest value of η. Figure 3 depicts the cor-responding propagation of the mean values of theposition and the norm. It can be observed thatwhile the Gaussian approximation becomes moreaccurate for larger values of η as expected, it al-ready qualitatively captures characteristic featuresof the evolution even for relatively small values of

-20 -10 0 10 20

x

0

10

20

30

40

50

z

0

0.2

0.4

0.6

0.8

1

-20 -10 0 10 20

x

0

10

20

30

40

50

z

0

0.5

1

1.5

-20 -10 0 10 20

x

0

10

20

30

40

50

z

0

0.2

0.4

0.6

0.8

1

-20 -10 0 10 20

x

0

10

20

30

40

50

z

0

0.5

1

1.5

2

0 10 20 30 40 50

z

-5

0

5

q

0 10 20 30 40 50

z

0

2000

4000

6000

8000

10000

12000

N2

Figure 4: The top row shows the (renormalised) propagationin the Gaussian approximation for η = 10, B0 = i, p0 = 0,and q0 = −4 on the left in comparison to the Gaussianapproximation for the equivalent Hermitian system on theright. The second row shows the same propagations for B0 =i2

. The bottom left figure shows the comparison betweenthe mean position in the presence (solid blue line for B0 = i,solid magenta line for B0 = i

2) and absence (dashed black

line) of gain and loss, corresponding to the figures in the toptwo rows. The right figure on the bottom shows the normof the propagated beam in the presence of gain and loss forB0 = i (blue) and B0 = i

2(magenta).

η. We shall return to a more detailed analysis andexplanation of some of these features later. In thefollowing we shall focus on the case η = 10, wherewe expect the Gaussian approximation to make rea-sonable predictions for propagation distances of theorder of a few oscillations.

Figure 4 shows the Gaussian approximation forthe propagation of initial Gaussian beams withB0 = i as well as B0 = i

2 and a relatively large ini-tial displacement from the central axis of q0 = −4(i.e. in the loss region) in comparison to the prop-agation of the same initial beam in the absence ofloss and gain, i.e. for γ = 0. The correspondingmean positions are depicted in direct comparisonin the lower left panel of the same figure. The dom-inant feature in the beam propagation even in thepresence of gain and loss in the current example arethe oscillations due to the real part of the potential.

4

-20 -10 0 10 20

x

0

10

20

30

40

50

z

0.1

0.2

0.3

0.4

0.5

0.6

-20 -10 0 10 20

x

0

10

20

30

40

50

z

0.1

0.2

0.3

0.4

0.5

0.6

-20 -10 0 10 20

x

0

10

20

30

40

50

z

0

0.2

0.4

0.6

0.8

-20 -10 0 10 20

x

0

10

20

30

40

50

z

0

0.2

0.4

0.6

0.8

0 10 20 30 40 50

z

-2

-1

0

1

2

q

0 10 20 30 40 50

z

0

20

40

60

80

100

120

140

N2

Figure 5: Same as figure 4, however, for q0 = −1.

We note a reduction of the symmetry and modula-tions on top of the periodic oscillations induced bythe loss-gain profile, partly related to the fact thatthe varying width of the wave packet influences thecentral motion. The norm of the propagated beamdepicted in the lower panel on the right, however, isstrongly influenced by the gain and loss, and showspronounced modulations as the beam oscillates be-tween the loss and the gain regions. The differencebetween the two initial widths is only a quantita-tive one here. We have found the Gaussian ap-proximation to be in good correspondence with thenumerically exact propagation for the propagationdistance depicted here.

From equations (5) we expect the influence ofthe gain-loss profile on the beam propagation to bemost pronounced where its first derivative with re-spect to the transversal coordinate is largest, thatis near the origin in our example. This can alreadybe observed in figures 2 and 3 where the propaga-tion for the smallest values of q0 shows the mostpronounced modulations. Figure 5 shows the samecomparisons as Figure 4, however, for a smaller ab-solute value of q0. We observe that while the dif-ferences induced by the gain-loss profile are morepronounced than for the larger initial value of q0 forboth B0 = i and B0 = i

2 , in the second case there

-15 -10 -5 0 5 10 15

x

0

2

4

6

8

10

z

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10

z

0

0.5

1

1.5

2

2.5

N2

-15 -10 -5 0 5 10 15

x

0

2

4

6

8

10

z

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8 10

z

0

0.5

1

1.5

2

N2

-15 -10 -5 0 5 10 15

x

0

2

4

6

8

10

z

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10

z

0

10

20

30

40

N2

0 2 4 6 8 10

z

-2

-1

0

1

2q

Figure 6: (Renormalized) beam propagation in the Gaus-sian approximation for η = 10, p0 = −1, q0 = 0 and threedifferent values of B0 (B0 = i

2at the top, B0 = i in the

middle, and B0 = 2i on the bottom.). The left panel depictsthe beam propagation, the right panel the norm. At thebottom we show the mean value of the position q(z) (solidlines) in the Gaussian approximation against the exact val-ues computed via the split operator method (dashed lines)for B0 = i (black), i

2(blue) and 2i (magenta).

are more striking differences developing. These aredue to the modulations in the width parameterthat are also present in the corresponding Hermi-tian case, but do not influence the central motionin that case.

In Figure 6 we depict another example wherethe difference in the initial width leads to drasticdifferences in the propagation, while all cases arewell described by the Gaussian approximation (ascan be seen by the direct comparison of the cen-ter propagation depicted at the bottom of the fig-ure). In the right panel we depict the propagationof three Gaussian beams with the same initial val-ues of position and momentum, but different initial

5

widths. The right column shows the correspondingnorm propagations. The initial momentum is non-zero (p0 = −1), nevertheless, for B = i, depictedin the middle row, the beam appears stationary.This is due to the fact that the beam is naturallydragged into the gain region, and we have chosenthe initial momentum such that it exactly coun-terbalances this dragging force for B = i. Notethat it is only possible to create such a stationarysolution for this particular value of the width pa-rameter, where the initial Gaussian beam is a verygood approximation for the ground state solutionof the potential. If the beam is initially wider, asdepicted for the example B0 = i

2 on the top, theinfluence of the gain-loss potential is stronger, andthe beam starts moving to the right into the gainregion and then starts oscillating due to the influ-ence of the real confining potential. If the beamis wider, on the other hand, as shown for the ex-ample B0 = 2i in the bottom panel of the figure,the influence of the gain-loss profile is effectivelydecreased. In this case, the initial momentum isstrong enough to make the beam move towards theloss region initially, before starting to oscillate.

These effects can be understood in more detailusing the approximative potential

V =1

2ω2x2 + iγx, (8)

which corresponds to the Taylor expansion of thepotential (7) around the origin. In this approxima-tion the equations of motion for q,B andN simplifyto

q =− ω2q +Re(B)

Im(B)γ,

B =−B2 − ω2,

N =1

~γqN,

(9)

and we can obtain p from p = q − 1Im(B)γ. The

equation for B does no longer depend on q, and theequation for q has the form of a forced harmonicoscillator, where the forcing term depends on B.Hence, the change of the shape of the wave packetacts like a forcing term in the evolution of the cen-tre of the packet. This is true for general systemswhenever the wave packet is in a region where theimaginary part of the potential is approximatelylinear. Once the equation for q is solved, the norm

-15 -10 -5 0 5 10 15

x

0

2

4

6

8

10

z

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10

z

-1.5

-1

-0.5

0

0.5

1

1.5

q

-15 -10 -5 0 5 10 15

x

0

2

4

6

8

10

z

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 2 4 6 8 10

z

-2

-1

0

1

2

q

-15 -10 -5 0 5 10 15

x

0

10

20

30

40

50

z

0.1

0.2

0.3

0.4

0.5

0 10 20 30 40 50

z

-5

0

5

q

Figure 7: (Renormalized) beam propagation in the Gaussianapproximation in the potential (8) for parameter values asin Figures 6 (top, B0 = i

2, middle: B0 = 2i) and 4 (bottom,

B0 = i). The left panel shows the renormalized beam prop-agation, the right panel shows the mean position q(z) of thispropagation (magenta line) in comparison to the Gaussianapproximation for the exact potential (7) (blue line).

N(z) is obtained by direct integration

N(z) = N0 expγ

~

∫ z

0

q(s) ds . (10)

The equation for B has one stationary solutionwith ImB > 0, namely B = iω, and for this choiceof B the forcing term in the equation for q vanishesand we arrive at a harmonic oscillation with a shiftby γ in the momentum

q(z) = q0 cos(ωz) +p0 + γ

ωsin(ωz)

p(z) = −ωq0 sin(ωz) + (p0 + γ) cos(ωz)− γ

N(z) = N0eγ~ ( q0ω sin(ωz)− p0+γ

ω2 cos(ωz)).(11)

This is exactly the propagation we have observed inthe top panel in Figure 5, as well as the stationarysolution in the middle panel of Figure 6. For ageneral initial condition B0 with ImB0 > 0 thesolution to the equation for B is given by

B(z) = ωB0 cos(ωz)− ω sin(ωz)

B0 sin(ωz) + ω cos(ωz), (12)

6

from which we obtain

Re(B(z))

Im(B(z))=|B0|2 − ω2

2ω ImB0sin(2ωz)+

ReB0

ImB0cos(2ωz),

(13)that is, the forcing in the q equation has frequency2ω and is small if ReB0 is small and ImB0 is closeto ω.

The evolution of the center of the wave packet isthus given by

q(z) = a cos(ωz) + b sin(ωz)− γ

ω2

Re(B(z))

Im(B(z))(14)

with a =(q0 + γ ReB0

ω2 ImB0

)and b =

(p0ω + γ

ω3

|B0|2ImB0

)and ReB/ ImB given by (13). The norm and themomentum can then be directly obtained as above.

This explains the modulations with twice the fre-quency that can be seen on the top and bottom ofFigure 2, and in Figures 5 and 6, for B0 6= i. Theapproximation (8) does not only provide a quali-tatively but also quantitatively good description ofthe propagation, as is demonstrated in Figure 7,which depicts the propagations of the two nontriv-ial cases in Figure 6 in the top two rows, using theapproximative potential (8). For a better compari-son we also plot the mean values of the position inthe right column of the figure. The bottom row de-picts the same comparison for a propagation witha larger initial displacement from the origin, as inthe top row of Figure 4. As expected the details ofthe propagation are not recovered in this case.

Let us finish with pointing out a possible appli-cation of the dependence of the propagation on thewidth of the wavepacket as a filtering device. Forshort propagation distances through a region wherethe potential is well described by the approximation(8) around the center of the initial wavepacket, theposition and momentum change according to

p(z) = p0 −(ω2q0 − γ

Re(B0)

Im(B0)

)z

q(z) = q0 +

(p0 + γ

1

Im(B0)

)z.

(15)

That is, for the same initial position and mo-mentum there is an extra shift in the position qthat is linear in γ, i.e., the slope of the gain-lossprofile at q0, and quadratic in the initial width:q(z) = q0 + p0z + 2γ(∆q)2z. This effect could be

used to spatially separate Gaussian beams with dif-ferent widths. If the width parameter of the beamalso has a non-vanishing real part, this leads to anadditional shift in the momentum, which translatesinto the angle in optical applications. At the sametime the overall norm is only changed linearly ac-cording to

N(z) = N0(1 + γq0z), (16)

for short propagation distances.

4 Summary and Conclusion

We have derived semiclassical equations of motionfor Gaussian beams propagating in waveguides inthe paraxial approximation in the presence of gainand loss. In the absence of losses this leads toHamiltonian motion of the center with a time de-pendent width. In the presence of gain and/orloss, however, the width of the beam influences thecentral motion. We have demonstrated that thisGaussian approximation can capture typical fea-tures of beam propagation in gain-loss wave guidessuch as power oscillations, and can accurately de-scribe the propagation if the refractive index is welldescribed by its Taylor expansion up to second or-der on length scales given by the typical widths ofthe Gaussian beam. Finally we have demonstratedhow the dependence on the width could be used asa filtering device.

Acknowledgment

E.M.G. acknowledges support from the RoyalSociety via a University Research Fellowship(Grant. No. UF130339), and a L’Oreal-UNESCOfor Women in Science UK and Ireland Fellow-ship. A.R. acknowledges support from the En-gineering and Physical Sciences Research Councilvia the Doctoral Research Allocation Grant No.EP/K502856/1.

References

[1] K. Hepp, Com. Math. Phys. 35 (1974) 265;E. J. Heller, J. Chem. Phys. 62 (1975) 1544

[2] R. G. Littlejohn, Phys. Rep. 138 (1986) 193

7

[3] E. J. Heller, J. Chem. Phys. 75 (1981) 2923;D. Huber and E. J. Heller, J. Chem. Phys. 89(1988) 4752; D. Huber, S. Ling, D.G. Imre,and E. J. Heller, J. Chem. Phys. 90 (1989)7317

[4] S. Longhi, Laser & Photon. Rev. 3 (2009) 243

[5] C. M. Bender, S. Boettcher, and P. N.Meisinger, J. Math. Phys. 40 (1999) 2201

[6] R. El-Ganainy, K. G. Makris, D. N.Christodoulides, and Z. H. Musslimani, Opt.Lett. 32 (2007) 2632; K. G. Makris,R. El-Ganainy, D. N. Christodoulides, andZ. H. Musslimani, Phys. Rev. Lett. 100,103904 (2008); S. Klaiman, U. Gunther, andN. Moiseyev, Phys. Rev. Lett. 101, 080402(2008); A. Guo, G. J. Salamo, D. Duchesne,R. Morandotti, M. Volatier-Ravat, V. Aimez,G. A. Siviloglou, and D. N. Christodoulides,Phys. Rev. Lett. 103, 093902 (2009); S.Longhi, Phys. Rev. Lett 103, 123601 (2009);C. E. Ruter, K. G. Makris, R. El-Ganainy,D, N, Christodoulides, and D. Kip, Nat. Phys.6, 192 (2010); E. M. Graefe and H. F. Jones,Phys. Rev. A 84 (2011) 013818; A. Regens-burger, C. Bersch, M-A Miri, G. Onishchukov,D. N. Christodoulides, and U. Perschel, Na-ture 488 (2012) 167

[7] E. M. Graefe and R. Schubert, Phys. Rev.A 83 (2011) 060101(R); E. M. Graefe andR. Schubert, J. Phys. A 45 (2012) 244033

8