Propagation of Gaussian beams in the presence of gain and loss

6 MB·

by  Eva-Maria Graefe

Propagation of Gaussian beams in the presence of gain and loss
Eva-Maria Graefe, Alexander Rush, and Roman Schubert ∗†
January 29, 2016
Abstract
ordertermsintheTaylorexpansionoftheHamilto-
6
nianarenegligiblethisisagoodapproximation. It
1
We consider the propagation of Gaussian beams is also the main ingredient of more powerful meth-
0
2 in a waveguide with gain and loss in the paraxial odstoapproximatedynamicsinthedeepquantum
approximation governed by the Schr¨odinger equa- regimesuchasthefrozenGaussianpropagatorand
n
tion. We derive equations of motion for the beam the method of hybrid mechanics [3].
a
J in the semiclassical limit that are valid when the
There are many analogies between the wave na-
8 waveguide profile is locally well approximated by
ture of light and the quantum mechanical wave
2 quadratic functions. For Hermitian systems, with-
function. In particular, the propagation of light
out any loss or gain, these dynamics are given by
] in wave guides in the paraxial approximation is
Hamilton’sequationsforthecenterofthebeamand
h described by a time-dependent Schr¨odinger equa-
p its conjugate momentum. Adding gain and/or loss
tion(wherethepropagationdirectionactsastime).
- to the waveguide introduces a non-Hermitian com-
t Thishasledtospectaculardemonstrationsofquan-
n ponent,causingthewidthoftheGaussianbeamto
tum dynamical effects, as well as to new applica-
a play an important role in its propagation. Here we
u tions in optics (see, e.g. [4] and references therein).
showhowthewidthaffectsthemotionofthebeam
q Inrecentyears,therehasbeenasurgeofinterestin
[ andhowthismaybeusedtofilterGaussianbeams
optical waveguides with gain and loss, mimicking
located at the same initial position based on their
1 non-Hermitian, and in particular PT-symmetric
width.
v [5], quantum systems [6]. It is an interesting ques-
2 tion what might be the fate of initial Gaussian
0
wave packets in the limit of large locally approx-
8 1 Introduction
imately harmonic potentials in the spirit of Hepp
7
0 and Heller. In [7] this question has been investi-
While quantum dynamics is in general very differ-
. gated,andanewtypeofclassicaldynamicsguiding
1 ent from classical dynamics, for short times ini-
0 the centers of Gaussian wave packets has been de-
tially Gaussian wave packets stay approximately
6 rived. In this dynamics the changing shape of the
Gaussianandfollowclassicaltrajectories. Thishas
1 wave packet couples back into the dynamics of the
: beenshownbyHeppandHellerusingthefactthat
v centre. That is, even in the limit where the beam
GaussianstatesstayexactlyGaussianforharmonic
i width is completely negligible compared to the ex-
X Hamiltonians, and Taylor expanding more general
ternal potential and where the dynamics is accu-
r Hamiltonians up to second order around the cen-
rately described by the Gaussian approximation,
a
tre of the wave packet [1] (see also [2] for a review
there are many classical trajectories for a given set
on the properties and dynamics of Gaussian wave
of initial position and momentum, depending on
packets). Aslongasthewavepacketstayswelllo-
the width of the wave packet.
calized compared to the scale on which the higher
Here we review the approximation and give ex-
∗Eva-MariaGraefeandAlexanderRusharewiththeDe- plicit equations of motion for the special case of
partment of Mathematics, Imperial College London, Lon-
a Hamiltonian with complex potential, and inves-
don,SW72AZ,UnitedKingdom
†Roman Schubert is with the School of Mathematics, tigate the resulting dynamics in an example of a
UniversityofBristol,BristolBS81TW,UnitedKingdom PT-symmetricmultimodewaveguide. Theapprox-
1
imation accurately describes the propagation and 50
10
effectssuchasthetypicalpoweroscillationsinPT-
5 25
symmetric systems if the potential is chosen large
enough compared to the beam width associated to V(x) 0 V(x) 0
the ground state. We demonstrate how the shape -5 -25
-10
of the beam affects the propagation, and how this -50
-15 -10 -5 0 5 10 15 -30 -20 -10 0 10 20 30
might be used as a filtering device. x x
Figure1: Real(solidbluelines)andimaginary(dashedma-
genta lines) parts of the potential (7) with γ = 1, ω = 1,
2 Gaussian beams in the and η = 5 (on the left) and η = 10 (on the right). The
imaginary part is scaled up by a factor of two, to make it
paraxial wave equation more visible. For comparison we also show the probability
distributionofaGaussianbeamwithB =i, corresponding
tothewidthoftheapproximategroundstateoftherealpart
For simplicity we consider the propagation of light
ofthepotential.
in 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- InthespiritofHeppandHellerwemakeaGaus-
sults are straight forwardly generalised to higher sian ansatz for the propagated beam
dimensions and refractive indices that are modu-
lated in z-direction as well. ψ(x,z)=N(z)(cid:16)Im(B(z))(cid:17)14ei(B(2z)(x−q(z))2+p(z)(x−q(z))+α(z)),
In the paraxial approximation the propagation π
(4)
oftheelectricfieldamplitudeψ isdescribedbythe
with q(z),p(z) ∈ R, B(z) ∈ C. Here N(z) ∈ R
Schr¨odinger equation
denotes the norm, and α(z) ∈ R is an additional
∂ψ (cid:126)2 ∂2ψ phase. We Taylor expand the potential around the
i(cid:126) =− +V(x)ψ, (1) center q(z) of the wave packet and equate terms of
∂z 2n ∂x2
0
the same orders of (x−q) to find the dynamical
where (cid:126):= λ , and λ is the wavelength, and where equations for the parameters [7]
2π
the effective potential
Re(B)
p˙ =−V(cid:48)(q)+ V(cid:48)(q),
n2−n2(x) R Im(B) I
V(x)= 0 ≈n −n(x) (2)
2n0 0 q˙=p+ 1 V(cid:48)(q),
Im(B) I
is approximately given by the refractive index pro-
B˙ =−B2−V(cid:48)(cid:48)(q)−iV(cid:48)(cid:48)(q), (5)
filen(x)ofthewaveguide,wheren isthereference R I
0 (cid:18) (cid:19)
refractive index of the substrate. Without loss of N˙ = 1V (q)+ 1 V(cid:48)(cid:48)(q) N,
generality we use rescaled units with n =1=(cid:126) in (cid:126) I 4Im(B) I
0
what follows. p2 1
α˙ =pq˙− −V (q)− Im(B).
We consider the propagation of an initial Gaus- 2 R 2
sian beam of the form
HereV andV denotetherealandimaginarypart
R I
ψ(x,0)=(cid:16)Im(B0)(cid:17)14 ei(B20(x−q0)2+p0(x−q0)), (3) ofthe(negative)refractiveindex,respectively. The
π dot denotes a derivative with respect to z, and the
prime a derivative with respect to q. To keep the
where q ∈ R is the position of the center of
0 notation compact we have dropped the explicit de-
the beam, p , the expectation value of the quan-
0 pendenceonz oftheparameters,andweshallcon-
tum momentum operator, describes the angle of
tinue to do so in what follows.
incidence, and B ∈ C encodes the shape of the
0 The first two equations in (5) are the generalisa-
beam. In particular it holds ∆q0 = √2I1mB0, and tions of the standard Hamilton’s equations for the
∆p0 = √2|BIm0|B0, and we demand Im(B0) > 0 such dynamicsofthepositionandthemomentumtosys-
that the wave packet is normalisable. tems with gain and loss. In stark contrast to the
2
well-known conservative case these equations de- 30 0.8 30 0.8
pend explicitly on the time-dependent parameter 25 25
0.6 0.6
B that describes the shape of the Gaussian beam 20 20
z15 0.4 z15 0.4
andwhosedynamicsisgovernedbythethirdequa-
10 10
tion in (5). The remaining two equations describe 0.2 0.2
5 5
thedynamicsoftheoverallnormandanadditional 0 0 0 0
-15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15
phase,andcanbetriviallyintegratedoncethefirst x x
three equations are solved. In the present paper 30 0.8 30 0.8
25 25
weshallnotbeconcernedwiththedynamicsofthe 0.6 0.6
20 20
phase α. z15 0.4 z15 0.4
We can eliminate p to derive a second order 10 10
0.2 0.2
differential equation for the position of the wave 5 5
0 0 0 0
packet center -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15
x x
Re(B) 1 30 0.8 30 0.8
q¨=−V(cid:48)(q)+ V(cid:48)(q)+ V(cid:48)(cid:48)(q)q˙, (6) 25 25
R Im(B) I Im(B) I 0.6 0.6
20 20
z15 0.4 z15 0.4
that couples to the dynamical equation for the
10 10
width parameter B, and both in turn determine 5 0.2 5 0.2
the dynamics of the norm N. Since the approxi- 0 0 0 0
-15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15
mation relies on the Gaussian to be well localized, x x
30 0.8 30 0.8
the imaginary part of B encodes the reliability of
25 25
the approximation. In what follows we shall con- 0.6 0.6
20 20
sider an example of a model for a PT-symmetric z15 0.4 z15 0.4
waveguide to demonstrate the quality of the ap- 10 10
0.2 0.2
proximation for initial Gaussian states, as well as 5 5
0 0 0 0
theinfluenceofthebeamshapeonthepropagation. -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15
x x
Figure 2: Renormalized beam propagation in the Gaussian
3 Gaussian propagation in approximation(left)andforanumericallyexactpropagation
using the split-operator method (right) for p0 = 0, B0 = i
a waveguide with PT- and different values of q0 and η (q0 =1 and η =5, q0 =2
and η = 10, q0 = 3 and η = 15, and q0 = 1 and η = 15
symmetric gain-loss profile respectively,fromtoptobottom).
We consider the complex potential
V =−(cid:18)1−iγ tanh(cid:18)x(cid:19)(cid:19)η2e−ω22ηx22 (7) wsciadlteho∆nqw=hic√h12ωth,ewphoicthenptrioavlisdheosutlhdebneatluorcaallllyenwgetlhl
η η
describedbythesecondorderTaylorexpansionfor
as a model for a multimode waveguide with PT- the Gaussian approximation to be valid. Figure 1
symmetric gain-loss profile. The parameter η de- depictstherealandimaginarypartofthepotential
termines the scale of the potential, which extends for γ = 1 and ω = 1 for two values of η. For com-
further and is deeper for larger values of η. Thus, parison the intensity profile for a Gaussian beam
the larger the value of η, the more classical (i.e. with B =i is also depicted.
described by equations (5)) we expect the propa- To illustrate how the Gaussian approximation
gation of Gaussian states to be. The strength of improves with increasing η we show examples of
the loss-gain profile is described by the parameter the propagation in the approximation as well as
γ. Close to the origin the real part of the potential the numerically exact solution of the Schr¨odinger
is approximately harmonic with frequency ω inde- equation (using a split-operator method) in Figure
pendentlyofthevalueofη. Thus,thefundamental 2 for three different values of η for B = i, p =0
0 2 0
mode of the real part of the potential is approx- and initial positions slightly off the central axis.
imately a Gaussian state with B = iω, i.e., with We have chosen a value of the width that is differ-
3
2 60 50 50
1 1.5
1 50 40 0.8 40
40 30 30 1
q0 2N30 z 0.6 z
20 20 0.4 20 0.5
-1 10 10 0.2 10
-2 0 0 0 0 0
0 5 10 15 20 25 30 0 5 10 15 20 25 30 -20 -10 0 10 20 -20 -10 0 10 20
z z x x
3 500 50 50
2 400 40 1 40 2
1 0.8 1.5
300 30 30
q0 2N200 z20 0.6 z20 1
-1 0.4
-2 100 10 0.2 10 0.5
-3 0 0 0 0 0
0 5 10 15 20 25 30 0 5 10 15 20 25 30 -20 -10 0 10 20 -20 -10 0 10 20
z z x x
4 3500 5 12000
3000 10000
2
2500 8000
q0 2N2000 q0 2N6000
1500
-2 1000 4000
-4 500 2000
0 -5 0
0 5 10 15 20 25 30 0 10 20 30 0 10 20 30 40 50 0 10 20 30 40 50
z z z z
2 100
Figure4: Thetoprowshowsthe(renormalised)propagation
1.5
1 80 in the Gaussian approximation for η =10, B0 =i, p0 =0,
q0.50 2N4600 aanpdproqx0im=at−io4nofonrtthhee leeqfutivinalecnotmHpaerrmisoitniatnostyhsteemGaounsstiahne
-0.5 20 right. ThesecondrowshowsthesamepropagationsforB0=
-1 i. The bottom left figure shows the comparison between
-1.50 5 10 15 20 25 30 00 5 10 15 20 25 30 t2hemeanpositioninthepresence(solidbluelineforB0=i,
z z solid magenta line for B0 = 2i) and absence (dashed black
Figure3: Gaussianapproximation(solidbluelines)andex- line)ofgainandloss,correspondingtothefiguresinthetop
act quantum propagation using the split-operator method two rows. The right figure on the bottom shows the norm
(dashedmagentalines)forthemeanposition(left)andthe ofthepropagatedbeaminthepresenceofgainandlossfor
norm(right)forthesameparametersasinfigure2. B0=i(blue)andB0= 2i (magenta).
entfromthatofthecoherentstateoftheharmonic η. We shall return to a more detailed analysis and
approximation around the origin, such that there explanation of some of these features later. In the
is a stronger contribution from the width. Since following we shall focus on the case η = 10, where
thepotential isnotcompletelyscale-invariantwith weexpecttheGaussianapproximationtomakerea-
respect to η we cannot directly compare the prop- sonablepredictionsforpropagationdistancesofthe
agation for different values of η and correspond- order of a few oscillations.
ing initial values of q . To provide an overview Figure 4 shows the Gaussian approximation for
0
we show three examples with different values of η the propagation of initial Gaussian beams with
for the same value of q0 as well as a fourth exam- B =i as well as B = i and a relatively large ini-
η 0 0 2
ple with a large value of η where the initial value tial displacement from the central axis of q = −4
0
of q is chosen the same as for the example with (i.e. in the loss region) in comparison to the prop-
0
the smallest value of η. Figure 3 depicts the cor- agation of the same initial beam in the absence of
responding propagation of the mean values of the loss and gain, i.e. for γ = 0. The corresponding
position and the norm. It can be observed that mean positions are depicted in direct comparison
while the Gaussian approximation becomes more inthelowerleftpanelofthesamefigure. Thedom-
accurate for larger values of η as expected, it al- inant feature in the beam propagation even in the
ready qualitatively captures characteristic features presenceofgainandlossinthecurrentexampleare
of the evolution even for relatively small values of theoscillationsduetotherealpartofthepotential.
4
50 0.6 50 0.6 10 2.5
0.7
40 0.5 40 0.5 8 0.6 2
30 0.4 30 0.4 6 0.5 1.5
z 0.3 z 0.3 z 0.4 2N
20 0.2 20 0.2 4 0.3 1
10 0.1 10 0.1 2 00..12 0.5
0 0 0 0
-20 -10 0 10 20 -20 -10 0 10 20 -15 -10 -5 0 5 10 15 0 2 4 6 8 10
x x x z
50 0.8 50 0.8 10 2
0.5
40 0.6 40 0.6 8 0.4 1.5
30 30 6
z 0.4 z 0.4 z 0.3 2N 1
20 20 4 0.2
10 0.2 10 0.2 2 0.1 0.5
0 0 0 0 0 0
-20 -10 0 10 20 -20 -10 0 10 20 -15 -10 -5 0 5 10 15 0 2 4 6 8 10
x x x z
2 140 10
120 0.7 40
1 100 8 0.6
80 6 0.5 30
q0 2N60 z 4 00..34 2N20
-1 2400 2 00..12 10
-2 0 0 0
0 10 20 30 40 50 0 10 20 30 40 50 -15 -10 -5 0 5 10 15 0 2 4 6 8 10
z z x z
2
Figure5: Sameasfigure4,however,forq0=−1.
1
q0
We note a reduction of the symmetry and modula-
-1
tions on top of the periodic oscillations induced by
the loss-gain profile, partly related to the fact that -20 2 4 6 8 10
z
thevaryingwidthofthewavepacketinfluencesthe
central motion. The norm of the propagated beam Figure 6: (Renormalized) beam propagation in the Gaus-
depictedinthelowerpanelontheright,however,is sian approximation for η =10, p0 =−1, q0 =0 and three
stronglyinfluencedbythegainandloss,andshows mdiiffdedrleen,tanvadluBe0s=of2Bio0n(tBh0eb=ot2tiomat.)t.hTehteople,ftBp0an=elidienpitchtes
pronouncedmodulationsasthebeamoscillatesbe- the beam propagation, the right panel the norm. At the
tween the loss and the gain regions. The difference bottom we show the mean value of the position q(z) (solid
lines)intheGaussianapproximationagainsttheexactval-
between the two initial widths is only a quantita-
ues computed via the split operator method (dashed lines)
tive one here. We have found the Gaussian ap- forB0=i(black), 2i (blue)and2i(magenta).
proximationtobeingoodcorrespondencewiththe
numerically exact propagation for the propagation
distance depicted here. are more striking differences developing. These are
From equations (5) we expect the influence of due to the modulations in the width parameter
thegain-lossprofileonthebeampropagationtobe that are also present in the corresponding Hermi-
most pronounced where its first derivative with re- tian case, but do not influence the central motion
spect to the transversal coordinate is largest, that in that case.
is near the origin in our example. This can already In Figure 6 we depict another example where
be observed in figures 2 and 3 where the propaga- the difference in the initial width leads to drastic
tion for the smallest values of q shows the most differences in the propagation, while all cases are
0
pronounced modulations. Figure 5 shows the same well described by the Gaussian approximation (as
comparisonsasFigure4,however,forasmallerab- can be seen by the direct comparison of the cen-
solute value of q . We observe that while the dif- ter propagation depicted at the bottom of the fig-
0
ferences induced by the gain-loss profile are more ure). In the right panel we depict the propagation
pronouncedthanforthelargerinitialvalueofq for of three Gaussian beams with the same initial val-
0
both B = i and B = i, in the second case there uesofpositionandmomentum,butdifferentinitial
0 0 2
5
widths. Therightcolumnshowsthecorresponding 10 1.5
norm propagations. The initial momentum is non- 8 0.7 1
0.6
zero (p0 = −1), nevertheless, for B = i, depicted 6 0.5 0.5
z 0.4 q 0
in the middle row, the beam appears stationary. 4 0.3 -0.5
This is due to the fact that the beam is naturally 2 00..12 -1
dragged into the gain region, and we have chosen 0 -1.5
-15 -10 -5 0 5 10 15 0 2 4 6 8 10
the initial momentum such that it exactly coun- x z
10 2
terbalances this dragging force for B = i. Note
0.7
that it is only possible to create such a stationary 8 0.6 1
6 0.5
solution for this particular value of the width pa- z 0.4 q0
rameter, where the initial Gaussian beam is a very 4 0.3
2 0.2 -1
good approximation for the ground state solution 0.1
0 -2
of the potential. If the beam is initially wider, as -15 -10 -5 0 5 10 15 0 2 4 6 8 10
x z
depicted for the example B0 = 2i on the top, the 50 5
influence of the gain-loss potential is stronger, and 40 0.5
0.4
the beam starts moving to the right into the gain 30
region and then starts oscillating due to the influ- z20 00..23 q0
ence of the real confining potential. If the beam 10 0.1
is wider, on the other hand, as shown for the ex- 0 -5
-15 -10 -5 0 5 10 15 0 10 20 30 40 50
ample B = 2i in the bottom panel of the figure, x z
0
the influence of the gain-loss profile is effectively Figure7: (Renormalized)beampropagationintheGaussian
decreased. In this case, the initial momentum is approximation in the potential (8) for parameter values as
strongenoughtomakethebeammovetowardsthe inFigures6(top,B0= 2i,middle: B0=2i)and4(bottom,
loss region initially, before starting to oscillate. B0=i). Theleftpanelshowstherenormalizedbeamprop-
agation,therightpanelshowsthemeanpositionq(z)ofthis
These effects can be understood in more detail propagation (magenta line) in comparison to the Gaussian
using the approximative potential approximationfortheexactpotential(7)(blueline).
1
V = ω2x2+iγx, (8) N(z) is obtained by direct integration
2
γ (cid:90) z
which corresponds to the Taylor expansion of the N(z)=N0exp (cid:126) q(s)ds . (10)
0
potential(7)aroundtheorigin. Inthisapproxima-
tiontheequationsofmotionforq,BandN simplify The equation for B has one stationary solution
to with ImB >0, namely B =iω, and for this choice
Re(B) ofB theforcingtermintheequationforq vanishes
q¨=−ω2q+ γ,
andwearriveataharmonicoscillationwithashift
Im(B)
by γ in the momentum
B˙ =−B2−ω2, (9)
p +γ
N˙ =1γqN, q(z)=q0cos(ωz)+ 0ω sin(ωz)
(cid:126)
p(z)=−ωq sin(ωz)+(p +γ)cos(ωz)−γ
0 0
and we can obtain p from p = q˙ − 1 γ. The
equationforB doesnolongerdependIomn(Bq,)andthe N(z)=N0eγ(cid:126)(qω0 sin(ωz)−p0ω+2γcos(ωz)).
(11)
equation for q has the form of a forced harmonic
Thisisexactlythepropagationwehaveobservedin
oscillator, where the forcing term depends on B.
the top panel in Figure 5, as well as the stationary
Hence, the change of the shape of the wave packet
solution in the middle panel of Figure 6. For a
acts like a forcing term in the evolution of the cen-
general initial condition B with ImB > 0 the
tre of the packet. This is true for general systems 0 0
solution to the equation for B is given by
whenever the wave packet is in a region where the
imaginary part of the potential is approximately B cos(ωz)−ωsin(ωz)
B(z)=ω 0 , (12)
linear. Once the equation for q is solved, the norm B sin(ωz)+ωcos(ωz)
0
6
from which we obtain usedtospatiallyseparateGaussianbeamswithdif-
ferent widths. If the width parameter of the beam
Re(B(z)) |B |2−ω2 ReB
= 0 sin(2ωz)+ 0 cos(2ωz), also has a non-vanishing real part, this leads to an
Im(B(z)) 2ωImB0 ImB0 additionalshiftinthemomentum,whichtranslates
(13)
into the angle in optical applications. At the same
that is, the forcing in the q equation has frequency
time the overall norm is only changed linearly ac-
2ω and is small if ReB is small and ImB is close
0 0 cording to
to ω.
The evolution of the center of the wave packet is N(z)=N (1+γq z), (16)
0 0
thus given by
for short propagation distances.
γ Re(B(z))
q(z)=acos(ωz)+bsin(ωz)− (14)
ω2Im(B(z))
4 Summary and Conclusion
(cid:16) (cid:17) (cid:16) (cid:17)
with a = q + γReB0 and b = p0 + γ |B0|2
0 ω2ImB0 ω ω3ImB0 We have derived semiclassical equations of motion
and ReB/ImB given by (13). The norm and the for Gaussian beams propagating in waveguides in
momentumcanthenbedirectlyobtainedasabove. the paraxial approximation in the presence of gain
Thisexplainsthemodulationswithtwicethefre- and loss. In the absence of losses this leads to
quency that can be seen on the top and bottom of Hamiltonian motion of the center with a time de-
Figure 2, and in Figures 5 and 6, for B0 (cid:54)= i. The pendent width. In the presence of gain and/or
approximation (8) does not only provide a quali- loss, however, thewidthofthebeaminfluencesthe
tatively but also quantitatively good description of central motion. We have demonstrated that this
the propagation, as is demonstrated in Figure 7, Gaussian approximation can capture typical fea-
which depicts the propagations of the two nontriv- tures of beam propagation in gain-loss wave guides
ial cases in Figure 6 in the top two rows, using the such as power oscillations, and can accurately de-
approximative potential (8). For a better compari- scribethepropagationiftherefractiveindexiswell
son we also plot the mean values of the position in described by its Taylor expansion up to second or-
therightcolumnofthefigure. Thebottomrowde- der on length scales given by the typical widths of
picts the same comparison for a propagation with the Gaussian beam. Finally we have demonstrated
a larger initial displacement from the origin, as in how the dependence on the width could be used as
the top row of Figure 4. As expected the details of a filtering device.
the propagation are not recovered in this case.
Let us finish with pointing out a possible appli-
Acknowledgment
cationofthedependenceofthepropagationonthe
width of the wavepacket as a filtering device. For
shortpropagationdistancesthrougharegionwhere E.M.G. acknowledges support from the Royal
thepotentialiswelldescribedbytheapproximation Society via a University Research Fellowship
(8)aroundthecenteroftheinitialwavepacket, the (Grant. No. UF130339), and a L’Or´eal-UNESCO
position and momentum change according to for Women in Science UK and Ireland Fellow-
ship. A.R. acknowledges support from the En-
(cid:18) (cid:19)
Re(B ) gineering and Physical Sciences Research Council
p(z)=p − ω2q −γ 0 z
0 0 Im(B ) via the Doctoral Research Allocation Grant No.
0
(15)
(cid:18) (cid:19) EP/K502856/1.
1
q(z)=q + p +γ z.
0 0 Im(B )
0
References
That is, for the same initial position and mo-
mentum there is an extra shift in the position q
[1] K. Hepp, Com. Math. Phys. 35 (1974) 265;
that is linear in γ, i.e., the slope of the gain-loss
E. J. Heller, J. Chem. Phys. 62 (1975) 1544
profile at q , and quadratic in the initial width:
0
q(z) = q +p z +2γ(∆q)2z. This effect could be [2] R. G. Littlejohn, Phys. Rep. 138 (1986) 193
0 0
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, and
Z. H. Musslimani, Phys. Rev. Lett. 100,
103904 (2008); S. Klaiman, U. Gu¨nther, and
N. 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. Ru¨ter, K. G. Makris, R. El-Ganainy,
D,N,Christodoulides,andD.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-AMiri,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 and
R. Schubert, J. Phys. A 45 (2012) 244033
8