Nonlinear Eigenmodes of a Polariton Harmonic Oscillator

1.5 MB·English

by  Florian Pinsker

Nonlinear Eigenmodes of a Polariton Harmonic Oscillator
Florian Pinsker1,∗ and Tristram J. Alexander2,†
1Department of Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom.
2School of Physical, Environmental and Mathematical Sciences,
UNSW Canberra, Canberra ACT 2600, Australia
(Dated: April 2, 2015)
We investigate theoretically the quantum oscillator-like states recently observed experimentally
in polariton condensates (Nat. Phys. 8, 190 (2012)). We consider a complex Gross-Pitaevskii type
model which includes the effects of self-interactions, and creation and decay of exciton-polaritons.
5 We develop a perturbation theory for approximate solutions to this non-equilibrium condensate
1
modelandcomparetheresultswithnumericallycalculatedsolutionsforbothrepulsiveandattractive
0
polariton-polaritoninteractions. Whilethenonlinearityhasaweakeffectonthemodeselectiontheir
2
density profiles are modified at moderate gain strengths and becomes more dominant when a very

n large gain of polaritons implies an extended cloud with high condensate densities. Finally, we
a identifytherelationoftheobservedpatternstotheinputpumpconfiguration,andsuggestthismay
J serve as a generalized NOR gate in the tradition of optical computing.

8
2 PACSnumbers: 03.65.-w,02.60.Cb,03.65.Sq,03.75.-b

]
s INTRODUCTION a local light field and decay after 1-100 ps mainly due to
a
g leaking of the cavity photons [6, 11, 18]. The decay rate
- significantlydependsonthequalityofconfinementofthe
t Bose-Einsteincondensates(BECs)areprimeexamples
n light field [18] within the microcavity and in this work it
ofnonlinearquantummany-bodysystems,becauseofin-
a is assumed to be constant over space. From a techno-
u teractionsbetweencoherentquantumparticles. Theyare
logical and experimental point of view polariton BECs
q regarded as a macroscopic quantum phenomenon due to
haveseveralpracticaladvantagesoveratomicBECs. For
.
t theirveryhighcoherenceandlong-rangeorderofallpar-
a instance, BECs of dilute atoms must be realized at tem-
ticles constituting this state of matter, a quantum state
m peratures within the nano-Kelvin range [19–21], whereas
that typically is formed at very low temperatures [1–6].
- polaritoncondensatescanbeobservedevenatroomtem-
d Many setups of atomic BEC are for fixed particle num- perature in polymers [22] and in organic materials [23].
n bers and can carry various topologically stable excita-
This is a consequence of the fact that polaritons have a
o tions ranging from solitons to giant vortices [3, 7–10].
verylightmassduetotheirinherentcouplingtophotons
c
More recently efforts have been made towards the real-
[ [6], and so have a much higher critical Bose condensa-
ization of coherent non-equilibrium many-body systems
tion temperature. Restrictions on the polariton density
1 that exchange particles and/or energy while at the same
v however have to be taken into account to ensure their
time obeying BEC properties e.g. in solid state systems
6 bosonic character is maintained [3].
[11, 12]. One such system is the polariton quasiparti-
2
0 cle Bose-Einstein condensate [2, 5, 6, 11, 12], which can In many settings a polariton condensate is an effec-
0 be effectively described by a non-conservative nonlinear tively2Dcoherentmany-bodysystemthatistightlycon-
0 Schr¨odinger-type equation [6, 13, 14] formally generaliz- fined with regards to the optical axis of the semiconduc-
4. ing the so called Gross-Pitaevskii equation (GPE) - the tor microcavity [2, 11, 12, 24]. Depending on additional
0 model for the condensate wave function of dilute and constraints the condensate may become essentially 1D
5 weakly interacting atomic BEC [4, 13, 14]. [2, 16]. The elementary excitations within a 1D system,
1 with repulsive polariton-polariton interactions, are dark
: Bose-condensed polaritons have become feasible in
v and grey solitons [6, 16]. Grey solitons are stable excita-
semiconductor microcavities in the last decade [2, 6, 11,
i tions which move over long distances without changing
X 12]. Polaritons in semiconductor microcavities are su-
their form and in principle with elastic scattering prop-
r perpositions between excitons (electron-hole pairs) and
a erties, but due to the non-equilibrium character obey a
cavity photons, and as such the combination of the com-
finite lifetime [25]. Dark solitons are the stationary spe-
ponents determines the particle statistics [2, 15]. These
cial case of the grey soliton when the density depletes
quasiparticles in a dilute regime show distinctive prop-
to zero at its minimum and with a π phase shift at the
erties such as Bose-Einstein condensation, superfluid-
nodal point [4].
ity, a finite nonzero speed of sound due to the non-
linearity and the emergence of elementary topologically In this work we are particularly interested in station-
stable excitations such as solitons or quantum vortices ary and stable patterns within a 1D polariton conden-
[2,6,13,16,17]. IntheirverynaturepolaritonBECsare sate,thatonceexcitedaresimilartodarksoliton[13]ar-
non-equilibrium systems - polaritons can be created via rays[16]withrespecttosingularitiesatlocalizedpoints-
2
the oscillator modes of a harmonically trapped quantum of recent experiments [13, 24],
state. Experimental observations of oscillator modes in
polariton condensates have been made in [26, 27], but a (cid:18) (cid:126)∇2
i∂ ψ = − +g N+
thorough theoretical description is lacking. The exper- t 2m∗ R
imentally observed non-equilibrium quantum oscillator (cid:19)
i
stateswereformedbetweentwospatiallyseparatedpump +g P +α |ψ|2+ (γ N −γ ) ψ. (1)
P 1 2 R C
spots of incoherently pumped polaritons, corresponding
to two localized laser beams shining on the semiconduc- where m∗ is the effective mass of the condensed polari-
tor microcavity and positioned within the 2D plane. Be-
tonsandP describesthegeometryandmagnitudeofthe
tween the pump spots stationary spatial patterns were
pumpspots,whichareinducedbylinearlypolarizedlaser
formedsimilartothoseobservedforaquantumharmonic
beams [26, 27]. The polariton reservoir is approximately
oscillator, hence suggesting an effective 1D problem.
given by [13, 24]
The spatial patterns can be understood in a dynam- (cid:18) (cid:19)
P β
ical picture as follows. The polaritons generated at the N (cid:39) 1− |ψ|2 . (2)
γ γ
spots move from the source peaks in all directions, be- R R
cause interactions of pump and reservoir with polaritons In Eq. 1 γ denotes the globally constant decay rate
C
are repulsive [24]. So in between the pump spots the of condensed polaritons, γ the relaxation of the reser-
R
bulk of polaritons are present and thus the condensate voir, β is the scattering rate of polaritons into the con-
and its patterns form predominantly there. Depending densate, α denotes the polariton-polariton interactions
1
on the pump strength, and the repulsiveness from the within the condensate, g is the repulsive interaction of
P
pump/reservoir, different modes can be observed; start- thecondensatewiththepumpandg therepulsiveinter-
R
ingfromthegroundstateofapproximatelygaussianform actions of the condensate with the reservoir. Parameter
(whennonlinearityplaysaminorrole),excitedstatesbe- values are chosen in accordance to most recent experi-
comefeasible. Experimentallyonlythecaseforrepulsive ments[13,16],i.e. effectivemassm∗ =5×10−5m where
0
self-interactions has been considered so far. In this work m isthemassofthefreeelectron,α =0.001·ps−1·µm2,
0 1
wepredict-basedontherecentexperimentaleffort[28]- g = 0.022·ps−1 ·µm2, g = 0.07·ps−1 ·µm2, β =
R P
thestatesforattractiveself-interactionsaswellandtheir 0.05·ps−1·µm2, γ =10·ps−1, R =0.06·ps−1·µm2,
R R
dependence on the gain. To elucidate the role of nonlin- γ =0.556·ps−1. The pump geometry is assumed to be
C
ear self-interactions for the pattern formation in quasi-
harmonically trapped polariton condensates we vary be- (cid:18)
P(x,y)=A exp(cid:0)−0.073·((x+x )2+y2)(cid:1)+
tween attractive and repulsive interactions and the non- 1
linear character of polariton BEC will become dominant (cid:19)
in a Thomas-Fermi (TF) type regime, i.e. when conden- +exp(cid:0)−0.073·((x−x )2+y2)(cid:1) (3)
2
sate densities are very high and pumping is strong.
with A being the amplitude and the position of the two
Our paper is organized as follows. First we will in-
gaussian pump spots is at x and x respectively, which
troduce the general model describing the polariton con- 2 1
modelsthegainduetolocalilluminationofthesemicon-
densate in 2D and a simplified model for the effective
ductor microcavity. If we neglect spatial extension in y
1D investigations. Then approximate analytical expres-
wesimplywriteP(x). Wenotethattheexactmathemat-
sionsforthecondensatewavefunctionsarepresentedand
ical form does not significantly change the physics under
compared with numerical simulations. Subsequently the
considerationand isanapproximation ofthe actualspot
dependence of the eigenvalues on pumping strength and
form used in recent experiments [13, 24].
thenonlinearityispresented. Finallyweintroduceagen-
Taking this into account, and as outlined in the next
eralizedopticalNORdeviceandsummarizeourfindings.
section in more detail, we can effectively reduce (1) to a
simplified model for the order parameter ψ of the polari-
ton condensate given by [30]
(cid:18) (cid:126)2∇2 (cid:19)
i(cid:126)∂ ψ = − +V(r)+U|ψ|2+i(γ −Γ|ψ|2) ψ.
t 2m∗ eff
POLARITON CONDENSATE MODEL (4)
Here U denotes the self-interaction strength, γ the lin-
eff
We describe the order parameter ψ of the polariton ear gain and Γ the nonlinear loss. The repulsive effect
condensatebythestateequationinthespincoherentcase on the polaritons of the pump P and the reservoir N,
[14, 29], which has proven to be an accurate description defined in (1), is assumed to be approximated by V. For
3
x
|ψ|2 average value of gain between both peaks assumed to
be at x = −10, x = 10 and 2Γ = β/γ . Con-
V 1 2 R
sequently the density dependent pump parameter be-
P
comes σ = Γ/U = 2.5 while the linear parameter is
2
µ =(0.165·A−0.278)/k with k =m∗ω2/2=0.092·A
2
as in [10] fixing the potential strength ω via the choice
of A. This identification yields the simplified equation
(5) once the corresponding rescaling has been applied.
Recallthatsgn(σ )dependsonthesignofthescattering
1
length between condensed polaritons.
Bytuningthepropertiesofthepump(inparticularthe
FIG.1. Effectiveharmonicconfiningpotential(dashedline)
with k = 0.05 vs. the pump profile P (dotted line) (3) for geometry and pumping strength), and also having con-
A = 5, x = 10µm and x = −10µm. A harmonic oscillator trol over the nonlinear interaction strength σ through
2 1 1
mode forms within the trapping (continuous line). the use of Feshbach resonances [28], we see that the gain
and loss coefficients (σ and µ ) can be varied widely.
2 2
In the remainder of this work we will set the density
our analysis we are interested in stationary states and so
dependent loss rate in rescaled units σ = 0.3 and con-
seti(cid:126)∂ ψ =µψ. Asasimplification,weneglectthetrans- 2
t sider values for the pumping strength µ from µ =0 to
1 2
versespatialdimension,andsowriteψ(x,y)=ψ(x),and
µ = 1, values that are in accordance with [30]. We re-
rescale(4)asin[30],i.e. V(x)=((cid:126)ω/2)(x2/l2)(however 2
mark that generally loss rates for polariton condensates
withouteffectivetransverseconfinement), whereω isthe
depend significantly on the actual experimental setup,
oscillator frequency and l = (cid:112)(cid:126)/(m∗ω) the correspond-
partly because lifetimes vary from 1ps to 100ps and cor-
inglength. Expressinglengthsinunitsofl andenergyin
respond to the quality of confinement within the micro-
units of (cid:126)ω, and by substituting ψ → (cid:112)(cid:126)ω/(2U)ψ, the
cavity[2,3,6,15,18],soevenaquasi-equilibriumscheme
result is
is feasible in polariton condensates [15]. This allows us
(cid:18) (cid:19) to examine the equilibrium limit in our nonlinear sys-
∂ −x2+(σ +iσ )|ψ|2−iµ ψ =−µ ψ (5)
xx 1 2 2 1 tem seamlessly (µ = 0), and explore the nature of the
2
system when gain and loss are of similar magnitudes but
with µ = (cid:0)2µ(cid:1), µ = 2γ /((cid:126)ω), σ = ±1 (+ corre- lessthanorequaltothemagnitudeofthenonlinearterm.
1 (cid:126)ω 2 eff 1
spondstoattractiveand−torepulsiveself-interactions), Setting σ1 = ±1 means an increase of real-valued ‘non-
σ = Γ/U and V(x) = x2 is the harmonic potential linear’ behavior than if we used the initially introduced
2
due to the approximation of the pumping spots left and ‘natural’ parameters, which is justified in physical terms
right of the condensate and the repulsive reservoir. We by utilizing Feshbach resonances [3, 10, 14].
notethat(5)isanonlineareigenvalueproblemwithreal-
valued eigenvalue µ , which we will determine numeri-
1
cally and analytically. THE POLARITON HARMONIC OSCILLATOR
Let us identify the Carusotto-Wouters model as ap-
proximatedin[13,24,29]withtheKeeling-Berloffmodel First we present results for simulations of the unre-
[30] by identifying the effective potential terms by duced state equation Eq. 1. The pump geometry is
assumed to be (3). In Fig. 2 (a) we present the den-
gRN +gPP +α1|ψ|2 ↔V(r)+U|ψ|2 (6) sity profile |ψ|2 showing the harmonic oscillator ground
state in-between two pump spots for A = 8.5 roughly
with V =kx2 and the effective gain and loss terms
approximated by a gaussian. Fig. 2 (b) shows the first
(γ N −γ )/2↔(γ −Γ|ψ|2) (7) excitedharmonicoscillator-typestateforA=11.5obey-
R C eff
ingtwodistinctpeaksofcondensatedensity. FinallyFig.
with N = P (1− β |ψ|2). To approximately match the 2 (c) shows the second excited harmonic oscillator-type
γR γR
potential terms, we neglect the density dependent part state for A=35 obeying three distinct peaks of conden-
of N (as it does not significantly affect the observed pat- satedensity. Increasingtheamplitudeallowsustoaccess
terns) and we use a harmonic function that passes the higher order excited states.
maxima of the two Gaussian spots of (3). In Fig. 1 we Although we mainly aim to explain the patterns ob-
show the correspondence between the pump spots and servedin[26], wenotethatthepolaritonharmonicoscil-
theharmonicapproximationandremarkthatk ∼A. To latorcanitselfbeimplementedwithinalong1Dnanowire
fit parameters in (6) we note that the density dependent similar to [16, 33] i.e. a microcavity with very strong
potential term can simply be matched α =U. transverse confinement. Additionally an external in-
1
To identify the pump terms we set P −γ = 2γ , plane trap can be imposed upon the effectively 1D con-
C eff
where P = 1/(|x − x |)(cid:82)x2Pdx = 0.33 · A is the densateasithasbeenexperimentallyshownforasample
1 2 x1
4
oscillator equation
(cid:0)∂ −x2(cid:1)φ =−E φ , (8)
xx n n n
which has Hermite-Gauss polynomials as eigensolution
φ (x)=c e−x2/2H (x) (9)
n n n
√
(a) with cn =(2nn! π)−1/2, where
dn
H (x)=(−1)nex2 e−x2 (10)
n dxn
with corresponding eigenvalues E = 1+2n with n ∈
n
{0,1,2,3,...}. These Hermite-Gauss polynomials are an
orthonormal basis (ONB) of a Hilbert space. On the
other hand any wave function in a Hilbert space has an
expansion in terms of Hermite polynomials of the form
∞
(cid:88)
(b) ψ(x)= B φ (x), (11)
n n
n=0
withcomplexcoefficientsandreal-valuedfunctionsφ (x)
n
providing the ONB. By inserting this expansion in (5)
and using (8) we get
∞ (cid:32) ∞ ∞ (cid:33)
(cid:88) (cid:88) (cid:88)
−E +(σ +iσ ) B φ B∗φ B φ =
n 1 2 k k l l n n
n=0 k=0 l=0
∞
(cid:88)
(c) =(−µ +iµ ) B φ . (12)
1 2 n n
n=0
By multiplying from the l.h.s. with φ and integrating
m
FIG. 2. Density profile |ψ(x,y)|2 of the harmonic oscillator over the whole space we obtain the formula
groundstateina2Dsettingforanareaof10×10µmisshown
in(a). In(b)|ψ(x,y)|2 ofthefirstexcitedharmonicoscillator ∞
(cid:88)
state in a 2D setting for an area of 10×10µm and in (c) the (σ +iσ ) B B∗B V =
1 2 k l n m,k,l,n
densityprofileofthesecondexcitedharmonicoscillatorstate
k,l,n=0
in a 2D setting for an area of 10×10µm.[34]
=(E −µ +iµ )B , (13)
m 1 2 m
where we have introduced the abbreviation V =
m,k,l,n
consistingofthreesetsoffourGaAs/AlAsquantumwells (cid:82) φ φ φ φ dx.
m k l n
embedded in a GaAs/AlGaAs microcavity in [35, 36]. Weassumethatforthejthnonlinearmodethelargest
On the other hand for the pump laser beams there is in coefficientinourexpansion(11)isB . WetakeB =0
j n<j
principle no restriction to the size of the area they are and assume that only basis wavefunctions of the same
illuminating [37] and thus they can be assumed to be symmetryofthenonlinearmodecontributeintheexpan-
extended over a wide range, including the area were the sion,thuswetakeB =0(wherek ∈N). Therefore
j+2k+1
condensateformswithintheharmonictrap. Soanalmost at leading order (13) becomes:
homogeneous pump distribution can be provided on the
1D geometry on top of a repulsive and approximately (σ +iσ )||B |2V =(E −µ +iµ ). (14)
1 2 j j,j,j,j j 1 2
harmonic trap as described by the simplified model (5).
Solving simultaneously the real and imaginary parts of
(14) gives us an expression for the eigenvalue µ ,
1
σ µ
µ =E − 1 2, (15)
PERTURBATION THEORY 1 j σ
2
and for |B |2:
We now derive analytical expressions approximately j
solving (5). Considering (5) in the parameter limit µ
|B |2 = 2 . (16)
σ1,σ2,µ2 →0 we get the well-known quantum harmonic j σ2Vj,j,j,j
5
Inourcomparison withthenumericalresults inthenext COMPARISON BETWEEN NUMERICAL AND
section we will use only these leading order results as ANALYTICAL RESULTS
they are relatively simple and easy to calculate, and im-
mediately give us an indication of how “nonlinear” the
rescaled wavefunction is. The eigenvalue (15) depends We now proceed to test our leading order perturba-
linearly on the nonlinear coefficient, but the amplitude tion theory by comparing it with numerically calculated
of the rescaled wavefunction, given by the square root results. Weuseamodifiedsquared-operatormethod[38]
of (16), depends just on the gain and loss parameters, to find numerical solutions φ(x) to the equation (5).
in a way which is in accordance with the results due
InFigure3weshowthefirstthreenonlinearmodesfor
toanon-conservativeLagrangianformalismpresentedin
both repulsive (Fig. 3(a-c)) and attractive (Fig. 3(d-e))
[13, 14], when back scaling of the perturbation series to
interparticle interactions for the case µ = 0.2. The nu-
the wave function satisfying (4) is made. Thus if we see 2
merical solutions are given by the thick lines, with real
that our numerical solution agrees well with this leading
and imaginary parts corresponding to the dashed and
order term then we can conclude that the nonlinearity
dash-dotted lines respectively, while the density is given
plays only a minor role for the rescaled condensate wave
bythesolidline. Weseethat,asinthecaseofthesingle-
function in this particular parameter regime.
componentatomicBEC[31],thejthmodehasj−1nodes
We see that at leading order we have no information
ofzerodensitywhereaπ phasechangeoccurs. However,
about the phase of the wavefunction. To proceed fur-
unliketheatomicBECcase,thepolaritoncondensatehas
ther and find simple expressions for the higher order
a spatially dependent phase away from the phase singu-
contributions we assume that B is purely real and that
j
larity (evident through the spatially varying ratio of real
B >> B . This allows us to find the following ex-
j j+2k
to imaginary part in the wavefunction). This variable
pression for the higher-order complex coefficients:
phase indicates that particle flow must occur in the con-
(cid:18) σ +iσ (cid:19) densate, with particles moving from regions where gain
B = 1 2 B3V . (17)
j+2k E −µ +iµ j 2k,j,j,j dominates, to regions where loss dominates.
j+2k 1 2
Note that despite considering a non-equilibrium system Wecancomparethenumericalresultsdirectlywiththe
the condensate wave function satisfies the normalization analytical expansion (11) assuming only the leading or-
condition (cid:82)∞ |ψ|2dx = (cid:80) |B |2 = c < ∞ and c cor- der term is present, as given by Eqs. (15) and (16). The
−∞ k k
responds to the number of condensate particles similarly analytical results are given by the thin solid lines in Fig.
to the case of fixed particle numbers [4]. 3. More precisely we can only compare the analytically
Thomas-Fermi type regime.–Startingfrom(5)wecon- predicted density with the numerical result, however we
sider the regime where the kinetic energy becomes neg- include also the real part and imaginary part assuming
ligible compared to the external potential, nonlinearities that the imaginary part is zero. At this leading order
and linear parameters, i.e. the nonlinearity plays no role in the form of the ana-
lytical wavefunction, so it is identical for the attractive
(cid:18) (cid:19)
−x2+(σ +iσ )|ψ|2−iµ ψ (cid:39)−µ ψ (18) and repulsive cases. Nevertheless, we see that it agrees
1 2 2 1
remarkably well with the numerical solutions. We see
that in the repulsive case the true solution lies outside
which to the leading order has the formal solution
the analytical result, while the inverse is true for the at-
(cid:115) tractive case, as expected. Interestingly we see that the
ψTF(x)=± −µ1+iµ2+x2. (19) agreement improves as we go to higher modes, suggest-
σ1+iσ2 ing the ground state is “more nonlinear” in the sense
that it deviates more strongly from the associated linear
Expression (19) is however not integrable for arbitrary wavefunction.
parameters. Besides we refer to [8, 9] for a rigorous dis-
cussionoftheThomas-Fermiapproximationinconserva- In Figure 4 we compare the analytical and numerical
tive GP theory. In contrast to the previous perturbation results at higher gain, given by µ = 0.5. We see that
2
theory, where the coefficients (16) and thus the conden- thepatternobservedforµ =0.2becomesmoreevident,
2
sate wave functions dependent solely on the parameters with the ground state showing the strongest deviation
µ and σ , the TF regime significantly depends on the from the linear prediction. The higher-order modes con-
2 2
nonlinearityσ andtheenergyµ asformula(19)shows. tinue to show remarkably good agreement.
1 1
The validity of the leading-order analytical prediction canbeseenataglancebycomparingthepredictedeigen-
6
FIG. 3. Comparison between numerical (thick lines) and analytical predictions (thin lines) for repulsive interactions (a-c) and
attractiveinteractions(d-e)atgainstrengthµ =0.2. (a,d)Groundstate;(b,e)Firstexcitedstate;(c,f)Secondexcitedstate.
2
Numerical results follow format: density – thick solid lines; real part – thick dashed lines; imaginary part – thick dash-dotted
lines. We see increasingly good agreement between the numerical and analytical predictions as we go to higher modes.
values (15) with those found numerically. We see in Fig- patternsobserved,iffbothpumpspotsareequal0. Sowe
ure 5 the dependence of the eigenvalues on gain parame- can identify all the harmonic oscillator patterns with an
terµ forthefirstthreemodes(pairsoflinesarearranged output 1, and thus we have established a simple feasible
2
upper to lower and correspond to second, first zeroth NOR gate by means of topologically stable excitations
modes respectively), for both repulsive (solid) and at- carriedbyanon-equilibriumpolaritoncondensatewithin
tractive(dashed)interactions. Theanalyticalpredictions asemiconductormicrocavitywhichstandinthetradition
are given by the associated thin lines. We see increas- of optical computing [17].
ingly good agreement with mode order, with the second ThisNORgatecanbegeneralizedinthesensethatwe
mode showing close agreement even up to µ = 1. The do not solely distinguish between ”on” corresponding to
2
ground state on the other hand is showing a discernible 0 and ”off” associated with 1 of both pump spots, but
discrepancy by µ = 0.5. The vertical dash-dotted lines in addition include the experimental and numerical ob-
2
correspond to the values of µ chosen to show the sta- servation that different pump strengths lead to different
2
tionary states in Figs. 3 and 4. Overall we see that the numbers of density lobes. Therefore we can enumerate
features of the stationary states are dictated predomi- thelogical0sleadingtodifferentlobesstates,i.e. ground
nantly by the gain and loss coefficients, rather than the state, first excited state, second excited state and so on.
interparticle interactions. That is, if the pump spots have a certain amplitude A it
leads to patterns of say B(A) : R → N lobes. However
whenonlyonepumpspotisturnedon, correspondingto
NOR GATE AND A nNOR GATE a logical input, 0 and 1, or boths pumps are off, corre-
sponding to 0 and 0 input, no lobes patterns form [40].
The pattern formation within a 1D nanowire can have
the following technological application as NOR gate. A
NOR gate [39] is defined by the logic that two or more CONCLUSIONS
zero inputs yield an output 1 in a boolean logic system.
Here we can identify each pump spot (3) with 0 or 1 Motivated by a recent experiment we considered a
by defining that, if the laser beam is turned on, A (cid:54)= 0, mean field model for polariton condensates to reproduce
the input corresponds to 0 and if not it corresponds to the observed patterns. Starting from this we could iden-
1. The output will be the quantum harmonic oscillator tify the general model with that of a simpler nonlin-
7
FIG. 4. Comparison between numerical (thick lines) and analytical predictions (thin lines) for repulsive interactions (a-c) and
attractiveinteractions(d-e)atgainstrengthµ =0.5. (a,d)Groundstate;(b,e)Firstexcitedstate;(c,f)Secondexcitedstate.
2
Format of the lines is the same as in Fig. 3. We see the ground state shows significant discrepancies with the analytical
predictions, however for the higher modes the agreement is still very good.
siangroundstatetothennodenon-equilibriumquantum
harmonic oscillator states. The analytical eigensolutions
were compared to corresponding numerical simulations
and showed excellent agreement. Due to the recent ex-
perimentalaccessibilityofFeshbachresonancesinpolari-
ton systems we further predicted the scenario of attrac-
tive self-interactions and outlined their role in nonlin-
ear pattern formation. Particularly the modification of
the density formation was explicitly illustrated. In the
course of our analysis we showed that the energy eigen-
valuesdependonthepumpingstrengthµ linearlytothe
2
leading order and conclude to provide a thorough expla-
nation for the experimentally observed patterns. Finally
wediscussedthepossibilityforageneralizedopticalNOR
device in polariton condensates utilizing the underlying
pattern formation logic.
FIG. 5. Family of eigenvalues, i.e. first, second and third
order modes as function of pumping strength µ . Solid lines
2
representthenumericallycalculatedeigenvaluesfortherepul- ACKNOWLEDGEMENTS
sivecaseandcorrespondinglythedottedlinesaretheanalyt-
ical prediction. For attractive condensates the dashed lines
FPhasbeenfinanciallysupportedthroughhisEPSRC
show numerically generated predictions and dotted lines are
doctoralprizefellowshipattheUniversityofCambridge.
the analytical predictions.
FP likes to thank Natasha Berloff, Hamid Ohadi and
Alexander Dreismann for stimulating discussions.
ear non-equilibrium quantum harmonic oscillator. For
this effective theory we have presented a perturbative
approach and so generalized the quantum harmonic os-
cillator theory to include the non-conservative character ∗ fl[email protected]
of the non-equilibrium polariton condensate. We intro- † [email protected]
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