Normal and superfluid fractions of inhomogeneous nonequilibrium quantum fluids

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by  Vladimir N. Gladilin

Normal and superfluid fractions of inhomogeneous nonequilibrium quantum fluids
Vladimir N. Gladilin1 and Michiel Wouters1
1TQC, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, Belgium
(Dated: January 6, 2016)
We present a theoretical analysis of the normal and superfluid fractions of quantum fluids de-
scribed bya nonequilibrium extension of theGross-Pitaevskii equation in thepresence of an exter-
nal potential. Both disordered and regular potentials are considered. The normal and superfluid
6 fractions are defined by the response of the nonequilibrium quantum fluid to a vector potential, in
1 analogy with the equilibrium case. We find that the physical meaning of these definitions breaks
0
downoutofequilibrium. Thenormalandsuperfluidfractionsnolongeradduptooneandforsome
2
typesof external potentials, they can even become negative.

n
a
J I. INTRODUCTION n, the normal fraction f can be defined as
n

5
m
f = lim χ (q). (1)
] The hydrodynamic behavior of multimode laser sys- n q→0 n T
s tems has been recognized long ago1,2, but it is on the
a Heremistheeffectivemassandχ (q)isthesusceptibil-
g platform of semiconductor exciton-polaritons that a sys- T
ity tensor relating the average transverse current to the
- tematic investigation of the hydrodynamics of photonic
t applied vector field in Fourier space.
n systems has been performed3. Microcavity polaritons
Themajorquestionisheretowhatextentthenonequi-
a arehybridlightmatterexcitationsinsemiconductorhet-
u librium character of polariton condensates affects their
erostructuresconsisting of a quantum wellembedded in-
q superfluid properties. A first theoretical study was per-
side an optical microcavity4. When the optical mode is
. formed by Keeling on the superfluid fraction of a ho-
t resonant with the quantum well exciton frequency, the
a mogeneous driven-dissipative polariton condensate with
m quasi-particles are the coherent superpositions of exci- a Keldysh functional integral approach11. Out of equi-
tonandphoton. Thankstothephotoniccomponent,the
- librium, a nonvanishing normal component is always
d microcavity polariton is much lighter than the bare ex-
present, due to nonequilibrium fluctuations.
n citon, which enhances the coherence properties. Thanks
The work by Janot et al.12 addressed superfluidity in
o totheexcitoniccomponent,polaritonsshowasignificant
c the disordered case. A reduction in superfluid fraction
interaction strength. Being bosonic particles, microcav-
[ due to disorder is due to the pinning of the fluid in the
ity polaritons can show long range phase coherence in
potential minima and a first theoretical estimate can be
1 thequantumdegenerateregime,givingrisetosuperfluid5
obtained at the mean field level13,14. In the theoretical
v properties, such as quantized vortices6.
6 studyofRef.12,theresponsewasstudiedtoasmalltwist
6 The photonic component also introduces losses,which θ ofthephasebetweentwoboundariesofthe condensate
7 makes it so far impossible to achieve fully equilibrated separated by its size L. This is equivalent to the pres-
0 Bose-Einstein condensation. In order to compensate for ence of a weak vector potential A = exθ/L. As argued
0
thefinitepolaritonlifetime,particleshavetobeinjected. inRef.12,thesuperfluidstiffnessinthiscasecanbeesti-
1. The twomainschemesareresonantandnonresonantex- matedinaccordancewiththeequilibriumdefinitions15,16
0 citation. Inthe formerregime,the velocityofthe polari- as
6
tonfluidcanbetunedbyvaryingtheangleoftheexcita-
1 ω(A) ω(0)
tion laser. With this pumping scheme, superfluidity ac- f = − , (2)
: s A2
v cordingtotheLandaucriterion(absenceofscatteringoff
Xi defects)7andthenucleationofquantizedvortices8aswell where ~ω is the chemical potential of the condensate.
as solitons9 have been experimentally observed. Despite The weak vector potential A = e θ/L corresponds to
r x
a the finite polariton life time, the observed phenomenol- a slow rotationat a velocity of less than one quantum of
ogy in those experiments was close to the equilibrium circulationv <2π/L. Duetothephasequantization,the
behavior. superfluidcannotrotate,sothatthecurrentissupported
by the normal phase only (Hess-Fairbank effect)5. This
From a conceptual viewpoint, the main disadvantage
allows us to define the normal fraction in terms of the
of resonant excitation is the fact that the pumping laser
dependence of the current density on the gauge field
fixes the phase of the quantum fluid, which excludes the
study of superfluidity in the sense of linear response or j (A )
x x
metastablesuperflow. Ithasbeenargued3 thatthe most fn = h i. (3)
nA
x
precise and quantitative definition of superfluid f and
s
normalf fractionsinvolvestheresponseofthequantum The application of (synthetic) gauge fields in optical
n
gas to a weak transverse vector potential A as analyzed systems has developed into an active field of research17,
inRef.10. Foraspatiallyhomogeneoussystemofdensity thankstoseveralexperimentalbreakthroughs18–20 anda
2
wealth of theoretical proposals. Seen from this context, case seems to be
we address the question of the effect of a small gauge
j (A ) j (0)
x x x
field on a the frequency and current in a nonequilibrium fn = h i−h i, (6)
nA
condensate. x
where the current density in the units used is given by
the expression
II. MODEL j=Im[ψ∗ ψ]. (7)
∇
We will apply the aforedescribed definitions of f and
s
f to the systems described by the generalized Gross- III. RANDOM POTENTIALS
n
Pitaevskii equation (gGPE)21,22, which is taken in the
form3 Irregular potential landscapes are highly relevant for
∂ψ ~2 2 experimental realisations of polariton condensation be-
i~ = ∇ +g ψ 2+V cause of growth imperfections in the semiconductor het-
∂t (cid:20)− 2m | |
erostructures. At equilibrium, the interplay between
i P Bose-Einsteincondensationanddisordergivesrisetorich
+ γ ψ, (4)
2(cid:18)1+ ψ 2/n − (cid:19)(cid:21) physics,withazerotemperaturesuperfluidtoBose-glass
s
| | quantum phase transition23.
with a static potential V(r) and a contact interaction We consider the case of a uniform pumping intensity
with the strength g. The imaginary term in the square (p(r) = 1) and a static potential, described by a ran-
brackets on the right hand side describes the saturable domdistributionv(r). Twoexamplesoftheusedrandom
pumping (with strength P and saturation density ns), distributions, vA(x,y) and vB(x,y), with the correlation
that compensates for the losses (γ). The physical origin length ξ = 5 and 3, respectively, are shown in Figs. 1a
v
ofthepumpingtermforexciton-polaritoncondensatesis and 1b. In Fig. 1c we plot the quantities f and f ,
s n
an excitonic reservoir that is excited by a nonresonant correspondingto these potentialdistributions,asa func-
laser. tion of the strength V of the potential. In a qualitative
0
In general, the pumping intensity P is coordinate de- agreement with the results of Ref. 12, for both random
pendent and can be represented as P(r) = P0p(r), with potentials the function fs decreases with V0 and falls to
P0 = maxP(r). Assuming that the interaction strength zero at a sufficiently strong potential. At the same time,
is positive, g > 0, it is convenient to rewrite Eq. (4) in the whole range of V under consideration, the calcu-
0
in a dimensionless form, by expressing the particle den- lated f remains close to zero or even becomes negative
n
~si/ty(g|nψ0|)2, ainndunleitnsgtohf nin0u≡nintss(oPf0~//γ√−2m1)g,nti0m: e in units of aistolabrvgioeuVs0ly, svoiotlhaattedt.heAetxfiprescttesdig“hstu,mthirsuvlei”olfasti+onfnco=uld1
be related exclusively to the calculated f (e.g., to lack
n
∂ψ
i = 2+ ψ 2+V v ofphysicalmeaningofexpression(6)oralackofnumeri-
0
∂t (cid:20)−∇ | | cal accuracy in the corresponding estimation). However,
1 ψ 2+(1+ν−1)(p 1) a deeper analysis shows that the problem actually has a
+ic −| | 1+ν ψ 2 − (cid:21)ψ. (5) moregeneralcharacter. Indeed,fromthe insettoFig.1c
| | onecanseenotonlyapronouncedasymmetryoff (A ),
n x
Equation (5) contains three dimensionless scalar param- which further illustrates inconsistencies in treating the
eters: V0 = max V(r)/gn0, c = γ/(2gns) and ν = found fn as the normal fraction of the condensate, but
| |
n0/ns. The dimensionless functions p(r) and v(r) = also the presence of a clear discontinuity of fs(Ax) at
V(r)/(V0gn0) describe the spatial distributions of the Ax 0. The latter allows us to put under question
| | →
pumping intensity and static potential, respectively. also the possibility to interpret the calculated fs as the
Our numerical simulations are performed for a region superfluid stiffness. In other words, in the presence of
of sizes L L with periodic boundary conditions in a random potential, neither Eq. (2) nor Eq. (6) seem to
x y
×
thexdirectionandtheNeumannboundaryconditionsat provide correct estimates for the superfluid and normal
y =0,L . AuniformgridwithN N nodesisused. A fractions of the condensate described by Eq. (5). This
y x y
×
vector potential, chosen as A = (A ,0,0) with constant fact can be attributed to the existence of non-negligible
x
A , is introduced by replacing ∂/∂x with ∂/∂x iA in in-planecurrentsinthecondensatesunderconsideration,
x x
−
Eq. (5). For electrically charged particles, our configu- even at A = 0. Below we will illustrate the above state-
ration would correspond to a 2D cylindrical shell in an ment by few simple examples with regular potentials.
axially symmetric magnetic field parallel to the cylinder
axis. We should keep in mind that in the case of an in-
IV. REGULAR POTENTIALS
homogeneous static potential and/or pumping intensity,
describedbythefunctionsv(r)andp(r),respectively,the
average current j can be nonzero even at A = 0. A First, let us consider a structure with a partial cut in
x
h i
naturalgeneralizationofthe expression(3)forf tothis thedirectionperpendiculartothevectorpotentiale A .
n x x
3
a -1 0 1 b -1 0 1 a 0 0.5 1 b -0.25 0 0.25
v v |y|2 j
A B x
60 60
40 40
y y
20 20
0 00 20 40 60 0 20 40 60
0 20 40 60 0 20 40 60 x x
x x
c
c 1.0
0.8
f +f
0.6 s n
f
n
f s
, 0.4 f
s
f n
0.2
0.0
2 4 6 8 10
n
Figure 2. Density |ψ|2 (panel a) and current jx (panel b)
Figure1. DistributionsvA(x,y)(panela)andvB(x,y)(panel distributions, calculated for a system with Lx = Ly = 64,
b)fortworandompotentialswiththecorrelationlengthξv = Nx = Ny = 32, Lcut = 0.4Ly, V0 = 0, p = 1 and c = 1 at
5 and 3, respectively. (c) fs and fn, calculated for v = vA, ν =2, Ax=0. (c) fs and fn,calculated for thesame system
Lx = Ly = 64, Nx = Ny = 32, Ax = 0.1 and for v = vB, at Ax=0.05, as a function of ν.
Lx =Ly =64,Nx =Ny =64,Ax =0.03,asafunctionofV0.
Inset: fs and fn, calculated for v=vA, Lx =Ly =64, Nx =
Ny =32,V0 =0.24,asafunctionofAx. Thecalculationsare
performed for ν =1, c=1, p(r)=1.
The cut is introduced through the additional condition
ψ =0. Anexampleofthedensityandcur-
|x=Lx/2,y≤Lcut
rent distributions in a system with such a cut, as given
in Figs. 2a and 2b, respectively, corresponds to V = 0,
0
p = 1, c = 1 and ν = 2. At those parameter values
the contribution of the pumping-loss term to gGPE is
relatively large [see Eq. (5)] leading to the appearance
of rather strong currents on both sides of the cut (see
Fig. 2b). The corresponding values of f and f as well
s n
astheirsumaresignificantlysmallerthanone(seeFig.2c
for ν 2). With increasing ν, the role of the pumping-
≤
losstermissuppressed,equation(5)approachesthestan- Figure3. fsandfnasafunctionofV0forLx =26,Nx =128,
dardGross-Pitaevskiiequation,andthequantitiesfsand c = 3, ν = 2, p = 1, Ax = 0.04. Inset: shape v(x) of the
f gradually restore their physical meaning, so that the static potential (solid line) and density distribution |ψ(x)|2
n
sum f +f tends to reach 1 at ν 10. at V0=0.45 (dashed line).
s n
∼
In Fig. 3 we show f (V ) and f (V ) calculated for a
s 0 n 0
system, where the aforementionedcut is replaced with a
potentialwell V v(x), uniformin the y direction(see the shown in Fig. 3, first manifests a decrease, qualitatively
0
insettoFig.3). Figure3correspondstoarelativelylarge similar to that in Fig. 1c. At larger V0, this decrease is
value of the parameter c (c = 3). In this case, the be- overridden by oscillations of fs(V0). Here, the definition
haviouroff (V )closelyresemblesthatinthepresenceof (1) clearly loses its physical meaning of normal fraction.
n 0
arandomstaticpotential(Fig.1c). Thequantityf (V ), The equality f +f =1 is seen to be satisfiedonly at
s 0 s n
4
a b
V 0. Like in our previous example, when suppress-
0
→
ing the pumping-loss term (in the present case this is
realised by decreasing the parameter c) so that the sys-
tem approaches an equilibrium regime, the behaviour of
x
y
f (V ) and f (V ) becomes more reasonable(see Fig.4).
s 0 n 0
In particular, as seen from the inset to Fig. 4, the range
ofpotentialstrengthsV ,wheretheconditionf +f =1
0 s n
is obeyed, increases clearly with decreasing c, so that at
c=0.03 this range includes V 1.
0
∼ c
a
b
d e
Figure 5. (a) Functions v(x) (solid line) and p(x) (dashed
line), which determine the spatial distributions of the static
potential and the pumping intensity, respectively, in a peri-
Ffco.irgIuLnrsxeet=4:.f2s6f,s+N(fpxna=naes1la2a8)f,uaνnnc=dtio2fn,npo(fp=Van10.,elAbx)=as0.a04fuanncdtiodniffoerfeVn0t odlfiodnerniec)sνistay=ytst|ν3ψe,m=|2cw(3=d,iotch4tt,=LeAdx4x,l=ian=ne5)d50,a.V0nN30d5x=,c=ua3rs.1r32ea.n8tf.(ucd()nbecfn)tssiDoiatniynstdojrxfifbVn(ud0,t.aicosaPhnlc-sadunoolefatltttseheddde
and e show fs and fn, respectively, as a function of Ax for
In our last example we consider a system where ν =3, c=4 and three different V0.
a strong asymmetry is induced by an inhomogeneous
pumping with the intensity maximum shifted with re-
specttotheextremaofthepotentialV v(x)(seeFig.5a).
0
The system is uniform in the y direction. As displayed have a pronounced discontinuity at A 0, which is
x
| | →
in Fig. 5b, the combined effect of the potential barrier similar to that shown in the inset to Fig. 1c for the case
andinhomogeneouspumpingleadstoaverynon-uniform of a random static potential. The origin of this discon-
density distribution with strong currents, mainly in the tinuity can be qualitatively understood by associating
positivexdirection. InFig.5c,atrelativelysmallV ,rea- the presence of a nonzero average current in the system
0
sonable, nearly zero values of f are accompanied by a at A = 0 with an effective vector potential A = 0.
s x 0x
6
counterintuitivedecreaseoff withincreasingtheheight Then, assuming like before that the chemical potential
n
V ofthepotentialbarrier. AtlargerV ,bothf (V )and of the superfluid is proportional to the vector potential
0 0 s 0
f (V ) demonstrate some oscillatory behaviour, which squared, ω(A) = αA2, one obtains from Eq. (2) the ex-
n 0
can hardly have any physical meaning. One can also pression f =(1+2A /A )α, which obviously diverges
s 0x x
notice that the condition f +f = 1 is not fully sat- at A 0. The existence of a nonzero average current
s n x
| |→
isfied even at V = 0. As seen from Fig. 5d, at suffi- in the absence of an externally applied vector potential
0
ciently large V , the calculated function of f (A ) can seemstoberesponsiblealsofortheasymmetryofthecal-
0 s x
5
culated function f (A ) at large V , both in the present At first, the definition of the normal fraction, based
n x 0
example(seeFig.5e)andinthe caseofarandompoten- on the current response of the system, could be thought
tial (see the inset in Fig. 1c). to be more robust with respect to the nonequilibrium
condition. Our numericalanalysishoweverstronglycon-
tradicts this. Already in the case of the disordered po-
V. CONCLUSIONS tential,wherethesuperfluidfractionbehavesreasonably,
thenormalfractiondoesnotshowtheexpectedbehavior.
Summarising our numerical findings, we can conclude Wealsoattributethefailureoftheequilibriumdefinition
that the definitions (1) and (2) lose their physical inter- of the superfluid fraction to the presence of currents in
pretation in terms of normal and superfluid fractions in the steady state. These currents invalidate the physical
thepresenceofdrivinganddecay. Forwhatconcernsthe picturethatthecurrentresponseisduetoalimitedstiff-
definition of the superfluid fraction (2), this may not be nessofthe condensatephase. Whencurrentsarepresent
thatsurprising,sincethestationarystateisnolongerthe in the steady state, also a redistribution of the density
one with minimal (free) energy. For the case of a disor- will contribute to the current response. The interplay
dered potential, however, the superfluid fraction shows between the density redistribution and the steady state
an expected behavior as a function of disorder strength, currents can then even lead to negative normal fraction,
starting from one at zero disorder and falling to zero for as we observed in the case of the disordered potential
stronger disorder. landscape (Fig. 1c). The conclusion of our numerical
For other potential profiles however, such as a single findings is negative: the equilibrium definitions for su-
potentialdip, theequilibriumdefinitionofthesuperfluid perfluid and normal fractions no longer give meaningful
fraction becomes problematic. It becomes negative for results out of equilibrium in the presence of external po-
certainvaluesofthepotential. Thismeansthefrequency tentials that induce currents.
ofthecondensateshowsadecreaseinsteadofanincrease
under the application of a phase twist in the boundary Itwillbeofinteresttoincludetheeffectoffluctuations
condition. In equilibrium, this is forbidden, since time inthe calculationof the superfluid andnormalfractions.
reversal invariance guarantees that the condensate wave Within the classical field formalism, this could be done
function in the ground state is real. Out of equilibrium, inthetruncatedWignerapproximation24. Ofspecificin-
wheretimereversalinvarianceisbroken,thesteadystate terestarethe superfluid andnormalfractionclose to the
also has currents for periodic boundary conditions. The phase transition. At equilibrium, it is of the BKT type
interplay between this existing current and the applied in two dimensions, with a characteristic jump in the su-
phase twist can then lead to a lower frequency. In the perfluid fraction. Indications that the BKT character of
case of a spatially inhomogeneous pump, the behavior the phase transition is preservedout of equilibrium were
of the superfluid fraction even differs more dramatically found recently in numerical simulations in the paramet-
from the equilibrium one, showing large oscillations as ric oscillation regime25, but the behavior of the normal
function of the external potential strength (see Fig. 5c). and superfluid fractions has not been addressed yet.
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