PHYSICAL REVIEW B 85, 075130 (2012)

Negative Bogoliubov dispersion in exciton-polariton condensates
Tim Byrnes,1 Tomoyuki Horikiri,1 Natsuko Ishida,1,2 Michael Fraser,1,3 and Yoshihisa Yamamoto1,4
12

National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
Department of Information and Communication Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
3
Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
4
E. L. Ginzton Laboratory, Stanford University, Stanford, California 94305 USA(Received 10 February 2012; published 29 February 2012)
Bogoliubov’s theory states that self-interaction effects in Bose-Einstein condensates produce a characteristic
linear dispersion at low momenta. One of the curious features of Bogoliubov’s theory is that the new quasiparticles
in the system are linear combinations of creation and destruction operators of the bosons. In exciton-polariton
condensates, this gives the possibility of directly observing the negative branch of the Bogoliubov dispersion
in the photoluminescence (PL) emission. Here we theoretically examine the PL spectra of exciton-polariton
condensates taking into account reservoir effects. At sufficiently high excitation densities, the negative dispersion
becomes visible. We also discuss the possibility for relaxation oscillations to occur under conditions of strong
reservoir coupling. This is found to give a secondary mechanism for making the negative branch visible.
DOI: 10.1103/PhysRevB.85.075130

PACS number(s): 71.36.+c, 67.10.−j, 74.78.Na

I. INTRODUCTION

Recent experimental advances have achieved the condensation of exciton-polaritons in semiconductor microcavity
structures.1–3 The short lifetime of the exciton-polaritonson the order of picoseconds means that the condensate israther different in nature to a traditional atomic Bose-Einsteincondensate. The condensate exists only resulting from thereplenishing of the polaritons from a reservoir of uncondensed
polaritons, which is in turn populated by illumination by alaser. Despite this difference, such condensates exhibit manyof the characteristics expected in equilibrium condensates,
ranging from superfluidity4 to vortex formation.5,6 In the workof Ref. 7, it was shown that self-interaction effects of thecondensate cause the dispersion characteristics of excitonpolariton condensates to follow a Bogoliubov dispersion relation. Although in the work of Ref. 7 no negative branch of the
Bogoliubov dispersion8,9 was detected, recently a four-wave
mixing experiment has revealed the presence of the negative
branch.10 The four-wave mixing experiment was originallytheoretically proposed in Ref. 11. The relative difficulty ofthe observation of the negative branch was attributed in thiswork to the bright condensate emission, which easily masks
the weaker negative branch.
In this paper, we present a detailed theoretical analysis of thephotoluminescence (PL) of the negative Bogoliubov branch.
In contrast to the four-wave mixing approach of Ref. 11, the PL
is calculated directly via two-time correlation functions of thepolariton equations of motion. In particular, we incorporate
the effect of the bottleneck polaritons, which is known tostrongly influence the dispersion of the polaritons.12 To thisend, we first reformulate the theory of Ref. 12 in a HeisenbergLangevin formalism. This step is analogous to the factthat standard Bogoliubov dispersion can be derived startingfrom a Gross-Pitaevskii (GP) equation or by diagonalizing a
microscopic Hamiltonian.15 Our version corresponds to theoperator approach, while the original derivation in Ref. 12 isderived by examining the fluctuations around the equilibrium
GP equation. The reformulation also makes it clear that the1098-0121/2012/85(7)/075130(5)theory is a modified Bogoliubov theory defining new bosonicexcitations in the system.
This paper is organized as follows. In Sec. II, we reformulate the theory of Ref. 11 in an operator formalism. In
Sec. III, the new quasiparticles of the system are discussed,
where we introduce “dissipative” Bogoliubov operators. The
photoluminescence is calculated in Sec. IV, where we examine
how the negative dispersion should become visible. In Sec. V
we give our conclusions.
II. HEISENBERG-LANGEVIN REFORMULATION OFWOUTERS-CARUSOTTO THEORY

The polaritons are assumed to obey the Hamiltonian 2 2h̄ ∇2†
H = d x p (x) −
p(x)
2m

gA †
†
†
+
p (x)p (x)p(x)p(x) + g̃Ap (x)p(x)nR (x) ,2(1)
where p† (x) is a creation operator for a polariton at position
x, nR (x) is the number operator of the reservoir polaritons,
m is the mass of a polariton, g is the interaction energy ofthe polaritons, and g̃ is the polariton-reservoir interaction.
For example, including only the exchange interaction g =a2
6e2|X|4 AB , where e is the electronic charge, aB is the Bohr
4πaB
radius, X is the excitonic Hopfield coefficient, and A is the
sample area, and we use SI units.13 The polaritons then obey
the Heisenberg-Langevin equations of motion
ih̄

i
dp
= [p(x),H ] + {R[nR (x)] − γ } p(x),
dt
2

(2)

where R[nR (x)] is the stimulated gain coefficient14 of thereservoir polaritons into the condensate and γ /h̄ is the decayrate of the polaritons through the microcavity mirrors. Following Ref. 12, we linearize Eq. (2) such that it only involves
terms linear in either the polariton or reservoir operators. To

075130-1©2012 American Physical Society

BYRNES, HORIKIRI, ISHIDA, FRASER, AND YAMAMOTO

PHYSICAL REVIEW B 85, 075130 (2012)

achieve this goal, we expand the reservoir operator into its
average and fluctuation components,(3)

tude of polaritons in the condensate. The prefactor of δn(x) is
chosen for later convenience. Substituting Eq. (3) into Eq. (1)and rewriting the operators in terms of their Fourier components p(x) = √1A k pk eikx and δn(x) = A1 k δnk eikx , weobtain+2m(8)where we have assumed a homogeneous pump G(x) = G0 .
The k = 0 components of Eqs. (6) and (8) can be conveniently summarized in the form


g  †
†
†
pk pk +p p  p k p k
2  k+q k −q

3

d i
ij j
Mk pk ,
ih̄ pk =dt
j =1

k,k ,qg̃n0R  †pk+q δnq pk .
ψ0where pki = (pk ,p−k e−2iμT t/h̄ ,δnk e−iμT t/h̄ ), andij

Mk

We note here that δnk should be interpreted as the amount ofdensity fluctuations in the reservoir with Fourier component k
and should not be confused with the number of reservoir polaritons with momentum k. We now follow the same procedure
to the Bogoliubov prescription to obtain a Hamiltonian that is†bilinear in the variables (pk ,p−k ,δnk ) by picking out terms inthe summation that involve polariton operators with k = 0 and
set these to their average values pk=0 → 0 = ψ0 e−iμT t/h̄15 ,where μT = g̃n0R + g|0 |2 is the condensate energy. Thisgives

+ gψ02

2m

⎜
−gψ02
=⎜
⎝
−i



⎞
g̃ + 2i R  n0R n0R ⎟−g̃ + 2i R  n0R n0R ⎟.
⎠


2
 0
−i γR + R nR ψ0

gψ02
2 2

− h̄2mk − gψ02

R(n0R )ψ02
n0R

−i

R(n0R )ψ02
n0R

The matrix Mk is identical to that given in Ref. 12.
In Fig. 1 we show the real part of the eigenvalues of Mk fortypical parameter values. We identify two regimes that depend
4

4
(b)

E ki /E 0

2

k

0

δmk

4
2

(5)

bk†

2

Meanwhile, the reservoir equation of motion obeys
(7)

1

10

2

|u ki1 | / |u ki2 |


 2 2
h̄ k
†
0
2
+ g̃nR + 2g|0 | pk + g02 p−k
2m


i  0
n0R 0g̃ + R nR δnk .
+
(6)
ψ0
2

8

0
ka

1

1

bk†
0
ka

1

10

4
0
2

δmk

4
2

2

6
2 δmk

0

bk

2

bk

(c)

1

2

8 (d)

bk†
0
ka

1

2

bk

6
4
2 δmk
0
2

1

bk†
0
ka

1

2

FIG. 1. (Color online) (a, b) Energy eigenvalues and (c, d)i2 2
2for the eigensolutions of theratio of coefficients |ui1
k | /|u k |matrix Mk (solid lines). Dashed lines show the corresponding
values for Bogoliubov
theory (no reservoir), with energy dispersion

EBog (k)/E0 =where G is the Langevin noise operator for the number
operator for reservoir polaritons16 and γR /h̄ is the decay rate of
the reservoir polaritons. The noise operator G originates fromthe coupling of the reservoir modes to high-energy excitations
induced by the laser pump. We have neglected the diffusion of
the reservoir polaritons since they have a negligible effect on
the dispersion characteristics. Substituting Eq. (3) into Eq. (7),

2

bk

2

The resulting equation of motion for the polariton operators isthus

dnR (x)= G(x) − γR nR (x) − R[nR (x)]p† (x)p(x),dt⎛ h̄2 k2(a)  h̄2 k 2
g
†
4
0
2
+ g̃nR + 2g|0 | pk pk
H = |0 | +
2
2m

dpk
ih̄
=
dt

(9)

†

(4)

k,q

g
† †
+ 02 pk p−k + 0∗ 2 pk p−k2g̃n0†+ R δnk (0 pk + 0∗ p−k ) .
ψ0†(0 pk + 0∗ p−k ) − R  n0R |0 |2 δnk ,E ki /E 0k+ g̃n0R

n0R

2

H =

  h̄2 k 2

ψ0 R n0R−

where n0R is the average reservoir number and ψ0 is the ampli-

2n0R
δn(x),
ψ0|u ki1 | / |u ki2 |nR (x) = n0R +expanding in Fourier space, and performing the Bogoliubov
linearization we obtainψ0dδnk = 0 G0 − γR n0R δ(k = 0) − γR δnk
dtnR(ka)2 [(ka)2 +
2 2

2gψ02
2
] and coefficients |ui1
k | − 1/2 =
E0

h̄ k
2
2
|ui2
k | + 1/2 = ( 2m + gψ0 )/2EBog (k). The energy scale is measuredh̄2
in units of E0 = 2ma2 , where a is the experimental length scale(e.g., in GaAs a = 10−4 cm and E0 = 0.68 meV). Parameters used
are g̃n0R = 1, gψ02 = 1, R  (n0R )n0R = 1, R(n0R ) = 1, g = g̃, and (a, c)
γR = 1 (corresponding to a flat dispersion regime), (b, d) γR = 0.1(corresponding to a relaxation oscillation regime). All parameters inunits of E0 .

075130-2

NEGATIVE BOGOLIUBOV DISPERSION IN EXCITON- . . .

PHYSICAL REVIEW B 85, 075130 (2012)

primarily on the relative magnitude of the scattering rate R
and the reservoir decay rate γR . In Ref. 12, it was generally
assumed that γR was large, giving the characteristic flat
dispersion regime in the vicinity of k = 0, as seen in Fig. 1(a).
However, considering that reservoir polaritons originate fromthe bottleneck region,17 which generally have a longer lifetime
than the condensate polaritons, it is possible to consider the
opposite regime, where γR is small and the scattering rate R islarge. In such a regime [see Fig. 1(b)] the eigenspectrum showsa split dispersion at k = 0, with a rounding of the Bogoliubov
spectrum, such that the particles once again acquire an effective
mass. This is caused by relaxation oscillations in the system,
where the reservoir and the condensate repeatedly exchange
population if displaced out of equilibrium.18 A simplified
description can be obtained by expanding the Gross-Pitaevskii
and reservoir equations in Ref. 12 around their steady-statevalues ψ(t) = ψ0 + δψ(t) and nR (t) = n0R + (n0R /ψ0 )δn(t)and ignoring interaction effects g = g̃ = 0. For small k we
obtain
⎞


 ⎛
1  0
R nR n0R
0
2
δψ
d δψ
⎠
h̄
.
= ⎝ 2R n0 ψ 2
dt δn
δn− n0R 0 −γR − R  n0R ψ02
R

The imaginary part of the eigenvalues gives the oscillation
frequency, which is

2

1
ω=
R  n0R R n0R ψ02 − γR + R  n0R ψ02 /4. (10)h̄In the limit of small scattering R(nR ) or large γR , the frequency
becomes purely imaginary, indicating that only damping
occurs.
III. QUASIPARTICLE OPERATORSThe eigenvalues of the matrix Mk give the new collective
excitations of the system. Specifically, new quasiparticle
operators
bik =

3


ij

j

(11)

uk pk

j =1

with [bk ,bk † ] = 1, which is a new “dissipative” Bogolii2 22ubov destruction operator. The solution with |ui1
k | < |u k | ,†meanwhile, corresponds to [b−k ,b−k ] = −1, the dissipative
Bogoliubov creation operator [see also Fig. 1(d)].
Putting in the correct time dependencies, outside the flat
dispersion regime we may associate the solutions to be bik ≡†(bk ,b−k e−2iμT t/h̄ ,δmk e−iμT t/h̄ ), corresponding to the positive,
negative, and dispersionless branches, respectively. Let us
examine this point in more detail so that a clear interpretationof the operators can be made. Looking at the “dissipative”
Bogoliubov destruction operator, the new quasiparticles canbe written in the form
bk = bk + Kk ,†12
−2iμT t/h̄
where bk = u11
is the standard Bogolik p k + u k p−k e
13
ubov operator and Kk = uk δnk e−iμT t/h̄ is a scalar quantityof which magnitude depends on the amount of reservoir
fluctuations δnk . This is simply a displaced boson operator
where the ground state is of the form
†

exp[−Kk bk ]|0 .

k

k

k

k

is zero or nonzero, respectively. Here we used the fact that†[δnk ,δnk ] = d xd y[δn(x),δn( y)]eik( y−x) = 0 (13)†and the identity δnk = δn−k . Operators with nonzero commu-tators may be normalized to ±1, defining new bosonic modesof the system. From Fig. 1(c) we see that no bosonic solutions
exist in the flat dispersion regime, since all eigenvectors
i2 2
i
i†
2
satisfy |ui1
k | = |u k | , which corresponds to [b k ,b k ] = 0 from
Eq. (12). In the regime beyond the flat dispersion, therei2 2
2is always one dispersionless solution with |ui1
k | = |u k |corresponding to a renormalized density fluctuation solution.
i2 2
2The solution with |ui1
k | > |u k | corresponds to a solution(15)

The role of the reservoir fluctuations is, therefore, to displace
the vacuum of the Bogoliubov particles, depending on theamount of fluctuations of the reservoir.
IV. PHOTOLUMINESCENCE

To calculate the PL spectrum, we apply the methods
presented in Ref. 13. The PL spectrum is given by†

P L(k,ω) ∝ |Ck |2 Re[ pk (t = 0)p̃k (ω) ]




j
2
1i ∗ 1j i †ūk ūk bk (t = 0)b̃k (ω) , (16)
= |Ck | Re
ij

where ūk is the inverse of uk (normalized to satisfy bosoniccommutation relations), Ck is the Hopfield coefficient forthe photonic component of the polaritons, and tildes denote
time Fourier transformed variables. In the diagonal basis theequations of motion Eq. (9) areij

may be defined where uk are the eigenvectors of Mk .
The matrix Mk allows for two types of solutions, either
dispersionless or bosonic, defined as whether the commutation
relation 2  2†(12)[bi ,bi ] = ui1  − ui2 

(14)

ih̄

dbik
= Eki bik + iFki ,
dt

(17)

where we have added a Langevin noise operator Fki =†(Fk ,F−k ,0) to account for a thermal population of dissipative
Bogoliubov particles,15 and Eki are the eigenvalues of Mk . The
noise operator is assumed to obey correlations of the form14
 i †
bk Fk = δi1 k nth(18)k,k /kB T
where nth− 1), k = Re[Ek1 ] = −Re[Ek2 ], k =k = 1/(e12Im[Ek ] = Im[Ek ], T is the temperature, and δij is the
Kronecker delta. This ensures that the dissipative Bogoliubov†
particles obey thermal statistics bk bk = nthk , with otheroff-diagonal correlations vanishing. Equation (18) is chosen
primarily because it is the simplest choice and is possible
to generalize to other distributions.19 The choice of Fk3 = 0
corresponds to that of no driving term for the reservoir
fluctuations, which corresponds to a homogeneous pumpprofile. This can be extended to the case of a spatially varying

075130-3

BYRNES, HORIKIRI, ISHIDA, FRASER, AND YAMAMOTO

PHYSICAL REVIEW B 85, 075130 (2012)

pump profile, which would lead to the displaced Bogoliubov
operators as discussed in Eq. (15).
After a Fourier transform of Eq. (17) and inserting thecorrelations, we obtain the expression for the PL: 2 th2 th
 n +1i ū12i|ū11kk
k | nk2
PL(k,ω) ∝ |Ck | Re+.
h̄ω − k − i kh̄ω + k − i k(19)Here the first term corresponds to the positive dispersion,weighted by the thermal population and the dissipative
Bogoliubov coefficient, while the second term is the negative
dispersion, which only appears when there is appreciable†
Bogoliubov mixing between the pk and p−k operators. In
Figs. 2(a) and 2(b), we plot the PL spectrum corresponding
to low- and high-density regimes, respectively. At low density
we see that only the positive branch is visible. Here thereis negligible mixing between the components of Eq. (11)due to the small off-diagonal components of Mk . The
positive branch is populated via the thermal noise field Fk ,while the negative branch remains dark. At sufficiently highdensity, the off-diagonal components of Mk become large
enough such that there is some mixing of the componentsof Eq. (11). This results in a visible negative dispersion in
the PL spectrum. A simple criterion for when the negative
branch becomes visible may be derived. As may be seen
by inspection of the Bogoliubov expression for the factor
-8

-6

-4

-2

15
10
5
0
5
10 (c)15
1

PL/PL max2
0(EgnR gΨ0 )/E 0

15
10
5
0
5
10 (a)
15
0
1

0

10

2
ka

3

4

15
10
5
0
5
10 (b)15
12
0(EgnR gΨ0 )/E 02
0(EgnR gΨ0 )/E 02
0(EgnR gΨ0 )/E 0

10 10 10 10

2
ka

3

4

15
10
5
0
5
10
15

2
ka

3

4

2

3

4

(d)
1

ka

FIG. 2. (Color online) Photoluminescence of the excitation
spectrum of exciton-polariton condensates for various parameters.
Regimes of (a) just above condensation threshold, (b) thermallydepleted high density, (c) high density and low temperature, and(d) zero interactions are shown. Parameters used are g̃n0R = 1,
R  (n0R )n0R = 1, R(n0R ) = 1, g = g̃, γR = 1, and (a) ψ02 = 0.001n0R ,kB T = 2, (b) ψ02 = n0R , kB T = 2, (c) ψ02 = n0R , kB T = 0.1, (d)g = g̃ = 0, ψ02 = n0R , kB T = 2. A chemical potential of μ = −1 wasassumed in the thermal distribution in order to account for finite-size
effects of the condensate. All parameters in units of E0 . Zero detuning
and a Rabi splitting of 10E0 was assumed to calculate the Hopfield
coefficient Ck .

2 2

h̄ k
2
2
|ū12
k | = ( 2m + gψ0 )/2EBog (k) − 1/2, the negative compo2 2
nent is only appreciable for momenta h̄2mk < gψ02 . Due tothe finite linewidth of the dispersion, the negative dispersion
2 2
only becomes resolvable beyond momenta h̄2mk > k . Thus
the negative dispersion only becomes visible for densities
exceeding the criterion gψ02 ∼ k .
It is interesting that for low temperatures where nth
k issmall, only the negative excitation branch is visible in the
PL spectrum [Fig. 2(c)]. The reason for this can be understoodby the following argument. According to the inverse relationof Eq. (11), the loss of a polariton out of the system (whichis the basic process underlying the PL) is described by theoperator†

12−2iμT t/h̄−iμT t/h̄
pk = ū11+ ū13.
k b k + ū k b−k ek δm k e(20)

The energy change of the system associated with the first term
is μT + k , which is nothing but the standard mechanism for
the PL with a positive dispersion. With no thermal populationof dissipative Bogoliubov particles, the first term automaticallygives zero; thus, the positive dispersion remains dark. The
second term is associated with the gain of a dissipative
Bogoliubov particle and the loss of two condensate particles,
which has an energy change of 2μT − (μT + k ) = μT − k .
Unlike the first term, where a dissipative Bogoliubov particle
needs to be originally present, the second term does not require
this condition and occurs regardless of the initial population.
We note that a similar effect has been observed in atom lasers.20
The third term causes a loss of a condensate particle with anenergy change of μT independent of k, giving a dispersionlessspectrum. This can, in principle, give rise to a flat PL emission
as seen in Fig. 1, if there are strong enough density fluctuationsin the reservoir, although we do not assume that this is typicallythe case in current experimental systems.
There is another mechanism for making the negative branchvisible, which is in a regime where reservoir scattering isstrong. To see this effect, we show in Fig. 2(d) the PL
spectrum for the illustrative case of zero interactions g = g̃ =0. We again see that the negative branch is visible along the
eigenvalues of Mk . In this case, the negative branch is visible
not because of self-interactions but due to mediation via thereservoir mode. From our simulation results, we find that thenegative branch becomes visible for the same condition asthat given to observe relaxation oscillation [i.e., that Eq. (10)is real]. The mechanism for this is that due to the relaxation†
oscillations there is an effective coupling of the pk and p−k
operators mediated by the δnk operator. In practice, both theself-interactions and the reservoir effect are likely to contributeto making the negative dispersion visible.
V. SUMMARY AND CONCLUSIONS

We have calculated the PL spectrum of the excitations of
exciton-polariton condensates. At sufficiently high densities
such that gψ02 ∼ k , the negative branch of the Bogoliubov
spectrum becomes visible. The negative dispersion should
be visible even if no thermal population is present, due tothe nature of the PL emission being associated with theloss of a polariton from the system. In the regime of longreservoir decay times, a regime of relaxation oscillations was

075130-4

NEGATIVE BOGOLIUBOV DISPERSION IN EXCITON- . . .

PHYSICAL REVIEW B 85, 075130 (2012)

identified, which was also found to be able to illuminate
the negative dispersion. One question that remains is why afour-wave mixing experiment was required in order to see thenegative dispersion in Ref. 10, instead of spontaneously. One
explanation is that the bright emission from the condensate
may be masking the negative branch, and only the faint tailat high momenta can be accessed.11 Another possibility is the
reduction of the condensate self-interactions at higher density
due to reduction of the Bohr radius.21,22 This may contributeto making the negative branch weaker than expected due tothe suppression of Bogoliubov mixing.

1

13

J. Kasprzak et al., Nature (London) 443, 409 (2006).
H. Deng et al., Science 298, 199 (2002).
3
H. Deng, H. Haug, and Y. Yamamoto, Rev. Mod. Phys. 82, 1489
(2010).
4
A. Amo et al., Nature (London) 457, 291 (2009).
5
K. Lagoudakis et al., Nat. Phys. 4, 706 (2008).
6
G. Roumpos et al., Nat. Phys. 7, 129 (2011).
7
S. Utsunomiya et al., Nat. Phys. 4, 700 (2008).
8
F. M. Marchetti, J. Keeling, M. H. Szymańska, and P. B. Littlewood,
Phys. Rev. B 76, 115326 (2007).
9
M. H. Szymańska, J. Keeling, and P. B. Littlewood, Phys. Rev. Lett.
96, 230602 (2006).
10
V. Kohnle, Y. Leger, M. Wouters, M. Richard, M. T. Portella-Oberli,and B. Deveaud-Pledran, Phys. Rev. Lett. 106, 255302 (2011).
11
M. Wouters and I. Carusotto, Phys. Rev. B 79, 125311 (2009).
12
M. Wouters and I. Carusotto, Phys. Rev. Lett. 99, 140402(2007).
2

ACKNOWLEDGMENTS

We thank J. Keeling for discussions. This work is supportedby the Special Coordination Funds for Promoting Science andTechnology, the FIRST program for JSPS, Navy/SPAWAR
Grant No. N66001-09-1-2024, MEXT, and NICT.

C. Ciuti, P. Schwendimann, and A. Quattropani, Phys. Rev. B 63,
041303(R) (2001).
14
M. Scully and M. Zubairy, Quantum Optics (Cambridge University
Press, Cambridge, 1997).
15
L. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford
University Press, Oxford, 2003).
16
W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley,
New York, 1973).
17
F. Tassone and Y. Yamamoto, Phys. Rev. B 59, 10830 (1999).
18
H. Haken, Light: Laser Light Dynamics, vol. 2 (North-Holland
Physics Publishing, Amsterdam, 1985).
19
D. Sarchi and V. Savona, Phys. Rev. B 77, 045304 (2008).
20
Y. Japha, S. Choi, K. Burnett, and Y. B. Band, Phys. Rev. Lett. 82,
1079 (1999).
21
T. Byrnes, T. Horikiri, N. Ishida, and Y. Yamamoto, Phys. Rev. Lett.
105, 186402 (2010).
22
K. Kamide and T. Ogawa, Phys. Rev. Lett. 105, 056401 (2010).

075130-5