2012 nphysics - sculpting oscillators with light within a nonlinear quantum fluid

LETTERSPUBLISHED ONLINE: 10 JANUARY 2012 | DOI: 10.1038/NPHYS2182

Sculpting oscillators with light within a nonlinearquantum fluidG. Tosi1,2, G. Christmann1, N. G. Berloff3, P. Tsotsis4, T. Gao4,5, Z. Hatzopoulos5,6, P. G. Savvidis4,5

and J. J. Baumberg1*Seeing macroscopic quantum states directly remains an elusivegoal. Particles with boson symmetry can condense into quan-tum fluids, producing rich physical phenomena as well asproven potential for interferometric devices1–10. However,direct imaging of such quantum states is only fleetingly pos-sible in high-vacuum ultracold atomic condensates, and notin superconductors. Recent condensation of solid-state polari-ton quasiparticles, built from mixing semiconductor excitonswith microcavity photons, offers monolithic devices capableof supporting room-temperature quantum states11–14 that ex-hibit superfluid behaviour15,16. Here we use microcavities ona semiconductor chip supporting two-dimensional polaritoncondensates to directly visualize the formation of a spon-taneously oscillating quantum fluid. This system is createdon the fly by injecting polaritons at two or more spatiallyseparated pump spots. Although oscillating at tunable THzfrequencies, a simple optical microscope can be used to directlyimage their stable archetypal quantum oscillator wavefunc-tions in real space. The self-repulsion of polaritons providesa solid-state quasiparticle that is so nonlinear as to modifyits own potential. Interference in time and space reveals thecondensate wavepackets arise from non-equilibrium solitons.Control of such polariton-condensate wavepackets demon-strates great potential for integrated semiconductor-basedcondensate devices.

Non-resonant optical pumping of a semiconductor microcavityin the strong coupling regime continuously injects incoherentcarriers which rapidly cool and scatter into the mixed light–matterstates known as polaritons. Above a certain pump threshold(10mW at T = 10K) these polaritons Bose condense11–13. Becauseof the extreme nonlinearities caused by strong repulsion betweenpolaritons, they are shifted to higher energies (blue-shifted)wherever the density is high, particularly at the pumped spot17,18.The polaritons thus feel an outward force and form an expandingpolariton condensate19. Here we explore the novel effects that occurwhen two neighbouring polariton condensates interact. Instead oftypical Josephson-junction coherent coupling phenomena20, neweffects arise because of the quasiparticle interactions.

The decreasing density (and hence blue-shifts) away from twopumped spots induces a two-peaked potential profile (Fig. 1a). Sur-prisingly, on the line between the pump spots this potential seemsparabolic, forming a potential trap like that of a simple harmonicoscillator, the quantum equivalent of a pendulum. The polaritonsexperiencing this potential redistribute in energy and space to

1NanoPhotonics Centre, Cavendish Laboratory, Department of Physics, JJ Thompson Ave, University of Cambridge, Cambridge, CB3 0HE, UK,2Departamento de Física de Materiales, Universidad Autonóma, E28049 Madrid, Spain, 3Department of Applied Mathematics and Theoretical Physics,University of Cambridge, Cambridge, CB3 0WA, UK, 4Department of Materials Science and Technology, University of Crete, PO Box 2208, 71003Heraklion, Crete, Greece, 5Foundation for Research and Technology - Hellas, Institute of Electronic Structure and Laser, PO Box 1527, 71110 Heraklion,Crete, Greece, 6Department of Physics, University of Crete, PO Box 2208, 71003 Heraklion, Crete, Greece. *e-mail: [email protected].

occupy the simple harmonic oscillator (SHO) states (Fig. 1b).Because the polaritons can slowly leak through their confiningmirrors into escaping photons, spatial images recorded on a camera(filtered at each emission energy) directly reveal the characteristicquantum wavefunctions (Fig. 1c), tens of micrometres across.Such extended coherent quantum states in a semiconductor areunprecedented to image in real time directly. The energy spacingbetween levels (Fig. 1d) is almost identical, En = hv(nSHO+ 1/2),and the increasing number of spatial nodes with ηSHO are clearlyresolved. Along the line between pump spots, the wavefunctions fitvery well the expected Hermite–Gaussian ψSHO(x) states (Fig. 1e).From such fits, the polariton potential is reconstructed (seeSupplementary Information) as superimposed in Fig. 1b.

By controlling the spacing between the pump spots the shapeand orientation of this SHO potential, V (r), can be directlymodified in real time, thus changing the energy level spacing hv(Fig. 2a,b). In contrast if only one pump is present, the condensatepolaritons flow out unhindered without relaxation (thus remainingat the blue-shifted polariton energy at the pump spot, Fig. 2c),whereas below threshold only incoherent emission is observedaround each pump spot. Plotting the quantized energy levels forseveral pump separations, L, confirms their equal energy spacingand the predicted inverse dependence on separation (Fig. 2d).Although recent 1D versions of coherent polariton states observedin microcavities etched into wires19 do not seem to possess energieslinked to any spatial scales, we suggest they arise in a similar fashionto here. As we show below, in 2D other significant features arise,including periodic oscillations of the polariton wavepackets andcontrol of the polariton potential (Supplementary Information).Our crucial advance is the ability to manipulate independentcondensates in 2D on a chip, using externally imprinted potentialsthat are not statically predefined.

If the pump separation is kept constant with the pump powersboth increased (Supplementary Fig. S2a,b), then the spacing hvonly increases slightly (Supplementary Fig. S2c,d). However theincreasingly deep potential (from the pump-induced blue-shifts)increases the number of SHO states trapped inside. From the SHOmodel, increasing the polariton density, |ψ |2, gives a linear blue-shift, Vmax = g|ψ |2 (Supplementary Fig. S2d, right axis) producingthe dependence as observed:

hv =

√2h2

m∗∂2V∂x2=

√2h2

m∗Vmax

L2'

hL

√2g|ψ |2

m∗. (1)

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NATURE PHYSICS DOI: 10.1038/NPHYS2182 LETTERSa Pump 1

Pump 1 Pump 2

Pump 2

x

y

V 21 30

c

7654

nSHO = 5

1050¬5¬10

1.544

1.543

1.542

1.541

1.540

86420nSHO

Pump 1 Pump 2

10 µm

Microcavity

10 µm

x (µm)100¬10

x (µm)

Ener

gy (

eV)

b d e

Figure 1 | Spatially mapped polariton-condensate wavefunctions. a, Experimental scheme with two 1 µm-diameter pump spots separated by 20 µmfocused on the planar microcavity. The effective potential V (red) produces multiple condensates (grey image shows nSHO= 3 mode). b, Real-spacespectra along line between pump spots. c, Tomographic images of polariton emission (repulsive potential seen as dark circles around pump spots).Labelled according to nSHO assigned from d. d, Extracted mode energies versus quantum number. e, Hermite–Gaussian fit of ψn=5

SHO(x) to imagecross-section, dashed in c.

P1 P2

1.541

1.540

1.539

151050nSHO

¬20 ¬10 0 10 20

L (μm)

¬20 ¬10 0 ¬20 ¬10 010 20

1.541

1.540SinglepumpL = 20 μm L = 40 μm

E (e

V)

1.541

1.540

E (e

V)

1.541

1.540

E (e

V)

a

d e

f

b c

0.3

0.2

4020

202740

Ener

gy (

eV)

ΔE (

mev

)

L (μm)

Figure 2 | Dependence of simple harmonic oscillator states on pump properties. a–c, Spatially resolved polariton energies on a line between pump spots(white arrows). In a and b, the pump separation L controls the SHO energy spacing. In c, no relaxation is observed. d, Ladder of SHO energies for threedifferent pump separations L, with average energy spacings extracted in the inset. e, Time-averaged 2D simulation of the cGL equation for a 20 µm pumpseparation. f, Resulting time-averaged spectra along the dotted line in e.

The SHO states are thus only resolved owing to the strong polaritonrepulsion (g) and the ultra-light polaritonmass,m∗=4.2×10−5me,(measured independently, see Supplementary Information). The

polariton SHO states are populated differently under differentconditions, with high pump powers and close pump separationsfavouring relaxation to the lower condensate SHO states. By using

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LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS2182

1 meV

80

60

40

20

020100¬10¬20

Delay time (ps)

tr

0 ps 6 ps 13 ps

0

100

10 μm

x (μm)

¬10 0 10

g(1) (%

)

Vis

ibili

ty (

%)

a

de

f

b c

Δt

Figure 3 | Coherence revivals in real-space condensate wavepacket interferometry. a–c, Real-space interference patterns for time delays shown. Pumpspots separated by 20 µm, just off the right and left sides of the image. Lower images show extracted first-order coherence, g(1)(r,r,t). d, Fringe visibilityaveraged over each image versus Michelson interferometer time delay. e, Real-space tomographic image of the n= 1 state for four pump spots (markedwith the actual spot size, and taken at the energy of the dashed line in f). f, Energies across dashed line in e.

non-equal pump powers, asymmetric potentials can also be created(see Supplementary Information).

Our theoretical explanation starts with the mean-fieldequation21–23 for the lower polariton wavefunction, ψ , in the pres-ence of a reservoir population,N , of optically injected hot excitons

ih∂ψ

∂t= [E(i∇)+g|ψ(r,t )|2+Vext(r)]ψ

+ ih2[P(r,t ,N )−Γc]ψ (2)

which includes the polariton dispersion E(k) = h2k2/2m∗, thestrength of the contact interaction potential g, an externalpotential Vext(r) that describes repulsive interactions with thereservoir, and the rate of polariton losses Γc. The details ofthe incoherent pump, reservoir excitons and relaxation due tocollisions between condensed and non-condensed particles areincluded in P (see Supplementary Information). The net polaritonpotential V (r) = g|ψ(r, t )|2 + Vext is produced by the repulsiveinteractions between polaritons themselves as well as with thereservoir excitons (N ) close to the pump spots. This extensionof the Gross–Pitaevskii equation is a complex Ginzburg–Landau(cGL) equation, a universal equation of mathematical physics

describing the behaviour of systems in the vicinity of an instabilityand symmetry breaking24 and capable of spontaneous patternformation. The non-equilibrium solution arises from the constantlocalized energy input at the pump spots together with polaritondecay. Relaxation of polaritons by acoustic phonon emission isvery slow25, and swamped by polariton–polariton and polariton–exciton scattering over the timescales discussed below. For twopolaritons initially in SHO states (n1,n2) their mutual scatteringto states (n′1,n

′

2) is only energy–phase matched if n′1+n′2 = n1+n2and E ′1 − E1 = −{E ′2 − E2}. Scattering is thus most rapid if theenergy separations between states are equal. At low powers, thepotential is not parabolic, giving unequal energy spacings, and thusscattering is slower. At higher polariton densities, the scatteringrate increases so that polariton relaxation populates lower energystates. The resulting new polariton density profile modifies thepolariton potential, leading to more parabolic and equally spacedenergies, thus speeding up scattering and feeding back positively.This contrasts with current theories for polariton condensatespostulating a phonon scattering energy threshold21 (for whichwe see no evidence in experiment). The self-organized natureof the highly nonlinear polaritons is to produce the most rapidnon-equilibrium energy flow through the system, by forming anSHO parabolic potential with an energy ladder that maximizes





NATURE PHYSICS DOI: 10.1038/NPHYS2182 LETTERSpolariton relaxation. To our knowledge this is the first solid-statequasiparticle that is so nonlinear as to modify its own potential inthisway, and links to current theories of nonequilibrium systems24.

We now demonstrate that as well as the individual SHO statesbeing coherent condensates, they are also mutually coherent witheach other. We compare first with the nonlinear Maxwell equationsfor optical pulses propagating inside a nonlinear medium withsusceptibilityχ so that iE={χ (1)

+χ (3)|E|2}E .Whereas propagating

optical pulses generate equally spaced sidebands through four-wavemixing26, condensate polaritons analogously parametrically scatterto give new coherent polariton states. In optics, such nonlineardispersions can produce solitons and lead to mode-locking of lasercavity modes to produce trains of coherent intense pulses26. Herepolaritonic wavepackets appear spontaneously, bouncing back andforth between the two pump spots, corresponding exactly to thespectral and spatial organization observed.

The real space images (not energy filtered) are split in aMichelson interferometer and recombined at an imaging camera(Fig. 3a–c), with the fringes aligned in the vertical direction. Wemap the fringe visibility and hence first order coherence g(1)(r,r,t )at the same spatial locations but separated in time (Fig. 3a–c, lower).As the time delay between the interfering images increases, thefringe visibility rapidly decreases everywhere but revives stronglyafter tr = ±13 ps (Fig. 3c,d). This revival implies not only thatthe individual SHO states are phase coherent but that they arecondensates with a stable phase relationship.

All observations above are accounted for by a single condensedcoherent state, ψ , describing a polariton wavepacket in the quan-tum liquid oscillating back and forth with period tr producingthe characteristic SHO sidebands observed. The oscillation periodscales as expected with pump spatial separation (SupplementaryFig. S5). Indeed from equation (1), tr = πL

√m∗/2g|ψ |2 (plotted

as open circles), again consistent with the wavepacket oscillations.These 0.3–1.0 THz SHO sidebands, generated by a propagating con-densate wavepacket moving through its own nonlinear potential,correspond to the frequency micro-combs recently observed incavity optomechanics27.

The revival fringe amplitude at increasing time delays decayswith a 40 ps lifetime, owing to coherent wavepacket dispersion,decay, dephasing and diffusion. This lifetime is, however, ingood agreement with the coherent lifetime of the condensatefound by converting the decay length of the ballisticallyexpanding condensate into time. Wavepacket dispersion arisesfrom the variation in the energy level spacing, 1(hv)rms ∼ 50 µeV,contributing 80 ps to the coherence decay. The temporal widthof the condensate wavepacket corresponds to the number ofSHO states observed, 1t ' tr/nSHO, (with nSHO ∼ 10 well abovethreshold). In this picture, when the interacting condensatesphase-lock, the resulting spatially modulated polaritons createpolariton sidebands at the SHO states. In turn this further sharpensup the wavepacket extent and enhances the nonlinear polaritonscattering. Such condensate polariton solitary waves are expected incGL equations, and resemble the spatial solitons recently observedin atomic Bose–Einstein condensates (refs 28,29). Higher-energysidebands escape the SHO potential (unlike atomic Bose–Einsteincondensates) leading to a modified solitary wave structure (notedark solitons are expected for the perfect 1D parabolic potential30).The behaviour observed is matched by our simulations of thecGL equation (Supplementary Information). Multiple wavepacketsare observed bouncing back and forth (Fig. 2e, SupplementaryFig. S4 and Supplementary Movie) which indeed produce spectracorresponding to the SHO sidebands (Fig. 2f).

This oscillating quantum liquid model explains another oth-erwise peculiar feature. Although polaritons are well-confined inbetween the pump spots, they are unconfined and even ejected bythe potential (Fig. 1a) in the perpendicular direction. In spite of this,

well-defined SHO wavefunctions (Fig. 1) are seen, which wouldnot be expected for non-interacting 2D quantum quasiparticles.This is explained by the oscillating wavepacket, which is repeatedlyamplified close to each pump spot on each pass.

The flexibility of the pump-induced polariton condensates easilyallow full 2D confinement using extra pump spots. For instance,the lowest energy state in the four-pump square spot potential isfully confined in the xy plane (Fig. 3e,f), and selective polariton-condensate beams can be extracted. Thus polariton-condensatecircuits can now be created on the fly merely by appropriatesculpting of the pumping geometry, which will lead to manyfuture developments.

MethodsTo produce the effects seen here, which persist all across the microcavity samples,high-quality growth is required. A 5λ/2 AlGaAs DBR microcavity is used for allexperiments, with four sets of three quantum wells placed at the antinodes of thecavity electric field31. The cavity quality factor is measured to exceed Q> 8,000,with transfer matrix simulations giving Q= 2×104, corresponding to a cavityphoton lifetime T = 9 ps. Strong coupling is obtained with a characteristic Rabisplitting between upper and lower polariton energies of 9meV. The microcavitywedge allows scanning across the sample to set the detuning between the cavityand photonic modes. All data presented here use a negative detuning of −5meV,although other negative detunings give similar results. Excitation is provided by asingle-mode narrow-linewidth CW laser, focused to 1 µm diameter spots through a0.7 numerical aperture lens, and tuned to the first spectral dip at energies above thehigh-reflectivity mirror stopband at 750 nm. To prevent unwanted sample heatingthe pump laser is chopped at 100Hz with an on/off ratio of 1:30.

The sample is held at cryogenic temperatures below 10K, although similareffects are seen at higher temperatures. Images are recorded on an uncooled SiCCD camera in the magnified image plane, while spectra are recorded through a0.55m monochromator with a liquid-nitrogen-cooled CCD. Tomography uses acomputer-controlled mirror selecting the line illumination of the front aperture ofthe monochromator. The Michelson interferometer uses retroreflectors mountedon delay stages to control the relative temporal separation of the interference. Thepump laser is spectrally filtered out of all images. The fringe visibility, and hencefirst-order coherence, is mapped by extracting the first-order diffraction fromthe Fourier transformed images produced in the image plane after the Michelsoninterferometer. These are normalized to the corresponding zeroth-order diffractionand then transformed back into real space images.

Received 15 September 2011; accepted 22 November 2011;published online 10 January 2012

References1. Anderson, M. H. et al. Observation of Bose–Einstein condensation in a dilute

atomic vapour. Science 269, 198–201 (1995).2. Davis, K. B. et al. Bose–Einstein condensation in a gas of sodium atoms.

Phys. Rev. Lett. 75, 3969–3973 (1995).3. Andrews, M. R. et al. Observation of interference between two Bose–Einstein

condensates. Science 275, 637–641 (1997).4. Anderson, B. P. & Kasevich, M. A. Macroscopic quantum interference from

atomic tunnel arrays. Science 282, 1686–1689 (1998).5. Cataliotti, F. S. et al. Josephson junction arrays with Bose–Einstein condensates.

Science 293, 843–846 (2001).6. Gupta, S. et al. Contrast interferometry using Bose–Einstein condensates to

measure h/m and α. Phys. Rev. Lett. 89, 140401 (2002).7. Wang, Y. J. et al. Atom Michelson interferometer on a chip using a

Bose–Einstein condensate. Phys. Rev. Lett. 94, 090405 (2005).8. Penrose, O. & Onsager, L. Bose Einstein condensation and liquid Helium.

Phys. Rev. 104, 576–584 (1954).9. Eisenstein, J. P. & MacDonald, A. H. Bose–Einstein condensation of excitons

in bilayer electron systems. Nature 432, 691–694 (2004).10. Lagoudakis, K. G. et al. Coherent oscillations in an exciton–polariton Josephson

junction. Phys. Rev. Lett. 105, 120403 (2010).11. Deng, H. et al. Condensation of semiconductor microcavity exciton polaritons.

Science 298, 199–202 (2002).12. Kasprzak, J. et al. Bose–Einstein condensation of exciton polaritons. Nature

443, 409–414 (2006).13. Balili, R. et al. Bose–Einstein condensation of microcavity polaritons in trap.

Science 316, 1007–1010 (2007).14. Christopoulos, C. et al. Room-temperature polariton lasing in semiconductor

microcavities. Phys. Rev. Lett. 98, 126405 (2007).15. Amo, A. et al. Collective fluid dynamics of a polariton condensate in a

semiconductor microcavity. Nature 457, 291–295 (2009).16. Amo, A. et al. Superfluidity of polaritons in semiconductor microcavities.

Nature Phys. 5, 805–810 (2009).

NATURE PHYSICS | VOL 8 | MARCH 2012 | www.nature.com/naturephysics 193© 2012 Macmillan Publishers Limited. All rights reserved.



LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS2182

17. Savvidis, P. G. et al. Angle-resonant stimulated polariton amplifier. Phys. Rev.Lett. 84, 1547–1550 (2000).

18. Kavokin, A., Baumberg, J. J., Malpuech, G. & Laussy, F. P. Microcavities(Oxford Univ. Press, 2007).

19. Wertz, E. et al. Spontaneous formation and optical manipulation of extendedpolariton condensates. Nature Phys. 6, 860–864 (2010).

20. Josephson, B. D. The discovery of tunnelling supercurrents. Rev. Mod. Phys.46, 251–254 (1974).

21. Wouters, M., Liew, T. C. H. & Savona, V. Energy relaxation in one-dimensionalpolariton condensates. Phys. Rev. B 82, 245315 (2010).

22. Wouters, M. & Carusotto, I. Excitations in a nonequilibrium Bose–Einsteincondensate of exciton polaritons. Phys. Rev. Lett. 99, 140402 (2007).

23. Keeling, J. & Berloff, N. G. Spontaneous rotating vortex lattices in a pumpeddecaying condensate. Phys. Rev. Lett. 100, 250401 (2008).

24. Aranson, I.S. & Kramer, L. The world of the complex Ginzburg–Landauequation. Rev. Mod. Phys. 74, 99–143 (2002).

25. Tartakovskii, A. I. et al. Relaxation bottleneck and its suppression insemiconductor microcavities. Phys. Rev. B 62, R2283–R2286 (2000).

26. Saleh, B. E. A. & Teich, M. C. Fundamentals of Photonics (Wiley, 1991).27. Kippenberg, T. J., Holzwarth, R. & Diddams, S. A. Microresonator-based

optical frequency combs. Science 332, 555–559 (2011).28. Burger, S. et al. Dark solitons in Bose–Einstein condensates. Phys. Rev. Lett. 83,

5198–5201 (1999).29. Strekker, K. E. et al. Formation and propagation of matter–wave soliton trains.

Nature 417, 150–153 (2002).

30. Pelinovsky, D.E., Frantzeskakis, D.J. & Kevrekidis, P.G. Oscillations ofdark solitons in trapped Bose–Einstein condensates. Phys. Rev. E 72,016615 (2005).

31. Tsintzos, S. I. A GaAs polariton light-emitting diode operating near roomtemperature. Nature 453, 372–375 (2008).

AcknowledgementsL. Vina for comments, and grants EPSRC EP/G060649/1, EU CLERMONT4 235114,EU INDEX 289968, Spanish MEC (MAT2008-01555) and Greek GSRT programIrakleitos II. G.T. acknowledges financial support from an FPI scholarship ofthe Spanish MICINN.

Author contributionsG.T. and G.C. performed the spectroscopy experiments, and together with J.J.B.analysed the data and wrote the manuscript. P.G.S. contributed to the preparation of themanuscript and together with P.T., T.G. and Z.H. designed and grew the microcavitysamples, providing characterization spectroscopy to sustain high-quality performance.N.G.B. devised, coded, and carried out themodelling simulations.

Additional informationThe authors declare no competing financial interests. Supplementary informationaccompanies this paper on www.nature.com/naturephysics. Reprints and permissionsinformation is available online at www.nature.com/reprints. Correspondence andrequests for materials should be addressed to J.J.B





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Sculpting oscillators with light within a nonlinear quantum fluid Supplementary Information: A: Sample details The semiconductor microcavity top (bottom) DBR is made of 32 (35) pairs of Al0.15Ga0.85As/AlAs layers of 57.2nm/65.4nm each. Each of the four sets of QWs contains three 10nm-thick GaAs quantum wells separated by 10nm-thick Al0.3Ga0.7As layers. The 5λ/2 (1200nm) cavity is made of Al0.3Ga0.7As.

B: Single spot condensation If just a single pump spot is excited, polariton condensation is observed at a pump power of Pt=10mW where the blueshift above the low power k=0 polariton far away from the pump spot is =1.4meV. This ejects polaritons from the centre which ballistically propagate outwards at a

velocity , set by

.[15] These polaritons decay with the normal polariton

lifetimes, mostly set by the cavity loss rate. Figure S1: Power dependence of single pump spot

a, The threshold for condensation observed is 10mW incident pump power at λ=750nm (with average power only 3% of this due to mechanical chopping of the beam). b, Measured dispersion relation at the pump spot ( =0μm), and far away where the blue shift is zero, showing the condensate ballistic wavevector . We confirm that the microcavity remains in the strong coupling regime until nearly 10 times above threshold, when we can see an abrupt transfer to photon lasing. The coherence of the single condensate is also confirmed above threshold by spatial interference. By fitting the dispersion relations (such as in Fig.S1b), we find the effective mass of the lower polariton branch as reported and that it is in good agreement with theoretical models. The condensates are linearly polarised, independent of the pumping polarisation, confirming the full relaxation of injected carriers. We have confirmed that the linear polarisation is aligned to the crystal axes of the GaAs substrate.

Sculpting oscillators with light within a nonlinear quantum fluid Supplementary Information: A: Sample details The semiconductor microcavity top (bottom) DBR is made of 32 (35) pairs of Al0.15Ga0.85As/AlAs layers of 57.2nm/65.4nm each. Each of the four sets of QWs contains three 10nm-thick GaAs quantum wells separated by 10nm-thick Al0.3Ga0.7As layers. The 5λ/2 (1200nm) cavity is made of Al0.3Ga0.7As.

B: Single spot condensation If just a single pump spot is excited, polariton condensation is observed at a pump power of Pt=10mW where the blueshift above the low power k=0 polariton far away from the pump spot is =1.4meV. This ejects polaritons from the centre which ballistically propagate outwards at a

velocity , set by

.[15] These polaritons decay with the normal polariton

lifetimes, mostly set by the cavity loss rate. Figure S1: Power dependence of single pump spot

a, The threshold for condensation observed is 10mW incident pump power at λ=750nm (with average power only 3% of this due to mechanical chopping of the beam). b, Measured dispersion relation at the pump spot ( =0μm), and far away where the blue shift is zero, showing the condensate ballistic wavevector . We confirm that the microcavity remains in the strong coupling regime until nearly 10 times above threshold, when we can see an abrupt transfer to photon lasing. The coherence of the single condensate is also confirmed above threshold by spatial interference. By fitting the dispersion relations (such as in Fig.S1b), we find the effective mass of the lower polariton branch as reported and that it is in good agreement with theoretical models. The condensates are linearly polarised, independent of the pumping polarisation, confirming the full relaxation of injected carriers. We have confirmed that the linear polarisation is aligned to the crystal axes of the GaAs substrate.


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SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHYS2182

C: Spatial Quantum Harmonic Oscillator We use the standard quantum simple harmonic oscillator model to give a preliminary fit to the spatial wavefunctions observed. The parabolic potential,

, has 1D wavefunctions

( ) { } ( √

)

where ( ) is the standard Hermite polynomial. By fitting these wavefunctions to the observed SHO states at , the scale of can be extracted, thus yielding the effective potential at that energy. This allows the complete potential to be reconstructed. D: Power dependence of SHO states While the spacing between the pump spots controls the energy spacing of the SHO states, increasing the pump power in both spots increases the local blue shifts. This increases the depth of the potential trap (Fig.S4) Figure S2: Dependence of SHO states on pump power

a-b, Spatially-resolved polariton energies on a line between the two pump spots (white arrows) as the equal pump powers are increased (as marked). c, Extracted energies of SHO states, together with the energy of each condensate on its own (Vmax) and the bottom of the polariton potential (Vmin). d, SHO ladder spacing for different pump powers above threshold (10mW, dashed), together with prediction (red line), see main text. Right axis: blueshift at each pump power (black points). E: Asymmetric pump powers When the two pump spots are pumped unequally a more complicated behaviour will result. As long as both condensates are above threshold, they produce different blue shifts, however even if they are non-degenerate the nearby condensates interact strongly with each other.

Figure S3: Condensate SHO energies under non-equal pumping

a, Emitted polariton energies along the line between the pump spots when P1=3Pt and P2=1.5Pt for 20μm separation. A nearly parabolic potential is still observed at these densities, allowing strong polariton interactions to populate the low energy states. Polaritons formed by the oscillating wavepacket are also formed at higher energies, but these are not confined by the second pump spot, and hence escape. Concentration of the density of these unconfined higher energy polaritons is observed as they travel across the low energy barrier at P2, since as expected they slow down. Also visible is the asymmetry in the polariton potential, whose centre is pushed towards the pump spot with lower power. This allows a further way to move the polariton condensates whose position can now be tuned simply using the different injected laser powers. F: Theoretical simulations of the complex Ginzburg Landau equation Following [21,22,23] we use for the pump terms in Eqn.(2),

( ) ( ) (S1) where the reservoir exciton density is governed by

( ) [ | ( )| ] ( ) ( ) (S2)

with , where , , are the rate of scattering polaritons from the reservoir, of decay of condensate polaritons, and of decay of reservoir polaritons, while parameter represents the energy relaxation. Microscopic justification of the term in the context of atomic condensates was provided by [Penckwitt et al PRL 89, 070409 (2002)] and [Gardiner et al PRL 79, 1793 (1997)] who showed that is proportional to the rate at which thermal atoms enter the condensate due to collisions. The coefficient depends on the temperature and the density of the thermal cloud. In our system the role of the thermal cloud is played by the reservoir of excitons, so it is suitable to assume that depends on linearly. We numerically integrated Eq.(2) of the main text with Eqs. (S1) and (S2) using the following parameters: ⁄ (inverse photon lifetime), (reservoir lifetime), pump spot ( ) with FWHM diameter 1 , (polariton repulsive interaction), (in-scattering interaction), ( ) (parameter representing the energy relaxation rate), and . The diffusion coefficient, , is optimised to give a good fit to the experimental reservoir shape. In this model we include only a single polariton population, despite the spin degree of freedom which has been shown to produce spin- and momentum-dependent scattering [Krizhanovskii et al,




C: Spatial Quantum Harmonic Oscillator We use the standard quantum simple harmonic oscillator model to give a preliminary fit to the spatial wavefunctions observed. The parabolic potential,

, has 1D wavefunctions

( ) { } ( √

)

where ( ) is the standard Hermite polynomial. By fitting these wavefunctions to the observed SHO states at , the scale of can be extracted, thus yielding the effective potential at that energy. This allows the complete potential to be reconstructed. D: Power dependence of SHO states While the spacing between the pump spots controls the energy spacing of the SHO states, increasing the pump power in both spots increases the local blue shifts. This increases the depth of the potential trap (Fig.S4) Figure S2: Dependence of SHO states on pump power

a-b, Spatially-resolved polariton energies on a line between the two pump spots (white arrows) as the equal pump powers are increased (as marked). c, Extracted energies of SHO states, together with the energy of each condensate on its own (Vmax) and the bottom of the polariton potential (Vmin). d, SHO ladder spacing for different pump powers above threshold (10mW, dashed), together with prediction (red line), see main text. Right axis: blueshift at each pump power (black points). E: Asymmetric pump powers When the two pump spots are pumped unequally a more complicated behaviour will result. As long as both condensates are above threshold, they produce different blue shifts, however even if they are non-degenerate the nearby condensates interact strongly with each other.

Figure S3: Condensate SHO energies under non-equal pumping

a, Emitted polariton energies along the line between the pump spots when P1=3Pt and P2=1.5Pt for 20μm separation. A nearly parabolic potential is still observed at these densities, allowing strong polariton interactions to populate the low energy states. Polaritons formed by the oscillating wavepacket are also formed at higher energies, but these are not confined by the second pump spot, and hence escape. Concentration of the density of these unconfined higher energy polaritons is observed as they travel across the low energy barrier at P2, since as expected they slow down. Also visible is the asymmetry in the polariton potential, whose centre is pushed towards the pump spot with lower power. This allows a further way to move the polariton condensates whose position can now be tuned simply using the different injected laser powers. F: Theoretical simulations of the complex Ginzburg Landau equation Following [21,22,23] we use for the pump terms in Eqn.(2),

( ) ( ) (S1) where the reservoir exciton density is governed by

( ) [ | ( )| ] ( ) ( ) (S2)

with , where , , are the rate of scattering polaritons from the reservoir, of decay of condensate polaritons, and of decay of reservoir polaritons, while parameter represents the energy relaxation. Microscopic justification of the term in the context of atomic condensates was provided by [Penckwitt et al PRL 89, 070409 (2002)] and [Gardiner et al PRL 79, 1793 (1997)] who showed that is proportional to the rate at which thermal atoms enter the condensate due to collisions. The coefficient depends on the temperature and the density of the thermal cloud. In our system the role of the thermal cloud is played by the reservoir of excitons, so it is suitable to assume that depends on linearly. We numerically integrated Eq.(2) of the main text with Eqs. (S1) and (S2) using the following parameters: ⁄ (inverse photon lifetime), (reservoir lifetime), pump spot ( ) with FWHM diameter 1 , (polariton repulsive interaction), (in-scattering interaction), ( ) (parameter representing the energy relaxation rate), and . The diffusion coefficient, , is optimised to give a good fit to the experimental reservoir shape. In this model we include only a single polariton population, despite the spin degree of freedom which has been shown to produce spin- and momentum-dependent scattering [Krizhanovskii et al,


4 NATURE PHYSICS | www.nature.com/naturephysics

SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHYS2182

Phys. Rev. B 73, 073303 (2006)]. In experiment we observe no strong cross-polarised components and suggest that this occurs because momentum transfers in polariton interactions do not produce large direction changes in this configuration. Using these equations (1,2) we elucidate theoretically the interaction between two condensates which are pumped at different separation as in the experiments. From the polariton wavefunction at each spatial position we can explore both the spatial dynamics and the steady state energies of the total condensate, which we extract by Fourier transform of the spatial dynamics at each point. We give here a few examples only, as they are the subject of a detailed paper in preparation. Figure S4: Theoretical simulation of interacting condensate dynamics

a, One frame of simulation for pump separation of 20μm, and b, spatial cross section along the line between pump spots (dashed in a) vs time, showing oscillating wavepackets with more complex dynamics. A full video of the simulated dynamics is available with the supplementary information. G: Revival times of coherence as a function of interferometric time delay Figure S5: Dependence of SHO states on pump power

a, Measured wavepacket round trip time (given by the time-delay of the coherence revival) for different pump separations plotted against the SHO energy separation (red dots). Predictions are

from Fourier relation (red line) and measured potential (red open circles, see main text). Right axis (blue): Decay of revival amplitude vs time delay, with exponential fit. Extracting the revival time delay of the visibility and its height gives information on the oscillating coherent wavepacket. The oscillation period scales as expected with pump spatial separation (Fig.S5) where for convenience these are plotted against the resulting energy spacing (cf Fig.2g). The revival fringe amplitude at increasing time delays (Fig.S5, blue points) corresponding to larger pump spot separations, decays with a 40ps lifetime. H: Comparison to 1D polariton wires Besides the direct visualisation of the polariton wavefunctions and the real-time construction of polariton potentials on the fly with selective pump laser spots, there are several other significant differences between the 2D experiments here and 1D etched polariton wires. Another degree of freedom arises from having two pumping spots rather than a single pump spot with the flow reflected at the end of the wire as in Ref. 19. Indeed, our findings and the analysis of the SHO levels implies the periodic motion of the polariton wave packet oscillating between two pumping spots and therefore, out of phase periodic beating of exciton reservoirs. In the reflective symmetry of Ref. 19 this degree of freedom is suppressed. In addition by using different pump intensities we create controllable asymmetric potentials (Fig.S3) which were impossible previously. Finally, topological excitations are completely different between 1D and 2D: for instance, a dark soliton of the GPE is unstable only in 2D as it is subject to the modulational (snake) instability [see e.g. E. A. Kuznetsov and J. J. Rasmussen, Phys. Rev. E 51, 4479 (1995)], and thus absent in 2D equilibrium condensates (unless confinement in the second direction is so small that this instability is suppressed). In our system there is no confinement, nevertheless, the periodic oscillations of solitary waves persist. In fact, we do observe the effect of the modulational instability at larger distances, where vortex pairs are created along the density dips. The formation of vortex pairs along the transverse direction to a dark soliton is a well-known outcome of modulational instabilities in the defocusing nonlinear Schrödinger equation. These features are even more evident when using larger numbers of pump spots, with different phase relationships emerging in the resulting vortex lattices.




Phys. Rev. B 73, 073303 (2006)]. In experiment we observe no strong cross-polarised components and suggest that this occurs because momentum transfers in polariton interactions do not produce large direction changes in this configuration. Using these equations (1,2) we elucidate theoretically the interaction between two condensates which are pumped at different separation as in the experiments. From the polariton wavefunction at each spatial position we can explore both the spatial dynamics and the steady state energies of the total condensate, which we extract by Fourier transform of the spatial dynamics at each point. We give here a few examples only, as they are the subject of a detailed paper in preparation. Figure S4: Theoretical simulation of interacting condensate dynamics

a, One frame of simulation for pump separation of 20μm, and b, spatial cross section along the line between pump spots (dashed in a) vs time, showing oscillating wavepackets with more complex dynamics. A full video of the simulated dynamics is available with the supplementary information. G: Revival times of coherence as a function of interferometric time delay Figure S5: Dependence of SHO states on pump power

a, Measured wavepacket round trip time (given by the time-delay of the coherence revival) for different pump separations plotted against the SHO energy separation (red dots). Predictions are

from Fourier relation (red line) and measured potential (red open circles, see main text). Right axis (blue): Decay of revival amplitude vs time delay, with exponential fit. Extracting the revival time delay of the visibility and its height gives information on the oscillating coherent wavepacket. The oscillation period scales as expected with pump spatial separation (Fig.S5) where for convenience these are plotted against the resulting energy spacing (cf Fig.2g). The revival fringe amplitude at increasing time delays (Fig.S5, blue points) corresponding to larger pump spot separations, decays with a 40ps lifetime. H: Comparison to 1D polariton wires Besides the direct visualisation of the polariton wavefunctions and the real-time construction of polariton potentials on the fly with selective pump laser spots, there are several other significant differences between the 2D experiments here and 1D etched polariton wires. Another degree of freedom arises from having two pumping spots rather than a single pump spot with the flow reflected at the end of the wire as in Ref. 19. Indeed, our findings and the analysis of the SHO levels implies the periodic motion of the polariton wave packet oscillating between two pumping spots and therefore, out of phase periodic beating of exciton reservoirs. In the reflective symmetry of Ref. 19 this degree of freedom is suppressed. In addition by using different pump intensities we create controllable asymmetric potentials (Fig.S3) which were impossible previously. Finally, topological excitations are completely different between 1D and 2D: for instance, a dark soliton of the GPE is unstable only in 2D as it is subject to the modulational (snake) instability [see e.g. E. A. Kuznetsov and J. J. Rasmussen, Phys. Rev. E 51, 4479 (1995)], and thus absent in 2D equilibrium condensates (unless confinement in the second direction is so small that this instability is suppressed). In our system there is no confinement, nevertheless, the periodic oscillations of solitary waves persist. In fact, we do observe the effect of the modulational instability at larger distances, where vortex pairs are created along the density dips. The formation of vortex pairs along the transverse direction to a dark soliton is a well-known outcome of modulational instabilities in the defocusing nonlinear Schrödinger equation. These features are even more evident when using larger numbers of pump spots, with different phase relationships emerging in the resulting vortex lattices.

caption for video file nphys2182-s2: Simulation of the time dependant evo-lution of the condensate density

2012 nphysics - sculpting oscillators with light within a nonlinear quantum fluid

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