LETTERS

Coherent zero-state and p-state in anexciton–polariton condensate arrayC. W. Lai1,2,3, N. Y. Kim1,2, S. Utsunomiya3,4, G. Roumpos1, H. Deng1, M. D. Fraser1, T. Byrnes2,3, P. Recher1,2,N. Kumada4, T. Fujisawa4 & Y. Yamamoto1,3

The effect of quantum statistics in quantum gases and liquidsresults in observable collective properties among many-particlesystems. One prime example is Bose–Einstein condensation,whose onset in a quantum liquid leads to phenomena such assuperfluidity and superconductivity. A Bose–Einstein condensateis generally defined as a macroscopic occupation of a single-particle quantum state, a phenomenon technically referred to asoff-diagonal long-range order due to non-vanishing off-diagonalcomponents of the single-particle density matrix1–3. The wave-function of the condensate is an order parameter whose phaseis essential in characterizing the coherence and superfluidphenomena4–11. The long-range spatial coherence leads to the exis-tence of phase-locked multiple condensates in an array of super-fluid helium12, superconducting Josephson junctions13–15 or atomicBose–Einstein condensates15–18. Under certain circumstances, aquantum phase difference of p is predicted to develop amongweakly coupled Josephson junctions19. Such a meta-stable p-statewas discovered in a weak link of superfluid 3He, which is character-ized by a ‘p-wave’ order parameter20. The possible existence of sucha p-state in weakly coupled atomic Bose–Einstein condensates hasalso been proposed21, but remains undiscovered. Here we report theobservation of spontaneous build-up of in-phase (‘zero-state’) andantiphase (‘p-state’) ‘superfluid’ states in a solid-state system; anarray of exciton–polariton condensates connected by weak periodicpotential barriers within a semiconductor microcavity. These in-phase and antiphase states reflect the band structure of the one-dimensional polariton array and the dynamic characteristics ofmetastable exciton–polariton condensates.

The strong coupling between quantum well excitons and micro-cavity photons allows studies of cavity quantum electrodynamiceffects in a solid-state system. The reversible energy exchangebetween the microcavity photons and quantum well excitons leadsto the anti-crossing of eigenstates, known as the upper and lowerexciton–polaritons (UPs and LPs)22. Unlike exciton–polaritons inbulk semiconductors, cavity exciton–polaritons involve no transportand are fully characterized by the in-plane wavenumber kjj, enabling adirect measurement of the exciton–polariton dispersion relation andmomentum distribution of particles through angularly resolvedspectroscopy23. The exciton–polaritons have been shown to condensein momentum space and exhibit spontaneous build-up of mac-roscopic spatial and temporal coherence24–26. Using Young’s dou-ble-slit interference experiment, we measured a spatial coherencelength of the condensate up to ,20 mm, limited by the pumping spotsize (see Supplementary Information).

In our experiments, an array of one-dimensional cigar-shapedexciton–polaritons is created by the deposition of periodic strips ofa metallic thin film on the top surface of the microcavity structure, as

shown in Fig. 1a (see also Methods). The GaAs-based microcavitystructure contains 12 quantum wells, exhibiting a vacuum Rabi split-ting of ,14 meV. For the given microcavity structure under themetallic layer, the cavity resonance energy EC (for kjj5 0) is expectedto increase by ,400 meV according to the transfer matrix method27

1E. L. Ginzton Laboratory, Stanford University, Stanford, California 94305, USA. 2Institute of Industrial Science, University of Tokyo, Meguro-ku, Tokyo 153-8505, Japan. 3NationalInstitute of Informatics, Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan. 4NTT basic research laboratories, NTT Corporation, Atugi, Kanagawa 243-0198, Japan.

aa

kz

k||

kθ

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DBR

DBR

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vity

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e st

acks

of f

our

GaA

s q

uant

um w

ells

Au/Ti strips

E

k||

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LPU0

Cavityphoton

0

~2U0

GaAIAsAIAs

30X/a

21

50

100

δE0

(µeV

)

0

b c Au/Ti

Cavity only

Cavity + Au

Figure 1 | Formation of an exciton–polariton array. a, A schematic of acavity polariton array formed by depositing periodic thin metallic strips (Au/Ti) on top of a microcavity structure. The 1.4-mm-wide strips are equallyspaced with a pitch distance a 5 2.8 mm. The microcavity structure consistsof a l/2 AlAs cavity (indicated by red lines) sandwiched by two distributedBragg reflectors with alternating GaAlAs/AlAs l/4 layers, where l is thecavity resonance wavelength (varying around the quantum well excitonresonance, ,776 nm with tapering). Three stacks of four GaAs quantumwells are positioned at the central three antinodes of the microcavity photonfield (the schematic black oscillatory curve). The distributed Bragg reflectors(DBR) consist of repeated GaAIAs/AIAs layers and are symbolized by thetwo short dotted vertical lines. b, Dispersion curves of the cavity photonmode (dashed lines) and LP (solid lines). The black dashed lines representthe uncoupled cavity photon mode and quantum-well heavy-hole excitonstate, which are in resonance (EC 5 EX) at k | | 5 0 in this case. The addition ofthe metallic layer shifts the LP energy (red solid line) by U0 < 200meV.c, Spatial LP energy modulation dE0 measured by scanning a pinhole acrossthe lattice at the first conjugate real-image plane (see Fig. 2 and Methods).

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529Nature ©2007 Publishing Group

(see Methods). When the cavity photon (EC) is near resonancewith the quantum well exciton (EX) at kjj5 0, the shift of thelower-polariton energy induced by the metallic layer is ,200 meV,approximately half of the cavity photon resonance shift (Fig. 1b).Therefore, the LPs are expected to be trapped in the gap region wherethe LP energy is lower. The measured spatial modulation of LP energyis ,100 meV, which is less than the theoretical prediction owing to thediffraction-limited spatial resolution (,2mm) of our optical detec-tion system (Fig. 1c). We confirm by an independent measurementthat the LP energy in a bare microcavity is ,200 meV lower than theLP energy under a uniform metallic layer of similar thickness forEC < EX.

The microcavity is excited by a mode-locked Ti:Sapphire laser witha ,2.5 ps pulse near the quantum well exciton resonance at an incid-ent angle of 60u, corresponding to an in-plane wavenumberkjj< 7 3 104 cm21 in air. The large kjj of the pump ensures that thecoherence of the pump laser is lost by multiple phonon emissionsbefore the polaritons scatter into kjj< 0 states. The LP distributionsin coordinate space (near-field) and momentum space (far-field) aremeasured by micro-photoluminescence spectroscopy and imaging,using the set-up shown in Fig. 2a (see Methods). The near-field imagesillustrate the spatial LP distribution, while the far-field images revealthe in-plane momentum LP distribution. The spectroscopy inmomentum space (that is, angularly resolved spectroscopy) providesa direct measurement of the LP dispersion relation and effective mass.

For a system of two condensates, the interference fringes can beobserved even in the absence of a locked relative phase in a single shotmeasurement7. In the case of an array of condensates, however, inter-ference fringes appear only if the relative phases between consecutivecondensates are locked28. The high degree of interference fringevisibility in momentum space can thus be used as an indication ofa long-range spatial coherence across the whole array of multiplecondensates. Such an interference measurement is commonly per-formed in atomic Bose–Einstein condensates by releasing conden-sates in an optical lattice.

Considering a periodic and coherent array of condensates alignedalong the x (where x is X or Y, see below) axis, we can approximate theorder parameter in momentum space as28:

y kxð Þ~y0 kxð ÞX

n~0,+1:::+nM

ei n kx azwð Þ

~y0 kxð Þsin N kxa=2ð Þ

sin kxa=2ð Þ

����w~0

ð1Þ

where n labels the condensate array element, N 5 2nM 1 1 is the totalnumber of periodic array elements, a is the pitch distance betweentwo neighbouring elements, and y0 kxð Þ is the momentum-spacewavefunction of an individual element. The overall momentum dis-

tribution, r kxð Þ~ y kxð Þj j2, reflects the coherence properties andnature of the condensate arrays through distinctive interferencepatterns. In the presence of the lattice, the momentum distributiondisplays narrow peaks at kx 5 m 3 2p/a (where m is an integer) with

an envelope function given by r0 kxð Þ~ y0 kxð Þj j2.If all the elements are locked in-phase (w 5 0), at a LP wavelength

l 5 780 nm and a grating pitch distance a 5 2.8 mm, a Fraunhoferdiffraction pattern has a central peak at h 5 0u and two side lobes ath 5 sin21(l/a) < 616u (kjj5 62p/a; m 5 61 first-order diffrac-tion). The LP emissions from an exciton–polariton condensate arrayare expected to display such an interference pattern in momentumspace, just like an atomic Bose–Einstein condensate experiment15–18.

The observed LP distribution in both coordinate and momentumspace of the condensate array are shown in Fig. 2b and c. Below thecondensation threshold (pump power P 5 20 mW), the near-fieldimage reveals the cigar-shaped LP emissions through the 1.4-mm-wide gaps between the metallic strips. The corresponding LPmomentum distribution is broad (DhFWHM < 36u) and isotropic,independent of the periodic intensity modulation and the ellipticallyshaped pumping spot. The result indicates that there is negligiblephase coherence among different cigar-shaped LPs. With an increas-ing pumping rate, two strong and two weak side lobes emerge at

∆Y =

50

µm

∆qY =

60º

∆qX = 60º, ∆kX = 2π/l = 8.1 µm–1

∆X = 60 µm 110–110–2

q

k|| = K sin q

0º 0º16º8º24º

Double slits/pinhole

L1 L2

qP

X

Y

k-spacer-spacek-space

CCD/spectrometer

Object

200 mW20 mW 70 mWb

c

a Figure 2 | Imaging and spectroscopy ofexciton–polariton distribution in coordinate andmomentum space. a, Schematic of the micro-photoluminescence set-up for imaging andspectroscopy in coordinate and momentumspace (see Methods). hP < 60u is the incidentangle of the pump beam. b, Near-field imagesshowing the LP distribution across a polaritonarray in coordinate space under pumping powersof 20, 70 and 200 mW (left to right) at T 5 20 K.Here the condensation threshold pumping poweris ,45 mW (intensity ,1,600 W cm22). Thewhite dashed lines indicate selected locations ofthe 1.4-mm-wide metallic strips. The colour scaleis the normalized LP emission intensity.c, Corresponding far-field images showing the LPdistribution in momentum space. When passingthrough the threshold, lobes at 68u and 624uemerge out of the isotropic background of thethermal polariton gas, which is indicative of thep-state. The strong central lobe at 0u and twoweak lobes at 616u appear at higher pumpingrates, indicative of the zero-state.

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h < 68u and 624u, which correspond to kjj5 6p/a andkjj5 63p/2a, respectively. The diffraction pattern suggests that thephase difference of adjacent condensates in the array is locked exactlyto p (see w 5p in equation (1)). Moreover, the corresponding near-field image (in Fig. 2b; 70 mW) reveals that strong LP emissions aregenerated from under the metallic strips rather than the gaps. Thecontrast is better observed near the boundary of the central condens-ate and the outside thermal LP emissions, which come through thegaps. The dominant LP emission through the metallic layers despitethe lower transmission indicates that LP condensates are stronglylocalized under the metallic strips. With a further increasing pump-ing rate, the intensity of the central peak near h 5 0u with weak sidelobes at h < 616u (Fig. 2c) gradually surpasses the peaks at h < 68uand 624u as shown in Fig. 2c. The corresponding near-field image ofthe condensate also recovers a standard spatial modulation (near thecentral area indicated by the white arrow in Fig. 2b).

These two distinct interference patterns suggest the transitionfrom an antiphase state to an in-phase state. We refer to the in-phasestate as the ‘zero-state’ and the antiphase state as the ‘p-state’ on thebasis of their relative phase differences between adjacent elements.

Both the zero-state and p-state can be observed for an exciton–polariton condensate array consisting of more than 30 strips ofcondensates (total array dimension ,100 mm). These collective statesare manifestations of the long-range spatial coherence as well as ofphase-locking across the array. Next, we examine the phase-lockingmechanism for the zero-state and p-state through the LP dispersionrelation.

In Fig. 3a, we show the energy versus in-plane momentumspectra for an LP array near resonance EC < EX and below threshold.From the dispersion curve, we deduce the LP kinetic energy atkjj5 6G0/2 5p/a (half of the primitive reciprocal lattice vector),Ek~B2 G0=2ð Þ2=2m�<500 meV, where the LP effective massm* 5 9 3 1025me (where me is the free electron mass). The kineticenergy Ek is larger than the spatial LP energy modulation,U0 < 200 meV. Therefore, we expect the exciton–polariton condens-ate to be only weakly perturbed by the periodic potential and thecoherence is maintained across the array. In such a large tunnellinglimit, we neglect the polariton–polariton interaction and deduce theband structure of the polariton array by assuming a ‘nearly freepolariton’ in the presence of the periodic square-well potential.Given a one-dimensional periodic potential U0(X) with a latticeconstant a, the band structure can be obtained using the standardBloch wavefunction formalism. Only those polaritons with in-planemomentum close to the Bragg condition (kjj5 m G0/2) are subject tobackscattering, owing to the periodic potential, and gaps appear atthe Bragg planes where standing waves are formed. Similar to thestandard extended-zone scheme29, the band structure can be con-structed by starting with an original parabola and displaced parabolasELP(kjj6 mG0), as shown in Fig. 3b. Here the size of the circles repre-sents the expected relative LP emission intensity from each band(Methods). We observe this in the LP energy versus in-planemomentum with a below-threshold pumping rate (Fig. 3a). The firstbandgap, between the first and second bands near the zone bound-aries (Bragg planes), is of the order of the barrier potential, jU0j. Thissmall gap is masked in the measured spectra by the large inhomo-geneous broadening of the LP emission lines (linewidth ,500 meV).

In Fig. 3c, we show the energy versus in-plane momentum of apolariton condensate array above threshold and EC 2 EX <16 meV.Above threshold, LP emissions occur in two states with an energydifference of about 1 meV. These two states with emission peaks atkjj5 0, 6G0 and at kjj5 6G0/2, 63G0/2 correspond to the zero-stateand p-state, respectively.

On referring to the multi-valley band structure (Fig. 3b), thedynamic condensation process of the exciton–polariton arraybecomes more transparent. Without the spatial modulation, thequasi-stationary state of the LP condensate is the lowest-energy stateat kjj5 0. Here the bottom of the LP dispersion parabola serves as a

‘trap’ of polaritons in momentum space. When the spatial potentialmodulation is introduced, a meta-stable dynamic condensate canoccur near the bottom of the second band (point A in Fig. 3b). Atthese points, polaritons experience a relaxation bottleneck, resultingin metastable condensates. Eventually, the LP system will relax to thelowest-energy in-phase state (point C in Fig. 3b) as the pumping rateis increased.

To understand the spatial distributions of LPs in the array (near-field images shown in Fig. 2b), we consider the Bloch wavefunctions.The Bloch wavefunctions for these eigenstates exhibit not onlyopposite relative phases between adjacent elements in the array butalso different characteristics. Schematic Bloch wavefunctions forselected eigenstates labelled as A, B and C in Fig. 3b are shown inFig. 3d. The zero-state at point C corresponds to the ‘s-like’ state withmaximal amplitude in potential wells and identical phase betweenadjacent elements across the array. The wavefunctions of A and B atkjj5 G0/2 both exhibit a relative p-phase difference between adjacentelements. The lower-energy state at point B corresponds to anti-bonding of ‘s-states’, while the higher-energy state at point Acorresponds to bonding of ‘p-states’. Like the atomic p-state, theLP density vanishes at the centre of the potential well for this p-state.This is fully confirmed by the near-field imaging shown in Fig. 2b. Wenote that under the weak potential modulation, the Bloch wavefunc-tions are superpositions of forward-and-backward travelling planewaves at the zone boundary and have a relatively broad distributionin space compared to the standard tight-binding model of a normalcondensed-matter system.

The metastability of the p-state is a unique property for dynamicpolariton condensates. Under strong pumping rates, the zero-statedominates eventually, owing to enhanced cooling by stimulatedpolariton–polariton scattering at high densities. A typical evolution

C

6m

eV

110–110–2

∆k|| = 3.5 × (2π/a) ≈ 7.85 µm–1

8 m

eV

k||/G0

3/2–3/2 –1/2 1/2 3/2 (G0)

~|U0|

011/20–1/2–3/2 –1

–16º –16º

a

1st BZ2

1.5955 eV

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plit

ude

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n = 2

n = 3

A

B

A

B

ba

Figure 3 | Band structure and ‘superfluid’ states in an exciton–polaritoncondensate array. a, Time-integrated energy versus in-plane momentum (Eversus k | | ) near resonance EC < EX below threshold (P 5 10 mW) at T < 7 K.The central bright parabola corresponds to the dispersion curve for thepolaritons in the absence of a periodic potential. Two additional parabolasdisplaced by 6G0 5 62p/a cross the central dispersion curve at k | | 5 6G0/2. b, Extended-zone scheme of the band structure. Anti-crossing occurs atthe boundary between the first and second Brillouin zones (BZ). The size ofthe open circles represents the expected relative emission intensity (logscale) of each energy band. Condensation occurs at the valleys labelled bysolid red (point A) and blue (point C) circles. c, E versus k | | at blue detuningEC 2 EX < 6 meV above threshold (P 5 40 mW). The lobes at k | | 5 0, 6G0

correspond to the in-phase zero-state, whereas the lobes at k | | 5 6G0/2,63G0/2 correspond to the antiphase p-state. d, Schematic Bloch-wavefunctions for states labelled as A, B or C in b.

NATURE | Vol 450 | 22 November 2007 LETTERS

531Nature ©2007 Publishing Group

from the metastable p-state to the stationary zero-state is illustratedby the LP momentum distribution profiles for various pumping ratesin Fig. 4. At a high pumping rate, the LP emission from the zero-statesurpasses that from the p-state by more than an order of magnitude.The transition between these dynamic condensates depends on thecoupling strength between adjacent elements in the array (seeSupplementary Information).

Thus we observed a metastable condensate in a p-state (anti-bonding of p-states) as well as a stable condensate in a zero-state(bonding of s-states) in an exciton–polariton array. Such a coherentlinear superposition state formed over many condensates in peri-odical sites is referred to as a ‘superfluid’ state. However, some com-mon superfluidity properties such as the frictionless flow over a longdistance beyond the pumping region are not necessarily meant. Ourexperiments constitute a first step towards the experimental studyof a dynamical Bose–Hubbard model in a solid-state system. Thepolariton system allows the direct observation of the spatial andmomentum distributions as well as the dispersion characteristics ofthe condensates. For instance, full control of the on-site interaction,barrier potential and occupation number may allow us to study thedynamical quantum phase transition properties of a Mott insulator.

METHODS SUMMARYThe periodic spatial modulation of LP energy ,200meV is induced by the

periodic thin metallic strips (a ,20-nm-thick Au /3-nm-thick Ti layer) with a

pitch distance a 5 2.8mm deposited on top of the microcavity. The relative

transmission through the top distributed Bragg reflectors (DBR) with and with-

out the metallic layer is ,40–50%.

The imaging and spectroscopy of LPs in the near-field (coordinate space) and

far-field (momentum space) are performed by positioning a charged-coupled-

device (CCD) camera or an imaging spectrometer in a plane conjugate to the

real-image or Fourier (back focal) plane of the objective. The collection efficiency

is approximately uniform to a full angle of ,60u.More details about the preparation of the metallic grating layer and its effects

on the far-field interference pattern, the experimental settings, including the

determination of the in-plane wavevector in momentum space, and the expected

LP diffraction patterns and intensities of the zero-state and p-state are given in

the Methods section and Supplementary Information.

Full Methods and any associated references are available in the online version ofthe paper at www.nature.com/nature.

Received 23 May; accepted 20 September 2007.

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3. Yang, C. N. Concept of off-diagonal long-range order and the quantum phases ofliquid He and of superconductors. Rev. Mod. Phys. 34, 694–704 (1962).

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6. Leggett, A. J. Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-matter Systems Chs 1–4 (Oxford Univ. Press, Oxford, 2006).

7. Andrews, M. R. et al. Observation of interference between two Bose condensates.Science 275, 637–641 (1997).

8. Hagley, E. W. et al. Measurement of the coherence of a Bose-Einstein condensate.Phys. Rev. Lett. 83, 3112–3115 (1999).

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11. Naraschewski, M. & Glauber, R. J. Spatial coherence and density correlations oftrapped Bose gases. Phys. Rev. A. 59, 4595–4607 (1999).

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13. Hansen, J. B. & Lindelof, P. E. Static and dynamic interactions between Josephsonjunctions. Rev. Mod. Phys. 56, 431–459 (1984).

14. Hadley, P. M., Beasley, R. & Wiesenfeld, K. Phase locking of Josephson-junctionseries arrays. Phys. Rev. B 38, 8712–8719 (1988).

15. Cataliotti, F. S. et al. Josephson junction arrays with Bose-Einstein condensates.Science 293, 843–846 (2001).

16. Anderson, B. P. & Kasevich, M. A. Macroscopic quantum interference fromatomic tunnel arrays. Science 282, 1686–1689 (1998).

17. Orzel, C. et al. Squeezed states in a Bose-Einstein condensate. Science 291,2386–2389 (2001).

18. Greiner, M. et al. Quantum phase transition from a superfluid to a Mott insulatorin a gas of ultracold atoms. Nature 415, 39–44 (2002).

19. Bulaevskii, L. N., Kuzii, V. V. & Sobyanin, A. A. Superconducting system with weakcoupling to current in ground state. JETP Lett. 25, 290–294 (1977).

20. Backhaus, S. et al. Discovery of a metastable pi-state in a superfluid 3He weak link.Nature 392, 687–690 (1998).

21. Smerzi, A. et al. Quantum coherent atomic tunneling between two trapped Bose-Einstein condensates. Phys. Rev. Lett. 79, 4950–4953 (1997).

22. Weisbuch, C. et al. Observation of the coupled exciton-photon mode splitting in asemiconductor quantum microcavity. Phys. Rev. Lett. 69, 3314–3317 (1992).

23. Houdre, R. et al. Measurement of cavity-polariton dispersion curve from angle-resolved photoluminescence experiments. Phys. Rev. Lett. 73, 2043–2046(1994).

24. Deng, H. et al. Polariton lasing versus photon lasing in a semiconductormicrocavity. Proc. Natl Acad. Sci. USA 100, 15318–15323 (2003).

25. Deng, H. et al. Quantum degenerate exciton-polaritons in thermal equilibrium.Phys. Rev. Lett. 97, 146402 (2006).

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27. Yeh, P. Optical Waves in Layered Media Ch. 5 (John Wiley, Hoboken, 2005).

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Supplementary Information is linked to the online version of the paper atwww.nature.com/nature.

Acknowledgements This work was supported by the JST/SORST programme andby Special Coordination Funds for Promoting Science and Technology in Japan. Thehigh-quality microcavity cavity sample is courtesy of G. S. Solomon, R. Hey, K. Ploogand A. Forchel. We thank T. Maruyama for support and S. Sasaki for the devicefabrication. C.W.L. thanks Y. R. Shen for comments and discussions.

Author Information Reprints and permissions information is available atwww.nature.com/reprints. Correspondence and requests for materials should beaddressed to C.W.L. (cwlai@stanford.edu) or Y.Y. (yyamamoto@stanford.edu).

a

–1.5 –1.0 –0.5 0 0.5 1.0 1.5

103

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100

10–1

20 mW25 mW30 mW40 mW

100 mW

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nsity

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itrar

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its)

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102

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itrar

y un

its)

k||/G0

–1.5 –1.0 –0.5 0 0.5 1.0 1.5k||/G0

30 mW35 mW40 mW60 mW

100 mW200 mW

Figure 4 | Evolution of the ‘zero-state’ and ‘p-state’. a, Profiles of the p-state as a function of in-plane momentum taken at a cross-section in thedispersion curve (exemplified by the white dottedlines in the inset) for an exciton–polaritoncondensate array for various pumping rates. Theinset shows the energy versus in-plane momentumspectra (axes and colour scale same as Fig. 3c, butonly the top 2 meV is shown in the inset) forP 5 100 mW, where relaxation from the p-state tozero-state is visible. The blue dashed line displaysthe uniform collection efficiency of the opticalsystem up to | k | | | 5 3G0/2. b, Profiles of the zero-state. The intensity of the zero-state surpasses thatof the p-state at around P 5 45 mW.

LETTERS NATURE | Vol 450 | 22 November 2007

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METHODSExciton–polariton array. The thin metallic film deposited on top of the micro-

cavity modifies the boundary condition of the top DBR structure. For a bare

microcavity structure, the resonant photon mode exhibits an anti-node at the

semiconductor/free-space interface. In the presence of a metallic layer, the anti-

node shifts towards the centre of the cavity, owing to suppression of the elec-

tromagnetic field inside the metallic layer. As a result, the energy of the cavity

photon resonance increases (blue shift). For a 20-nm-thick Au/3-nm-thick Ti

layer on top of the microcavity structure, the cavity resonance energy increases

by ,400meV according to the calculation by the transfer matrix method27. Theassociated LP energy shift is of the order of ,200meV near-resonance EC < EX,

approximately half of the cavity resonance shift. This is consistent with the

experimentally measured energy shift under a uniform metallic layer of similar

thickness.

The relative transmittance through the top DBR with and without the metallic

layer is ,40–50%, so the optical pumping and LP emission through the periodic

metallic strips are spatially modulated. The initial spatial modulation of LP

generation by the metallic strips is mostly washed out by internal diffusion of

polaritons through the cooling process. However, the detected LP emission is

subject to intensity modulation by the periodic metallic strips. This effect must

be separately considered for understanding the far-field interference pattern

from the spontaneously formed ‘zero-state’ and ‘p-state’ under the built-in

periodic potential.

Experimental techniques. The exciton–polaritons are excited by a p-polarized

Ti:sapphire laser of 2.5 ps duration (0.5 meV spectral width) at a 76 MHz repe-

tition rate. The pulses are focused to an elliptical spot (,60 mm 3 30mm or

,100mm 3 50 mm) onto the sample at an oblique incidence angle of ,60u(kjj< 7 3 104 cm21 in air). The elliptical pumping spot covers approximately20–30 periodic elements.

LPs are trapped in the cavity mostly until they relax to states of lower energy

and in-plane momentum. The emitted luminescence is collected by a micro-

photoluminescence set-up composed of a 503 objective lens (effective focal

length, fobj 5 4 mm and a numerical aperture, NA 5 0.55, corresponding to a

full collection angle of ,67u in air) and a pair of relay lenses of focal length

fL 5 10 cm (L1, which is removable, and L2) as shown in Fig. 2. The collection

efficiency is approximately uniform up to a full angle Dh < 60u (Dkjj< 2p/l).

The imaging and spectroscopy of LPs in the near-field (coordinate space)

and far-field (momentum space) are performed by positioning a charged-

coupled-device (CCD) camera or an imaging spectrometer in a plane

conjugate to the real-image or Fourier (back focal) plane of the objective. The

optical axis of the collection optical system is perpendicular to the surface of

the sample. For imaging and spectroscopy in coordinate space (near-field),

the repositionable lens L1 is removed. The Fourier (back focal) plane of the

sample located in the object plane, with coordinates X and Y, is thus imaged

onto to a CCD camera directly or through an imaging spectrometer. This

arrangement allows determination of the LP distribution in momentum space(kX, kY) (far-field imaging) as well as the LP dispersion relation ELP(kjj).

Coordinate space is correlated to momentum space by a Fourier transform:

X , Yð Þ~ u=M , v=Mð Þ< kX , kYð Þ~ k|u=fobj, k|v=fobj

� �, where M is the mag-

nification (M 5 fL/fobj < 25), (u, v) are coordinates of the near-field or far-field

images, and kjj~kX ,Y ~k sin h~ 2p=lð Þ sin h in air. Near-field and far-field

images are measured with a CCD with a pixel size of ,8 3 8 mm2, giving a

diffraction-limited spatial resolution of ,2mm and a pixel-size-limited angular

resolution of dh 5 0.11u. The energy versus in-plane momentum relation of LPs

and LP arrays are determined by spectroscopy in momentum space (angle-

resolved spectroscopy). The spectroscopic system is a 75-cm spectrometer

equipped with an 1,800 g mm21 grating and a liquid-nitrogen-cooled CCD with

a pixel size of 26 3 26 mm2. The system has an angular resolution of dh < 0.4u and

a spectral resolution of ,20 meV, limited by the pixel size.

LP emission intensity from the array. The probability amplitude of LPs in the

presence of a periodic potential is represented by the Bloch wave function yn,kjj,

where kjj is the in-plane lattice momentum, and n the index of the band. The

expected leakage LP emission intensity is proportional to the overlapping

probability density r~ wkjjjyn,kjj

D E������

2

, where wkjj~ exp i kjj:r

� �is the plane wave

corresponding to the free-moving polaritons (that is, photons leaking out from

the cavity) with in-plane momentum kjj. The calculated relative r of each energy

band is represented by the size of the open circles (log scale) in Fig. 3b.

doi:10.1038/nature06334

Nature ©2007 Publishing Group