LETTERS
PUBLISHED ONLINE: 28 NOVEMBER 2010 | DOI: 10.1038/NPHYS1841

Single vortex–antivortex pair in an
exciton-polariton condensate
Georgios Roumpos1 *, Michael D. Fraser2,3 , Andreas Löffler4 , Sven Höfling4 , Alfred Forchel4
and Yoshihisa Yamamoto1,2 *In a homogeneous two-dimensional system at non-zerotemperature there can be no ordering of infinite range1,2 .
However, for a Bose liquid under such conditions, a superfluid
phase is predicted3–5 . Bound vortex–antivortex pairs dominate
the thermodynamics and phase coherence properties in thissuperfluid regime. It is believed that several systems share
this behaviour when the parameter describing their orderedstate has two degrees of freedom6 . This theory has beentested for some of them7–12 , but there has been no direct
experimental observation of a quasi-condensate that includes abound vortex–antivortex pair. Here we present an experimental
technique that can identify a single vortex–antivortex pair in a
two-dimensional exciton-polariton condensate. The pair is generated through the inhomogeneous spot profile of the pumping laser, and is revealed in the time-integrated phase maps
acquired using Michelson interferometry. Numerical modellingbased on the open-dissipative Gross–Pitaevskii equation suggests that the pair evolution is distinctly different in thisnon-equilibrium system compared with atomic condensates13 .
Microcavity exciton polaritons14 behave as a system of strictly
two-dimensional bosons when their density is below the exciton
saturation density. As a result of their half-light half-matter
nature, their effective mass is extremely small, so that quantum
many-body effects are important at relatively high temperatures,
even up to room temperature15 . In particular, dynamic polariton
condensation is observed16–18 , and its signatures are similar to
Bose–Einstein condensation, namely massive occupation of theground state and phase coherence up to large distances. However,the short lifetime allows only the formation of a quasi-equilibrium
steady state, in which polaritons escaping from the cavity arecontinuously replenished by the external pump through anintermediate reservoir state.
Here, we show that single vortex–antivortex pairs can beobserved in an exciton-polariton condensate. We have createda pumping spot that generates a minimum of the condensate
density at the centre. A zero in density can be thought of as asuperposition of a vortex and antivortex that can be separated
by an external perturbation19 . Thus, the centre of the condensate
acts as a source of vortex–antivortex pairs. We have found thatfor a particular condensate size, there is on average one pair at
any time. A Michelson interferometer is used to reconstruct the
time-integrated phase map of the system. When the sample disorder
potential is stronger than the blueshift induced by polariton–polariton interactions, pinned pairs appear at certain locations.
When the sample disorder potential is weak, on the other hand,

they are mobile. They appear along a fixed axis, because of asmall asymmetry in the pumping spot, and are created with arandom polarization, namely the vortex can appear on the rightside of the spot and the antivortex on the left, or vice versa. In
the time-integrated measurement, two distinct characteristic phase
defects appear in both cases.
We have applied an open-dissipative Gross–Pitaevskii equation
model similar to that in ref. 20, which reproduces the observed
density minimum at the condensate centre. We also found thatvortex–antivortex pair motion is significantly modified because of
the dissipative nature of the polariton condensate and the repulsive
interactions between condensate and reservoir particles. The vortex
pair is found to migrate perpendicular to its dipole moment and,in our experimental parameter space, recombines before reaching
the condensate edge (see Supplementary Information). Despite theshort polariton lifetime, a vortex pair survives for a long enoughtime to be observed.
Our experiment is carried out in a GaAs-based microcavity
sample21,22 . The pump laser at above bandgap energy creates
free electon–hole pairs, which form excitons and finally relax
towards the lower polariton branch, as shown in the schematicof Fig. 1a. When a lower polariton decays, it emits one photon,
which we observe. We employ a commercial beam shaper tocreate a flat-top pumping spot (see Supplementary Fig. S5).
Figure 1b shows measured luminescence below threshold, whichfeatures Airy-function-like patterns because of diffraction effects.
We assume that the shape is the same as the laser pumping
spot, and use it as an input to our numerical calculation. Abovethe condensation threshold, a population dip develops at thecondensate centre, as shown in the measured luminescence imageof Fig. 1d. Our numerical simulation reproduces the observed
population dip (Fig. 1d), and suggests that the inhomogeneity of thepump spot profile, notably the ring near the centre, generates thiscondensate shape. The condensate density minimum is stabilized
by the repulsive reservoir–condensate interaction, because thereservoir shows a corresponding density maximum at the same
point (Fig. 1c). This density minimum can be considered as anoverlapping vortex and antivortex.
Phase fluctuations23 of the condensate are responsible for the
spontaneous vortex pair generation19 , and are locally maximized at
the condensate centre, where the condensate density is minimum
and the non-condensate density (reservoir polaritons) is maximum(see Supplementary Section S2.4). The mean-field interactionenergy due to the reservoir is maximum at the centre, where
the reservoir density is maximum. As power fluctuations of our

1 E. L. Ginzton Laboratory, Stanford University, Stanford, California, 94305, USA, 2 National Institute of Informatics, Tokyo, 101-8430, Japan, 3 Institute for

Nano Quantum Information Electronics, University of Tokyo, Tokyo, 153-8505, Japan, 4 Technische Physik, University of Würzburg, Physikalisches Institut
and Wilhelm-Conrad-Röntgen-Research Center for Complex Material Systems, Würzburg, 97074, Germany. *e-mail: roumpos@stanford.edu;
yyamamoto@stanford.edu.
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200k||Figure 1 | Pump and condensate profile. a, Schematic of the pumping mechanism (see text). b, Lower-polariton luminescence image at half the threshold
power. The profile along x = 0 is shown by the yellow line. We assume that it has the shape of the laser pumping spot, and use it as an input to ournumerical simulation. c, Numerically calculated reservoir density above threshold. d, Calculated and measured condensate density images for pumping
power three times above threshold.

multimode pump laser modulate the reservoir population strongly,
this results in modulation of the local interaction energy and thus of
the local phase of the condensate, which induces a separation of thevortex and antivortex. We confirmed that a vortex–antivortex pairis not regularly generated when a quiet single-mode pump laser isused22 . We also note that thermal effects are not strong enough tospontaneously create a vortex–antivortex pair in our system.
We use a Michelson interferometer to reconstruct the phase mapof the condensate. A schematic of the set-up is shown in Fig. 2a.
We can overlap the real-space image with its reflected version
on the camera, and because the two phase fronts are tilted withrespect to each other, an interference pattern is observed (Fig. 2b).
Changing the difference in the interferometer arm lengths shifts
the fringes on the camera. Therefore, the intensity measured at
one particular pixel shows a sinusoidal modulation (Fig. 2c). Thisreveals two pieces of information: the relative phase of the twooverlapping images at this pixel point as well as the fringe visibility,
which corresponds to the first-order spatial correlation function.
Repeating this procedure for every pixel point, we can constructboth phase and fringe visibility maps.
The expected phase map for a pinned vortex–antivortex pair isshown in Fig. 3a. If we carry out an interferometry measurement,in the way described above, with this phase map as an input, theexpected result is Fig. 3b. The two double-dislocation patterns, one
at the upper part, and the other at the lower part of the figure, mark130

the position of the pair in the original and in the reflected images.
Figure 3c shows a phase map measured experimentally. It features
the double-dislocation pattern characteristic of a pinned vortex–
antivortex pair. In Fig. 3d, we have subtracted the global phase slope
to reveal the actual phase map of a single vortex–antivortex pair,corresponding to the expected one (Fig. 3a).
In this case (Fig. 3c,d), owing to the sample disorder potential,
the pair has a pinned position and polarization, defined as the vector
pointing from the antivortex to the vortex. This is similar to theexperiment in ref. 20, where single vortices pinned by the disorder
potential were observed. However, the disorder potential in our
sample is very weak generally, much smaller than the blueshift
due to polariton–polariton interactions (see Supplementary Figs
S10 and S6), so the disorder is screened when the condensate isformed24 . Therefore, there are only a few locations where pinned
vortex–antivortex pairs can be observed. If we mount the sample
carefully, to minimize strain (see the Methods section), we can nolonger find any pinned vortex pair.
We next numerically study the evolution of a mobile vortex
pair imprinted along the horizontal axis, using a time-dependentopen-dissipative Gross–Pitaevskii equation25 . In a harmonicallytrapped conservative condensate, a single vortex pair will undergo
linear motion with velocity vv−av = h̄/mdv , which is inversely
proportional to the vortex–antivortex separation dv , and on
interaction with the boundary will wrap back on itself with a

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NATURE PHYSICS DOI: 10.1038/NPHYS1841a

LETTERS

M2

Camera

M1

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PBS

Sample

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1Figure 2 | Michelson interferometer. a, Schematic of the set-up formeasurement of phase and fringe visibility maps. It employs a mirror (M1)and a right-angle prism (M2), which creates the reflection of the original
image along one axis, depending on the prism orientation. A two-lensmicroscope set-up overlaps the two real-space images of the polariton
condensate on the camera. PBS: polarizing cube beam splitter; NPBS:
non-polarizing cube beam splitter. b, Typical interference pattern observed
above the polariton condensation threshold (at 60 mW) along with aschematic showing the orientation of the two overlapping images. c, Bluecircles: intensity on one pixel of the camera as a function of the prism (M2)
position in normalized units. Red line: fit to a sine function.

cyclical motion. However, a polariton condensate with considerable
repulsive condensate–reservoir interaction (due to the large
reservoir population present in the experiment) experiences dragforces strongly perturbing the vortex pair motion26 , causing it torecombine after only a short travel distance. For these types of
micro-motion, despite the vortex pair motion, a signature of thepair in the time-integrated interferogram is preserved. There are
two areas where the phase is shifted by π, one at the top and the otherat the bottom of the figure, and they are surrounded by minima
in the fringe visibility. The measured positions of the phase defects
depend on the time-averaged positions of the vortex and antivortex.
The calculated time-integrated phase and visibility maps for such amoving pair are shown in Fig. 4a and b respectively.
We indeed observed such unique patterns experimentally when
the sample strain was removed, as shown in Fig. 4c,d, which
illustrates typical experimental data. In Fig. 4e, we have notsubtracted the global phase slope, so that the direct experimental
phase map is given. In Fig. 4f, we plot the cross-section of the phasealong two (solid and dashed) lines passing through the two areasof π-phase shift, where this phase shift is shown quantitatively. We
also plot the measured fringe visibility along the same lines, and findthat the visibility minima coincide with the phase jumps. This result
demonstrates that the vortex–antivortex pairs are created with arandom polarization after the removal of strain. Moreover, when
we rotate the prism by 90◦ , so that the reflected image is along thevertical axis, no phase defect is observed (Fig. 4g). The difference
between Fig. 4e and g can be understood as follows: the vortex pairalways remains on the horizontal axis and we fold the reflected
image along the vertical axis. In this case, the phase rotation aroundthe vortex by 2π and that around the antivortex by −2π cancel
out; thus, there is no phase defect in the interference pattern (seeSupplementary Fig. S13). The corresponding visibility map without
any defect is shown in Fig. 4h.
The free pair of Fig. 4 shows a distinct signature compared withthe pinned pair of Fig. 3. Indeed, in the interferogram of Fig. 3dthere are two singularity points, one at the location of the vortex,
and the other at the location of the antivortex, and the measured
phase is continuous everywhere else. The experimental data in
Fig. 4 suggest that such vortex–antivortex pairs are not pinned andtravel a distance comparable to the spot size in a process similar
to the one described in ref. 27. When the pair can form withrandom polarization and also move, the measured time-integrated
interferogram is a superposition of many interferograms of the same
type as Fig. 3d with varying positions for the vortex and antivortex.
This results in a pattern with two π-phase shift areas (Fig. 4c), whichis consistent with our theory (Fig. 4a).
We consistently observe pairs along the same axis, even after werotate the sample by 90◦ . However, when we rotate the pump laser
spot by 90◦ , the polarization axis of the vortex pair is also rotated by90◦ as shown in Supplementary Fig. S9. These results suggest thatthe pair polarization direction is determined by a small asymmetryof the pump laser spot, rather than by the disorder potential
landscape in the sample. Although the pair orientation follows therotation of the laser spot, the random switching of pair polarization
confirms that the pair generation mechanism is non-deterministic.
We found that there is no noticeable dependence of the measured
phase maps on the laser pumping power, up to ∼5 times thethreshold power. Finally, we have created a Gaussian pumping spotand did not observe any phase defects (Supplementary Fig. S7).
This observation supports our argument that the minimum of thecondensate density at the centre of the spot acts as a source ofvortex–antivortex pairs. Given that the condensate size is not muchlarger than the size of the vortex pair, it is not possible for morethan one pair to enter into the system. Indeed, no phase defects areobserved in a smaller condensate (Supplementary Fig. S8).
Our results suggest that vortex–antivortex pairs can be createdin a two-dimensional condensate without rotation or stirring,
although thermal fluctuations are not believed to play any rolein pair generation in our system. Further measurements onlarger condensates containing more vortex–antivortex pairs should
clarify the influence of pair dynamics on phase coherence andpolariton superfluidity20,21,28–31 .

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131

NATURE PHYSICS DOI: 10.1038/NPHYS1841

LETTERSab

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Figure 3 | Phase map of a pinned pair. a, Expected phase map of a condensate including a single vortex–antivortex pair. The arrows show the direction of
the phase increase around the vortex and antivortex. b, Simulation of the experimentally measured phase map when a interferes with its reflection along
the horizontal (dashed) line. A global phase slope along the vertical direction is added. c, Experimentally measured phase map at 55 mW above thecondensation threshold of 20 mW. The blue square marks the position of a double-dislocation pattern. d, Expanded view of the blue square in c, where theglobal slope along the horizontal direction is subtracted. c and d are rotated by 90◦ with respect to all other experimental data.

c
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Figure 4 | Phase and fringe visibility map of a free pair. a, Theoretical time-integrated phase map for a vortex–antivortex pair imprinted onto a condensate
and evolving according to the dissipative Gross–Pitaevskii equation. b, Corresponding fringe visibility map. c,d, Measured phase and fringe visibility maps.
e, The same as in c, but now the global phase slope is not subtracted. f, Blue: phase cross-section along the solid and dashed lines in c. Red: fringe visibility
cross section along the same lines. g, Measured phase map when the prism is rotated by 90◦ , along with a schematic showing the orientation of theinterfering images. h, Corresponding fringe visibility map. Experimental data are taken at 55 mW, above the condensation threshold of 20 mW.

Methods
Michelson interferometer. We employ a commercial refractive beam shaper(Newport) to create a flat-top pumping spot. The laser pump is vertically polarized
and is focused on the sample through a polarizing cube beam splitter (seeSupplementary Information). Luminescence of the orthogonal linear polarization132

is detected and split into two parts by a 50–50 non-polarizing cube beam splitter.
We use a dielectric mirror for the first arm of the interferometer, and an uncoatedglass right-angle prism for the second one. The position of the prism is controlled
by a combination of a translation stage and a piezoelectric actuator. Our imaging
system combines a high-numerical-aperture objective lens (NA = 0.55) with a

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NATURE PHYSICS DOI: 10.1038/NPHYS1841

LETTERS

plano-convex lens and offers a magnification of ×50. The image is focused on ahigh-resolution electron-multiplying CCD (charge-coupled device) camera with apixel size of 8 × 8 µm2 (Andor). In front of the camera, we use a combination of
two interference filters, one long-pass at 750 nm and one band-pass at 770 ± 5 nm,
which block the laser wavelength without distorting the signal spectrum.
Sample. Our microcavity sample features a λ/2 AlAs cavity with three stacks
of 4 GaAs quantum wells at the central three antinodes of the resonant electricfield. It shows a Rabi splitting of 2h̄R = 14 meV. Measurements are carriedout for photon–exciton detuning δ ∼ 0 meV, which gives an effective mass forlower polaritons of m∗LP ∼ 5 × 10−5 me . From the linewidth of the reflection dip
as well as from measurement of time-resolved luminescence, we estimate thelower-polariton lifetime to be τLP ∼ 4 ps. The sample is mounted using silver
paint on a copper sample holder, attached to the cold finger of a helium flow
cryostat. The energy splitting between orthogonally polarized luminescence at 0◦
collection angle (ground-state splitting 1Egr ) can be up to 150 µeV. However,when special care is taken to reduce strain during cooldown, by applying ahomogeneous layer of silver paint between the sample and the sample holder, theground-state splitting can be reduced below 50 µeV. Data in Fig. 3 are taken with
1Egr = 150 µeV, whereas all other data, including Supplementary Information,
are taken with 1Egr ≤ 50 µeV. In the latter case, luminescence above threshold isweakly linearly polarized along the direction of minimum energy, with a degree of
linear polarization ≤20%.
Laser. We pump the sample with a mode-locked Ti:sapphire laser (Coherent)
operated in the continuous-wave mode. To avoid sample heating, we use an opticalchopper to modulate the laser in 0.5 ms pulses at 100 Hz repetition rate. The laser
is incident from the normal direction and its wavelength is at the first reflectivity
minimum of the microcavity above the stop band (723 nm). The incident power
is controlled by a polarization filtering set-up employing a variable retarder, as inref. 22. The threshold power of 20 mW (see Supplementary Information) gives athreshold particle density of ∼102 µm−2 for our 20-µm-diameter pumping spot,
given the 4 ps lower-polariton lifetime, and estimated pumping efficiency of 15%.
Theoretical model. The open-dissipative Gross–Pitaevskii equation20,25 consistsof two coupled equations describing the time evolution of the condensate orderparameter ψ(r,t ) and reservoir polariton density nR (r,t )∂ψ(r,t )  h̄2 ∇ 2 ih̄2= −− [γC − R(nR (r,t ))] + gC |ψ(r,t )| + gR nR (r,t )ψ(r,t )
ih̄
∂t
2mLP
2
∂nR (r,t )= Pl (r,t ) − γR nR (r,t ) − R(nR (r,t ))|ψ(r,t )|2∂tnR (r,t ) is controlled by laser pumping gain Pl (r) and reservoir loss γR . Interaction
between condensate and reservoir is assigned a coupling constant gR = 2gC , where
the condensate coupling constant is gC = 6 × 10−3 meV µm2 . Other parameters are
the condensate polariton loss rate γC = 0.33 ps−1 , and stimulated scattering rateR(nR (r,t )). The strong influence of the reservoir population on the condensate
spatial profile allows us to indirectly probe the reservoir population density anduniquely determine the parameter space. In particular, we obtain best fits toexperimental sequences of real-space data as a function of Pl (see Supplementary
Fig. S14) for pumping threshold Pth = 25 mW with 24% pumping efficiency andγR ≈ 1.5γC and R(nR (r,t ))/nR (r,t ) ≈ γC γR /Pth . To study the vortex pair dynamics,
a vortex and antivortex are imprinted onto a steady-state condensate directly
through a phase factor eiθ along the horizontal axis, and are subsequently left free to
evolve in time. The temporal frames are stored and then cumulatively subjected tothe same time-averaging mechanism as in the experiment.

Received 30 April 2010; accepted 8 October 2010;
published online 28 November 2010; corrected online
2 December 2010

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Acknowledgements
We thank M. Kasevich and A. L. Fetter for stimulating discussions. This work was
supported by Special Coordination Funds for Promoting Science and Technology,
National Science Foundation ECCS-09 25549, Navy/SPAWAR Grant N66001-09-1-2024,
JSPS through its Funding Program for World-Leading Innovative R and D on Scienceand Technology (FIRST Program), and MEXT.

Author contributions
G.R. designed and built the experimental set-up, took and analysed the data and wrote
the paper. M.D.F. carried out the numerical calculations and wrote the paper. A.L., S.H.
and A.F. fabricated the microcavity sample. Y.Y. conceived the experiment. All authors
contributed to data interpretation.

Additional information
The authors declare no competing financial interests. Supplementary informationaccompanies this paper on www.nature.com/naturephysics. Reprints and permissions
information is available online at http://npg.nature.com/reprintsandpermissions.
Correspondence and requests for materials should be addressed to G.R. or Y.Y.

NATURE PHYSICS | VOL 7 | FEBRUARY 2011 | www.nature.com/naturephysics© 2011 Macmillan Publishers Limited. All rights reserved.

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