Abstract
Einstein’s 1925 paper predicted the occurrence of Bose–Einstein condensation (BEC) in an ideal gas of noninteracting bosonic particles^{1}. However, particle–particle interaction and peculiar excitation spectra are keys for understanding BEC and superfluidity physics. A quantum fieldtheoretical formulation for a weakly interacting Bose condensed system was developed by Bogoliubov in 1947, which predicted the phononlike excitation spectrum^{2} in the lowmomentum regime. The experimental verification of the Bogoliubov theory on the quantitative level was carried out for atomic BEC^{3} using the twophoton Bragg scattering technique^{4}. Excitonpolaritons in a semiconductor microcavity, which are elementary excitations created by strong coupling between quantumwell excitons and microcavity photons, were proposed as a new BEC candidate in solidstate systems^{5}. Recent experiments with excitonpolaritons have demonstrated several interesting signatures from the viewpoint of polariton condensation, such as quantum degeneracy at nonequilibrium conditions^{6,7,8}, the polaritonbunching effect at the condensation threshold^{9}, long spatial coherence^{10,11,12} and quantum degeneracy at equilibrium conditions^{13}. The particle–particle interaction and the Bogoliubov excitation spectrum are at the heart of BEC and superfluidity physics, but have only been studied theoretically for excitonpolaritons^{14,15}. In this letter, we report the first observation of interaction effects on the excitonpolariton condensate and the excitation spectra, which are in quantitative agreement with the Bogoliubov theory.
Similar content being viewed by others
Main
In a semiconductor microcavity with single or multiple quantum wells (QWs), eigenstates are altered to the new normal modes, called excitonpolaritons, when the cavity photon/QWexciton coupling rate exceeds the decay rates of the photon and exciton. The excitonpolariton is a promising solidstate system for studying the dynamical condensation phenomena in solids^{5,16}. Because its effective mass is eight orders of magnitude smaller than that of a hydrogen atom and four orders of magnitude smaller than exciton mass, the critical temperature of the polariton Bose–Einstein condensation (BEC) transition is expected to be up to room temperature. The leakage photons carry identical energy and inplane momentum to the internal polaritons, thus it is possible to directly measure the energy–momentum dispersion relation and population distribution of the polaritons by an angleresolved spectroscopy technique. This important information is available for liquid^{4}He systems only through ‘quantum evaporation’^{17} and for gaseous atoms only through ‘Feshbach resonance controlled free expansion’.
The excitonpolariton trap used in our experiment is shown in Fig. 1a. Three stacks of four GaAs QWs are embedded at the central three antinode positions of an AlAs/AlGaAs distributed Bragg reflector planar microcavity. The normalmode splitting is 2g_{0}∼15 meV and the cavity photon lifetime is 2 ps. This leads to a k=0 lowerpolariton (LP) lifetime of ∼4 ps at zero detuning, Δ≡E_{C}−E_{X}=0, where k is the inplane wavenumber and E_{C} and E_{X} are the cavity and QW exciton energies at k=0. The trap potential of ∼200 μeV is provided by a hole surrounded by a thin metal (Ti/Au) film^{18}. The cavity resonant field normally has an antinode at the AlGaAs–air interface. However, the antinode position is shifted inside the AlGaAs layer with the metal film as shown in Fig. 1a, which results in a blueshift of the cavity resonance and the LP energy. In our experiment, the LPs were confined in circular holes of varying diameters from 5 μm to 100 μm. In a trap with 5–10 μm diameter, a single fundamental transverse mode dominates the condensation dynamics over other higherorder modes owing to the relatively weak confining potential.
The total number n of LPs injected into a trap by the pump pulse and the number n_{0} of the LP centred at k=0 and within , corresponding approximately to the trapped ground state, are plotted in Fig. 1b as a function of normalized pump rate P/P_{th}. The fractional ratio n_{0}/n increases nonlinearly at the condensation threshold and reaches a maximum of ∼0.6 at P/P_{th}∼6.
The top panels in Fig. 1c show the nearfield emission patterns from a trap with 8 μm diameter at pump rates below, just above and well above the condensation threshold. The measured standard deviations for the polariton position Δx and the wavenumber Δk are plotted as functions of P/P_{th} in Fig. 1c. Sudden decreases in Δx and Δk were clearly observed at . Just above threshold, the measured uncertainty product ΔxΔk is ∼0.98, which may be compared to the Heisenberg limit (ΔxΔk∼0.5) for a minimumuncertainty wavepacket. The monotonic increase in Δx and ΔxΔk at higher pump rates stems from the repulsive interaction among LPs in a condensate and is well reproduced by theoretical analysis using the Gross–Pitaevskii (GP) equation as shown in Fig. 1c (Supplementary Information, S1 for detailed theoretical analysis and Supplementary Information, S2 for complete experimental data.)
The k=0 LP energy is blueshifted with the number of polaritons as shown in Fig. 1d. This is a direct manifestation of the aforementioned repulsive interaction among LPs in a condensate. The k=0 LP energy shift U(n) is calculated by the relation
where δ E_{X}=E_{B}(n/n_{s}) and g(n)=g_{0}(1−(n/n_{s}′)) represent the blueshift of the QW exciton energy due to fermionic exchange interaction^{19,20} and the reduced normalmode splitting due to phasespace filling and fermionic exchange interaction^{21}, respectively. n_{s}=(N_{QW}S/2.2πa_{B}^{*}^{2}X^{2}) and n_{s}′=(N_{QW}S/4πa_{B}^{*}^{2}X^{2}) are the saturation polariton numbers for the above two nonlinear processes, respectively. a_{B}^{*} is the QW exciton Bohr radius, N_{QW}=12 is the number of QWs, g_{0}≈7.5 meV is the photon–exciton coupling strength, S is the crosssectional area of the condensate and is the exciton fraction of the k=0 LP.
If we ignore the pumprate dependency of the condensate crosssectional area and assume a constant crosssectional area determined by the trap area S=π(4×10^{−4} cm)^{2}, the theoretical energy shift is shown by the lightblue line in Fig. 1d. We can numerically solve the GP equation to incorporate the pumpratedependent condensate size (see Supplementary Information, S1). The results are shown by red circles in Fig. 1d. These two theoretical predictions are compared with the experimental results (blue diamonds). We note that the abovementioned nonlinear model based on weakly interacting bosons^{19,20,21} can reproduce the experimental data only in lowpolaritondensity regimes. Therefore, we use the experimental values (not theoretical values) for U(n) as the interaction energy in subsequent discussions for the universal feature of the Bogoliubov excitations.
The dispersion relations between the LP energy E and inplane wavenumber k obtained by the angleresolved spectroscopy are shown for the pump rate above threshold P/P_{th}=3 in Fig. 2. Figure 2a represents a linear plot of the intensity, whereas Fig. 2b–d use logarithmic plots of the intensity to magnify the excitation spectra. Here we show an ‘untrapped’ case, where the pump spot size (diameter ∼30 μm) is considerably smaller than the trap size (diameter∼90 μm). In such a case, the LP condensate is formed in an area determined by the pump spot size and the pump rate rather than the trap size owing to the limited lateral diffusion and varying spatial density of the LPs^{12,18}. Above threshold, two drastic changes are noticed compared with the standard quadratic dispersion observed far below threshold. One is the blueshift of the k=0 LP energy and the other is the phononlike linear dispersion relation in the lowmomentum regime k ξ<1, where is the healing length. White and black lines in Fig. 2b represent the two quadratic dispersion relations, E_{LP}=−U(n)+((ℏk)^{2}/2m) and E_{LP}′=((ℏk)^{2}/2m), where m is the effective mass of the k=0 LP. Here we choose the zero energy as the condensate energy for convenience. Neither of the two theoretical curves can explain the measurement result. A solid pink line in Fig. 2b is the Bogoliubov excitation energy, given by^{22,23}
The measured dispersion relation for the excitation energy versus inplane wavenumber is in good agreement with the Bogoliubov excitation spectrum without any fitting parameter.
A circularly polarized pump laser beam was used to inject spinpolarized LPs in this experiment. In such a case, the LP condensate preserves the original spin polarization of optically injected polaritons^{24}. The result shown in Fig. 2b was taken for detecting the leakage photons with the same circular polarization as that of the pump (cocircular detection). If leakage photons with crosscircular polarization were detected, the standard quadratic dispersion was obtained but with slightly blueshifted energy, as shown in Fig. 2c. Even though the circularly polarized pump beam is injected, a small number of crosscircularly polarized photons was detected because of the spinflip relaxation during a cooling process^{24}. On increasing the pump rate, the number of LPs with opposite spin is also increased, so their energy is blueshifted compared with the singlepolariton energy E_{LP} (white line) in Fig. 2c. In Fig. 2d we intentionally mixed a small number of the cocircular polarized photons to be detected with the crosscircular polarized photons. The difference between the Bogoliubov excitations with cocircular polarization and the standard quadratic excitation with crosscircular polarization is clearly seen.
The dispersion curves for the LP condensate in a trap with 8 μm diameter are shown in Fig. 3 for below and above threshold. The increasing blueshift of the condensate energy with the pump rate is seen from Fig. 3b to d. The Bogoliubov excitation energy is modified if the density of a condensate is not homogeneous in space owing to trapping in a finite size^{25}. The lightblue dotted line in Fig. 3b is the modified Bogoliubov excitation energy based on the local densitydependent excitation energy and the spatial average of E_{B}(r), where U(n) varies with position (Supplementary Information, S3). In the phononlike regime (k ξ<1), the Bogoliubov dispersion is still linear, E≈C′k. However, the speed of sound is not equal to as in the homogeneous case, but is slightly reduced. In the freeparticle regime (k ξ>1), the dispersion is still quadratic, E_{B}(k ξ>1)≈U′(n)+((ℏk)^{2}/2m). Here, the offset energy U′(n) is also given by the spatial average of U(n). Just above threshold, the experimental result agrees with such an inhomogeneous model (lightblue dotted line) rather than a homogeneous model (pink solid line). However, as shown in Fig. 3c,d, the excitation spectra of the trapped condensates well above threshold are well reproduced by the homogeneous model. According to the local density approximation, at relatively high pump rates, the trapped condensate spreads over the entire trap owing to the repulsive interaction among condensate polaritons and thus interacts with the excitations uniformly. Such a system can be well described by the homogeneous model (Supplementary Information, S3).
As indicated by equation (2), the Bogoliubov excitation energy normalized by the interaction energy E_{B}/U(n) is a universal function of the wavenumber normalized by the healing length k ξ. In Fig. 4a, this universal relation (green solid line) is compared with the experimental results for four different untrapped condensate systems. The experimental data shown in Fig. 4a are taken by numerical search for the intensity maximum wavenumber for varying E values. In both the phononlike regime at k ξ<1 and the freeparticle regime at k ξ>1, the experimental results agree well with the universal curve. On the other hand, at a pump rate far below threshold, the measured dispersion relation is completely described by the singlepolariton energy E_{LP} (red solid line).
The sound velocity deduced from the phononlike linear dispersion spectrum is of the order of ∼10^{8} cm s^{−1}. This value is eight orders of magnitude larger than that of atomic BEC. This enormous difference comes from the fact that the polariton mass is eight orders of magnitude smaller than the atomic mass and the polariton interaction energy is seven orders of magnitude larger than the atomic interaction energy. According to the Landau criterion^{26}, the observation of this linear dispersion in the lowmomentum regime is an indication of superfluidity in the excitonpolariton system. However, we note that a polariton system is a dynamical system with a finite lifetime, so the Landau criterion might be modified on a quantitative level.
In the freeparticle regime (k ξ>1), the excitation energy associated with the condensate is larger by 2U(n) than that of a single LP for the same wavenumber. In Fig. 4b,c, this important prediction of the Bogoliubov theory is compared with the experimental results for four different untrapped and trapped condensate systems, respectively. The experimental data were determined as the difference between the measured excitation energy with the presence of the condensate and the standard quadratic dispersion for a single LP state, which is determined by the experimental data obtained for a pump rate far below threshold (P/P_{th}≪1). The experimental data were taken for varying pump rates in the range of P/P_{th}≫1 so that the homogeneous model can be applied to both untrapped and trapped cases. The experimental data are in good agreement with the theoretical curve (grey dashed line) for both untrapped and trapped cases.
Figure 4d shows the normalized LP number n_{k}/n_{k}^{0} versus the normalized wavenumber k ξ for a trapped condensate, where n_{k}^{0} is evaluated at k ξ=0.5 for convenience. The LP occupation number n_{k} in the excitation spectrum can be calculated by applying the Bose–Einstein distribution for the Bogoliubov quasiparticles and subsequently taking the inverse Bogoliubov transformation^{22},
The first and second terms of the righthand side of equation (3) represent the real particles (excitonpolaritons) created by the quantum depletion and the thermal depletion, respectively. In the present polariton condensate system, the thermal depletion is much stronger than the quantum depletion, so the second term of the righthand side of equation (3) dominates over the first term (Supplementary Information, S4). In such a case, the LP population is approximated by n_{k}≈(m k_{B}T/(ℏk)^{2}) in the smallk ξ regime, whereas the LP population is given by if the quantum depletion is dominant^{22}. This theoretical prediction of 1/k^{2} dependency of n_{k} for thermal depletion is compared with the experimental data in Fig. 4d and reasonable agreement was obtained.
References
Einstein, A. Quantentheorie des einatomigen idealen Gases: Zweite Abhandlung. Sitzungber. Preuss. Akad. Wiss. 1, 3–14 (1925).
Bogoliubov, N. N. On the theory of superfluidity. J. Phys. USSR 11, 23–32 (1947).
Anderson, M. H. et al. Observation of Bose–Einstein condensation in a dilute atomic vapor. Science 269, 198–201 (1995).
StamperKurn, D. M. et al. Excitation of phonons in a Bose–Einstein condensate by light scattering. Phys. Rev. Lett. 83, 2876–2879 (1999).
Imamoglu, A., Ram, R. J., Pau, S. & Yamamoto, Y. Nonequilibrium condensates and lasers without inversion: Excitonpolariton lasers. Phys. Rev. A 53, 4250–4253 (1996).
Dang, L. S. et al. Stimulation of polariton photoluminescence in semiconductor microcavity. Phys. Rev. Lett. 81, 3920–3923 (1998).
Senellart, P. & Bloch, J. Nonlinear emission of microcavity polaritons in the low density regime. Phys. Rev. Lett. 82, 1233–1236 (1999).
Savvidis, P. G. et al. Angleresonant stimulated polariton amplifier. Phys. Rev. Lett. 84, 1547–1550 (2000).
Deng, H. et al. Condensation of semiconductor microcavity exciton polaritons. Science 298, 199–202 (2002).
Kasprzak, J. et al. Bose–Einstein condensation of exciton polaritons. Nature 443, 409–414 (2006).
Balili, R. et al. Bose–Einstein condensation of microcavity polaritons in a trap. Science 316, 1007–1010 (2007).
Deng, H. et al. Spatial coherence of a polariton condensate. Phys. Rev. Lett. 99, 126403 (2007).
Deng, H. et al. Quantum degenerate excitonpolaritons in thermal equilibrium. Phys. Rev. Lett. 97, 146402 (2006).
Sarchi, D. & Savona, V. Spectrum and thermal fluctuations of a microcavity polariton Bose–Einstein condensate. Phys. Rev. B 77, 045304 (2008).
Shelykh, I. A., Malpuech, G. & Kavokin, A. V. Bogoliubov theory of Bosecondensates of spinor excitonpolaritons. Phys. Status Solidi A 202, 2614–2620 (2005).
Keeling, J., Marchetti, F. M., Szymanska, M. H. & Littlewood, P. B. Collective coherence in planar semiconductor microcavities. Semicond. Sci. Technol. 22, R1–R26 (2007).
Wyatt, A. F. G. Evidence for a Bose–Einstein condensate in liquid ^{4}He from quantum evaporation. Nature 391, 56–59 (1997).
Lai, C.W. et al. Coherent zerostate and πstate in an excitonpolariton condensate array. Nature 450, 529–532 (2007).
Ciuti, C. et al. Role of the exchange of carriers in elastic exciton–exciton scattering in quantum wells. Phys. Rev. B 58, 7926–7933 (1998).
Rochat, G. et al. Excitonic Bloch equations for a twodimensional system of interacting excitons. Phys. Rev. B 61, 13856–13862 (2000).
SchmittRink, S., Chemla, D. S. & Miller, D. A. B. Theory of transient excitonic optical nonlinearities in semiconductor quantumwell structures. Phys. Rev. B 32, 6601–6609 (1985).
Pitaevskii, L. P. & Stringari, S. Bose–Einstein Condensation (Clarendon, Oxford, 2003).
Ozeri, R., Katz, N., Steinhauer, J. & Davidson, N. Colloquium: Bulk Bogoliubov excitations in a Bose–Einstein condensate. Rev. Mod. Phys. 77, 187–205 (2005).
Deng, H. et al. Polariton lasing versus photon lasing in a semiconductor microcavity. Proc. Natl Acad. Sci. USA 100, 15318–15323 (2003).
Stenger, J. et al. Bragg spectroscopy of a Bose–Einstein condensate. Phys. Rev. Lett. 82, 4569–4573 (1999).
Landau, L. D. & Lifshiëtís, E. M. Fluid Mechanics 2nd edn (Pergamon, Oxford, 1987).
Acknowledgements
This work was supported by the JST/SORST programme and Special Coordination Funds for Promoting Science and Technology in Japan. We thank T. Maruyama for support and S. Sasaki for device fabrication.
Author information
Authors and Affiliations
Contributions
S.U. carried out the experiments, analysed the data and wrote the paper, L.T. theoretically studied the data, G.R. carried out the experiments and analysed the data, C.W.L. conceived, designed and carried out the experiments, N.K., T.F., M.G., A.L., S.H. and A.F. prepared materials and experimental tools and Y.Y. conceived the project in this paper.
Corresponding authors
Supplementary information
Supplementary Information
Supplementary Information and Supplementary Figures 1—9 (PDF 432 kb)
Rights and permissions
About this article
Cite this article
Utsunomiya, S., Tian, L., Roumpos, G. et al. Observation of Bogoliubov excitations in excitonpolariton condensates. Nature Phys 4, 700–705 (2008). https://doi.org/10.1038/nphys1034
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nphys1034
This article is cited by

Topological unwinding in an excitonpolariton condensate array
Communications Physics (2024)

Ultrafast imaging of polariton propagation and interactions
Nature Communications (2023)

Collective excitations of a boundinthecontinuum condensate
Nature Communications (2023)

Nonequilibrium Bose–Einstein condensation in photonic systems
Nature Reviews Physics (2022)

Directional Goldstone waves in polariton condensates close to equilibrium
Nature Communications (2020)