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Bose-Einstein Condensation of Trapped Polaritons in a Microcavity

Bose-Einstein Condensation of Trapped Polaritons in a Microcavity. Oleg L. Berman 1 , Roman Ya. Kezerashvili 1 , Yurii E. Lozovik 2 , David W. Snoke 3 , R. Balili 3 , B. Nelsen 3 , L. Pfeiffer 4 , and K. West 4

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Bose-Einstein Condensation of Trapped Polaritons in a Microcavity

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  1. Bose-Einstein Condensation of Trapped Polaritons in a Microcavity Oleg L. Berman1, Roman Ya. Kezerashvili1, Yurii E. Lozovik2, David W. Snoke3, R. Balili3, B. Nelsen3, L. Pfeiffer4, and K. West4 1Physics Department, New York City College of Technology of City University of New York (CUNY), Brooklyn NY, USA 2Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow Region, Russia 3Department of Physics and Astronomy, Univeristy of Pittsburgh, Pittsburgh PA, USA 3Bell Labs, Lucent Technologies, Murray Hill NJ, USA

  2. OUTLINE • 2D EXCITONS AND POLARITONS IN QUANTUM WELLS EMBEDDED IN A MICOCAVITY • EXPERIMENTS DEVOTED TO TRAPPED POLARITONS IN A MICROCAVITY • BEC AND SUPERFLUIDITY of 2D POLARITONS IN A HARMONIC POTENTIAL IN A MICOCAVITY • GRAPHENE • BEC OF TRAPPED QUANTUM WELL AND GRAPHENE POLARITONS IN A MICROCAVITY IN HIGH MAGNETIC FIELD • CONCLUSIONS

  3. Semiconductor microcavity structure Bragg refractors:

  4. Trapping Cavity Polaritons cavity photon: quantum well exciton:

  5. upper polariton lower polariton Mixing leads to “upper polariton” (UP) and “lower polariton” (LP) LP effective mass ~ 10-4me Tune Eex(0) to equal Ephot(0): cavity photon exciton Exciton life time ~ 100 ps Polariton life time ~ 10 ps

  6. Spatially traped polaritons (using applied stress)R. B. Balili, D. W. Snoke,L. Pfeiffer and K. West, Appl. Phys. Lett. 88, 031110 (2006)

  7. Spatially trapped polaritons(using applied stress)1. R. B. Balili, D. W. Snoke, L. Pfeiffer and K. West, Appl. Phys. Lett. 88, 031110 (2006) 2. R. B. Balili, V. Hartwell, D. W. Snoke,L. Pfeiffer and K. West, Science 316, 1007 (2007). Theory: O. L. Berman, Yu. E. Lozovik, and D. W. Snoke, Phys Rev B 77, 155317 (2008). Starting with the quantum well exciton energy higher than the cavity photon mode, stress was used to reduce the exciton energy and bring it into resonance with the photon mode. At the point of zero detuning, line narrowing and strong increase of the photoluminescence are seen. An in-plane harmonic potential was created for the polaritons, which allows trapping, potentially making possible Bose-Einstein condensation of polaritons analogous to trapped atoms. Drift of the polaritons into this trap was demonstrated.

  8. Spatial profiles of polariton luminescence

  9. Spatial profiles of polariton luminescence- creation at side of trap

  10. x p  Angle-resolved luminescence spectra 50 mW 400 mW 600 mW 800 mW

  11. Hamiltonian of trapped polaritons Total Hamiltonian: Htot= Hexc+ Hph+ Hex-ph Exciton Hamiltonian: Trapping potential:V(r)=1/2 γr2 Exciton spectrum:εex(P) = P2/2M Photon Hamiltonian: Photon spectrum: λ is a spacing of the Bragg refractors (size of the cavity) Hamiltonian of exciton-photon interaction: Rabi splitting 15 meV

  12. Hamiltonian of non-interacting polaritons (after unitary Bogoliubov transformations) After the diagonalization of the total Hamiltonian applying the unitary transformations we get

  13. Effective Hamiltonian of lower polaritons Heff(after unitary Bogoliubov transformations) small parametersαandβ (very low temperature, small momentum and cloud size) Effective mass of a polariton Meff: Effective external trapping potential: Veff(r)=1/4 γr2

  14. BEC in Popov’s approximation Anomalous averages are neglected at temperatures O.L. Berman, Yu. E. Lozovik, and D. W. Snoke, Physical Review B 77, 155317 (2008). 1 γ=960 eV/cm2 γ=860 eV/cm2 0.9 γ=10 eV/cm2 γ=760 eV/cm2 0.8 Condensate fraction N0/N as a function of temperature T Condensate profiile n0(r) in the trap γ=100 eV/cm2 n(r=0) = 109 cm-2

  15. Superfluidity O.L. Berman, Yu. E. Lozovik, and D. W. Snoke, Physical Review B 77, 155317 (2008). NS=N-Nn Superfluid component Normal component Linear Response on rotation

  16. Superfluid fraction O.L. Berman, Yu. E. Lozovik, and D. W. Snoke, Physical Review B 77, 155317 (2008). γ=100 eV/cm2 γ=50 eV/cm2 γ=10 eV/cm2 γ→0 Superfluid fraction Ns/N as a function of temperature T Normal density was calculated as a linear response of the total angular momentum on rotation with an external velocity

  17. Conclusions: O.L. Berman, Yu. E. Lozovik, and D. W. Snoke, Physical Review B 77, 155317 (2008). • Thecondensate fractionand • the superfluid componentare • decreasingfunctions oftemperature, • andincreasing functions of the curvature of theparabolic potential.

  18. Graphene Graphene was obtained and studied experimentally for the first time in 2004 by K. S. Novoselov, S. V. Morozov, A. K. Geim,et.al. from the University of Manchester (UK). 2D atomic honeycomb crystal lattice of carbon (graphite) Perfect Graphene crystal and resultant Band Structure. The effective masses of electrons and holes and the energy gap in graphene equals 0

  19. Landau Levels in Graphene: Landau Levels in 2DEG: Landau levels in Gallium Arsenide and in Graphene: Quantum Hall Effect in Graphene: ωC is cyclotron frequency, nS is the number of an energy level and rB is magnetic length

  20. Magnetoexcitons in graphene Hamiltonian (A. Iyengar, J. Wang, H. A. Fertig, and L. Brey, Phys. Rev.B 75, 125430 (2007)) Conserving quantity magnetic momentum (instead of momentum without magnetic field) Generalization of the approach used by Lerner and Lozovik, JETP (1980) Wave function of e-h pair

  21. Semiconductor microcavity structure or graphene layers

  22. Graphene in an optical microcavity in high magnetic field in the potential trap • Magnetoexcitons: • All LLs below 1 are filled. All other LLs are empty. • All LLs below 0 are filled. All other LLs are empty. • Magnetoexcitons: • Electron: Landau level 1; Hole: Landau level 0 • Electron: Landau level 0; Hole: Landau level -1 graphene O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Physical Review B 80, 115302 (2009).

  23. Graphene in an optical microcavity in high magnetic field in the potential trap • The potential trap can be produced by applying an external inhomogeneous electric field. • The trap is caused by the inhomogeneous shape of the cavity. O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Physical Review B 80, 115302 (2009). O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Nanotechnology 21, 134019  (2010).

  24. Effective Hamiltonian of trapped magnetoexcitons and cavity photons O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Physical Review B 80, 115302 (2009). Htot= Hmex+ Hph+ Hmex-ph εex(P) = P2/2mB V(r)=1/2 γr2 index of refraction of cavity potential of a trap length of cavity Rabi splitting Rabi splitting

  25. Effective Hamiltonian of lower polaritons in a trap(after unitary Bogoliubov transformations) O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Physical Review B 80, 115302 (2009). Effective mass of a magnetopolariton: Critical temperature of a magnetopolariton BEC:

  26. Rabi splitting corresponding to the criation of magnetoexciton in graphene: matrix term of the Hamiltonian of the electron-photon interaction corresponding to magnetoexciton generation transition volume of microcavity Rabi splitting

  27. The ratio of the BEC critical temperature to the square root of the total number of magnetopolaritons as function of the magnetic field B and different spring constants γ. O.L. Berman, R.Ya. Kezerashvili and Yu. E. Lozovik, Physical Review B 80, 115302 (2009).

  28. The ratio of the BEC critical temperature to the square root of the total number of magnetopolaritons as function of the magnetic field B and the spring constant γ. O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Physical Review B 80, 115302 (2009). O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Nanotechnology 21, 134019  (2010).

  29. Conclusions O.L. Berman, R.Ya. Kezerashvili and Yu. E. Lozovik, Physical Review B 80, 115302 (2009). O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Nanotechnology 21, 134019  (2010). • The BEC critical temperature for graphene and quantum well polaritons in a microcavity Tc(0) decreases as B-1/4and increases with the spring constant as γ1/2. • We have obtained the Rabi splitting related to the creation of a magnetoexciton in a high magnetic field in graphene which can be controlled by the external magnetic field B.

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