1. 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 1 Physics Department, New York City College of Technology of City University of New York (CUNY), Brooklyn NY, USA 2 Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow Region, Russia 3 Department of Physics and Astronomy, Univeristy of Pittsburgh, Pittsburgh PA, USA 3 Bell 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. Tune E ex (0) to equal E phot (0): cavity photon exciton Mixing leads to “upper polariton” (UP) and “lower polariton” (LP) upper polariton lower polariton LP effective mass ~ 10 -4 m e 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. Angle-resolved luminescence spectra 50  W400  W 600  W800  W  x  p 

11. Hamiltonian of trapped polaritons H tot = H exc + H ph + H ex-ph 15 meV Trapping potential: V(r)=1/2 γr 2 Exciton spectrum: ε ex (P) = P 2 /2M λ is a spacing of the Bragg refractors (size of the cavity) Exciton Hamiltonian: Total Hamiltonian: Photon Hamiltonian: Photon spectrum: Hamiltonian of exciton- photon interaction: Rabi splitting

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 H eff (after unitary Bogoliubov transformations) small parameters α and β (very low temperature, small momentum and cloud size) Effective mass of a polariton M eff : Effective external trapping potential: V eff (r)=1/4 γr 2

14. BEC in Popov’s approximation are neglected Condensate fraction N 0 /N as a function of temperature T γ=10 eV/cm 2 γ=100 eV/cm 2 n(r=0) = 10 9 cm -2 Condensate profiile n 0 (r) in the trap 0.8 0.9 1 O.L. Berman, Yu. E. Lozovik, and D. W. Snoke, Physical Review B 77, 155317 (2008). γ=760 eV/cm 2 γ=860 eV/cm 2 γ=960 eV/cm 2 Anomalous averages at temperatures

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

16. Superfluid fraction Superfluid fraction N s /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 γ=100 eV/cm 2 γ=50 eV/cm 2 γ=10 eV/cm 2 γ→0 O.L. Berman, Yu. E. Lozovik, and D. W. Snoke, Physical Review B 77, 155317 (2008).

17. Conclusions: The condensate fraction and the superfluid component are decreasing functions of temperature, and increasing functions of the curvature of the parabolic potential. O.L. Berman, Yu. E. Lozovik, and D. W. Snoke, Physical Review B 77, 155317 (2008).

18. 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). Graphene Perfect Graphene crystal and resultant Band Structure. The effective masses of electrons and holes and the energy gap in graphene equals 0 2D atomic honeycomb crystal lattice of carbon (graphite)

19. Landau Levels in 2DEG: Landau Levels in Graphene: Landau levels in Gallium Arsenide and in Graphene: Quantum Hall Effect in Graphene: ω C is cyclotron frequency, n S is the number of an energy level and r B is magnetic length

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

21. Semiconductor microcavity structure or graphene layers

22. Graphene in an optical microcavity in high magnetic field in the potential trap Magnetoexcitons: 1.Electron: Landau level 1; Hole: Landau level 0 2.Electron: Landau level 0; Hole: Landau level -1 Magnetoexcitons: All LLs below 1 are filled. All other LLs are empty. All LLs below 0 are filled. All other LLs are empty. O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Physical Review B 80, 115302 (2009). graphene

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 H tot = H mex + H ph + H mex-ph V(r)=1/2 γr 2 ε ex (P) = P 2 /2m B length of cavity potential of a trap Rabi splitting index of refraction of cavity Rabi splitting O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Physical Review B 80, 115302 (2009).

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

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

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 The BEC critical temperature for graphene and quantum well polaritons in a microcavity T c (0) decreases as B -1/4 and increases with the spring constant as γ 1/2. 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. 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).