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.

Bose-Einstein Condensation of Trapped Polaritons in a MicrocavityOleg 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, USA2Institute of Spectroscopy, Russian Academy of Sciences,

Troitsk, Moscow Region, Russia3Department of Physics and Astronomy,

Univeristy of Pittsburgh, Pittsburgh PA, USA 3Bell Labs, Lucent Technologies, Murray Hill NJ, USA

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

Semiconductor microcavity structure

Bragg refractors:

Trapping Cavity Polaritons

cavity photon:

E c kz2 k||

2 c ( / L)2 k||2

quantum well exciton:

E Egap bind

h2N 2

2mr (2L)2

2k||2

2m

Tune Eex(0) to equal Ephot(0):

cavity photon

exciton

Mixing leads to “upper polariton” (UP) and “lower polariton” (LP)

upper polariton

lower polariton

LP effective mass ~ 10-4 me Exciton life time ~ 100 ps

Polariton life time ~ 10 ps

Spatially traped polaritons (using applied stress)R. B. Balili, D. W. Snoke, L. Pfeiffer and K. West,

Appl. Phys. Lett. 88, 031110 (2006)

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.

Spatial profiles of polariton luminescence

Spatial profiles of polariton luminescence- creation at side of trap

Angle-resolved luminescence spectra

50 W 400 W

600 W 800 W

x p

Hamiltonian of trapped polaritons

Htot= Hexc+ Hph+ Hex-ph

15 meV

Trapping potential: V(r)=1/2 γr2

Exciton spectrum: εex(P) = P2/2M

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

Hamiltonian of non-interacting polaritons

(after unitary Bogoliubov transformations) After the diagonalization of the total Hamiltonian

applying the unitary transformations we get

Effective Hamiltonian of lower polaritons Heff

(after unitary Bogoliubov transformations)

small parameters α and β (very low temperature, small momentum and cloud size)

Effective mass of a polariton Meff:

Effective external trapping potential: Veff(r)=1/4 γr2

BEC in Popov’s approximation are neglected

Condensate fraction N0/N as a function of temperature T

γ=10 eV/cm2

γ=100 eV/cm2

n(r=0) = 109 cm-2

Condensate profiile n0(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/cm2

γ=860 eV/cm2

γ=960 eV/cm2

Anomalous averages

at temperatures

Superfluidity

NS=N-Nn

Superfluid component

Normal component

Linear Response on rotation

O.L. Berman, Yu. E. Lozovik, and D. W. Snoke, Physical Review B 77, 155317 (2008).

Superfluid fraction

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

γ=100 eV/cm2

γ=50 eV/cm2

γ=10 eV/cm2

γ→0

O.L. Berman, Yu. E. Lozovik, and D. W. Snoke, Physical Review B 77, 155317 (2008).

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).

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)

12( )ns C sE n f

ns s f sB

VE n V e B n

r

Landau Levels in 2DEG: Landau Levels in Graphene:

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

m

BeC

2

1

BerB

Magnetoexcitons in grapheneHamiltonian

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 byLerner and Lozovik, JETP (1980)

Semiconductor microcavity structure

or graphene layers

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

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).

Effective Hamiltonian of trapped magnetoexcitons and cavity photons

Htot= Hmex+ Hph+ Hmex-ph

V(r)=1/2 γr2

εex(P) = P2/2mB

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).

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).

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

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).

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).

Conclusions

• The BEC critical temperature for graphene and quantum well polaritons in a microcavity Tc

(0) decreases as B-1/4 and 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.

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).