Ref: Bikash Padhi, and SG, Phys. Rev. Lett, 111, (2013) HRI, Allahabad,Cold Atom Workshop, February, 2014

Transcription

1 Cavity Optomechanics with synthetic Landau Levels of ultra cold atoms: Sankalpa Ghosh, Physics Department, IIT Delhi Ref: Bikash Padhi, and SG, Phys. Rev. Lett, 111, (2013)! HRI, Allahabad,Cold Atom Workshop, February, 2014

2 Cold IIT Delhi Cold Delhi Chaitanya Joshi and Sankalpa Ghosh, European Physics Journal B, 68, 467 (2009). Rashi Sachdeva, Sonika Johri and Sankalpa Ghosh, Physical Review, A, 82, (2010) Rashi Sachdeva and Sankalpa Ghosh, Physical Review A, 85, (2012) Adhip Agarwala, Madhurima Nath, Jasleen Lugani, K. Thyagarajan and Sankalpa Ghosh, Physical Review A Vol (2012) Bikash Padhi and Sankalpa Ghosh, Physical Review A, Vol 111, (2012) Rashi Sachdeva and Sankalpa Ghosh, arxiv: cond-mat ( communicated) Jasleen Laguna, K. Thyagarajan and Sankalpa Ghosh, Journal of Physics B : At. Mol. and Opt. Physics, Vol 47, (2014).!2

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4 CAVITY OPTOMECHANICS:BASICS Marquardt, Les Houches Lectures, 2011 P. Meystre, arxiv: (quant-ph) H = ~!(x)â â + ˆp2 2m mˆx 2 Upto a Linear approximation, the position dependent resonant frequency is!(x)! c (1 ˆx L ) This leads to H = ~! c â â + ˆp2 2m m 2 mˆx 2 ~g 0 â â(ˆb + ˆb Last Term is the optomechanical coupling by which a mode in the Fabry Perot cavity is coupled with the vibrating mode of a mechanical object

5 ENTERS ATOMS IN A CAVITY

6 Basic Theory = 3 +2gcos(!t + ) 1 2 second term! ( + + ) (e i + e i ) ating wave approximation ignores fast oscillating term near resonant condi second term ( + e i + e i ) The interaction between a single two level atom and a photon (single mode) in dipole approximation is given by Jaynes Cumming Model Ĥ JC = ~! at 2 ˆz + ~! c â â + ~g 0 (ˆ â +ˆ a )+Ĥapple + Ĥ!6

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8 atoms Ultracold quantum gases in a nutshell Atomic wave packets overlap Macroscopic wave function Superfluid behaviour i.e. flow without friction W. Ketterle, Nobel Lecture (2001)

9 Time of Flight imaging Common Technique: Time of Flight imaging

10 CAVITY OPTOMECHANICS WITH MANY ATOMS Slide from T. Esslinger s Talk

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12 Excited modes of BEC:realization of a two level macro atom e 2ikx + e 2ikx Ψ ( x) = c ( t) p = 0 + c ( t) p = ± 2! k Ψ( x) 2 = 0 N + c 2 N 2 cos(2kx)cos(4 ω r t)

13 CAVITY OPTOMECHANICS WITH A BEC ( ZURICH EXPERIMENT)

14 Quantum Simulation: There is plenty room at bottom -R. P. Feynman

15 First Step towards simulation (Non Linear Schroedinger Equation) Feynman Lecture- Vol 3 A seminar on superconductivity Bose Einstein condensation leads to Macroscopic quantum state The interaction strength can be controlled from positive to negative and can be made 0 as well as infnity

16 Idea of Quantum Simulation 4He absorbed in! porous medium and! granular superconductor Mott Insulator ( U(V)>>t(J)) integer atoms per site No phase coherence No number fluctuations! H = t ( â i+â j + h.c)+ U 2 <i, j> i ˆn i ( ˆn i 1) µ i ˆn i Bose Hubbard Model Ultracold 87 Rb atoms in! optical lattice : much! Simpler and controllable Superfluid (t(j)>>u) Delocalized atoms Long range phase coherence (macroscopic wave function) Number fluctuation present

17 Cold atomic systems as Quantum simulators Use optical lattices V 0 M.Greiner et. al Nature (2002) Counter- propagating laser beams phase coherence phase incoherence

18 electron in a magnetic field:cyclotron motion v R R c

19 THE CURIOUS TALE OF LANDAU LEVELS B=0 KF

20 PRL 109, (2012) N type InSb (110) surface Experiment Theory

21 J Solyom : Fundamental of Solids Vol. 2 (Springer)

22 Density of states and oscillation of thermodynamic quantities J Solyom : Fundamental of Solids Vol. 2 (Springer)

23 SHUBNIKOV de HAAS OSCILLATION AND DE HAAS VAN ALPHEN EFFECT Ashcroft and Mermin: Solid state Physics

24 Interaction under control, can we simulate a magnetic field OF NEUTRAL ULTRA COLD ATOMS? Motivation: Quantum Hall Effect, more recently Topological Insulator Our modest motivation: simulation of the atomic analogue of SdH oscillation

25 Artificial magnetic field for neutral atoms. Rotation of neutral atoms trapped in a harmonic trap causes appearance of an effective magnetic field. H = p2 2m m!2 r 2 L z Introduce A = r The hamiltonian in the co-rotating frame is H = 1 2m (p ma ) m(!2 2 )r 2 Rotating trapped Bose-Einstein condensates Rev. Mod. Phys. 81, 647 Published 18 May 2009! Alexander L. Fetter 12 February 2014!25

26 z Experiments on rotating condensates Ω The condensate is stirred with a moving laser beam and image is taken after ballistic expansion K. Madison, F. Chevy, V. Bretin, P. Rosenbuch, J. Dalibard (ENS Group) 200 µm Clear proof of superfluidity Evidence for quantum vortices ENS 0 Ω c Rotation frequency Ω Coddington et al. PRL 91 (2003), JILA Towards quantum Hall effect?

27 NIST WAY (Spielman et al.) Y. J. Lin et al. PRL 102, (2009) 2 H =! x x + 2m [( k aa ) 2 k 2 ] y

28 Observation of vortices and vortex lattice(nist) Y J. Lin et al, Nature 462, 628 (2009)

29 Ultra cold fermions (From Debi Jin s slide)

30 Natur

31 The System under consideration Phys. Rev. Lett. 111, (2013), Bikash Padhi and SG 12 February 2014!31

32 For cavity opto-mechanics with degenerate Fermi gas see Kanamoto, Meystre (PRL, 2010)

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35 A schematic introduction to Bosonization for one dimensional fermions In one dimension kf=πn0 The low energy long wavelength theory for a one dimensional system of fermions Can be written in terms of particle-hole excitation about the Fermi points that are bosonic in nature and obeys linear dispersion C. Kane, Boulder Lectures

36 More mathematical introduction to Bosonization There are many resources on this theory: just type review articles on bosonization for one dimensional fermions in Google.

37 Landau Level Bosonization : Schematics n=3 +kf B n=1 n=2 H Westfahl et Al. PRB, 55, R7347 (1997) -kf n=0!37

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40 Analogue of Shubnikov de Hass oscillation for neutral atoms

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42 Oscillations again

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45 BISTABLE CAVITY TRANSMISSION WITH SYNTHETIC MAGNETIC FIELD Cavity relaxation rate Padhi and Ghosh, Phys. Rev. Lett. 111, (2013) Phys. Rev. Lett. 111, (2013)

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47 Thank You! Ph. D. positions are available!47