Ana Maria Rey. Okinawa School in Physics 2016: Coherent Quantum Dynamics. Okinawa, Japan, Oct 4-5, 2016

Transcription

1 Ana Maria Rey Okinawa School in Physics 016: Coherent Quantum Dynamics Okinawa, Japan, Oct 4-5, 016

2 What can we do with ultra-cold matter?

3 Quantum Computers Lecture II-III Clocks and sensors Synthetic Materials Lecture I-II Quantum Simulators

4 Brief overview of dilute ultra-cold gases The Bose Hubbard and Hubbard models Exploring quantum magnetisms with ultra-cold atoms

5 Room temperature Velocity~ 300 m/s He condensation T=4K Velocity~ 90 m/s Kelvin atoms T ~100 nk Density: cm -3 Velocity~ cm/s Laser cooling: microk 1997 Bose Einstein Condensation 001

6 High temperature T: Thermal velocity v Density d -3 billiard balls Low temperature T: De Broglie wavelength DB =h/mv~t -1/ Wave packets T=T crit : Bose Einstein Condensation De Broglie wavelength DB =d ~ / Matter wave overlap Ketterle T=0 : Pure Bose Einstein Condensate Giant matter wave

7 In 1995 (70 years after Einstein s prediction) teams in Colorado and Massachusetts achieved BEC in super-cold gas.this feat earned those scientists the 001 Nobel Prize in physics. S. Bose, 194 Light A. Einstein, 195 Atoms E. Cornell W. Ketterle C. Wieman Using Rb and Na atoms ~ / T ~100 nk

8 At T<T f ~T c fermions form a degenerate Fermi gas ~ / 1999: 40 K JILA, Debbie Jin group

9 Thank you for all what you did for us!!! We will always remember you..

10 1. Laser cooling: Nobel Prize 1997 S. Chu Cohen-Tannoundji W. Phillips By bombarding the atoms with laser light, it is posible to slow them down and thus cool them down

11 An atom with velocity V is illuminated with a laser with appropriate frequency To cool Atom the atoms down we need to apply lasers in all three directions Atoms absorb light and reduce their speed Laser cooling is not enough to Slower atom cool down the atoms to quantum degeneracy and other tecniques are required As the atoms slow down the gas is cooled down Temperature limited by recoil energy: ~1-10K laser Trapping and cooling: MOT

12 Temperatures down to nk This process is similar to what happens with your cup of coffee: the hottest molecules escape from the cup as vapor In a magnetic field atoms can be trapped: magnetic By changing the trap depth the hot atoms escape and only the ones that are cold enough remain trapped Key: Collision re-thermalize atoms

13 (1) Particles behave like waves (T 0) ΨΨ=EΨ () Angular momentum is quantized: Ultra cold atoms collide via the lowest partial waves ll A ultra-cold temperatures l=0 collisions dominate!! Centrifugal barrier p-wave, l=1 Spatially anti-symmetric ~ K s-wave, l=0, Spatially symmetric r(a 0 )

14 Goal: find scattered wave f: scattering amplitude (r) = l l, l cos. l = 0, 1, s-, p-, waves, l 1 sin l l There is only a phase shift at long range!!

15 Get phase shift l l l l l 0 = scattering length Characterize s-wave collisions =b 3 scattering volume Characterize p-wave collisions Quantum statistics matter Pauli Exclusion principle Identical bosons: even Identical fermions: odd Non-identical species: all

16 Two interaction potentials V and V are equivalent if they have the same scattering length So: after measuring a for the real system, we can model with a very simple potential. Actually, to avoid divergences you need Huang and Yang, Phys. Rev. 105, 767 (1957) Also E. Fermi (1936), Breit (1947), Blatt and Weisskopf (195)

17 Ultra cold gases are dilute E E int. kin an n m / 3 an 1/ 3 ~ : Scattering Length n: Density Cold gases have almost 100% condensate fraction. In contrast to other superfluids like liquid Helium which have at most 10% Quantum phenomena on a macroscopic scale.

18 Field operator ˆ ˆ t [ ( r ), ( r' )] ( r r ') Many body Hamiltonian Equation of Motion In general Ψ =0

19 Dilute ultracold bosonic atoms are easy to model 1. Short range interactions. At T=0 all share the same macroscopic wave function Gross-Pitaevskii Equation We can understand the many-body system by a single non linear equation

20 A BEC is a coherent collection of atomic debroglie waves laser light is a coherent collection of electromagnetic waves laser

21 Vortices JILA 00 MIT (1997). Coherence Weakly interacting Bose Gas Superflow Non Linear optics MIT (1997). NIST (1999).

22 How can increase interactions in cold atom systems? E E int kin. an n m / 3 * * man 1 / 3 1 : Scattering Length n: Density 1. increase : Using Feshbach resonances. Increase the effective mass m m* One way to achieve. is with an optical lattice

23 Artificial crystals of light When atoms are illuminated by laser beams they feel a force which depends on the laser intensity. Two counter-propagating beams form a standing wave atoms

24 Periodic light shift potentials for atoms created by the interference of multiple laser beams. e g h d ~ 4 ~Intensity Standing wave a=/ V ( x ) 4 o sin ( kx )

25 Single particle in an Optical lattice Solved by Bloch Waves Periodic function q: Quasi-momentum k/ q k/ n: Band Index k= /a Reciprocal lattice vector Recoil Energy: ћ k /(m) Effective mass m * d E dq 1 k k k k k m* grows with lattice depth k

26 Single particle in an Optical lattice Bloch Functions V=0 V=0.5 Er V=4 Er V=0 Er Wannier Functions localized wave functions:

27 R o i sp i E V x x w H x x w dx J 4 exp ) ( ) ( And expand in lowest band Wannier states Assuming: Lowest band, Nearest neighbor hopping ) ( ) ( ) ( ] [ sin x t i x x V kx V x m o We start with the Schrodinger Equation j j j j j x V J i ) ( ( 1) 1 iqja j Ae a] cos[ ) ( q J q E If V=0 Plane Bloch waves Cosine spectrum ) ( 0 i i i x x w Band width = 4 J

28 We start with the full many-body Hamiltonian and expand the field operator in Wannier states ˆ aˆ jw 0 ( x x j ) jaˆ n n n 1 j j Assuming: Lowest band, Short -range interactions, Nearest neighbor hopping H=-J <i,j> â i â j Hopping Energy + U/ j â j â j â j â j + j V j â j â j Interaction Energy J External potential kt, U, J W 0 (x) j U j+1 4a m U jd. Jaksch et. al., PRL 81, 3018 (1998) V dx 3 w 0 ( x) 4 av 3/4 0

29 Superfluid Mott Insulator Quantum phase transition: Competition between kinetic and interaction Shallow potential: U<<J energy Deep potential: U>>J U Weakly interacting gas Strongly interacting gas Superfluid Mott insulator Lattice depth : Laser Intensity M.P.A. Fisher et al,prb40:546 (1989) M. Greiner et al.nature: 415, (00);

30 U 1 0 n 0 =3 n 1 n 0 = n 0 =1 Mott Mott Mott Superfluid J M.P.A. Fisher et al., PRB40:546 (1989) U n 0 =1 Step 1: Use the decoupling approximation Step : Replace it in the Hamiltonian Step 3: Compute the energy using as a perturbation parameter and minimize respect to E () = 0 Critical point

31 t=0 Turn off trapping potentials Imaging the expanding atom cloud gives important information about the properties of the cloud at t=0: Spatial distribution -> Momentum distribution after time of flight at t=0

32 In the lattice at t=0 x a After time of flight σ(t)= tħ/(mσ o ) x th am ( x, t) j ne i w0 ( x, t) e iq( x) ja n x m t ( ) ~ n( Q) t 0 Q nk,0

33 Superfluid Quantum Phase transition Mott insulator Lattice depth : Laser Intensity Markus Greiner et al. Nature 415, (00); shallow deep shallow The loss of the interference pattern demonstrates the loss of quantum phase coherence.

34 Optical lattice and parabolic potential U i 0 i 4 n o =3 n 1 n o = Mott Mott Superfluid n o =1 Mott 0 J U ultracold.uchicago.edu

35 Quantum gas microscopes: shell structure High-resolution imaging just resolves atoms in adjacent lattice sites. S. Waseems et al Science, 010 S. Waseems et al Science, 010

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37 The Hubbard model is a minimal model for interacting fermions in a lattice. It was invented to study magnetism in strongly correlated systems. H= i -J (cˆ i, cˆ i+1, +h.c.) Hopping Energy + i U nˆ i nˆ i Interaction Energy J + i i nˆ i Parabolic potential W(x) i i+1 U A Mott insulator made of fermions first observed 008: at ETH (Esslinger group ) and Mainz (Bloch group). Many groups now

38 Its phase diagram in and 3 dimensions remains unknown Possible phase diagram Ferromagnetism? Ferro U/J Paramagnetic n AF 0 1/ doping Used to model cuprate superconductors: High Temperature superconductivity Can cold atoms help to identify the phase diagram? Known Super-exchange

39 Exchange interactions Effective spin-spin interactions can arise due to the interplay between the SPIN-INDEPENDENT forces and EXCHANGE SYMMETRY Exchange Direct overlap Basic Idea Triplet Energy Singlet

40 Experimental Control of Exchange Interactions Spin : = F=1,m F =0 H ex V ex M. Anderlini et al. Nature 448, 45 (007) = F=1,m F =-1 Orbitals: Two bands g and e ( S S 1 s 3/4) Singlet < 1 ( g e ) t Triplet 1 w 0 w 1 ( ) V ex 8a m dx 3 w 0 ( x) w ( x) 1

41 Experimental Control of Exchange Interactions M. Anderlini et al. Nature 448, 45 (007) Prepare a superposition of singlet and triplet two level system: t= and s= t, cos, sin Measured spin exchange: Number of e and g atoms in ground band s

42 Super-Exchange Interactions Spin order can arise even though the wave function overlap is practically zero. Super- Exchange Virtual processes E.g. Two electrons in a hydrogen molecule, MnO fermions or bosons Singlet Mn O Energy Triplet P.W. Anderson, Phys. Rev. 79, 350 (1950)

43 Super-exchange in optical lattices Consider a double well with two atoms At zero order in J, the ground state is Mot insulator with one atom per site and all spin configurations are degenerated J lifts the degeneracy: An effective Hamiltonian can be derived using second order perturbation theory via virtual particle hole excitations 0, J J m: Virtual particle-hole excitations,,,0 HJef exi. j S- Bosons, + Fermions JUJexi Sj

44 Two bosons in a Double Well with S z =0 Only 4 states: singly occupied configurations: (1,1) s, (1,1) t (1,1) s 1 ( ) Singlet t 1 ( ) Triplet doubly occupied configurations: (,0) t, (0,) t (0,) S (,0) (0,)

45 Energy levels in symmetric DW:»U Good basis: (1,1) t, s, 0 0 t s, - are not coupled by J. They have E=0,U for any J. t, + are coupled by J: Form a level system ħ ħ ++ t - U t + + s In the U>>J limit ħ 1 ~ 4J /U: Super-exchange ħ ~ U

46 Magnetic field gradient In the limit U>>J, only the singly occupied states are populated and they form a two level system: s= and t= HJB e B0 Magnetic field gradient couples and B ( BR BL ) zˆ xa If B «J ex then s and t are the eigenstates If B» J ex then and are the eigenstates

47 Experimental Observation S. Trotzky et. al, Science, 319,95(008) Prepare Turn of B B >0 t t s Evolve two level system: s= and t= s M Measure spin imbalance N z : # atoms - # atoms N z ( t) f ( 1, ) In the limit J<<U, t Nz(t) cos( J ex t) Simple Rabi oscillations

48 Measuring Super-exchange Two frequencies V=6Er V=11 Er V=17 Er Almost one frequency: superexchange

49 ? Short-Range Magnetism in Quantum gas microscopes (015): Harvard, Munich, MIT, Toronto, Princenton, Glasgow, Short-Range Magnetism in an optical lattice (ETH, (013), Rice (014) Propagation of a two excitations in a 1D chain (Max-Planck, München, 013) Propagation of a single excitation in a 1D chain (Max-Planck, München, 013) Double-exchange in double-wells (Mainz, 008) Exchange in a single site (JQI, 007) Long -Range Magnetism in Quantum gas microscopes (016): Harvard

50 Interpretation of scattering length (a) (R) See below Na+Na 1 K 1 m 3 different short-range potentials with 3 different scattering lengths Internuclear separation R (a 0 ) V ˆ 4 m A( R ) R R Huang and Yang, Phys. Rev. 105, 767 (1957) Also E. Fermi (1936), Breit (1947), Blatt and Weisskopf (195) (R) sin[k(r A)] (R) k 1/ as R x 0 A = -100 a 0 A = 0 a 0 A= +100 a Internuclear separation R (a 0 )

51 Superimpose two lattices: one with twice the periodicity of the other Adjustable bias and barrier depth by changing laser intensity and phase