Adiabatic trap deformation for preparing Quantum Hall states

Marco Roncaglia, Matteo Rizzi, and Jean Dalibard Adiabatic trap deformation for preparing Quantum Hall states Max-Planck Institut für Quantenoptik, München, Germany Dipartimento di Fisica del Politecnico, Torino, Italy Laboratoire Kastler Brossel, Paris, France Quantum Technologies Conference Torun, September 30th, 2010

Outline Quantum Hall Effect & Rotating traps The strongly correlated regime Experimental proposal One-body physics Many-body physics Ground state phase diagram Adiabatic trap deformation Detection

Quantum Hall effect 2D electron gas Plateaus: ρ H = 1 ν h e 2 Integer: 1-body effect Fractional: strong correlations Filling factor: no. particles / no. flux quanta Φ 0 = hc e

Rotating traps Ω Harmonic trap quasi-2d limit H trap = 2 2m N ( 2 i=1 x 2 i + 2 y 2 i ) + mω2 2 N (x 2 i + yi 2 ) i=1 A. Fetter, RMP 81, 647 (2009).

Rotating traps Ω Harmonic trap quasi-2d limit H trap = 2 2m N ( 2 i=1 x 2 i + 2 y 2 i ) + mω2 2 N (x 2 i + yi 2 ) i=1 A. Fetter, RMP 81, 647 (2009). 4 dω 1. 3 EL 2 1 na 0 4 2 0 2 4 b b

Rotating traps Harmonic trap quasi-2d limit H trap = 2 2m N ( 2 i=1 x 2 i + 2 y 2 i ) + mω2 2 N (x 2 i + yi 2 ) i=1 Fast rotation with frequency offset H rot = ΩL z = iω N i=1 ( x i y i y i x i ) A. Fetter, RMP 81, 647 (2009). dω ω Ω 4 dω 1. 3 EL 2 1 na 0 4 2 0 2 4 b b

Rotating traps A. Fetter, RMP 81, 647 (2009). H trap = ( p A ) 2 2m + m 2 (ω2 Ω 2 )(x 2 + y 2 ) A = mωẑ r B = (2mΩ/q)ẑ dω = ω Ω = 0 deconfinement 4 dω 1. 3 EL 2 1 na 0 4 2 0 2 4 b b

Rotating traps A. Fetter, RMP 81, 647 (2009). H trap = ( p A ) 2 2m + m 2 (ω2 Ω 2 )(x 2 + y 2 ) A = mωẑ r B = (2mΩ/q)ẑ dω = ω Ω = 0 deconfinement Recasted into easy oscillators H tot = ω (2a a + 1) + dω (b b a a) a = z + z/2 b = z + z/2 EL 44 33 22 11 dω dω 0.05 1. 00 4 4 2 0 2 4 L n b n a n b na

Rotating traps A. Fetter, RMP 81, 647 (2009). H trap = ( p A ) 2 2m + m 2 (ω2 Ω 2 )(x 2 + y 2 ) A = mωẑ r B = (2mΩ/q)ẑ dω = ω Ω = 0 deconfinement Recasted into easy oscillators H tot = ω (2a a + 1) + dω (b b a a) a = z + z/2 b = z + z/2 Landau Level structure & analogy with Quantum Hall Effect: analiticity of LLL states in z only! ψ LLL m = z m e z z/2 EL 44 33 22 11 LLL dω dω 0.05 1. 00 4 4 2 0 2 4 L n b n a n b na

Quantum Hall regime

Quantum Hall regime Without rotation, trapped cold bosonic atoms are in a BEC Over a certain rotation, vortex nucleation starts and triangular vortex patterns appear In analogy with QHE, filling factor is ν = N N 2 l max L l max N v L N

Quantum Hall regime Without rotation, trapped cold bosonic atoms are in a BEC Over a certain rotation, vortex nucleation starts and triangular vortex patterns appear In analogy with QHE, filling factor is ν = N N 2 l max L l max N v L N Laughlin Pfaffian Read-Rezayi Vortex melting Vortices 1/2 1 3/2 2 ~ 6 ν

Quantum Hall regime Without rotation, trapped cold bosonic atoms are in a BEC Over a certain rotation, vortex nucleation starts and triangular vortex patterns appear In analogy with QHE, filling factor is ν = N N 2 l max L l max N v L N Laughlin Pfaffian Read-Rezayi Vortex melting Vortices 1/2 1 3/2 2 ~ 6 ν QHE regime appears when mean-field breaks down Up to now, requirement dω c 2 N is too strong: typical ν 10 3 I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008)

The idea We start from a rotating gas confined in a ring by using a Mexican-hat potential. Harmonic trap + U h (r) = 1 2 mω2 r 2 Blue-detuned laser U w (r) = α exp α ( 2r2 w 2 w ) = intensity = waist

The idea We start from a rotating gas confined in a ring by using a Mexican-hat potential. Harmonic trap + U h (r) = 1 2 mω2 r 2 Experimental proposal: Blue-detuned laser U w (r) = α exp ( 2r2 w 2 (I) We initially prepare an ultracold gas of N particles in a giant vortex ring with a given angular momentum L by using a repulsive plug laser at the center; (II) The plug is adiabatically switched off in absence of rotation; (III) In the final harmonic trap, we obtain the ground state with the initially imparted angular momentum L, thanks to rotational simmetry. We get the bosonic Laughlin state if α w ) = intensity = waist L Lau = N(N 1)

One-body physics We can identify a LLL structure also in presence of the central plug. m E m 0 5 10 15 20 0 2 4 6 8 10 Λ 4 dω 0.1 Α 2.00 w 2 = λ LLL Approximation: use the unperturbed wavefunctions ψ m (z) = 1 πm! z m e z 2 /2 m U w m = α (1 + 2w 2 ) (m+1) Matrix elements of the perturbation

One-body physics We can identify a LLL structure also in presence of the central plug. LLL Approximation: use the unperturbed wavefunctions Matrix elements of the perturbation ψ m (z) = 1 πm! z m e z 2 /2 m U w m = α (1 + 2w 2 ) (m+1) 10 E m 8 6 4 2 0 Λ 4 dω 0.1 Α 2.00 w 2 = λ 0 5 10 15 m 20 If we want a minimum in m = λ then the optimal waist is w 2 = λ For having a bump we must set the laser intensity high enough α > λ/4 For a minimum in m = λ dω c λ = 2α λ + 1 λ 2 set rotation inside an interval centered in ( 1 + 2 λ) (λ+2) α λ

Many-body physics 2-body interaction H 2 = c 2 δ( x i x j ) i<j c2 proportional to s-wave scattering length (tunable with a Feshbach resonance) Laughlin state = exact ground for pure 2-body interactions c 2 = 8πa/ξ z Ψ Lau = i<j(z i z j ) 2 ν = 1/2 L Lau = N(N 1) Units: energy ω, length ξ = /(mω)

Many-body physics 2-body interaction H 2 = c 2 δ( x i x j ) i<j c2 proportional to s-wave scattering length (tunable with a Feshbach resonance) Laughlin state = exact ground for pure 2-body interactions c 2 = 8πa/ξ z Ψ Lau = i<j(z i z j ) 2 ν = 1/2 L Lau = N(N 1) In the rotating Mexican-hat potential the single-body physics dominates for weak interactions. Laser stirring is used to induce a rotating quadrupolar deformation H ɛ = ɛ(x 2 y 2 ) Giant Vortex Ψ gv = N i=1 We want the same total angular momentum as the final QHE state. z λ i λ Lau = N 1 w 2 = N 1 Units: energy ω, length ξ = /(mω)

Ground-state diagram For investigating the competition between collisions and trap potential, we have to diagonalize the many-body problem. Total angular momentum of the many-body ground state dω!"!!%! )) dω N = 9 L Lau = 72 w 2 = 8!!"!!$#!"!!$!!"!!!#!"!!!! #( '% Laughlin "&! c 2 = 0.1 } Laughlin Enlarged window of stability dω (0.08, 0.1) Also with interactions a giant vortex is created All the circulation of velocity is around the center.

Ground-state diagram For investigating the competition between collisions and trap potential, we have to diagonalize the many-body problem. Total angular momentum of the many-body ground state dω!"!!%! )) dω N = 9 L Lau = 72 w 2 = 8!!"!!$#!"!!$!!"!!!#!"!!!! #( '% Laughlin "&! c 2 = 0.1 } Laughlin Enlarged window of stability dω (0.08, 0.1) Also with interactions a giant vortex is created All the circulation of velocity is around the center. N. K. Wilkin and J. M. F. Gunn, PRL 84, 6 (2000)

Trap deformation The intensity of the plug laser is switched off Ρr 0.35 0.30 0.25 0.20 0.15 N 6 Λ 5 c 2 0.1 Α 0.0 0.2 0.5 2.0 The gas, initially confined in a ring, expands radially 0.10 0.05 0.00 0 1 2 3 4 5 r N = 9 Giant vortex Laughlin

Adiabatic condition The adiabatic condition determines the rate of change of laser intensity α Ψ 0 ( H/ t) Ψ 1 2 = E 1 E 0 gap T = F α dα F α 2 Ψ 0 ( U w / α) Ψ 1 Total minimal time for adiabaticity Inverse rate of change of α For 87 Rb in realistic traps: c 2 = 0.1 N = 9 w 2 = 8 dω (0.08, 0.1) α = 3 0 In this case T 200ω 1

Scaling with N w 2 = N 1 α I = N/2! / N 1/2 0.2 0.15 0.1 4 5 6 7 8 9 10 Constant chemical potential µ N c 2 0.05 c 2 N 1/2 = 0.3 c 2 N 1/2 0 0 0.1 0.2 0.3 0.4 0.5 " / N From a Bogoliubov analysis in the ring N 1/2 There is a slow-down in the last part of the adiabatic path

Scaling with N w 2 = N 1 α I = N/2! / N 1/2 0.2 0.15 0.1 4 5 6 7 8 9 10 Constant chemical potential µ N c 2 0.05 c 2 N 1/2 = 0.3 c 2 N 1/2 0 0 0.1 0.2 0.3 0.4 0.5 " / N From a Bogoliubov analysis in the ring N 1/2 There is a slow-down in the last part of the adiabatic path

Scaling with N w 2 = N 1 α I = N/2! / N 1/2 0.2 0.15 0.1 4 5 6 7 8 9 10 Constant chemical potential µ N c 2 0.05 c 2 N 1/2 = 0.3 c 2 N 1/2 0 0 0.1 0.2 0.3 0.4 0.5 " / N From a Bogoliubov analysis in the ring N 1/2 There is a slow-down in the last part of the adiabatic path

Exploring different trap deformations Faster adiabatic path

Detection The interaction energy goes to zero, since the Laughlin state is in the kernel of 2-body collisions. 0.010 It is directly related to the 2-body correlation function at zero distance g (2) (0) Detectable by photoassociation H2 / N 0.008 0.006 0.004 0.002 0.000 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5! / N The maximal number of particles N is determined by the total adiabatic time T the experimenter can keep the system under control

Outline Quantum Hall Effect & Rotating traps The strongly correlated regime Experimental proposal One-body physics Many-body physics Ground state phase diagram Adiabatic trap deformation Detection

Setup in optical lattices 3D lattice of highly decoupled quasi-2d wells, equally populated, individually rotated d Ω N. Gemelke, et al., ArXiv 1007.2677 N = 4 d = 3.5µm ω 3 Khz ω 30 Khz c 2 10 1 Ω ω