tunneling theory of few interacting atoms in a trap

tunneling theory of few interacting atoms in a trap Massimo Rontani CNR-NANO Research Center S3, Modena, Italy www.nano.cnr.it Pino D Amico, Andrea Secchi, Elisa Molinari G. Maruccio, M. Janson, C. Meyer, T. Matsui, C. Heyn, W. Hansen, A. Schramm, R. Wiesendanger University of Hamburg

cold atoms in the few-body limit 87 Rb, Bloch group bosons S Will et al. Nature 465, 197 (2010) 6 Li, Jochim group fermions (yesterday talk) F. Serwane et al. Science 332, 336 (2011)

motivation control key parameters of clean correlated system particle number interaction strength confinement potential ideal lab for few-fermion physics comparison with exact theoretical models building block of many-body systems

few-fermion vs bulk physics bulk: kinetic energy + interaction homogeous (cf. many fermions at trap center) few particles in a trap: quantum confinement inhomogeous analogies with atoms, nuclei, quantum dots tunneling spectroscopy, shell structure, Hund s rule, pairing for few fermions

outline tunneling spectroscopy in interacting cold atoms & condensed-matter nano-objects: quasiparticle theory repulsive interactions between 6 Li atoms wave function imaging in interacting quantum dots pairing for a few 6 Li atoms & pair tunneling

tunneling spectroscopy à la Heidelberg talk by G. Zuern yesterday prepare a 6 Li atom in the trap ground state

tunneling spectroscopy à la Heidelberg switch off the potential barrier

tunneling spectroscopy à la Heidelberg 1 the atom escapes

tunneling spectroscopy à la Heidelberg 1 after hold time t, switch on the barrier

tunneling spectroscopy à la Heidelberg 2 second possibility

tunneling spectroscopy à la Heidelberg 2 after hold time t

tunneling spectroscopy à la Heidelberg 2 the atom remains in the trap

tunneling spectroscopy à la Heidelberg 1 P(1) + P(2) = 1 at any t (a) check if atom in the trap (b) repeat many times (c) vary hold time t (d) iterate 2

measure decay time t [Zuern et al. PRL 2013] noninteracting M. Rontani, PRA 88, 043633 (2013) N WKB t = N 0 e t/τ

many fermions: energy spectroscopy e 2 e 1 e 0 t(e 0 ) two-species fermions no interaction

many fermions: energy spectroscopy e 2 e 1 e 0 t(e 1 ) decay as resonance

many fermions: energy spectroscopy e 2 e 1 e 0 t(e 2 ) dominant decay channel at Fermi energy

what if atoms interact? e 2 e 1 e 0 t =?

what if atoms interact? g = 0 Heidelberg setup e 1 V int = g d(x 1 x 2 )

what if atoms interact? M. Rontani, PRL 108, 115302 (2012) g 0 e e 1 e = chemical potential

interaction blockade measure chemical potential e ( N) E0( N) E0( N 1) equivalent to addition energy in quantum dots and ionization potential in atoms and molecule prediction: Kapelle, Reimann & coworkers [PRL 99, 010402 (2007)] exp confirmation (bosons): Bloch group [PRL 101, 090404 (2008)] compute t using e in WKB formula (mean-field theory)?

chemical potential e two interacting atoms (Zuern et al., PRL 2012) after T. Busch et al. (1998) V int = g d(x 1 x 2 )

chemical potential e two interacting atoms (Zuern et al., PRL 2012) after T. Busch et al. (1998) exp B V int = g d(x 1 x 2 )

chemical potential e two interacting atoms (Zuern et al., PRL 2012) after T. Busch et al. (1998) B V int = g d(x 1 x 2 )

mean-field theory fails M. Rontani, PRL 108, 115302 (2012)

Scanning Tunneling Spectroscopy single molecule Repp et al., PRL 94, 026803 (2005) well-isolated nano-objects quantized energy levels Coulomb blockade short nanotube Venema et al., Science 283, 52 (1999) Maltezopoulos et al., PRL 91, 196804 (2003) quantum dot wave function imaging

STS in electrically insulated molecules Repp et al., PRL 94, 026803 (2005) NaCl NaCl layer layer metallic metallic substrate substrate tip 0 pentacene z J. Repp et al., Phys. Rev. Lett. 94, 026803 (2005). P. Liljeroth et al., Nano Lett. 10, 2475 (2010). C. J. Villagomez et al., Surf. Sci. 603, 1526 (2009). A. Bellec et al., Nano Lett. 9, 144 (2009).

STS in electrically insulated molecules 0 tip N NaCl NaCl layer layer pentacene metallic metallic substrate substrate z molecule z

STS in electrically insulated molecules N apply tip-gate bias voltage molecule z

STS in electrically insulated molecules di / dv PRL 94, 026803 (2005) N y x molecule z peak in di / dv from N to N - 1

STS in electrically insulated molecules ev di / dv PRL 94, 026803 (2005) N y x molecule z peak in di / dv from N to N + 1

mean-field theory e 5 N e e 4 3 e 2 di / dv ( r) ( r) d ( e EF) energy-resolved local density of states (LDOS) = single-particle level 2 e 1 molecule z Tersoff & Hamann 1985

mean-field theory di / dv PRL 94, 026803 (2005) exp y x DFT molecule z di / dv r HOMO ( ) 2

mean-field theory di / dv PRL 94, 026803 (2005) ev exp y x DFT molecule z di / dv r LUMO ( ) 2

ambiguities of mean-field theory N or N + 1 to compute self-consistent orbitals? molecule

ambiguities of mean-field theory N or N + 1 to compute self-consistent orbitals? N wave function is NOT a single N electron configuration : N N N ci i i i generic question what is actually measured in tunneling experiments in the presence of manybody interactions?

answer 1/t di /dv 2 2 M n( e f ) T = 0 density of final states Bardeen s viewpoint (1961): many-body matrix element connecting states with N and N-1 fermions in the trap / molecule / Quantum Dot * k, N k, N 1

resonant many-body states of the whole system barrier continuum dot N * k, N N tot - N z bar N - 1 k, N 1 N tot N + 1

M k k k r r r c c ˆ ) ( ) ˆ ( ) ( ˆ field operator single-particle QD state single-particle continuum state, *, ˆ 1, N k N k M M t d d ˆ ˆ ˆ ˆ * 2 ˆ bar 2 z z z z m M

M k k k r r r c c ˆ ) ( ) ˆ ( ) ( ˆ field operator single-particle QD state single-particle continuum state, *, ˆ 1, N k N k M M t d d ˆ ˆ ˆ ˆ * 2 ˆ bar 2 z z z z m M 1 / t 2

key quantity QP ( r) ˆ 1 ( r) N N hole quasiparticle wave function in practice: QP i, j N 1* N ( r) ci c j ( i, j) ( r) expansion coefficients: N N c j N j single-particle orbital

mean-field theory fails M. Rontani, PRL 108, 115302 (2012)

quasiparticle wave function theory M. Rontani, PRL 108, 115302 (2012)

evidence for fermionization M. Rontani, PRL 108, 115302 (2012)

evidence for fermionization g = 0 + g = M. Rontani, PRL 108, 115302 (2012) g = 0 g =

change of spectral weight M. Rontani, PRL 108, 115302 (2012) t 0 = WKB mean-field A QP = norm

initial + - - final - -

so far quasiparticle theory for decay time of 6 Li atoms works 1/t = (1/t 0 ) X (many-body factor) measure chemical potential and spectral weight

density of states density of states density of states density of states growth direction energy quantum confinement in nanostructures simplest case: quantum well A B A AlGaAs GaAs semiconductor A B A conduction band 2D confinement growth direction valence band 3D 2D 1D 0D energy energy energy energy bulk quantum well quantum wire quantum dot

experiment: InAs quantum dots x strong anisotropy x y y STM images Nano Lett. 7, 2071 (2007)

(di/dv)/(i/v) STS spectroscopy 5 4 3 2 Nano Lett. 7, 2071 (2007) Center side WL A B C D A: s state B: p y state C: p y state D: d state? 1 0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Voltage (V) N 0 N 1 first guess: ground- and excited-states A B C D x y

envelope function approximation, semiconductor effective parameters Hamiltonian full configuration interaction ground & excited states y x V(x,y) ) ( 2 * 2 2 2 0 2 2 0 * 2 1 2 i y i x i N i y x m m H single-particle term j i j i r r r e 1 2 2 many-body term compute the wavefunction as a superposition of Slater determinants ij i N i N i N H H c 0 ' l m N i c c Rontani et al., J. Chem. Phys. 124, 124102 (2006) compute ) ( QD r

correlation in a circular QD e d H N i1 2 2m* 2 i * 2 m 0 2 ( x 2 i y 2 i ) p s single-particle term e many-body term large e mean-field theory holds small e correlation effects

N 0 N 1 simulating the STS image di 2 2 QD ( r) 1 s( ) / dv r lengths in units of l QD

N 1 N 2 2 2 di / dv r QD ( r) 1 s( ) e no correlation lengths in units of l QD

N 1 N 2 di / dv r 100 e QD ( ) 2 mev lengths in units of l QD

N 1 N 2 di / dv r 50 e QD ( ) 2 mev lengths in units of l QD

N 1 N 2 di / dv r 10 e QD ( ) 2 mev lengths in units of l QD

N 1 N 2 di / dv r 5 e QD ( ) 2 mev lengths in units of l QD

N 1 N 2 di / dv r 1 e QD ( ) 2 mev lengths in units of l QD

N 1 N 2 di / dv r 0.5 e QD ( ) 2 mev lengths in units of l QD

N 1 N 2 di / dv r 0.1 e QD ( ) 2 mev lengths in units of l QD

N 1 N 2 di / dv r 0.05 e QD ( ) 2 mev lengths in units of l QD

N 1 N 2 di / dv r 0.01 e QD ( ) 2 mev lengths in units of l QD

effect of anisotropy

N 1 N 2 effect of anisotropy di / dv r QD ( ) 2 / 0x 0 y 1 lengths in units of l QD

N 1 N 2 effect of anisotropy di / dv r QD ( ) 2 / 0x 0 y 1.21 lengths in units of l QD

N 1 N 2 effect of anisotropy di / dv r QD ( ) 2 / 0x 0 y 1.69 lengths in units of l QD

N 1 N 2 effect of anisotropy di / dv r QD ( ) 2 / 0x 0 y 2.25 lengths in units of l QD

predictions STS image distortion driven by: confinement strength dot anisotropy dielectric environment

theo-exp comparison

theo-exp comparison B A = theo-exp matching

theo-exp comparison C = theo-exp matching

theo-exp comparison = theo-exp matching

theo-exp comparison Nano Lett. 7, 2071 (2007)

is the C state really due to double charging? C B low I stab in out 1 e - in the dot high I stab in out 2 e - in the dot Nano Lett. 7, 2071 (2007)

origin of the C-state image? A B C D

the origin of the C-state image 76 % 11 % 6 % s = + C state d

wires and carbon nanotubes A. Secchi, M. Rontani, PRB 85, 121410(R) (2012) N = 2 N = 3 Wigner localization observed in carbon nanotubes S. Pecker et al., Nature Phys. 2013

conclusions theo and exp of quasiparticle wave function imaging sensitivity to correlation effects

evidence of pairing in 1D P. D Amico and MR, arxiv:1404.7762 exp = G. Zuern et al., PRL 2013

correlated pair tunneling MR, PRA 88, 043633 (2013) P. D Amico and MR, arxiv:1404.7762 exp = G. Zuern et al., PRL 2013

http://web.nano.cnr.it/ispie2014/

http://web.nano.cnr.it/ispie2014/ Lecturers J. Baumberg: Polaritons L. Butov: Indirect excitons M. Dyakonov: Spin physics of excitons L. Keldysh*: Fundamentals of excitons L. Levitov: Exciton physics in bilayers and graphene A. Pinczuk: Optics of two-dimensional electron gases quantum Hall bilayers D. Ritchie: Electrically formed electronhole systems 2 hours + 1 hour discussion / exercise Speakers K. Bolotin: TBA F. Dubin: Dark excitons A. Holleitner: Few-exciton physics V. Pellegrini: TBA L. Pfeiffer: Semiconductor bilayers M. Polini: TBA L. Ponomarenko: Transport in bilayer graphene M. Rontani: Anomalous magnetization of carbon nanotubes as excitonic insulators P. Savvidis: TBA C. Tejedor: TBA M. Vladimirova: Spin dynamics in coupled quantum wells