Quantum tunnelling and coherence of mesoscopic spins B. Barbara, Institut Néel, CNRS, Grenoble Brief history From classical to quantum nanomagnetism Quantum nanomagnetism From relaxation to coherence Ensemble single molecules magnets, single ions magnets Conclusion
Brief history 70 s: Search for «macroscopic quantum tunnelling» phenomena ( Schrödinger, Leggett) 1981 First evidence of MQT in J - J (R. Voss & R. Webb, IBM Yorktown-Heights) 1973-1988 Rare-earths with «narrow domain walls»: Dy 3 Al 2, SmCo 3.5 Cu 1.5 80 s-90 s T-independent relaxation Films, nanoparticles ensembles: a-smco, a-tbfe, (TbCe)Fe 2, Theory: T. Egami R. Schilling, J.L. van Hemmen, P. Stamp, E. Chudnovsky, L. Gunther, N. Prokof ev, 90 s Two directions: 1) single particule Micro-SQUIDs 2) ensembles of identical nanoparticles Single Molecules Magnets
1993-1996 1986-1995 1973-198 ingle Molecule Magnetic Protein Cluster Nanoparticle 1 nm 2 nm 3 nm 20 nm K. Ziemelis, Nature, «Milestones on Spin», S19, March 2008 (Produced by Nature Physics)
Quantum nano-magnetism Mn12 acetate (very schematic) Mn(III) S=2 Mn(IV) S=3/2 Total Spin = 10 T. Lis, Acta. Cryst. 1980
Single molecule magnets of Mn 12 -ac The molecules are regularly arranged in the crystal Macroscopic quantum magnet 1 mm From Kunio Awaga, Nagoya university
Macroscopic quantum magnet of Fe 8 1 mm From Satoru Maegawa, Kyoto university
Mn12 acetate (very schematic) Mn(III) S=2 1 M Mn(IV) S=3/2 0 H A H -1-100 0 100 Total Spin = 10
Resonant Tunneling of Magnetization (in Mn 12 -ac) Quantum tunneling and classical hysteresis NATO ASI workshop «Quantum Tunnelling of Magnetization», 1994 Grenoble (Organization: B.B., L.Gunther, N.Garcia, and A.J. Leggett). 1,0 1 M / M s Hysteresis loop 0,5 0,5 M/M S 0,0 0-0,5-1 -1,0 0.55 K 0.65 K 0.7 K 0.75 K 0.45 K 1.5K 1.6K 1.9K 2.4K -3-2 -1 0 1 2 3-5,5-5,0-4,5-4,0-3,5-3,0-2,5-2,0 H n = nd/gµ B ~ nh A /2S B L (T) B 0 (T) 0.8 K 0.85 K 0.9 K 0.95 K 1 K 1.1 K 1.3 K 1.37 K I. Chiorescu et al L. Thomas et al, Nature (1996); Friedman et al, PRL (1996). B.B. et al, JMMM (1995)
Classical barrier and tunnelling of a collective spin (S=10) 0 0 0 0 H = - DS z2 -BS z4 - gµ B S z H z -7-8 -9 Low Temperature regime ω ²M = ±1-10 10 0-1 -0.5 0 0.5 1 µ 0 H z (T) 0 - gµ B (S + + S - )H x /2 + E(S +2 + S -2 ) - C(S +4 + S -4 ) 7 8 9 Energy (K) Thermally activated reversal S, m-n > S, -m > S,-S+2> S,S-2> 1 P S,-S+1> S, -m > S,-S> spin down H z = 0 S, m-n > Ground state tunneling S Z magnetic field S,S-1> S,S> spin up S + Landau-Zener Γ 2 (TS model n /DS 2 ) 4S/n H 0 (TS n /DS H = 2 0) 2S/n, n 2S H A Probability: Resonances P LZ = 1 exp[-π( /ħ) «under 2 the /γc] barrier ~ 2 /c» 1 - P Thermally activated tunnelling ħ, v H /
Usual double-well energy barrier E = Dm 2 E(θ) with θ = Cos -1 (m/s) ; m = <S z >
Effect of long-range dipolar interactions lassical barrier E(m) = - Dm 2 E(θ) with θ = cos -1 (m/s) ; m = <S z > From «zero-kelvin» tunneling to Equilibrated superparamagnetism B n (T) 5,0 10-0 10-1 4,5 4,0 3,5 3,0 9-0 9-1 9-2 8-0 8-1 8-2 7-0 7-1 7-2 6-0 6-1 6-2 20 (n-p) : -S+p S-n-p E (K) 10 0-10 -20-30 9-2 10-1 8-2 9-1 10-0 7-2 8-1 9-0 6-2 7-1 8-0 6-1 7-0 6-0 N(E) 0,4 0,6 0,8 1,0 1,2 1,4 T(K) 3,0 3,5 4,0 4,5 5,0 B 0 (T) Local fields small shifts of spin levels Inhomogeneous distribution of states Thermally activated tunneling P ~ e B - E/kT
From a single spin to an ensemble of spins Effects of the magnetic environment (spin-bath) Long range dipolar interactions at T=0 Kelvin LZ probability: P LZ = 1 exp[-π( /ħ) 2 /γc] ~ 2 /c Spin-bath (Prokofiev & Stamp, 2000) P SB ~ ( 2 /ω 0 )e - ξ /ξ 0 n(ed ) >> P LZ energy 1/2,1/2> 1/2,-1/2> S, m-n > ² 0 S, -m > 1-P 1 - P _ hω H = 0 2 +(2µ B B 0 ) 2 ξ 0 = hyperfine tunnel window >>> T < T c 1 1 P energy 1/2,-1/2> S, -m > P 1/2,1/2> S, m-n > Non-equilibrium hole magnetic field 0,0 applied field Allows observation of mesoscopic tunnelling Strong decoherence
Fom molecules to simple paramagnetic ions Nuclear spins! Molecules of Mn 12 ac Ho 3+ ions in YLiF 4 1 1,0 M/M S M/M S 0,5 0-0,5-1 -3-2 -1 0 1 2 3 B L (T) 1.5K 1.6K 1.9K 2.4K 0,5 0,0 dh/dt=0.55 mt/s -0,5-1,0 200 mk 150 mk 50 mk 300 200-80 -40 0 40 80 120 µ 0 H z (mt) 1/µ 0 dm/dh z (1/T) n=1 dh/dt > 0 n=2 100 n=0 n=-1 n=3 0-20 0 20 40 60 80 L.Thomas, F. Lionti, R. Ballou, R. Sessoli, R. Giraud, W. Wernsdorfer, D. Mailly, A.Tkachuk, D. Gatteschi, and B. Barbara, Nature, 1996. and B. Barbara, PRL, 2001. Steps at B n = 450.n (mt) Steps at B n = 23.n (mt)
Ising CF Ground-state + Hyperfine Interactions H = H CF-Z + A{J z I z + (J + I - + J - I + )/2} The ground-state doublet 2(2 x 7/2 + 1) = 16 states E (K) -178,5-179,0-179,5 I = 7/2 5/2-180,0 3/2-7/2-200 -150-100 -50 0 50 100 150 200 µ 0 H z (mt) g J µ B H n = n.a/2-7/2-5/2 5/2 7/2 7/2 1,0 0,5 0,0-0,5-1,0 M/M S 200 mk 150 mk 50 mk 300 200-80 -40 0 40 80 120 µ 0 H z (mt) 1/µ 0 dm/dh z (1/T) A = 38.6 mk n=1 dh/dt > 0 n=2 100 n=0 n=-1 n=3 0-20 0 20 40 60 80 Avoided Level Crossings between Ψ, I z > and Ψ +, I z > if I= (I z -I z )/2= odd Co-Tunneling of electronic and nuclear momenta
1,0 0,5 E (K) 0,0-0,5 40 20 0 a) M/M S 50 mk -179,0 0.3 T/s -1,0 n -300-200 -100 0 100 200 300 µ 0 H z (mt) 60 b) 8 n = 6-179,5 n=0 1/µ 0 dm/dh z (1/T) 50 mk 200 mk 0.3 T/s -180,0 Additional steps at intermediate fields n=1 µ 0 H n (mt) 4 0 240 180 120-60 120 160 200 240-150 -75 0 75 150 225 µ 0 H z (mt) 60 0 linear fit µ 0 H n = n x 23 mt -120 integer n -180 half integer n -8-6 -4-2 0 2 4 6 8 10 n = 7 n = 8 n = 9 dh/dt<0 Giraud & B.B et al, Phys. Rev. Lett. (2001) Fast measurements (τ meas ~ τ bott > τ 1 >> τ s ) Simultaneous tunneling of Ho 3+ pairs (4-bodies tunnelling) Detailed studies in ac-susceptibility Accurate fits with spins-spins, spin-phono bottleneck, weak CF disorder (B.Malkin):
Ho: YLiF 4 Er: CaWO 4 X band spectrometer (9-10GHz) Continuous wave (CW) Time resolved (TR) or pulsed Temperature 2.5K to 300K Bruker Elexys E580 Copyright CEA-Grenoble
Calculated energy spectrum 167 Er 3+ :CaWO 4 I=7/2 7/2 5/2 3/2 1/2-1/2-3/2-5/2-7/2 φ +, m I > m J = ±1 m I =0 H c-axis -7/2-5/2-3/2-1/2 1/2 3/2 5/2 7/2 m I = ±1 φ -, m I > φ 1 > = α 13/2> + δ 11/2> + β 5/2> + γ 3/2>
CW-EPR (9.7 GHz) EPR sequence used Electro-nuclear Rabi oscillations Narrow lines t (µs) I=0 H//b Er (0.001%):CaWO 4 I=7/2 ( 167 Er) I =7/2 ( 167 Er) I=0 Excitation π/2 π Echo <S z > Pulsed EPR 9 decoupled 9 decoupled qubits, qubits, adressed adressed with with small small fields fields (large (large moments) moments) S. Bertaina, S. Gambarelli, A. Tkachuk, B. Malkin, A. Stepanov, and B.Barbara, Nature nanotechnology, (2007) A new class of spins qubits
Effect of anisotropic hyperfine interaction H c Ω R (n,m,µ)~ g J h µ <φ n S + φ m > Calculated in the electro-nuclear 128 dimension basis
Rabi frequency Ω nm measured vs microwave ac field for three different orientations Lines: Calculated Rabi frequency: Ω µ =g µ µ B h µ with g µ = g J <φ 1µ S + φ 2 µ >
Rabi oscillations on Er (0.001%):CaWO 4, H=0.522 T //c, 0.15 mt //b, at 3.5 K Ω/2π ~ ±17 MHz (halfwidth ~2 MHz ~ π/τ R ) Varie avec The phase of the wave function Ψ( t) = a( t) Φ Φ 1 + b( t) is preserved at the timescales of µs φ1 > = α 13/2> + δ 11/2> + β 5/2> + γ 3/2> φ2 > = α 13/2> δ 11/2> β 5/2> γ 3/2> 2 <S z > = S 0 e -t/τ R sin(ωt/2) τ R = 0.2 µs << τ 2 ~ 7µs
Effect of the microwave power (0.05% Er:CaWO 4 ) h ac = 0.6 mt h ac =0.05mT The damping rate decreases with concentration and power Spin-bath decoherence Stochastic noise, interferences
The damping rate scales with the Rabi frequency Decoherence in spin + electromagnetic baths Stamp and Prokof ev Spin-bath Stochastic noise, interferences 1/τ R = 1/τ 2 g(ω R τ 2 ) 1/τ 2
Coherence times τ 2 vs T «Pairwise decoherence» T cosh 2 Gd 3+ T 2 = 1ms, 4K
Coherent multilevel manipulations in Gd:CaWO 4 2 close transtions: 1 st excite and 2 nd probe ν 1 ν 2 ν 2 τ 2 1ms at 4K ν 2 3 2 10 ms at 80 K ν 1 1
From ions to molecules Molecular and supramolecular chemistry Achim Müller (Bielfeld)
V ( H ) [ ] IV III 6 V As O 15 15 6 42 2O S=1/2 Huge Hibert space! D H = 2 15 6 nm
( ) ( ) = = = < = + + + = 3 1 3 1 12,13,31 3 1, 0 i i B i j i ij j i ij j i j i j i g A J H S H S I S S D S S µ Complex Hamiltoninan
Well separated molecules Random orientation {( CH 2 ) 17 Me} N CH 3 CH 3 H 3 C N CH 3 V 15 H 3 C N H 3 C + H 3 C N H 3 C CH 3 N CH 3 H 3 C CH 3 N H 3 C H 3 C N {( CH 2 ) 17 Me} DODA + ~ 13 nm Surfactants DODA = Me 2 N{(CH2)17Me} 2 + Embedding material for anionic clusters
First Rabi Oscillations in a Molecular Magnet Doublets Entanglement of 15 spins with photons Quartet D H = 2 15 Factor of merit ~ 1 See also: P. Stamp, Nature News & Views May 2008 R. Winpenny, Angew. Chem. Highlights Sept. 2008 Ardavan et al, PRL (2007) Factor of merit ~ 10 S. Bertaina, S. Gambarelli, T. Mitra, B. Tsukerblat, A. Müller, & B. Barbara
T 2 = 1 tanh Inter-molecular décohérence (V 15 ) / kt 2 P. Stamp et al PRL, 2006 Pairwise decoherence i j Ai, j 2 100 µs Well separated molecules! Protons 4π = 18 s ~ 9 µs E T2 µ 2 CW-EPR ( 1 H, 75 As, 51 V) First time that decoherence can be explained quantitatively (Any system!) Intra-molecular decoherence Nuclear spins ( 1 H, 75 As, 51 V) W ~ 34 mt 4π E= W/2 ~17 mk T2 = 2 1µ s E Main decoherence: Nuclear Spins of V 15
CONCLUSION Entanglement of photons with a complex molecule with huge Hilbert space Self-organized 2D supra-molecular depositions become possible From: I. Chiorescu, Y. Nakamura, K. Hartmans, H. Mooij et al, Delft University of Technology M. Ruben, J. V. Barth et. al., INT Karlsruhe, TU Munich 100 µs expected
Collaborations Quantum coherence (rare-earth ions, V 15 ) S. Bertaina (Grenoble,Tallahassee), S. Gambarelli (Grenoble), A. Stepanov (Marseille), B. Malkin (Kazan), A.M. Tkachuk (St. Petersbourg). A. Müller and his group (Bielefeld). Thank You!! Quantum relaxation (initial studies in Mn 12 -ac, rare-earth ions) L. Thomas (IBM-Almaden), Chiorescu (Tallahassee), W. Wernsdorfer (Grenoble) R. Giraud (LPN-Marcoussis), D. Gatteschi and his group (Florence).