Books (NMR): Spin dynamics: basics of nuclear magnetic resonance, M. H. Levitt, Wiley, 2001. The principles of nuclear magnetism, A. Abragam, Oxford, 1961. Principles of magnetic resonance, C. P. Slichter, Springer, 1990. Reviews (NMR QIP): NMR techniques for quantum control and computation, LMK Vandersypen and I. Chuang, Reviews of Modern Physics (2004) vol. 76 (4) pp. 1037 1069. Quantum information processing using nuclear and electron magnetic resonance: review and prospects, et al., 2007. arxiv: 0710.1447. NMR Based Quantum Information Processing: Achievements and Prospects, D. G. Cory et al., 2000. arxiv:quant ph/0004104.
What is Spin? Stern-Gerlach Experiment (1922) (s=1/2) E = µ B = µ z B z
Zeeman Energy E = µ B 1 0 = µ z B z H = µb z σ z /2 = µb 2 z 0 1 2 E = hν = µb z H = µb z 2 H = +µb z 2 Electron: µ 2µ B = e m Nucleus: µ = γ n gyromagnetic ratio of isotope n
Spin 1/2 nucleus NMR freq (MHz) (at 10 T) natural abundance 1 H 426 99.9% 13 C 107 1.1% 15 N 43 0.4% 19 F 401 100% 28 Si 85 4.7% 31 P 175 100% Nuclei with an odd (even) number of nucleons possess half integer (integer) spin Nuclei with an even number of protons and neutrons are spinless
1 H NMR Hamiltonian: 2 coupled spins in liquid state NMR (e.g. 13 C labelled Chloroform) ( =1) H 0 = 1 ( 2 ω 1σ 1z + ω 2 σ 2z + πj 12 σ 1z σ ) 2z 13 C Chloroform molecule, CHCl 3 Larmor frequency (rad/s): External magnetic field (T): Coupling strength (Hz): J kl ω k = γ k B B Chemical shielding: For the same isotope, γ j = γ ( 0 1 δ ) j for nucleus at the j th chemical site. Ability to address different nuclei as qubits
Larmor precession of a single spin H 0 = ω 1σ 1z 2 U(t) = e iω 1tσ 1z 2 = Take ψ = + 2 U(t) ψ = e iω 1t 2 + e +iω 1t 2 2 + e +iω =e iω 1 t 2 1 t 2 + i = +
1 H Eigenstates of the Hamiltonian H 0 = 1 ( 2 ω 1σ 1z + ω 2 σ 2z + πj 12 σ 1z σ ) 2z state 1: 1 H 2: 13 C eigenvalue 1 ( 2 ω 1 + ω 2 + πj 12 ) 1 2 ω 1 + ω 2 πj 12 1 ( 2 ω 1 ω 2 πj 12 ) ( ) 1 ( 2 ω ω + πj 1 2 12) ω 1 = 2π 426MHz ω 2 = 2π 107MHz J 12 200Hz B =10T 1 ( 2 ω 1 + ω 2 ) 1 ( 2 ω + ω 1 2) 1 ( 2 ω 1 ω 2 ) 1 ( 2 ω 1 ω 2 ) + π 2 J 12 π 2 J 12 π 2 J 12 + π 2 J 12 13 C Chloroform molecule, CHCl 3
NMR transitions (spectrum) 1 ( 2 ω 1 + ω 2 ) + π 2 J 12 2 2 1 1 1 ( 2 ω 1 + ω 2 ) 1 ( 2 ω 1 ω 2 ) π 2 J 12 π 2 J 12 2πJ 2πJ 1 ( 2 ω 1 ω 2 ) + π 2 J 12 ω 1 ω 2 frequency These are magnetic dipole transitions, and are thus driven by an oscillating (Radiofrequency, RF ) magnetic field Need transverse spin operators to drive them. σ x,σ y ( )
RF Hamiltonian: H rf (t) = Ω(t) 2 θ(t) = ω rf t + φ [ σ 1x cos(θ(t)) + σ 1y sin(θ(t)) ] RF amplitude RF frequency B x cos(ω rf t) ω rf +ω rf
Rotating reference frame transformation: two isotopes (doubly rotating frame) R(t) = e iω 1rf tσ 1z 2 e iω 2rf tσ 2z 2 ψ R = R(t) ψ lab id ψ lab dt = H lab ψ lab id ψ R dt = H R ψ R H lab = H 0 + H rf (t) H R = 1 2 ω 1 ω 1rf [( )σ 1z + ( ω 2 ω 2rf )σ 2z + πj 12 σ 1z σ 2z ] + 1 [ 2 Ω 1(t) ( σ 1x cos(φ 1 ) + σ 1y sin(φ 1 ))] + 1 [ 2 Ω 2(t) ( σ 2x cos(φ 2 ) + σ 2y sin(φ 2 ))]
Rotating frame = We had Larmor precession in Lab frame: + e +iω1t U(t) ψ Lab = e iω 1 t 2 2 But in the rotating frame (on resonance): + i R(t) U(t) + R(t) ψ Lab (t) = R(t) ( U(t) ψ Lab ) = + 2 =
Resonant pulses as Bloch sphere rotations Consider a single spin at resonance: ω 1 = ω 1rf H R = 1 [( 2 ω 1 ω 1rf )σ 1z + Ω 1 (t)( σ 1x cos(φ 1 ) + σ 1y sin(φ 1 ))] In NMR, we can control both RF phase ( ) and amplitude ( ) in φ Ω time, so we can perform arbitrary single spin rotations U(t) = e iω 1 t ( σ 1x cos(φ 1 )+σ 1y sin(φ 1 ) ) / 2 R y (π /2) φ = π /2 Ωt = π /2 R x (π /2) φ = π Ωt = π /2 Y X ( 0 + 1 ) 2 Y Y X Y ( 0 + i 1 ) 2
Example of two qubit logic: the CNOT gate H(t) = 1 ( 2 ω 1σ 1z + ω 2 σ 2z + πjσ 1z σ 2z ) + H rf (t) Zeeman coupling RF pulses (control)
q 1 q 2 1 2 3 4 5 R z (90) R y (90) R y ( 90) R x (90) U 1 = e i π 2 I 1 σ 1y 2 t = π /2J time U 2 = e i π 2 σ 1z σ 2z / 2 U 3 = e iπ 2 σ 1z 2 I 2 U 4 = e +iπ 2 I 1 σ 2y 2 U CNOT = U 5 U 4 U 3 U 2 U 1 = 0 0 1 I 2 + 1 1 1 σ 2x U 5 = e iπ 2 I 1 σ 2x 2
The thermal state A general mixed state: ρ = k a k ψ k ψ k Thermal equilibrium (Boltzmann distribution): ρ th = e H 0 kt Tr e H 0 ( kt ) In the high temperature limit ( ) H 0 << kt ρ th 1 2 I H 0 I kt 2 1 4kT ω 1σ 1z + ω 2 σ 2z ( ) (two spin case) ω kt ~ 10 5 at room temperature in NMR
The thermal state ρ th 1 2 I H 0 I kt 2 1 4kT ω 1σ 1z + ω 2 σ 2z ( ) The identity term cannot be changed by pulses or observed So we define a deviation density matrix (note it is not a proper density matrix, since ) Tr(ρ) 1 ρ dev = H 0 kt ω 1σ 1z + ω 2 σ 2z indicates a population difference σ z between spin up and spin down states (the J coupling term is ~10 6 times smaller than Zeeman energy)
Pseudopure states To initialize an NMR quantum info. processor, we want a pure state or at least one that behaves like a pure state: ρ pp = I 2 α ψ ψ Example: making a 1 H 13 C pseudopure state 5 0 0 0 0 3 0 0 ρ dev 4σ 1z + σ 2z = 0 0 3 0 0 0 0 5
Example: making a 1 H 13 C pseudopure state Apply three permutation circuits and sum results: ρ dev ρ dev 5 0 0 0 0 3 0 0 0 0 3 0 0 0 0 5 5 0 0 0 0 5 0 0 0 0 3 0 0 0 0 3 15 0 0 0 0 5 0 0 = 20 00 00 5 I 0 0 5 0 0 0 0 5 ρ dev 5 0 0 0 0 3 0 0 0 0 5 0 0 0 0 3 ρ dev 00 00
Measurement in NMR Bulk magnetization of j th isotope: M z j = µ j ( n j n ) j 90 degree ( readout ) pulse rotates magnetization to x y plane: Z M z B // ˆ z Ensemble of identical, noninteracting processor molecules Y X M x Y
Measurement in NMR In the lab frame, magnetization precesses in the x y plane, at the Larmor frequency Z Y X M x Y the oscillating magnetic flux induces an emf in the solenoid coil via Faraday s law.
Measurement in NMR This corresponds to an expectation value measurement, with observables: M x = Tr ρ σ x,k = Tr ρ dev k M y = Tr ρ σ y,k = Tr ρ dev k k k σ x,k σ y,k We call these observables transverse magnetization
Measurement in NMR Example: 1 qubit tomography of an arbitrary state ρ dev Expt. 1 ρ dev M x, M y p x, p y Expt. 2 ρ dev R y (π /2) M x p z ρ dev p xσ x + p y σ y + p z σ z p x 2 + p y 2 + p z 2 Unfortunately, the # of expts needed grows exponentially with # qubits, regardless of the physical system we use for implementation!
Measurement in NMR : FT spectrum V (0) M x (0) NMR transitions Voltage T 2 V (0) /1.6 Fourier transform Δω ~ T 2 1 Time amp M x (0) Frequency free induction signal ( ) M x (t) = Tr( ρ(t) σ x ) = Tr e ih 0 t ρ(0)e +ih 0 t σ x
NMR Spectrometer
Deutch-Jozsa Algorithm balanced constant
Quantum circuit to compute binary function f. Or another representation:
Hadamard gate H 1 ( ) f (x) 0 1 2
NMR pulse sequence 1 H 1 0 R y 90 R x 180 R x 180 R z 90 1 H 2 0 R 180 90 x R y R y 90 τ τ Rx 180 R x 180 R x ±90 ( U 01,U 10 )
NMR results f 00 f 01 f 10 f 11 Jones and Mosca, 1998
Factoring 15 Vandersypen et al, 2001
Current experiments at IQC: 12-qubit NMR QIP
Relaxation and decoherence General Kraus operator sum: ρ A k ρa k + k k A k + A k = I Z Relaxation Dephasing Spin operators σ x,σ y σ z X Y Example: pure dephasing A = e iφσ z ρ dφ e iφσ z ρe+iφσ z f (φ,t,t 2 ) Lorentzian with time dependent FWHM ( ) a be t /T 2 b * e t /T 2 Dephasing on the Bloch sphere 1 a
Pure dephasing Or in another form: Z Λ(ρ) = ( 1 p) ψ ψ + pσ z ψ ψ σ z With Kraus operators: A 0 = 1 pi Y A 1 = pσ z p = 1 e t /T 2 2 Dephasing on the Bloch sphere
Relaxation A 0 = p 1 0 0 1 η A 1 = p 0 η 0 0 A 2 = 1 η 0 1 p 0 1 A 3 = 1 p 0 0 η 0 see e.g. Nielsen & Chuang ( Λ(ρ) = a a 0)e t /T 1 + a 0 be t / 2T 1 b * e t / 2T 1 ( a 0 a)e t /T 1 + (1 a 0) For pure relaxation, T 2 = 2T 1 In liq. state NMR, typically T 2 ~ 0.5 5 s T 1 ~ 5 20 s