INTRODUCTION TO NMR and NMR QIP

Transcription

1 Books (NMR): Spin dynamics: basics of nuclear magnetic resonance, M. H. Levitt, Wiley, The principles of nuclear magnetism, A. Abragam, Oxford, Principles of magnetic resonance, C. P. Slichter, Springer, Reviews (NMR QIP): NMR techniques for quantum control and computation, LMK Vandersypen and I. Chuang, Reviews of Modern Physics (2004) vol. 76 (4) pp Quantum information processing using nuclear and electron magnetic resonance: review and prospects, et al., arxiv: NMR Based Quantum Information Processing: Achievements and Prospects, D. G. Cory et al., arxiv:quant ph/

2 What is Spin? Stern-Gerlach Experiment (1922) (s=1/2) E = µ B = µ z B z

3 Zeeman Energy E = µ B 1 0 = µ z B z H = µb z σ z /2 = µb 2 z 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

4 Spin 1/2 nucleus NMR freq (MHz) (at 10 T) natural abundance 1 H % 13 C % 15 N % 19 F % 28 Si % 31 P % Nuclei with an odd (even) number of nucleons possess half integer (integer) spin Nuclei with an even number of protons and neutrons are spinless

5 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

6 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 e +iω =e iω 1 t 2 1 t 2 + i = +

7 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 ) ω 1 = 2π 426MHz ω 2 = 2π 107MHz J Hz 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 C Chloroform molecule, CHCl 3

8 NMR transitions (spectrum) 1 ( 2 ω 1 + ω 2 ) + π 2 J ( 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 ( )

9 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

10 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 ))]

11 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 =

12 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 ( ) 2 Y Y X Y ( 0 + i 1 ) 2

13 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)

14 q 1 q 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 = I σ 2x U 5 = e iπ 2 I 1 σ 2x 2

15 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

16 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)

17 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 ρ dev 4σ 1z + σ 2z =

18 Example: making a 1 H 13 C pseudopure state Apply three permutation circuits and sum results: ρ dev ρ dev = I ρ dev ρ dev 00 00

19 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

20 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.

21 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

22 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!

23 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

24 NMR Spectrometer

25 Deutch-Jozsa Algorithm balanced constant

26 Quantum circuit to compute binary function f. Or another representation:

27 Hadamard gate H 1 ( ) f (x) 0 1 2

28

29 NMR pulse sequence 1 H 1 0 R y 90 R x 180 R x 180 R z 90 1 H 2 0 R x R y R y 90 τ τ Rx 180 R x 180 R x ±90 ( U 01,U 10 )

30 NMR results f 00 f 01 f 10 f 11 Jones and Mosca, 1998

31 Factoring 15 Vandersypen et al, 2001

32 Current experiments at IQC: 12-qubit NMR QIP

33 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

34 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

35 Relaxation A 0 = p 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 ~ s T 1 ~ 5 20 s