Control of quantum two-level systems

Transcription

1 Control of quantum two-level sstems

2 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. General concept AS-Chap. 0 - How to control a qubit? Does the answer depend on specific realization? No, onl its implementation Qubit is pseudo spin General concept eists Independent of qubit realization Methods from nuclear magnetic resonance (NMR) Nuclear magnetic resonance Method to eplore the magnetism of nuclear spins Important application Magnetic resonance imaging (MRI) in medicine MRI eploits the different nuclear magnetic signatures of different tissues

3 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. General concept AS-Chap. 0-3 Brief MRI histor earl suggestions H. Carr (950) and V. Ivanov (960) 97 MRI imaging machine proposed b R. Damadian (SUNY) 973 st MRI image b P. Lauterbur (Urbana-Champaign) Late 970ies Fast scanning technique proposed b P. Mansfield (Nottingham) 003 Nobel Prize in Medicine for P. Lauterbur and P. Mansfield

4 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. NMR techniques AS-Chap. 0-4 Overview: Important NMR techniques Basic idea Rotate spins b static or oscillating magnetic fields Static fields parallel to quantization ais free precession changes φ on Bloch sphere Oscillating fields perpendicular to quantization ais change population changes θ on Bloch sphere Important protocols Rabi population oscillations Relaation measurement T Ramse fringes T Spin echo (Hanh echo) T corrected for reversible dephasing

5 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3. Control of quantum two-level sstems.. Quantum two-level sstem AS-Chap. 0-5 Hamiltonian of a quantum two-level sstem (TLS) Arbitrar TLS H = H H is Hermitian matri H H H, H R and H = H we choose H > H Smmetrize Subtract global energ offset H +H 4 H = ε Δ iδ = ε Δ + iδ ε σ z + Δ σ + Δ σ ε H H > 0 Δ, Δ R Natural or phsical basis φ +, φ H 0 = ε 0 0 ε = ε σ z H φ ± = ± ε

6 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Quantum two-level sstem Diagonalization of H H = ε Δ + iδ Δ iδ ε Eigenvalues det ε E ± Δ iδ = 0 Δ + iδ ε E ± ε E ± ε E ± Δ iδ Δ + iδ = 0 E ± = ± ε + Δ + Δ ± ħω q Eigenvectors Ψ + = +e iφ cos θ φ + + e +iφ sin θ φ transition energ convenientl epressed in frequenc units Ψ = e iφ sin θ φ + + e +iφ cos θ φ H Ψ ± = E ± Ψ ± tan θ Δ +Δ tan φ Δ Δ ε with 0 θ π with 0 φ π C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics Volume One, Wile-VCH Basis Ψ +, Ψ energ eigenbasis (because H has energ units) AS-Chap. 0-6

7 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Quantum two-level sstem AS-Chap. 0-7 Visualization on Bloch sphere z φ + Ψ + = +e iφ cos θ φ + + e iφ sin θ φ Ψ = e iφ sin θ φ + + e iφ cos θ φ θ Ψ + tan θ Δ +Δ tan φ Δ Δ ε with 0 θ π with 0 φ π Ψ φ Basis rotation on Bloch sphere! φ Vectors Ψ and Ψ + define new quantization ais H = ε +Δ +Δ σ z in basis Ψ, Ψ + Ψ and Ψ + become new poles

8 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Fictitious spin ½ AS-Chap. 0-8 Analog to spin ½ in static magnetic field Fictitious spin ½ in fictitious magnetic field B B z H = γħ B S = γħ B + ib γ is the gromagnetic ratio B ib B z B = B, B, B z T is the magnetic field vector Fictitous spin in fictitious B-field quantum TLS φ + φ u Ψ + u Ψ (u denotes the quantization ais along which H is diagonal) ħ γ B E + E = ħω q Polar angles of B θ, φ γħb z ε γħb Δ γħb Δ Orientation with respect to the quantization ais depends on ε, Δ, Δ An quantum TLS has a builtin static magnetic field NMR situation!

9 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Free precession AS-Chap. 0-9 Free precession Ψ = cos θ e + eiφ sin θ g In the energ eigenbasis g, e, the qubit state vector Ψ 0 is parallel to built-in field for Ψ 0 = g antiparallel to the built-in field for Ψ 0 = e Built-in field points along z-ais on Bloch sphere e z When qubit state vector Ψ 0 is not parallel or antiparallel to built-in field free evolution corresponds to free precession about the z-ais Ψ(t) θ 0 φ(t) Ψ 0 Also called Larmor precession φ 0 In absence of decoherence, onl φ t evolves linearl with time t g

10 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Free precession AS-Chap. 0-0 Larmor precession formal calcualtion Ψ = cos θ e + eiφ sin θ g Energ eigenbasis H = ħω q σ z fictitious field aligned along quantization ais Time-independent problem Develop into stationar states Ψ t = e Ψ 0 e iω qt/ e + g Ψ 0 e iω qt/ g Ψ 0 = cos θ 0 e + e iφ 0 sin θ 0 g Ψ(t) e z θ 0 φ(t) Ψ 0 Ψ t = e iω qt/ cos θ 0 e + eiφ 0e iω qt/ sin θ 0 g = cos θ 0 e + ei(φ 0+ω q t) sin θ 0 g φ(t) φ 0 g

11 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Rabi Oscillations AS-Chap. 0 - Rotating drive field H = γħ B z B + ib B ib B z Consider the qubit state vector Ψ epressed in energ eigenbasis g, e Appl a drive field with amplitude ħω d rotating about the z-ais at frequenc ω B z = ħω q z H t = ħ ω q ω d e iωt ω d e iωt ω q ωt B B 0 ω d 0 π 4 ω d ω d π 0 ω d 3π 4 ω d ω d π ω d 0 5π 4 ω d ω d 3π 0 ω d 7π 4 ω d ω d

12 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Rabi Oscillations AS-Chap. 0 - Driven quantum TLS H t = ħ ω q ω d e iωt ω d e iωt = ħω q ω q σ z + ħω d σ + e +iωt + σ e iωt H 0 H d Operators σ e g and σ + e g create or annihilate an ecitation in the TLS Here, our phsics convention is not ver intuitive, but consistent! Matri representation σ + = and σ = Drive rotating about arbitrar equatorial ais on the Bloch sphere H d = ħω d σ + e +i ωt+φ i ωt+φ + σ e without loss of generalit we choose φ = 0

13 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Rabi Oscillations Time evolution of driven quantum TLS H t = ħ ω q ω d e iωt ω d e iωt ω q Qubit state Ψ t = a e t e + a g t g obes Schrödinger equation iħ d dt Ψ t = H Ψ t i d dt a e t i d dt a g t = ω q a e(t) + ω d e iωt a g (t) = ω d eiωt a e (t) ω q a g(t) Time-dependent Difficult to solve Move to rotating frame! b e t e iωt a e t b g t e iωt a g t Schrödinger equation looses eplicit time dependence i d dt b e t i d dt b g t = ω q ω b e (t) + ω d b g(t) = ω d b e(t) ω q ω b g (t) AS-Chap. 0-3

14 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Rabi Oscillations AS-Chap. 0-4 Interpretation of the rotating frame i d dt b e t i d dt b g t = ω q ω b e (t) + ω d b g(t) = ω d b e(t) ω q ω b g (t) H t = ħ ω q ω d e iωt ω d e iωt ω q The frame rotates at the angular speed ω of the drive Driving field appears at rest Drive can be in resonance with Larmor precession frequenc ω q Awa from resonance ω ω q 0 red terms dominate no g e transitions induced b drive Near resonance ω ω q blue terms dominate g e transitions induced b drive Formal treatment Effective Hamiltonian H describing the same dnamics as H(t) H = ħ Δω ω d ω d Δω iħ d dt Ψ t = H Ψ t with Δω ω ω q Ψ t b e t e + b g t g

15 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Rabi Oscillations AS-Chap. 0-5 Dnamics of the effective Hamiltonian H = ħ Δω ω d ω d Δω Diagonalize H = tan θ = ω d Δω tan φ = 0 ħ Δω + ω d σ z New eigenstates Ψ + = +e iφ cos θ φ + + e +iφ sin θ φ Ψ = e iφ sin θ φ + + e +iφ cos θ φ Δω ω ω q Ψ + = + cos θ e + sin θ g Ψ = sin θ e + cos θ g Epand into stationar states Ψ t = Ψ Ψ 0 e +it Δω +ω d Ψ + Ψ + Ψ 0 e it Δω +ω d Ψ+ Initial state Ψ 0 = g (energ ground state) Ψ t = cos θ e+it Δω +ω d Ψ + sin θ e it Δω +ω d Ψ+

16 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Rabi Oscillations AS-Chap. 0-6 Probabilit P e to find TLS in e Ψ t = cos θ e+it Δω +ω d Ψ + sin θ e it Δω +ω d Ψ+ P e e Ψ t = e Ψ t Ψ + = + cos θ e + sin θ g = Ψ = sin θ e + cos θ g = sin θ cos θ e it Δω +ω d e + it Δω +ω d t Δω + ω d = sin θ sin Δω ω ω q tan θ = ω d Δω P e = ω d Δω + ω sin d t Δω + ω d Driven Rabi oscillations TLS population oscillates under transversal drive

17 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Rabi Oscillations AS-Chap. 0-7 Rabi Oscillations on the Bloch sphere Ψ t = cos θ e+it Δω +ω d Ψ + sin θ e it Δω +ω d Ψ+ On resonance ω = ω q Rotating frame cancels Larmor precession State vector Ψ t has no φ-evolution Ψ t = cos ω dt g + i sin ω dt e Rotation about -ais Ψ(t) z Δω ω ω q tan θ = ω d Δω Finite detuning Δω > 0 Additional precession at Δω Population oscillates faster Reduced oscillation amplitude ω d t P e = ω d t Δω + ω d Δω + ω sin d Ψ 0 = g

18 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Rabi Oscillations AS-Chap. 0-8 Rabi Oscillations Graphical representation P e P e = ω d Δω + ω sin d t Δω + ω d On resonance ω = ω q P e t Detuning Δω = 3ω d 0. t First eperimental demonstration with superconducting qubit Y. Nakamura et al., Nature 398, 786 (999)

19 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Rabi Oscillations Oscillating vs. rotating drive Microwave pulses π/ω P e = ω d Δω + ω sin d t Δω + ω d ω d Oscillating drive ħω d cos ωt = ħω d e +iωt + e iωt in frame rotating with +ω the e iωt -component rotates fast with ω For ω d ω this fast contribtion averages out on the timescale of the slowl rotating component Rotating wave approimation P e On resonance (ω = ω q ) t AS-Chap. 0-9

20 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Rabi Oscillations Important drive pulses on the Bloch sphere π/ω e z ħω d Δt z e i g e + i g g π-pulse (ω d Δt = π) g e flips, refocus phase evolution g π/-pulse (ω d Δt = π/) Rotates into equatorial plane and back AS-Chap. 0-0

21 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Rabi Oscillations Rabi Oscillations in presence of decoherence P e = ω d Δω + ω sin d t Δω + ω d Effect of decoherence (qualitative approach) Loss of coherent properties to environment at a constant rate Γ dec = π/t dec Change of coherence (time derivative) proportional to amount of coherence Eponential deca of coherent propert with factor e Γ dec t π = e t T dec Argument holds well for population deca (energ relaation, T ) Loss of phase coherence more diverse depending on environment (eponential, Gaussian, or power law) Eperimental timescales range from few ns to 00 μs Deca envelope P e Δω = e 0. 5 T dec t AS-Chap. 0 -

22 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Rabi Oscillations Rabi deca time Complicated interpla between T, T, and the drive P e = At long times, small oscillations persist Nevertheless useful order-of-magnitude check for T dec Important tool for single-qubit gates ω d Δω + ω sin d t Δω + ω d To determine T, T, T correctl, more sophisticated protocols are required energ relaation measurements, Ramse fringes, spin echo P e Δω = e 0. 5 T dec t AS-Chap. 0 -

23 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Rabi Oscillations AS-Chap. 0-3 Energ relaation on the Bloch sphere z Ψ θ, φ θ φ g Environment induces energ loss State vector collapses to g Implies also loss of phase information Intrinsicall irreversible T -time rate Γ = π T Golden Rule argument Γ S ω q S ω is noise spectral densit High frequenc noise Intuition: Noise induces transitions Quantum jumps Single-shot, quantum nondemotiton measurement ields a discrete jump to g at a random time Probabilit is equal for each point of time Eponential deca with e t T

24 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Rabi Oscillations Dephasing on the Bloch sphere e z θ 0 Ψ(t) Ψ θ 0, φ 0 Environment induces random phase changes Γ S ω 0, T = π Γ Low-frequenc noise is detuned No energ transfer /f-noise S ω ω Eample: Two-level fluctuator bath To some etent reversible Deca laws e T, t e t t β T, T φ 0 g Visualization Phase φ becomes more and more unknown with time Classical probilit, no superposition! Phase coherence lost when arrows are distributed over whole equatorial plane AS-Chap. 0-4

25 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Energ Relaation Qubit dnamics Relaation Init π Δt Measurement Rotating frame & no detuning (Δω = ω ω q = 0) no -evolution P e e T Δt AS-Chap. 0-5

26 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Ramse fringes Qubit dnamics Ramse fringes (T ) Init π/ free evolution time Δt π/ Measurement Δω > 0 Δω = e P e 0. 5 Beating ΔT = π/δω T = T + T Energ deca alwas present! T Δt AS-Chap. 0-6

27 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Spin echo Qubit dnamics Spin echo (T E ) Init π/ free evolution time Δt / π Refocussing time Δt / π/ Measurement Δω = 0 P e Refocussing pulse reverses low-frequenc phase dnamics T E > T 0. 5 e 0. 5 T E Δt AS-Chap. 0-7

28 R. Gross, A. Mar & F. Deppe, Walther-Meißner-Institut (00-03) 0.3 Control of quantum two-level sstems.. Qubit Control: Ramse vs. Spin echo sequence F(ωt/) Ramse vs. spin echo sequence Spin echo cancels the effect of low-frequenc noise in the environment Pulse sequences act as filters! Environment described b noise spectral densit S(ω) Deca envelope e t F R ωt F E ωt + ωt = sin ωt ωt = sin4 4 ωt 4 Sequence length t is important! Spin echo sequence S ω F R,E ωt dω Ramse filter F R spin echo filter F E ωt/ Filters low-frequenc noise for ωt 0 ωt Noise field fluctuates snchronousl with π-pulse No effect AS-Chap. 0-8