Control of quantum two-level systems

Transcription

1 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) AS-Chap Control of quantum two-level sstems

2 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) AS-Chap General concept 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, and K. Fedorov Walther-Meißner-Institut (00-08) AS-Chap 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, and K. Fedorov Walther-Meißner-Institut (00-08) AS-Chap 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 Controlled ecitations Relaation measurement T Ramse fringes T Spin echo (Hahn echo) corrects T for reversible dephasing T NMR reminder Spin-lattice (longitudinal) relaation time τ Spin-spin (transversal) relaation time τ τ Free induction deca (FID) in presence of magnetic field inhomogeneities τ τ In traditional NMR Hahn echo does not alwas ield τ Multiple echoes (CPMG)

5 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) The 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 i Δ = ε Δ + i Δ ε σ z + Δ σ + Δ σ ε H H > 0 and Δ, Δ R Without loss of generalit, we usuall assume Δ = 0 Natural or phsical basis φ +, φ H 0 = ε 0 0 ε = ε σ z H φ ± = ± ε A quantum TLS is called qubit in information processing! C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Volume One (Wile VCH) AS-Chap

6 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) Diagonalization of H Eigenvalues det ε E ± Δ i Δ = 0 Δ + i Δ ε E ± H = ε E ± ε E ± Δ i Δ Δ + i Δ = 0 E ± = ± Eigenvectors H Ψ ± = E ± Ψ ± ε + Δ + Δ ± ħω q Ψ = e i φ sin θ φ + + e +i φ cos θ φ Ψ + = +e i φ cos θ φ + + e +i φ sin θ φ ε Δ + i Δ Transition energ convenientl epressed in frequenc units Δ i Δ ε Recovers Bloch sphere picture! tan θ Δ + Δ tan φ Δ Δ ε with 0 θ π with 0 φ π Bloch angles! 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

7 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) AS-Chap Visualization on the Bloch sphere z φ + Ψ + = +e i φ cos θ φ + + e i φ sin θ φ Ψ = e i φ sin θ φ + + e i φ cos θ φ Ψ φ θ Ψ + tan θ Δ + Δ tan φ Δ Δ ε with 0 θ π with 0 φ π Basis rotation on Bloch sphere! Vectors Ψ and Ψ + define new quantization ais φ H = ε +Δ + Δ σ z in basis Ψ, Ψ + Ψ and Ψ + become new poles Practical relevance ε or Δ (and therefore the rotation angles) often depend on an eternal control parameter!

8 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) AS-Chap 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 An quantum TLS has a builtin static field NMR situation! Field orientation with respect to the quantization ais depends on ε, Δ, Δ

9 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) AS-Chap Dnamics of the qubit state Analog to spin ½ in static magnetic field Dnamics as known from NMR! Intrinsic dnamics Free evolution Precession about z-ais Driven evolution Population (Rabi) oscillations Pulsed driving schemes Ramse fringes Hahn echo NMR-tpe control and pulsed control of ħω q Single qubit gates

10 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) AS-Chap Free precession Ψ = cos θ e + eiφ sin θ g In the energ eigenbasis Free evolution corresponds to free precession about the z-ais 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 When qubit state vector Ψ 0 is not parallel or antiparallel to built-in field Ψ(t) e z 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

11 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) Larmor precession formal calculation Ψ = cos θ e + eiφ sin θ g Energ eigenbasis H = ħω q σ z fictitious field aligned along quantization ais Time-independent H Develop into stationar states Ψ(t) = e Ψ 0 e iω q t e + g Ψ 0 e iω q t g e z Ψ 0 = cos θ 0 Ψ t e + e iφ 0 sin θ 0 = e iω q t cos θ 0 e + eiφ 0e i ω q t sin θ 0 g Global phase arbitrar g Ψ(t) θ 0 Ψ 0 φ 0 φ(t) Ψ t = cos θ 0 e + ei φ 0+ω q t sin θ 0 g φ(t) g AS-Chap

12 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) AS-Chap 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 power ħω d rotating around 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

13 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) AS-Chap 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 g e create or annihilate an ecitation in the TLS Note: σ + and σ are unintuitive in the IT convention, because of the unintuitive assosiation of a negative eigenvalue of σ z with state! Matri representation σ = and σ + = σ + = e g σ = g e

14 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) 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 = ω q a e(t) + ω d e iωt a g (t) i d dt a g t = ω d eiωt a e (t) ω q a g(t) Time-dependent coupled differetial equations 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 = ω q ω b e (t) + ω d b g(t) i d dt b g t = ω d b e(t) ω q ω b g (t) AS-Chap

15 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) AS-Chap Interpretation of the rotating frame i d dt b e t = ω q ω b e (t) + ω d b g(t) i d dt 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

16 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) AS-Chap 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 i φ cos θ φ + + e +i φ sin θ φ Ψ = e i φ sin θ φ + + e +i φ cos θ φ Ψ + = + cos θ e + sin θ g Ψ = sin θ e + cos θ g Δω ω ω q 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+

17 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) AS-Chap 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 with Rabi frequenc ω R Δω + ω d under transversal drive

18 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) 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 Ψ(t) Rotation about -ais z Δω ω ω q tan θ = ω d Δω Finite detuning Δω > 0 Additional precession of 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 Arbitrar equatorial ais H d = ħω d σ e +i ωt+φ + σ + e i ωt+φ AS-Chap

19 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) AS-Chap Rabi Oscillations Graphical representation P e = ω d Δω + ω sin d t Δω + ω d P e On resonance ω = ω q P e t Detuning Δω = 3 ω d 0. 5 t

20 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) t t Rabi Oscillations Graphical representation P e = ω d Δω + ω sin d t Δω + ω d Γ dec = ω d π ω d π ω d Δω ω d Color code P e (0 =black, =white) Δω ω d AS-Chap

21 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) 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 contribution averages out on the timescale of the slowl rotating component Rotating wave approimation Interaction picture (removing eigenenergies rotating frame with ω q qubit-drive interaction) σ ± σ ± e ±iωqt cos ωt σ cos ωt σ + e +iωqt + σ e iω qt ħω d e +iωt + e iωt σ + e +iωqt + σ e iω qt = ħω d σ + e +i ω+ω q t + σ e i ω+ω q t + σ + e iδω t + σ e +iδω t ħω d σ + e iδω t + σ e +iδω t (if ω d ω, ω q ) Rabi oscillations just as for rotating drive Δω ω ω q AS-Chap

22 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) 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

23 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) 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

24 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) Rabi deca time Complicated interpla between T, T, and the drive P e = At long times, small oscillations persist for oscillator drive 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

25 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) Pure and mied states on the Bloch sphere z No decoherence Unitar evolution State vector Ψ alwas well defined Pure state Describe via densit matri ρ Ψ Ψ Hermitian ρ = ρ Epectation value Ψ A Ψ = Tr ρ A Tr ρ = Bloch vector length Decoherence Loss of unitar evolution State vector Ψ onl known probabilisticall φ θ ρ Ψ Ψ Mied state ρ n i= p i Ψ i Ψ i Tr ρ < for n > Bloch vector length < p i are classical probabilities! Center of Bloch sphere Completel depolarized state ρ = / 0 0 / General state ρ = + a σ with σ σ T, σ, σ z Bloch vector a a, a, a z T with a Eigenvalues of ρ are ± a AS-Chap

26 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) AS-Chap Eample: Pure state densit matri on the Bloch sphere Goal Get intuition for the Bloch vector a Ψ = cos θ e + eiφ sin θ g ρ = Ψ Ψ = cos θ e + eiφ sin θ g cos θ e + e iφ sin θ g = = cos θ e e + sin θ g g + eiφ sin θ cos θ e g + e iφ sin θ cos θ g e σ + = e g σ = g e = cos θ e iφ sin θ cos θ e iφ sin θ cos θ sin θ = + cos θ e iφ sin θ e iφ sin θ cos θ sin θ = sin θ cos θ + cos θ = cos θ cos θ = sin θ Bloch vector ρ = + a σ = + a z a ia a + ia a z Compare coefficients a = sin θ cos φ sin θ sin φ cos θ Vector on suface of a sphere Unit length & polar angles θ, φ

27 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) AS-Chap Dnamics of the state vector on the Bloch sphere with decoherence Equations of motion for spin ½ under influence of the control fields B(t) dr t dt = B t r t T r z t r z 0 z T r t + r t r(t) is state vector on the Bloch sphere Introduced in 946 b Bloch for NMR sstems Describe simple eponential decas with time constants T and T In simple scenarios confirmed b quantum master equations

28 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) AS-Chap Energ relaation on the Bloch sphere φ z θ Ψ θ, φ 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 g Quantum jumps Single-shot, quantum nondemolition 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

29 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) Dephasing on the Bloch sphere e z Ψ(t) Environment induces random phase changes Γ φ S ω 0, T φ = π Γ φ, (for Markovian bath) Low-frequenc noise is detuned from ω q No energ transfer g θ 0 φ 0 Ψ θ 0, φ 0 Visualization /f-noise S ω ω Eample: two-level fluctuator bath To some etent reversible Deca laws e t Tφ, e Phase φ becomes more and more unknown with time Classical probabilit, no superposition! Phase coherence lost when arrows are distributed over whole equatorial plane t Tφ, t T φ β AS-Chap

30 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) Qubit dnamics Relaation (T ) Init π Δt Measurement Rotating frame & no detuning (Δω = ω ω q = 0) no -evolution P e e T Δt AS-Chap

31 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) Qubit dnamics Ramse fringes (T ) Init π/ free evolution time Δt π/ Measurement Δω > 0 P e Beating ΔT = Δω = 0 π/δω e 0. 5 T = T + T φr Energ deca alwas present! TR = T Δt AS-Chap

32 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) 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 = T R ) Energ deca unaffected T E = T + T φe Spin echo dephasing approaches true dephasing T φe T φ 0. 5 e 0. 5 TE Δt AS-Chap

33 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) F(ωt/) Ramse vs. spin echo sequence Simple picture T E T More precise Spin echo cancels the effect of low-frequenc noise Pulse sequences act as filters! Environment described b noise spectral densit S(ω) Deca envelope e t F R ωδt + ωδt S ω FR,E ωδt = sin ωδt ωδt F E ωδt = sin4 4 ωδt 4 Sequence length Δt is important! dω Ramse filter F R spin echo filter F E Spin echo sequence ωδt ωt/ Filters low-frequenc noise forωδt 0 ωδt Noise field fluctuates snchronousl with π-pulse No effect F ωδt AS-Chap

34 R. Gross, A. Mar, F. Deppe, and K. Fedorov Walther-Meißner-Institut (00-08) AS-Chap Superconducting qubits Improvement of decoherence times M. H. Devoret and R. J. Schoelkopf, Science 339, 69 (03); DOI:0.6/science.3930