Probing inside quantum collapse with solid-state qubits

Transcription

1 Ginzburg conf., Moscow, 5/31/1 Probing inside quantum collapse with solid-state qubits Alexander Korotkov Outline: What is inside collapse? Bayesian framework. - broadband meas. (double-dot qubit & QPC) - narrowband meas. (circuit QED setup) Realized experiments - partial collapse (null-result & continuous) - uncollapse (+ entanglement preservation) - persistent Rabi oscillations, quantum feedback

2 Quantum mechanics = Schrödinger equation (evolution) + collapse postulate (measurement) 1) Probability of measurement result p r = ψ ψ r ) Wavefunction after measurement = ψ r State collapse follows from common sense Does not follow from Schrödinger Eq. (contradicts) What is inside collapse? What if collapse is stopped half-way?

3 What is the evolution due to measurement? (What is inside collapse?) controversial for last 8 years, many wrong answers, many correct answers solid-state systems are more natural to answer this question Various approaches to non-projective (weak, continuous, partial, generalized, etc.) quantum measurements Names: Davies, Kraus, Holevo, Mensky, Caves, Knight, Walls, Carmichael, Milburn, Wiseman, Gisin, Percival, Belavkin, etc. (very incomplete list) Key words: POVM, restricted path integral, quantum trajectories, quantum filtering, quantum jumps, stochastic master equation, etc. Our limited scope: (simplest system, experimental setups) Alexander Korotkov solid-state qubit detector classical output

4 Quantum Bayesian framework (slight technical extension of the collapse postulate) 1) Quantum back-action (spooky, physically unexplainable) simple: update the state using information from measurement and probability concept (Bayes rule) ) Add classical back-action if any (anything with a physical mechanism) 3) Add noise/decoherence if any 4) Add Hamiltonian (unitary) evolution if any (Practically equivalent to many other approaches: POVM, quantum trajectory, quantum filtering, etc.)

5 Typical setup: double-quantum-dot qubit + quantum point contact (QPC) detector 1Ò Ò H e I(t) 1Ò Ò Ò 1Ò I(t) Gurvitz, 1997 H = H QB + H DET + H INT ε H = σ + QB H σ Z X ΔI I() t = I + z() t + ξ () t const + signal + noise Two levels of average detector current: I 1 for qubit state 1, I for Response: ΔI = I 1 I Detector noise: white, spectral density S I For low-transparency QPC + DET = l l l r r r + l r l, r r l + l r H Eaa Eaa Taa ( aa ) INT lr, 1 1 r l H = ΔT( cc cc) aa+ h.c. Alexander Korotkov SI = ei ( broadband )

6 Bayesian formalism for DQD-QPC system H QB = 1Ò Ò H e I(t) Qubit evolution due to measurement (quantum back-action): ψ () t = α() 1 t + β() t or ρ ij () t Bayes rule (1763, Laplace-181): posterior probability P( A ) (res ) ( res) i P A P A i i = P( A ) P(res A ) k 1) α(t) and β(t) evolve as probabilities, i.e. according to the Bayes rule (same for ρ ii ) ) phases of α(t) and β(t) do not change (no dephasing!), ρ ij /(ρ ii ρ jj ) 1/ = const k k 1 τ τ I() t dt (A.K., 1998) prior measured probab. likelihood I 1 I So simple because: 1) no entanglement at large QPC voltage ) QPC is ideal detector 3) zero qubit Hamiltonian

7 1Ò Ò Δ I = I1-I noise S I 1 τ I() t dt τ D = S /τ Im I Now add classical back-action and decoherence H qb = quantum backaction (non-unitary, Ò spooky, unphysical ) 1Ò ρ11( τ) ρ11() exp[ -( Im -I1) / D] = no self-evolution I(t) ρ( τ) ρ() exp[ -( Im -I) / D] of qubit assumed ρ11( τ) ρ( τ) ρ1( τ) = ρ1() exp( iki m τ)exp( γτ) ρ11() ρ() decoherence classical backaction (unitary) Example of classical ( physical ) backaction: D D I 1 I Each electron passed through QPC rotates qubit + DET = l l l r r r + l r l, r r l + l r H Eaa Eaa Taa ( aa) * arg( T ΔT) INT lr, 1 1 r l H = ΔT( cc cc) aa+ h.c.

8 H e I(t) e Now add Hamiltonian evolution Vg V I(t) Time derivative of the quantum Bayes rule Add unitary evolution of the qubit H ΔI ρ =- ρ =- Im ρ + ρ ρ [ I() t -I ] SI H ΔI ρ = iερ + i ( ρ -ρ ) + ρ ( ρ -ρ ) [ I() t -I ]-γ ρ SI ΔI=I 1 -I, I =(I 1 +I )/, S I detector noise γ = for QPC For simulations: 11 (A.K., 1998) ΔI I = I + ( ρ ρ ) + ξ noise S ξ = Evolution of qubit wavefunction can be monitored if γ= (quantum-limited) S I

9 Relation to conventional master equation ΔI ρ =- ρ =- H Im ρ + ρ ρ [ I() t -I ] SI ΔI ρ = iερ + ih( ρ -ρ ) + ρ ( ρ -ρ ) [ I( t) -I SI + ik[ I( t) - I ] ρ -γρ 1 1 ] response ΔI noise S I Averaging over measurement result I(t) leads to usual master equation: ρ =- ρ / dt =-H Imρ 11 1 ρ 1 = iερ1 + ih( ρ11 -ρ)-γρ1 ( ) /4 I I /4 Γ ensemble decoherence, Γ= Δ I S + K S + γ Quantum efficiency: η = ( ΔI) /4S I Γ spooky physical dephasing or η = 1 γ Γ

10 Quantum measurement in POVM formalism Davies, Kraus, Holevo, etc. (Nielsen-Chuang, pp. 85, 1) Measurement (Kraus) operator M r (any linear operator in H.S.) : Probability : P r Completeness : r M M r = r = M ψ r ψ 1 or system M r M ψ r ψ ancilla or ρ r = r ρ r P Tr( M M ) projective measurement M r ρ M r r Tr( M ρ M ) (People often prefer linear evolution and non-normalized states) r Relation between POVM and quantum Bayesian formalism: decomposition r = r r r M U M M unitary Bayes (almost equivalent)

11 Narrowband linear measurement (circuit QED setup) Difference from broadband: two quadratures (two signals: A(t)cosωt + B(t)sinωt) microwave generator ω d resonator ω r qubit (transmon) paramp quantum signal ( quadratures) mixer Q(t) I(t) Schoelkopf et al. output (two quadratures) = 1 H a a + aa ω qb σ z + ω r χ σ z qubit state changes resonator freq., number of photons affects qubit freq. Blais et al., 4 Gambetta et al., 6, 8 Alexander Korotkov carries information about qubit ( quantum back-action) 1Ò Ò cos( ω d t ) sin( ω d t ) carries information about fluctuating photon number in the resonator ( classical back-action)

12 μwave gen. ω resonator d ω r qubit Phase-sensitive (degenerate) paramp paramp mixer Q(t) I(t) cos( ω d t ) 1Ò Ò ϕ I(t) amplified sin( ω d t ) cos(ω d t +ϕ) is amplified: I(t) sin(ω d t +ϕ) is suppressed get some information (~cos ϕ) about qubit state and some information (~sin ϕ) about photon fluctuations ρgg( τ) ρgg() exp[ -( I -Ig ) / D] = ρ ( τ) ρ () exp[ -( I -I ) / D] ee ee e Alexander Korotkov 1 I() t dt D = S / I τ ρgg( τ) ρee( τ) ρge( τ) = ρge() exp( ikiτ) I ρgg() ρee() ( ΔI cos ϕ ) SI ΔI 8χ n Γ = + K = = unitary Bayes 4SI 4 4SI κ (rotating frame) Same as for QPC, but ϕ controls trade-off A.K., arxiv: between quantum & classical back-actions (we choose if photon number fluctuates or not) I τ τ I I =ΔIcosϕ g e ΔI K = sinϕ S

13 μwave gen. ω resonator d ω r qubit Alexander Korotkov paramp mixer Q(t) I(t) Phase-preserving (nondegenerate) paramp ρ gg ( τ) ρ () exp[ -( - ) / ] = gg I I g D ρ ( τ) ρ () exp[ -( I -I ) / D] ee ee e Ò ϕ = δω t 1 1 I() t dt Q () D = Q t dt ρgg( τ) ρee ( τ) ρge( τ) = ρge() exp( ikqτ) ΔI ΔI 8χ n ρgg() ρee() Γ = + = 8SI 8SI κ Bayes unitary Equal contributions to ensemble dephasing Understanding important from quantum & classical back-actions for quantum feedback A.K., arxiv: I τ I g τ I = e ΔI K = I(t) 1Ò ΔI S I cos( ω d t ) ϕ sin( ω d t ) Q(t) δω t Now information in both I(t) and Q(t). Choose I(t) cos(ω d t) (qubit information) Q(t) sin(ω d t) (photon fluct. info) Small δω can follow ϕ(t) Large δω (>>Γ) averaging over ϕ (phase-preserving) τ τ S I τ

14 Why not just use Schrödinger equation for the whole system? qubit detector information Impossible in principle! Technical reason: Outgoing information makes it an open system Philosophical reason: Random measurement result, but deterministic Schrödinger equation Einstein: God does not play dice (actually plays!) Heisenberg: unavoidable quantum-classical boundary

15 Superconducting experiments inside quantum collapse UCSB-6 Partial collapse UCSB-8 Reversal of partial collapse (uncollapse) Saclay-1 Continuous measurement of Rabi oscillations (+violation of Leggett-Garg inequality) Berkeley-1 Quantum feedback of persistent Rabi osc. (phase-sensitive paramp) Yale-1 Partial (continuous) measurement (phase-preserving paramp)

16 Partial collapse of a Josephson phase qubit 1 Γ N. Katz, M. Ansmann, R. Bialczak, E. Lucero, R. McDermott, M. Neeley, M. Steffen, E. Weig, A. Cleland, J. Martinis, A. Korotkov, Science-6 What happens if no tunneling? Main idea: out, if tunneled -Γ t / iϕ ψ = α + β 1 ψ( t) = α + β e e 1 -Γt α + β e, if not tunneled Non-trivial: amplitude of state grows without physical interaction finite linewidth only after tunneling continuous null-result collapse (idea similar to Dalibard-Castin-Molmer, PRL-199)

17 Polar angle Azimuthal angle Visibility Partial collapse: experimental results p= 1-e -Γt lines - theory dots and squares expt. no fitting parameters in (a) and (b) probability p pulse ampl. in (c) T 1 =11 ns, T =8 ns (measured) probability p N. Katz et al., Science-6 In case of no tunneling phase qubit evolves Evolution is described by the Bayesian theory without fitting parameters Phase qubit remains coherent in the process of continuous collapse (expt. ~8% raw data, ~96% corrected for T 1,T ) quantum efficiency η >.8 Good confirmation of the theory

18 Uncollapsing for qubit-qpc system (theory) I(t) Detector (QPC) Qubit (double-dot) Alexander Korotkov r(t) First accidental measurement r ΔI t rt () = [ It (') dt' It ] S - I A.K. & Jordan, 6 Uncollapsing measurement Simple strategy: continue measuring until r(t) becomes zero! Then any unknown initial state is fully restored. (same for an entangled qubit) It may happen though that r = never happens; then uncollapsing is unsuccessful. Somewhat similar to quantum eraser of Scully and Druhl (198) t

19 Experiment on wavefunction uncollapse partial state measure p preparation I μw I dc 1 ns 7 ns p π partial measure p 1 ns 7 ns p tomography & final measure time If no tunneling for both measurements, then initial state is fully restored iφ Γt / α + e β e 1 α + β 1 Norm iφ Γt/ iφ Γt/ e αe + e β e 1 = iφ e ( α + β 1 ) Norm N. Katz, M. Neeley, M. Ansmann, R. Bialzak, E. Lucero, A. O Connell, H. Wang, A. Cleland, J. Martinis, and A. Korotkov, PRL-8 Nature News phase is also restored ( spin echo ) Uncollapse protocol: - partial collapse - π-pulse - partial collapse (same strength) 1 p= 1-e -Γt Γ

20 Initial state Experimental results on the Bloch sphere i N. Katz et al. Partially collapsed Uncollapsed uncollapsing works well Both spin echo (azimuth) and uncollapsing (polar angle) Difference: spin echo undoing of an unknown unitary evolution, uncollapsing undoing of a known, but non-unitary evolution

21 Suppression of T 1 -decoherence by uncollapse Protocol: ρ 11 storage period t (zero temperature) partial collapse towards ground state (strength p) Ideal case (T 1 during storage only) for initial state ψ in =α +β 1 ψ f = ψ in with probability (1-p)e -t/t 1 ψ f = with (1-p) β e -t/t1 (1-e -t/t1 ) procedure preferentially selects events without energy decay π Alexander Korotkov π uncollapse (measurem. strength p u ) Uncollapse seems to be the only way to protect against T 1 -decoherence without encoding in a larger Hilbert space (QEC, DFS) QPT fidelity (F av s, Fχ ) Realistic case (T 1 and T ϕ at all stages) QPT fidelity, probability Ideal fidelity (1-p u ) κ 3 κ 4 = (1-p) κ 1 κ κ without =.3 } uncollapsing probability A.K. & Keane, PRA-1 p u = 1- e -t/t1 (1-p) p u = p without uncollapsing e -t/t 1 = measurement strength p κ ϕ = 1,.95 κ 1 = κ 3 = κ 4 = 1,.999, ti / T1 κ. i = e tσ / Tϕ. κϕ = e measurement strength p Trade-off: fidelity vs. probability almost full suppression as in expt.

22 Realization with photons J.-C. Lee, Y.-C. Jeong, Y.-S. Kim, and Y.-H. Kim, Opt. Express-11 probability QPT fidelity p=.9 γ is purity Alexander Korotkov Works perfectly (optics, not solid state!) Amplitude damping ( energy relaxation ) decoherence is imitated in a clever way

23 Uncollapsing preserves entanglement Y.-S. Kim, J.-C. Lee, O. Kwon, and Y.-H. Kim, Nature Phys.-1 decoherence strength Extension of 1-qubit experiment Revives entanglement even from sudden death D=.6 measurement strength A.K., Sleeping beauty approach, Nature Phys.-1

24 qubit Recent experiment in Michel Devoret s group Courtesy of Michel Devoret MEASUREMENT PROTOCOL State preparation Variable strength measur ment varies (Yale Univ., manuscript in preparation) Tomography or cavity outcome (I m,q m ) (X f,y f,z f ) (phase-preserving paramp) Alexander Korotkov =±1 or =±1 or repeat 1,, times =±1

25 Q m /σ Courtesy of Michel Devoret (manuscript in preparation) histogram of measurement outcomes I m /σ Q m /σ tomography along X, Y, Z after measurement I m /σ 6 3 <X> Q m /σ <Y> I m /σ 1 Q m /σ Max Counts Cavity Drive = 5.e-4 photons g e (I m - Im )/(σ) =.46 Alexander Korotkov -3-6 <Z> I m /σ -1

26 Q m /σ Courtesy of Michel Devoret (manuscript in preparation) histogram of measurement outcomes I m /σ Q m /σ tomography along X, Y, Z after measurement I m /σ 6 3 <X> Q m /σ <Y> I m /σ 1 Q m /σ Max Counts Cavity Drive = 1.1e-1 photons g e (I m - Im )/(σ) =.543 Alexander Korotkov -3-6 <Z> I m /σ -1

27 Q m /σ Courtesy of Michel Devoret (manuscript in preparation) histogram of measurement outcomes I m /σ Q m /σ tomography along X, Y, Z after measurement I m /σ 6 3 <X> Q m /σ <Y> I m /σ 1 Q m /σ Max Counts Cavity Drive = 5.1e-1 photons g e (I m - Im )/(σ) = 1.3 Alexander Korotkov -3-6 <Z> I m /σ -1

28 Q m /σ Courtesy of Michel Devoret (manuscript in preparation) histogram of measurement outcomes I m /σ Q m /σ tomography along X, Y, Z after measurement I m /σ 6 3 <X> Q m /σ <Y> I m /σ 1 Q m /σ Max Counts Cavity Drive = 5.e+ photons g e (I m - Im )/(σ) = 4.1 Alexander Korotkov -3-6 <Z> I m /σ -1

29 Non-decaying (persistent) Rabi oscillations left excited ground z left right ( ΔI ) S e g 4 I right << Ω Direct experiment is difficult - Relaxes to the ground state if left alone (low-t) - Becomes fully mixed if coupled to a high-t (non-equilibrium) environment - Oscillates persistently between left and right if (weakly) measured continuously 1. ρ 11 Reρ 11 Imρ ( reason : attraction to left/right states) to verify: stop& check time A.K., PRB-1999

30 S I (ω)/s S I Indirect experiment: spectrum of persistent Rabi oscillations C qubit detector I(t) peak-to-pedestal ratio = 4η 4 ( ω) C= = S + Ω =H C = ΔI HS ( ) / I ω/ω Ω - Rabi frequency Ω ( ΔI ) Γ ( ω Ω ) +Γ Alexander Korotkov ω ΔI I() t = I + z() t + ξ () t (const + signal + noise) amplifier noise fl higher pedestal, poor quantum efficiency, but the peak is the same!!! A.K., LT 1999 A.K.-Averin, S I (ω) integral under the peak variance z Ú ηü1 perfect Rabi oscillations: z Ò= cos Ò=1/ imperfect (non-persistent): z ÒÜ1/ quantum (Bayesian) result: z Ò = 1 (!!!) (demonstrated in Saclay expt.) z is Bloch coordinate 1 1 ω/ω

31 Saclay experiment n=,1,,5,1, photons Zeno effect A.Palacios-Laloy, F.Mallet, F.Nguyen, P. Bertet, D. Vion, D. Esteve, and A. Korotkov, Nature Phys., 1 superconducting charge qubit (transmon) in circuit QED setup microwave reflection from cavity: full collection, only phase modulation driven Rabi oscillations (z-basis is g>& e>) Standard (not continuous) measurement here: ensemble-averaged Rabi starting from ground state

32 Now continuous measurement n= n =.3 n = 1.56 Palacios-Laloy et al., 1 η = ΔS 4S ~1 Pre-amplifier noise temperature T N =4K 1 T + ω 1 N.3 Theory by dashed lines, very good agreement

33 Violation of Leggett-Garg inequalities In time domain Rescaled to qubit z-coordinate K( τ ) z( t) z( t+ τ ) 1 1 Palacios-Laloy et al., 1 K( τ ) + K( τ ) K( τ + τ ) 1 K( τ) K( τ) 1 f () = K() = z LG z = 1.1 ±.15 flg ( τ ) K ( τ ) K ( τ ) f (17 ns) = 1.44±.1 Ideal f LG,max =1.5 LG Standard deviation σ =.65 violation by 5σ

34 Quantum feedback control of persistent Rabi oscillations In simple monitoring the phase of persistent Rabi oscillations fluctuates randomly: z() t = cos[ Ω t + ϕ ()] t for η=1 phase noise fl finite linewidth of the spectrum Goal: produce persistent Rabi oscillations without phase noise by synchronizing with a classical signal t integral 6 S I (ω)/s z = + = 1 integral 6 1 ω/ω 4 1 z = 1 ω/ω z () = cos( Ωt ) desired ΔI I () t = I + z() t + ξ () t I I S S Δ Δ = + S + S 4 ξ I zz z synchronized cannot synchronize

35 qubit H e Several types of quantum feedback Bayesian Direct Simple control stage (barrier height) environment D (feedback fidelity) Best but very difficult (monitor quantum state and control deviation) C<<1 C=C det =1 τ a = detector feedback signal I(t) comparison circuit Bayesian equations C env /C det = Ruskov & A.K., desired evolution Δ H fb / H = F Δϕ F (feedback strength) ρ ij (t) Alexander Korotkov (apply measurement signal to control with minimal processing) D (feedback fidelity) as in Wiseman-Milburn (1993) Δ H / H = Fsin( Ωt) fb I() t I cosωt ΔI / C=1 η =1 averaging time τ a = (π/ω)/ F (f db k t th) F (feedback strength) Ruskov & A.K., C<<1 qubit detector D (feedback fidelity) Berkeley-1 experiment: direct and simple Imperfect but simple (do as in usual classical feedback) ΔHfb = F φm H I(t) η eff =.5..1 control cos (Ω t), τ-average local oscil. sin (Ω t), τ-average 1.5 ε/h = 1.1 ΔΩ/CΩ=. C =.1 X Y rel. phase τ [(ΔI) /S I ] = 1 φ m F/C (feedback strength) A.K., 5

36 Quantum feedback of Rabi oscillations R. Vijay, C. Macklin, D. Slicher, S. Weber, K. Murch, R. Naik, A. Korotkov, and Irfan Siddiqi, 1 (unpub.) (phase-sensitive paramp) Paramp BW 1 MHz, Cavity LW 8 MHz, Rabi freq. 3 MHz, Meas. dephasing.5 MHz, Env. dephasing.5 MHz Alexander Korotkov Courtesy of Irfan Siddiqi

37 STABILIZED RABI OSCILLATIONS Feedback OFF Feedback ON Courtesy of Irfan Siddiqi

38 STILL GOING Single quadrature measurement Operate with measurement dephasing dominant Appearance of narrow peak when PLL operational Courtesy of Irfan Siddiqi

39 STATE TOMOGRAPHY Observe expected rotation in the X,Z plane Observe Bloch vector reduced to 5% of maximum Courtesy of Irfan Siddiqi

40 FEEDBACK EFFICIENCY D = 1 F η Γ / Ω R Γ / Ω + F R D: feedback efficiency F: feedback strength η: detector efficiency (-1) Γ: dephasing rate Ω R : Rabi frequency Analytics do not include delay time, finite bandwidth, T 1 Numerics include delay and bandwidth good agreement Courtesy of Irfan Siddiqi

41 Conclusions It is easy to see what is inside collapse: simple Bayesian framework works for many solid-state setups Measurement backaction necessarily has a spooky part (informational, without a physical mechanism); it may also have a classical part (with a physically understandable mechanism) Five superconducting experiments so far: - partial collapse, - uncollapse, - monitoring of non-decaying Rabi oscillations, - quantum feedback of persistent Rabi oscillations, - partial measurement with continuous result Hopefully something useful in future