Non-projective measurement of solidstate qubits: collapse and uncollapse (what is inside collapse)

Transcription

1 Non-projective measurement of solidstate qubits: collapse and uncollapse (what is inside collapse) Alexander Korotkov IQC, 4/1/10 Outline: Introduction (weirdness of collapse) Bayesian formalism for quantum measurement Persistent Rabi oscillations (+ expt.) Wavefunction uncollapse (+ expts.) Acknowledgements: Theory: R. Ruskov, A. Jordan Expt.: UCSB (J. Martinis, N. Katz et al.), Saclay (D. Esteve, P. Bertet et al.) Funding:

2 Quantum mechanics is weird Niels Bohr: If you are not confused by quantum physics then you haven t really understood it Richard Feynman: I think I can safely say that nobody understands quantum mechanics Weirdest part is quantum measurement

3 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?

4 Einstein-Podolsky-Rosen (EPR) paradox (1935) ψ ( x, x ) = ψ ( x ) u ( x ) 1 n n n 1 1 = 1 δ 1 ψ ( x, x ) exp[( i/ )( x x ) p] dp ~ ( x x ) x1 x a Alexander Korotkov (nowadays we call it entangled state) Measurement of particle 1 cannot affect particle, while QM says it affects (contradicts causality) => Quantum mechanics is incomplete Bell s inequality (John Bell, 1964) (setup by David Bohm) b ψ = 1 ( 1 ) 1 Advantage: choice of meas. directions Is it possible to explain the QM result assuming local realism and hidden variables (without superluminal collapse)? No!!! Experiment (Aspect et al., 198; photons instead of spins, CHSH): yes, spooky action-at-a-distance

5 What about causality? Actually, not too bad: you cannot transmit your own information by choosing a particular measurement direction a a or Result of the other measurement does not depend on direction a Randomness saves causality Collapse is still instantaneous: OK, just our recipe, not an objective reality, not a physical process Consequence of causality: No-cloning theorem Proof: You cannot copy an unknown quantum state Otherwise get information on direction a (and causality violated) Application: quantum cryptography Wootters-Zurek, 198; Dieks, 198; Yurke Information is an important concept in quantum mechanics 5/41 Alexander Korotkov

6 What is the evolution due to measurement? (What is inside collapse?) controversial for last 80 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, Plenio, 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 I(t), noise S

7 Typical setup: double-quantum-dot (DQD) 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 = I0 + 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 + SI = ei DET = l l l r r r + l r l, r r l + l r H Eaa Eaa Taa ( aa ) INT = Δ, 1 1 lr r l + l r H T ( cc cc )( aa aa )

8 Bayesian formalism for DQD-QPC system H QB =0 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 0 So simple because: (A.K., 1998) prior measured probab. likelihood I 1 I 1) no entaglement at large QPC voltage ) QPC happens to be an ideal detector 3) no Hamiltonian evolution of the qubit

9 Bayesian formalism for a single qubit H e I(t) e Vg V I(t) Time derivative of the quantum Bayes rule Add unitary evolution of the qubit Add decoherence γ (if any) ΔI ρ =- ρ =- H Im ρ + ρ ρ [ It ()-I ] SI ΔI ρ = iερ + ih( ρ -ρ ) + ρ ( ρ -ρ ) [ It ()-I ]-γ ρ SI γ = Γ-( Δ ) /4, Γ ensemble decoherence γ = 0 I S I for QPC detector Alexander Korotkov (A.K., 1998) Averaging over result I(t) leads to conventional master equation with Γ Evolution of qubit wavefunction can be monitored if γ=0 (quantum-limited) Natural generalizations: add classical back-action entangled qubits

10 Assumptions needed for the Bayesian formalism: Detector voltage is much larger than the qubit energies involved ev >> ÑΩ, ev >> ÑΓ, Ñ/eV << (1/Ω,1/Γ), Ω=(4H +ε ) 1/ (no coherence in the detector, classical output, Markovian approximation) 10/41 Simpler if weak response, ΔI << I 0, (coupling C ~ Γ/Ω is arbitrary) Derivations: 1) logical : via correspondence principle and comparison with decoherence approach (A.K., 1998) ) microscopic : Schr. eq. + collapse of the detector (A.K., 000) n ρ ij () t nt ( k ) qubit detector pointer quantum interaction Alexander Korotkov quantum frequent collapse classical information n number of electrons passed through detector 3) from quantum trajectory formalism developed for quantum optics (Goan-Milburn, 001; also: Wiseman, Sun, Oxtoby, etc.) 4) from POVM formalism (Jordan-A.K., 006) 5) from Keldysh formalism (Wei-Nazarov, 007)

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

12 Fundamental limit for ensemble decoherence ensemble decoherence rate Γ = (ΔI) /4S I + γ γ 0 γ ( ΔI) /4SI 1 η = - = Γ Γ single-qubit decoherence ~ information flow [bit/s] Γ (ΔI) /4S I detector ideality (quantum efficiency) η 100% Translated into energy sensitivity: (ε O ε BA ) 1/ / ε O, ε BA : sensitivities [J/Hz] limited by output noise and back-action Γτ m A.K., 1998, 000 S. Pilgram et al., 00 A. Clerk et al., 00 D. Averin, 000,003 Known since 1980s (Caves, Clarke, Likharev, Zorin, Vorontsov, Khalili, etc.) (ε O ε BA - ε O,BA ) 1/ / Γ (ΔI) /4S I + K S I /4 measurement time (S/N=1) τ m = SI /( ΔI ) 1 Danilov, Likharev, Zorin, 1983

13 POVM vs. Bayesian formalism General quantum measurement (POVM formalism) (Nielsen-Chuang, p. 85,100): Measurement (Kraus) operator M Mrρ M rψ r ρ M r (any linear operator in H.S.) : ψ or Mrψ Tr( Mr ρ Mr) Probability : Pr = Mrψ or Pr = Tr( Mr ρ Mr) Completeness : r r = (People often prefer linear evolution 1 and non-normalized states) r M M POVM is essentially a projective measurement in an extended Hilbert space Easy to derive: interaction with ancilla + projective measurement of ancilla For extra decoherence: incoherent sum over subsets of results Relation between POVM and quantum Bayesian formalism: decomposition r = r r r M U M M unitary Bayes Mathematically POVM and quantum Bayes are almost equivalent We focus not on the mathematical structures, but on particular setups and experimental consequences

14 Can we verify the Bayesian formalism experimentally? Direct way: prepare partial measur. control (rotation) projective measur. A.K.,1998 However, difficult: bandwidth, control, efficiency (expt. realized only for supercond. phase qubits) Tricks are needed for real experiments

15 V A (t) QPC A on τ A off Q A = I A dt I A (t) Bell-type measurement correlation Ω I B (t) qubit (DQD) QPC B τ on off τ B Q B = I B dt detector A qubit detector B V B (t) δ B = (<Q B > - Q 0B )/ΔQ B Q A is fixed (selected) (Q A /τ A - I 0A )/Δ I A = 0.6, 0, -0.3 δq A > 0 δq A = 0 δq A > 0 τ A (Δ I A ) /S A = 1 conventional (A.K., PRB-001) τ Ω /π τ Ω /π Idea: two consecutive measurements of a qubit by two detectors; probability distribution P(Q A, Q B, τ) shows effect of the first measurement on the qubit state. Advantage: solves the bandwidth problem Same idea with another averaging weak values (Romito, Gefen, Blanter, PRL-008) after π/ pulse 15/41 Alexander Korotkov

16 Non-decaying (persistent) Rabi oscillations left excited ground right - 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 z left 1.0 ρ 11 to verify: stop& check e g right Reρ 11 Imρ Direct experiment is difficult time A.K., PRB-1999

17 S I (ω)/s 0 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 0 Alexander Korotkov ω ΔI I() t = I + z() t + ξ () t 0 (const + signal + noise) amplifier noise fl higher pedestal, poor quantum efficiency, but the peak is the same!!! A.K., LT 1999 A.K.-Averin, 000 S I (ω) integral under the peak variance z Ú ηü1 How to distinguish experimentally persistent from non-persistent? Easy! 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 ω/ω

18 z left e g right How to understand z Ò =1? Alexander Korotkov ΔI I() t = I0 + z() t + ξ () t First way (mathematical) We actually measure operator: z σ z z σ z =1 (What does it mean? Difficult to say ) Second way (Bayesian) I I SI ( ω Δ ) Sξξ Szz( ) S ( ) 4 ω Δ = + + ξ z ω quantum back-action changes z in accordance with the noise ξ qubit detector I(t) Equal contributions (for weak coupling and η=1) what you see is what you get : observation becomes reality Can we explain it in a more reasonable way (without spooks/ghosts)? +1 z(t)? No (under assumptions of macrorealism; -1 Leggett-Garg, 1985) or some other z(t)?

19 qubit Leggett-Garg-type inequalities for continuous measurement of a qubit detector I(t) Assumptions of macrorealism (similar to Leggett-Garg 85): It () = I + ( Δ I/) zt () +ξ () t 0 zt ( ) 1, ξ ( t) zt ( + τ ) = 0 Then for correlation function K( τ ) = I( t) I( t+ τ ) K( τ ) + K( τ ) K( τ + τ ) ( ΔI/) 1 1 and for area under narrow spectral peak Alexander Korotkov [ SI ( f) S0] df (8/ π )( ΔI/) η is not important! Leggett-Garg,1985 K ij = Q i Q j Ò if Q = ±1, then 1+K 1 +K 3 +K 13 0 K 1 +K 3 +K 34 -K 14 Ruskov-A.K.-Mizel, PRL-006 Jordan-A.K.-Büttiker, PRL-006 S I (ω)/s 0 quantum result 3 ( /) ΔI ( ΔI /) 6 4 4S 0 S I (ω) ω/ω violation 3 Experimentally measurable violation (Saclay experiment) π 8

20 Recent experiment (Saclay group, unpub.) spectral density <n>=0.34 <n>=0.78 <n>=1.56 A. Palacios-Laloy et al. (unpublished) courtesy of Patrice Bertet 0.0 S z (σ z / MHz) 0/ Raw data Detector BW corrected Multiplied by f Δ Δ = 5MHz Area f (MHz) Alexander f (MHz) Korotkov frequency (MHz) 1 8/π 0.66 superconducting charge qubit (transmon) in circuit QED setup (microwave reflection from cavity) driven Rabi oscillations perfect spectral peaks LGI violation (both K and S)

21 D (feedback fidelity) Next step: quantum feedback Goal: persistent Rabi oscillations with zero linewidth (synchronized) Types of quantum feedback: Bayesian Direct Simple Best but very difficult (monitor quantum state and control deviation) qubit H e control stage (barrier height) C<<1 environment C=C det =1 τ a =0 detector feedback signal I(t) desired evolution comparison circuit Bayesian equations C env /C det = Ruskov & A.K., 00 ρ ij (t) Δ H fb / H = F Δϕ F (feedback strength) Alexander Korotkov a la Wiseman-Milburn (1993) (apply measurement signal to control with minimal processing) D (feedback fidelity) Δ H / H = Fsin( Ωt) fb I() t I0 cosωt ΔI / C=1 η =1 averaging time τ a = (π/ω)/ F (f db k t th) F (feedback strength) Ruskov & A.K., 00 qubit C<<1 detector D (feedback fidelity) Imperfect but simple (do as in usual classical feedback) ΔH H I(t) η eff = control cos (Ω t), τ-average local oscil. sin (Ω t), τ-average fb (Useful?) = F φ 1 0 m 0.5 ε/h 0 = ΔΩ/CΩ=0. C = 0.1 X Y rel. phase τ [(ΔI) /S I ] = 1 φ m F/C (feedback strength) A.K., 005

22 Quantum feedback in optics First experiment: Science 304, 70 (004) First detailed theory: H.M. Wiseman and G. J. Milburn, Phys. Rev. Lett. 70, 548 (1993)

23 Quantum feedback in optics First experiment: Science 304, 70 (004) paper withdrawn PRL 94, 0300 (005) also withdrawn First detailed theory: H.M. Wiseman and G. J. Milburn, Phys. Rev. Lett. 70, 548 (1993) Recent experiment: Cook, Martin, Geremia, Nature 446, 774 (007) (coherent state discrimination)

24 Undoing a weak measurement of a qubit ( uncollapse ) A.K. & Jordan, PRL-006 It is impossible to undo orthodox quantum measurement (for an unknown initial state) Is it possible to undo partial quantum measurement? (To restore a precious qubit accidentally measured) Yes! (but with a finite probability) If undoing is successful, an unknown state is fully restored ψ 0 (unknown) weak (partial) measurement ψ 1 (partially collapsed) successful unsuccessful uncollapse (information erasure) ψ 0 (still unknown) ψ

25 Quantum erasers in optics Quantum eraser proposal by Scully and Drühl, PRA (198) 5/41 Our idea of uncollapsing is quite different: we really extract quantum information and then erase it Alexander Korotkov Interference fringes restored for two-detector correlations (since which-path information is erased)

26 Uncollapse of a qubit state Evolution due to partial (weak, continuous, etc.) measurement is non-unitary, so impossible to undo it by Hamiltonian dynamics. How to undo? One more measurement! = need ideal (quantum-limited) detector (similar to Koashi-Ueda, PRL-1999) Alexander Korotkov (Figure partially adopted from Jordan-A.K.-Büttiker, PRL-06)

27 Uncollapsing for DQD-QPC system A.K. & Jordan, 006 I(t) Qubit (DQD) H=0 r(t) first ( accidental ) measurement r 0 uncollapsing measurement Detector (QPC) Simple strategy: continue measuring until result r(t) becomes zero! Then any initial state is fully restored. (same for an entangled qubit) However, if r = 0 never happens, then uncollapsing procedure is unsuccessful. Probability of success: P S = Meas. result: ΔI t rt () = [ It (') dt' It 0 0 ] SI - If r = 0, then no information and no evolution! e e - r r - r ρ (0) + e ρ (0) t

28 POVM formalism (Nielsen-Chuang, p.100) General theory of uncollapsing Uncollapsing operator: Alexander Korotkov Measurement operator M r : C 1 Mr max( C) = min p, p eigenvalues of Probability of success: P = Tr( M ρ M ) Probability : r r r i i i P S min Pr P ( ρ ) r in r r Mrρ M ρ Tr( Mr ρ M ) M r M r r = 1 (to satisfy completeness, eigenvalues cannot be >1) P r (ρ in ) probability of result r for initial state ρ in, min P r probability of result r minimized over all possible initial states Averaged (over r) probability of success: Completeness : min P av r r (cannot depend on initial state, otherwise get information) (similar to Koashi-Ueda, 1999) P r M M r A.K. & Jordan, 006

29 1 0 Partial collapse of a Josephson phase qubit Γ N. Katz, M. Ansmann, R. Bialczak, E. Lucero, R. McDermott, M. Neeley, M. Steffen, E. Weig, A. Cleland, J. Martinis, A. Korotkov, Science-06 How does a qubit state evolve in time before tunneling event? (What happens when nothing happens?) Main idea: out, if tunneled -Γ t / iϕ ψ = α 0 + β 1 ψ( t) = α 0 + β e e 1, if not tunneled -Γt α + β e amplitude of state 0 grows without physical interaction finite linewidth only after tunneling continuous null-result collapse (better theory: Pryadko & A.K., 007) (similar to optics, Dalibard-Castin-Molmer, PRL-199) Alexander Korotkov

30 Superconducting phase qubit at UCSB Courtesy of Nadav Katz (UCSB) I μw X Y 10ns Flux bias Qubit I μw I z Reset Z 3ns Compute Meas. Readout I dc +I z SQUID I S V S I dc time I s V s ω 01 Repeat 1000x prob. 0,1 1 Φ 0 30/41 Alexander Korotkov

31 Experimental technique for partial collapse Nadav Katz et al. (John Martinis group) Protocol: 1) State preparation (via Rabi oscillations) ) Partial measurement by lowering barrier for time t 3) State tomography (microwave + full measurement) Measurement strength p = 1 - exp(-γt ) is actually controlled by Γ, not by t p=0: no measurement p=1: orthodox collapse

32 ψ in = θy Experimental tomography data Nadav Katz et al. (UCSB, 005) p =0 p =0.14 p =0.06 θx p =0.3 p =0.3 p =0.43 π π/ p =0.56 p =0.70 p =0.83

33 Partial collapse: experimental results N. Katz et al., Science-06 Polar angle Azimuthal angle Visibility lines - theory dots and squares expt. no fitting parameters in (a) and (b) probability p pulse ampl. in (c) T 1 =110 ns, T =80 ns (measured) probability p 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. ~80% raw data, ~96% corrected for T 1,T ) quantum efficiency η 0 > 0.8

34 Uncollapse of a phase qubit state 1) Start with an unknown state ) Partial measurement of strength p 3) π-pulse (exchange 0Ú 1Ú) 4) One more measurement with the same strength p 5) π-pulse iφ If no tunneling for both measurements, then initial state is fully restored! Γt/ iφ Γt/ Γt / α 0 + e β e 1 α 0 + β 1 Norm e αe 0 + e β e 1 = iφ e ( α 0 + β 1 ) Norm iφ A.K. & Jordan, p= 1-e -Γt phase is also restored (spin echo) Γ

35 Experiment on wavefunction uncollapse partial state measure p preparation I μw π partial measure p tomography & final measure N. Katz, M. Neeley, M. Ansmann, R. Bialzak, E. Lucero, A. O Connell, H. Wang, A. Cleland, J. Martinis, and A. Korotkov, PRL-008 I dc p p 10 ns 7 ns 10 ns 7 ns time Nature News Nature-008 Physics Uncollapse protocol: - partial collapse - π-pulse - partial collapse (same strength) ψ in = State tomography with X, Y, and no pulses Background P B should be subtracted to find qubit density matrix 35/41 Alexander Korotkov

36 Initial state Experimental results on the Bloch sphere 0 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

37 Quantum process tomography N. Katz et al. (Martinis group) p = 0.5 uncollapsing works with good fidelity! Why getting worse at p>0.6? Energy relaxation p r = t/t 1 = 45ns/450ns = 0.1 Selection affected when 1-p ~ p r Overall: uncollapsing is well-confirmed experimentally

38 Recent experiment on uncollapsing using single photons Kim et al., Opt. Expr.-009 Alexander Korotkov very good fidelity of uncollapsing (>94%) measurement fidelity is probably not good (normalization by coincidence counts)

39 Protocol: ρ 11 partial collapse towards ground state (strength p) Suppression of T 1 -decoherence by uncollapsing storage period t (low temperature) Ideal case (T 1 during storage only) for initial state ψ in =α 0 +β 1 ψ f = ψ in with probability (1-p)e -t/t 1 ψ f = 0 with (1-p) β e -t/t1 (1-e -t/t1 ) procedure preferentially selects events without energy decay π π uncollapse (measurem. strength p u ) best for 1 - pu = (1- p)exp( -t/ T1 ) QPT fidelity (F av s, Fχ ) Ideal Korotkov & Keane, arxiv: p u = 1- e -t/t1 (1-p) p u = p without uncollapsing e -t/t 1 = measurement strength p almost complete suppression Trade-off: fidelity vs. probability

40 QPT fidelity, probability Realistic case (T 1 and T ϕ at all stages) fidelity (1-p u ) κ 3 κ 4 = (1-p) κ 1 κ } κ without = 0.3 uncollapsing κ ϕ = 1, 0.95 κ 1 = κ 3 = κ 4 = 1, 0.999, probability 0. ti / T1 κ i = e tσ / Tϕ κϕ = e measurement strength p as in expt. Easy to realize experimentally (similar to existing experiment) Improved fidelity can be observed with just one partial measurement Tϕ-decoherence is not affected fidelity decreases at p 1 due to T 1 between 1st π-pulse and nd meas. Uncollapse seems the only way to protect against T 1 -decoherence without quantum error correction Trade-off: fidelity vs. selection probability A.K. & Keane, arxiv: /41 Alexander Korotkov

41 Conclusions It is easy to see what is inside collapse: simple Bayesian formalism works for many solid-state setups Rabi oscillations are persistent if weakly measured Collapse can sometimes be undone if we manage to erase extracted information (uncollapsing) Continuous/partial measurements and uncollapsing may be useful Three direct solid-state experiments have been realized, many interesting experimental proposals are still waiting 41/41 Alexander Korotkov