Non-projective measurement of solid-state qubits

Transcription

1 Outline: Ackn.: Non-projective measurement of solid-state qubits (what is inside collapse) Bayesian formalism for quantum measurement Persistent Rabi oscillations (+expt.) Wavefunction uncollapse (+expts.) New experimental proposals - decoherence suppression by uncollapsing - persistent Rabi oscillations revealed via noise Theory: R. Ruskov, A. Jordan, K. Keane Expt.: UCSB (J. Martinis, N. Katz et al.), Saclay (D. Esteve, P. Bertet et al.) Funding: Bad Honnef, 11/0/09

2 Quantum mechanics = Schrödinger equation + measurement postulate 1) Probability of measurement result r : p r = where Aˆ ψ = r ψ r r r ψ ψ r ψ ) Wavefunction after measurement = (collapse) Instantaneous collapse is surely an approximation (though often OK in optics, also the main point in Bell s ineq.), especially obvious for solid-state systems What is the evolution due to measurement? (What is inside collapse?) (controversial for last 80 years, many wrong answers, many correct answers) Our limited scope: (simplest system, experimental setups) solid-state qubit detector I(t), noise S /5

3 Superconducting charge qubit Y. Nakamura, Yu. Pashkin, and J.S. Tsai (Nature, 1998) E - J ˆ ( e) H = ( nˆ -ng ) C ( n n+ 1 + n+ 1 n ) e Vg island Josephson junction n g Single Cooper pair box n n+1 Vion et al. (Saclay group); Science, 00 Q-factor of coherent (Rabi) oscillations = 5,000 ( quantronium ) Quantum coherent (Rabi) oscillations Δt (ps) E J n: number of Cooper pairs on the island

4 Charge qubits with SET readout Vg V I(t) Duty, Gunnarsson, Bladh, Delsing, PRB 004 Guillaume et al. (Echternach s group), PRB 004 e Cooper-pair box measured by singleelectron transistor (rf-set) Setup can be used for continuous measurements All results are averaged over many measurements (not single-shot )

5 Some other superconducting qubits Flux qubit Mooij et al. (Delft) Phase qubit J. Martinis et al. (UCSB and NIST) Charge qubit with circuit QED R. Schoelkopf et al. (Yale) 5/5

6 Some other superconducting qubits Flux qubit J. Clarke et al. (Berkeley) Quantronium qubit I. Siddiqi, R. Schoelkopf, M. Devoret, et al. (Yale)

7 Semiconductor (double-dot) qubit T. Hayashi et al., PRL 003 Detector is not separated from qubit, also possible to use a separate detector Rabi oscillations

8 Some other semiconductor qubits Spin qubit (QPC meas.) C. Marcus et al. (Harvard) Spin qubit L. Kouwenhoven et al. (Delft) Double-dot qubit Gorman, Hasko, Williams (Cambridge)

9 qubit detector I(t) The system we consider: qubit + detector 1Ò Ò H e 1Ò Ò I(t) Double-quantum-qot and quantum point contact Ò 1Ò I(t) e Vg V I(t) Cooper-pair box and single-electron transistor Yale Charge qubit with circuit QED readout H = H QB + H DET + H INT H QB = (ε/)(c 1 + c 1 c + c )+H(c 1 + c +c + c 1 ) ε asymmetry, H tunneling Ω = (4H +ε ) 1/ /Ñ frequency of quantum coherent (Rabi) oscillations 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 DQD and QPC (setup due to Gurvitz, 1997) DET = l l l r r r + l r l, r r l + l r INT = Δ (, 1 1 )( r l + l r) lr H Eaa Eaa Taa ( aa) + H T cc cc aa aa S I = ei

10 What happens to a qubit state during measurement? Start with density matrix evolution due to measurement only (H=ε=0 ) Orthodox answer Decoherence answer exp( Γt) exp( Γt) > or >, depending on the result no measurement result! (ensemble averaged) Decoherence has nothing to do with collapse! applicable for: single quant. system continuous meas. Orthodox yes no Decoherence (ensemble) no yes Bayesian, POVM, quant. traject., etc. yes yes 10/5 Bayesian (POVM, quant. traj., etc.) formalism describes gradual collapse of a single quantum system, taking into account measurement result

11 Bayesian formalism for DQD-QPC system 1Ò H=0 ψ () t = α() 1 t + β() t or ρ ij () t H e Ò 1) α(t) and β(t) evolve as probabilities, i.e. according to the Bayes rule (same for ρ ii ) I(t) Qubit evolution due to measurement (quantum back-action): Bayes rule (1763, Laplace-181): posterior probability P( A ) (res ) ( res) i P A P A i i = P( A ) P(res A ) k ) phases of α(t) and β(t) do not change (no dephasing!), ρ ij /(ρ ii ρ jj ) 1/ = const prior probab. k likelihood k I 1 I measured So simple because: (A.K., 1998) 1) QPC happens to be an ideal detector ) no Hamiltonian evolution of the qubit Similar formalisms developed earlier. Key words: Imprecise, weak, selective, or conditional measurements, POVM, Quantum trajectories, Quantum jumps, Restricted path integral, etc. Names: Davies, Kraus, Holevo, Mensky, Caves, Gardiner, Carmichael, Plenio, Knight, Walls, Gisin, Percival, Milburn, Wiseman, Habib, etc. (very incomplete list)

12 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 (if any) ρ =- ρ =- ( H / )Im ρ + ρ ρ ( ΔI / S )[ It ()-I ] 11 Hˆ = ( ε /)( cc cc) + Hcc ( + cc) (A.K., 1998) QB Ò ÆI 1, ÒÆI, ΔI=I 1 -I, I 0 =(I 1 +I )/, S I detector noise Averaging over result I(t) leads to conventional master equation: 1 11 I 0 ρ 1 = i( ε / ) ρ1 + i( H / )( ρ11 -ρ) + ρ1( ρ11 -ρ)( ΔI / SI )[ It ()-I0]-γρ1 γ = Γ-( Δ ) /4, Γ ensemble decoherence I S I Evolution of qubit wavefunction can be monitored if γ=0 (quantum-limited) dρ11/ dt =- dρ/ dt =-( H / )Imρ1 dρ / dt = i( ε / ) ρ + i( H/ )( ρ -ρ )-Γρ Ensemble averaging includes averaging over measurement result!

13 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) (no coherence in the detector, classical output, Markovian approximation) 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 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)

14 Why not just use Schrödinger equation for the whole system? qubit detector information Impossible in principle! Technical reason: 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

15 15/5 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/ / where ε O is output-noise-limited sensitivity [J/Hz] and ε BA is back-action-limited sensitivity [J/Hz] Γτ m A.K., 1998, 000 S. Pilgram et al., 00 A. Clerk et al., 00 D. Averin, 000,003 Sensitivity limitation is known since 1980s (Caves, Clarke, Likharev, Zorin, Vorontsov, Khalili, etc.); also Zorin-1996, Averin-000, Clerk et al.-00, 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 ) (Shnirman & Schön, 1998) 1 Danilov, Likharev, Zorin, 1983 /4 = = η η ε ε O BA opt

16 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 decomposition quantum Bayesian formalism: So, mathematically, POVM and quantum Bayes are very close (Caves was possibly first to notice) r = r r r M U M M unitary Bayes We emphasize not mathematical structures, but particular setups (goal: find a proper description) and experimental consequences

17 Experimental predictions and proposals from Bayesian formalism Direct experimental verification (1998) Measured spectrum of Rabi oscillations (1999, 000, 00) Bell-type correlation experiment (000) Quantum feedback control of a qubit (001, 004, 009) Entanglement by measurement (00) Measurement by a quadratic detector (003) Squeezing of a nanomechanical resonator (004) Violation of Leggett-Garg inequality (005) Partial collapse of a phase qubit (005) Undoing of a weak measurement (006, 008) Decoherence suppression by uncollapsing (009)

18 left ground 1.0 ρ 11 Reρ 11 Imρ 11 excited Persistent Rabi oscillations z right left e g 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 ( reason : attraction to two points on the Bloch sphere great circle) to verify: stop& check time A.K., 1999 Phase of Rabi oscillations fluctuates (dephasing) Direct experiment is difficult (good quantum efficiency, bandwidth, control)

19 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 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

20 How to understand z Ò =1? ΔI I() t = I0 + z() t + ξ () t First way (mathematical) We actually measure operator: z σ z z σ z =1 Second way (Bayesian) I I SI ( ω Δ ) Sξξ Szz( ) S ( ) 4 ω Δ = + + ξ z ω 0/5 quantum back-action changes z in accordance with the noise ξ (what you see becomes reality) (What does it mean? Difficult to say ) Equal contributions (for weak coupling and η=1) Can we explain it in a more reasonable way (without spooks/ghosts)? +1-1 z left e g right or some other z(t)? z(t)? qubit detector I(t) No (under assumptions of macrorealism; Leggett-Garg, 1985)

21 qubit Leggett-Garg-type inequalities for continuous measurement of a qubit detector I(t) Assumptions of macrorealism (similar to Leggett-Garg 85): I() t = I + ( Δ I/) z() t +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 [ 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

22 May be a physical (realistic) back-action? qubit detector I(t) S I (ω)/s 0 1 S I (ω) ηü ω/ω ΔI I() t = I0 + z() t + ξ () t OK, cannot explain without back-action ξ () t z( t + τ ) 0 But may be there is a simple classical back-action from the noise? In principle, classical explanation cannot be ruled out (e.g. computer-generated I(t); no non-locality as in optics) Try reasonable models: linear modulation of the qubit parameters (H and ε) by noise ξ(t) No, does not work! Our (spooky) back-action is quite peculiar: ξ () t dz( t + 0) > 0 what you see is what you get : observation becomes reality

23 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) 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)

24 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) 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) ΔHfb = F φm H I(t) η eff = control cos (Ω t), τ-average local oscil. sin (Ω t), τ-average 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

25 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) 5/5

26 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)

27 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 undoing (information erasure) ψ 0 (still unknown) ψ

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

29 Uncollapse of a qubit state Evolution due to partial (weak, continuous, etc.) measurement is non-unitary (though coherent if detector is good!), therefore it is impossible to undo it by Hamiltonian dynamics. How to undo? One more measurement! = need ideal (quantum-limited) detector (similar to Koashi-Ueda, PRL-1999, also Nielsen-Caves-1997, Royer-1994, etc.) (Figure partially adopted from Jordan-A.K.-Büttiker, PRL-06)

30 Uncollapsing for DQD-QPC system A.K. & Jordan, PRL-006 I(t) Qubit (DQD) H=0 r(t) First accidental Undoing measurement measurement r 0 Detector (QPC) t 30/5 Simple strategy: continue measuring until result r(t) becomes zero! Then any unknown initial state is fully restored. (same for an entangled qubit) However, if r = 0 never happens, then undoing 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! - r0 e r - r e ρ (0) + e ρ (0)

31 POVM formalism (Nielsen-Chuang, p.100) General theory of uncollapsing Uncollapsing operator: 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

32 1 0 Partial collapse of a Josephson phase qubit Γ How does a qubit state evolve in time before tunneling event? 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 continuous null-result collapse N. Katz, M. Ansmann, R. Bialczak, E. Lucero, R. McDermott, M. Neeley, M. Steffen, E. Weig, A. Cleland, J. Martinis, A. Korotkov, Science-06 (What happens when nothing happens?) Qubit ages in contrast to a radioactive atom! amplitude of state 0> grows without physical interaction finite linewidth only after tunneling (better theory: Pryadko & A.K., 007) (similar to optics, Dalibard-Castin-Molmer, PRL-199)

33 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

34 Experimental technique for partial collapse Nadav Katz et al. (John Martinis group) Protocol: 1) State preparation by applying microwave pulse (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

35 ψ in = Experimental tomography data Nadav Katz et al. (UCSB, 005) p =0 p =0.14 p =0.06 θy θx p =0.3 p =0.3 p =0.43 π π/ p =0.56 p =0.70 p = /5

36 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 (null-result measurement) phase qubit evolves This evolution is well described by a simple Bayesian theory, without fitting parameters Phase qubit remains fully coherent in the process of continuous collapse (experimentally ~80% raw data, ~96% after account for T1 and T) quantum efficiency η 0 > 0.8

37 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

38 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

39 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

40 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 40/5

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

42 Protocol: ρ 11 partial collapse towards ground state (strength p) Suppression of T 1 -decoherence by uncollapsing storage period t (zero temperature) Ideal case (T 1 during storage only, T=0) 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 ) (almost same as existing experiment!) Trade-off: fidelity vs. selection probability 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 Unraveling of energy relaxation t/ T1 * t/t1 β e αβ e * t/ T1 t/ T = 1 αβe 1 β e = p (1 p ) ψ ψ t/ T β (1 e 1 ) where p = t/t ψ =( α 0 + βe 1 1 )/ Norm - t / T 1 fl optimum: 1 - p = e (1- p) u

43 An issue with quantum process tomography (QPT) QPT fidelity is usually where χ is the QPT matrix. However, QPT is developed for a linear quantum process, while uncollapsing (after renormalization) is non-linear. The two ways practically coincide (within line thickness) F χ = Tr( χdesired χ ) A better way: average state fidelity F = Tr( ρ U ψ ψ ) d ψ av Without selection f 0 in in in s ( d + 1) Fav -1 Fχ = Fav =, d = d Another way: naïve QPT fidelity (via 4 standard initial states) F av s, Fχ Analytics for the ideal case Average state fidelity 1 1 ln(1 + C) Fav = + + C C Naïve QPT fidelity C Fχ = (1 + C) ( + C) where C = (1 p)(1 e Γt ) Γt p = 1 e (1 p) Ideal u p u = 1- e -t/t1 (1-p) p u = p without uncollapsing e -t/t 1 = measurement strength p

44 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. Protocol: meas protected Easy to realize experimentally (similar to existing experiment) Increase of fidelity with p can be observed experimentally Improved fidelity can be observed with just one partial measurement π not π meas decoherence due to pure dephasing is not affected T 1 -decoherence between first π-pulse and second measurement causes decrease of fidelity at p close to 1 Trade-off: fidelity vs. selection probability Uncollapse seems to be the only way to protect against T 1 -decoherence without encoding in a larger Hilbert space (QEC, DFS) A.K. & Keane, arxiv:

45 One more experimental proposal: Persistent Rabi oscillations revealed in low-frequency noise Hopefully, simple enough for semiconductor qubits Goal: something easy for experiment, but still with a non-trivial measurement effect 45/5

46 V A (t) Setup: one qubit & two detectors QPC A I A (t) Ω qubit (DQD) I B (t) QPC B V B (t) Single-shot measurements are not yet available fl use train (comb) of meas. pulses in QND regime One-detector stroboscopic QND measurement Δ t =π/ω (one pulse per Rabi period) V(t) 1 z(t) -1 time on off τ on off τ A τ B For single-shot measurements partial collapse can be revealed via correlations of I A and I B. (Korotkov, PRB-001) Same idea with another averaging weak values (Romito et al., PRL-008) Stroboscopic QND: Braginsky, Vorontsov, Khalili, 1978 Jordan, Buttiker, 005 Jordan, Korotkov, 006 Stroboscopic QND measurement synchronizes (!) phase of persistent Rabi oscillations (attracts to either 0 or π)

47 V A (t) QPC A I A (t) Ω qubit (DQD) I B (t) QPC B V B (t) Idea of experiment Perfect QND fl correlation/anticorr. between currents in two detectors Imperfect QND fl random switching between two Rabi phases (0 and π) fl low-frequency telegraph noise same combs on V A and V B π-shifted combs on V A and V B V A (t) t V A (t) t 1 z(t) -1 V B (t) t 1 z(t) -1 V B (t) t anticorrelation between I A and I B correlation (still QND!) correlation/anticorrelation between low-frequency (telegraph) noises indicates presence of persistent Rabi oscillations

48 V A (t) S S IA QPC A Analytical results for current noise I A (t) Ω qubit (DQD) I B (t) QPC B V B (t) 1/ΓS ( ΔI A) ω S δt ( ) A δta ω SA + T T 1 + ( / Γ ) shot noise telegraph noise δ t 1/, ( ) A δ tb ΓS ω ± ΔIA ΔIB T T 1 + ( ω / Γ ) IA IB (fully correlated/anticorrelated in first approx.) A B δ A A δ B B A + B 1 M M t M t M Γ S + Ω [ ϕ + + ] 4T 4π M M 1T ΔI M t ( ) /4S, S ei (1 T ) S V A (t) V B (t) det. noise noise correlation δt A T =π/ω ϕ/ω (phase shift ϕ) S IA S IB -π 0 π ϕ (phase shift -π S IA,IB π ϕ - AB, = δ AB, AB, AB, AB, = AB, AB, ϕ 1, δt T, δt 4 S/( ΔI), T T Correlat. factor K ± 1 Assumed: t t between combs) ϕ

49 S AB (0), S A (0) (arb.units) Calculation based on numerical solution of the master equation Numerical results Low-frequency telegraph noise (dashed) and cross-noise (solid) solid - cross-noise (correlation) dashed - one-detector telegraph noise 1/T =0 T(ΔI A,B ) /4S A,B = ϕ/π (phase shift) δt/t=0.05 δt/t=0.1 δt/t=0. harmonic

50 I A (t) I B (t) Estimates Assume: V A (t) QPC A Ω qubit (DQD) Then: QPC B V B (t) QPC current I =100 na response ΔI/I =0.1 duty cycle δt/t=0. (symmetric) Rabi frequency ~ GHz attraction (collapse) time 1.5 ns (few Rabi periods) switching rate S telegraph S shot Γ 1 1 ϕ S 4T + 1μs + 13ns T 600 min(, 1) 50ns (many Rabi periods) need T > 10ns seems to be reasonable and doable (relatively large noise signal) 50/5

51 Useful modification V A (t) t V A (t) t V B (t) t V B (t) t V A noise correlation QPC A I A (t) I B (t) Ω qubit Any alternative explanation? QPC B V B ϕ (zero average, easier for rf) 1) no oscillations then no corr./anticorr. ) unsynchronized Rabi oscillations then different dependence on ϕ (cos ϕ instead of ϕ - ); also S telegr (f) df at least twice smaller 3) resonant frequency - driven Rabi? Then oscillations between gú and eú (both do not give a signal) with different frequency. Driven Rabi decreases corr./anticorr. (not an alternative explanation, but should be avoided) Good news: both phases insensitive to driven Rabi (harmonic rf is also OK)

52 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 (uncollapsing) Three direct solid-state experiments have been realized Many interesting experimental proposals are still waiting Two last proposals: - suppression of T 1 -decoherence by uncollapsing - persistent Rabi oscillations revealed via noise correlation in two detectors 5/5