Continuous quantum measurement of solid-state qubits

Transcription

1 Wayne State University, Detroit, 02/14/18 Continuous quantum measurement of solid-state qubits Outline: Introduction (quantum measurement) Quantum Bayesian theory of qubit measurement Experiments on partial and continuous measurement of superconducting qubits Simultaneous measurement of non-commuting observables of a qubit Arrow of time in continuous qubit measurement

2 Orthodox (Copenhagen) quantum mechanics Schrödinger equation + collapse postulate 1) Fundamentally random measurement result rr (out of allowed set of eigenvalues). Probability: 2) State after measurement corresponds to result: Instantaneous, single quantum system (not ensemble) Contradicts Schr. Eq., but follows from common sense Needs observer to get information Why so strange (unobjective)? - Shut up and calculate - May be QM founders were stupid? - Use proper philosophy? pp rr = ψψ ψψ rr 2 ψψ rr

3 Werner Heisenberg Books: Physics and Philosophy: The Revolution in Modern Science Philosophical Problems of Quantum Physics The Physicist's Conception of Nature Across the Frontiers Immanuel Kant ( ), German philosopher Critique of pure reason (materialism, but not naive materialism) Nature - Thing-in-itself (noumenon, not phenomenon) Humans use concepts (categories) of understanding ; make sense of phenomena, but never know noumena directly A priori: space, time, causality Niels Bohr A naïve philosophy should not be a roadblock for good physics, quantum mechanics requires a non-naïve philosophy Wavefunction is not a reality, it is only our description of reality

4 Bell s inequality (EPR paradox, CHSH) a instantaneous (superluminal) b or ψψ = rotationally invariant Experiments (1982--present, photons): yes, spooky action-at-a-distance What about causality? Not too bad: only useless (quantum) information is transmitted faster than light, you cannot transmit useful (classical) information by choosing meas. direction aa The other meas. result does not depend on aa Randomness saves causality Collapse is still instantaneous: not a physical process Consequence of causality: No-cloning theorem (1982) You cannot copy an unknown quantum state Proof: Otherwise get information on direction aa (and causality is violated) Application: quantum cryptography Information is an important concept in quantum mechanics

5 Causality principle in quantum mechanics objects a and b observers A and B (and C) time A space light cones a b C Our focus: continuous collapse 0 B observers have free will ; they can choose an action A choice made by observer A can affect evolution of object b back in time However, this retroactive control cannot pass useful information to B (no signaling) Randomness saves causality (even C cannot predict result of A measurement) Ensemble-averaged evolution of object b cannot depend on actions of observer A 1

6 What is inside collapse? What if collapse is stopped half-way? Various approaches to non-projective (weak, continuous, partial, generalized, etc.) quantum measurements Names: Davies, Kraus, Holevo, Mensky, Caves, Knight, Walls, Carmichael, Milburn, Wiseman, Aharonov, Molmer, Gisin, Percival, Belavkin, (very incomplete list) Key words: POVM, restricted path integral, quantum trajectories, quantum filtering, quantum jumps, stochastic master equation, etc. Limited scope: (simplest system, experimental setups) solid-state qubit detector classical output

7 Quantum Bayesian formalism for qubit meas. qubit (double Qdot) V I(t) detector (quantum point contact) Bayes rule (1763, Laplace-1812): posterior probability prior probab. Qubit evolution due to measurement (informational back-action) 1) αα tt 2 and ββ tt 2 evolve as probabilities, i.e. according to the Bayes rule (same for ρρ iiii ) 2) phases of αα tt and ββ tt do not change (no dephasing!), ρρ iiii ρρ jjjj = const likelihood PP AA ii res = PP AA ii PP(res AA ii ) norm ψψ tt = αα tt 0 + ββ tt 1 or ρρ iiii (tt) ρρ iiii II = tt 0 II tt ddddd tt PP PP( II 0) II = ρρ 00 0 PP (A.K., 1998) II 0 II 1 measured So simple because: 1) no entanglement at large QPC voltage 2) QPC is ideal detector 3) no other evolution of qubit II PP( II 1) II 0 + ρρ 11 0 PP( II 1)

8 Two derivations 1. Logical derivation - Probabilities must evolve classically (quantum-classical correspondence) - Lower bound for ensemble dephasing since ρρ 01 ρρ 00 ρρ 11 - Comparison with ensemble-averaged evolution shows ρρ 01 = ρρ 00 ρρ Microscopic derivation - Solve combined quantum evolution, qubit+detector - Apply textbook collapse to detector qubit double Qdot V I(t) detector (quantum point contact) Informational quantum back-action: amplitude likelihood ψψ tt = PP II 0 αα PP II 1 ββ 0 1 norm

9 Further steps in quantum Bayesian formalism 1 PP( II 0) 0 II PP( II 1) αα tt 0 + ββ tt 1 I(t) II = 0 tt II tt ddddd tt II 0 II 1 measured ρρ iiii tt 1. Informational ( quantum ) back-action, likelihood ψψ tt = PP II 0 αα PP II 1 ββ 0 1 norm 2. Add unitary (phase) back-action, physical mechanisms for QPC and cqed ψψ tt = PP II 0 exp iiii II II 0 + II 1 2 norm 3. Add detector non-ideality (equivalent to dephasing) ρρ iiii tt = PP II ii ρρ iiii 0 norm, αα PP II 1 ββ(0) 1 ρρ 01 tt ρρ 00 tt ρρ 11 tt = ee iiii( II II 0+II 1 2 ) ρρ 01 0 ρρ 00 0 ρρ 11 0 γγ = Γ ΔII 2 4SS II exp( γγγγ) KK2 SS II 4

10 Further steps in quantum Bayesian formalism 4. Take derivative over time (if differential equation is desired) Simple, but be careful about definition of derivative dddd tt dddd = ff tt + dddd 2 ff(tt dddd/2) dddd Stratonovich form preserves usual calculus dddd tt dddd = ff tt + dddd ff(tt) dddd Ito form requires special calculus, but keeps averages 5. Add Hamiltonian evolution (if any) and additional decoherence (if any) Standard terms Steps 1 5 form the quantum Bayesian approach to qubit measurement (A.K., )

11 Generalization: measurement of operator AA Informational quantum Bayesian in differential (Ito) form: ρρ = AAAAAA (AA2 ρρ + ρρaa 2 ) 2 2ηηηη + AAAA + ρρρρ 2ρρTr (AAAA) 2SS ξξ(tt) II tt = Tr AAAA + SS 2 ξξ(tt) noisy detector output SS: spectral density of the output noise ξξ tt ξξ tt = δδ(tt tt ) normalized white noise With additional unitary (Hamiltonian) back-action BB and additional evolution ρρ = L ρρ + AAAA + ρρρρ 2ρρTr (AAAA) 2SS ηη: quantum efficiency ξξ tt ii BB, ρρ 1 2SS ξξ tt L[ρρ]: ensemble-averaged (Lindblad) evolution The same as in the Quantum Trajectory theory (Wiseman, Milburn, ) Nowadays quantum trajectories often mean Bayesian real-time monitoring

12 Quantum measurement in POVM formalism Davies, Kraus, Holevo, etc. (Nielsen-Chuang, pp. 85, 100) system ancilla Measurement (Kraus) operator M r (any linear operator in H.S.) : Probability : Completeness : P r 2 M r r M r = ψ = M ψ or r 1 M ψ M r or ρ rψ r = r ρ r P Tr( M M ) 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 polar decomposition: r = r r r M U M M unitary Bayes

13 Circuit QED setup for superconducting qubits Idea: qubit state shifts resonator frequency, this affects amplitude and phase of microwave passed through (reflected from) resonator Narrowband setup, so two signals (quadratures): AA tt cos(ωω dd tt) + BB tt sin(ωω dd tt) microwave generator ω d resonator ω r qubit (transmon) paramp mixer output (two quadratures) HH = 1 2 ωω qqqqσσ zz + ωω rr aa aa + χχσσ zz aa aa resonator frequency: qubit frequency: ωω rr ± χχ ωω qqqq + 2χχ nn (ac Stark shift) Theory: A. Blais et al., PRA-2004 First expt.: A. Wallraff et al., Nature-2004 carries information on qubit state (causes informational back-action) e g cos(ωω dd tt) sin(ωω dd tt) carries information on fluctuating photon number in the resonator (causes phase back-action)

14 Phase-sensitive amplifier µwave gen. ω resonator d ω r qubit paramp mixer cos( ω d t ) e g ϕ sin( ω d t ) get some information (~cos 2 ϕ) about qubit state and some information (~sin 2 ϕ) about photon fluctuations ρgg ( τ) ρgg (0) exp[ -( I -Ig ) / 2 D] = 2 ρ ( τ) ρ (0) exp[ -( I -I ) / 2 D] ee ee e ρge( τ) = ρge(0) ρgg ( τ) ρee( τ) ρgg (0) ρee(0) exp( ikiτ) Bayes unitary (rotating frame) A.K., arxiv: II = ττ 1 0 ττ II tt dddd Γ = PP DD = SSII /2ττ II gg II ee = ΔII cos φφ KK = ΔII sin φφ SS II ΔII cos φφ 2 4SS II PP( II gg) II = ρρ gggg 0 PP + KK 2 SS II 4 = ΔII2 = 8χχ2 nn 4SS II κκ Amplified phase ϕ controls trade-off between informational & phase back-actions (we choose if photon number fluctuates or not) II gg II ee PP( II ee) II gg + ρρ eeee 0 PP II ee

15 Non-trivial causality Ensemble-averaged evolution cannot be affected retroactively, but single realizations can be affected back in time µwave gen. ω resonator d ω r qubit paramp mixer e A.K., arxiv: We can choose direction of qubit evolution to be either along parallel or along meridian or in between (delayed choice) g Expt. confirmation: K. Murch et al., Nature-2013

16 µwave gen. Phase-preserving amplifier resonator mixer ω d ω r paramp qubit Now information in both II(tt) and QQ(tt) e g I(t) cos(ωω dd tt) sin(ωω dd tt) Q(t) II tt : qubit information QQ(tt): photon fluct. info ρ gg ( τ) ρ (0) exp[ -( - ) / 2 ] = gg I I g D 2 ρ ( τ) ρ (0) exp[ -( I -I ) / 2 D] ee ee e ρge( τ) = ρge(0) ρgg ( τ) ρee( τ) ρgg (0) ρee(0) exp( ikqτ) Bayes unitary Similar to phase-sensitive case, but separate I and Q channels 2 II = 1 ττ ττ 0 II tt dddd II gg II ee = ΔII 2 KK = Γ = ΔII 2 8SS II + ΔII 2 8SS II QQ = 1 ττ ττ 0 QQ tt dddd ΔII 2SS II = 8χχ2 nn κκ Equal contributions to ensemble dephasing from informational & phase back-actions A.K., arxiv: DD = SS II 2ττ

17 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

18 Experiments

19 Partial collapse of a Josephson phase qubit 1 0 Γ N. Katz, M. Ansmann, R. Bialczak, E. Lucero, R. McDermott, M. Neeley, M. Steffen, E. Weig, A. Cleland, J. Martinis, A. Korotkov, Science-2006 What happens if nothing happens? Main idea: ψψ 0 = αα 0 + ββ 1 ψψ tt = out, if tunneled αα 0 + ββee Γtt/2 ee iiii 1, ifnot tunneled norm Non-trivial: amplitude of state 0 grows without physical interaction finite linewidth only after tunneling continuous null-result collapse (idea similar to Dalibard-Castin-Molmer, PRL-1992)

20 Polar angle Azimuthal angle Partial collapse: experimental results lines - theory dots and squares expt. p= 1-e -Γt no fitting parameters in (a) and (b) probability p pulse ampl. N. Katz et al., Science-2006 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 2 ) Visibility in (c) T 1 =110 ns, T 2 =80 ns (measured) probability p Good confirmation of the theory

21 Uncollapse for qubit-qpc system (theory) I(t) Detector (QPC) Qubit (double-dot) First accidental measurement r 0 r(t) I t r( t) = [ I( t ') dt ' I 0 0t] S - I Uncollapsing measurement Simple strategy: continue measuring until r(t) becomes zero. Then any unknown initial state is fully restored. If r = 0 never occurs, then uncollapsing is unsuccessful. A.K. & A. Jordan, PRL-2006 t ψ 0 (unknown) weak (partial) measurement ψ 1 (partially collapsed) uncollapse (information erasure) ψ 0 (still unknown) ψ 2 Somewhat similar to quantum eraser of Scully and Druhl (1982)

22 Experiment on wavefunction uncollapse partial state measure p preparation I µw I dc 10 ns 7 ns p π partial measure p 10 ns 7 ns p tomography & final measure time If no tunneling for both measurements, then initial state is fully restored iφ Γt /2 α 0 + e β e 1 α 0 + β 1 Norm iφ Γt/2 iφ Γt/2 e αe 0 + e β e 1 = iφ e ( α 0 + β 1 ) Norm phase is also restored ( spin echo ) N. Katz, M. Neeley, M. Ansmann, R. Bialzak, E. Lucero, A. O Connell, H. Wang, A. Cleland, J. Martinis, and A. Korotkov, PRL-2008 Uncollapse protocol: - partial collapse - π-pulse - partial collapse (same strength) 1 0 p= 1-e -Γt Γ

23 Initial state Partially collapsed Experimental results on the Bloch sphere 0 i N. Katz et al., 2008 Uncollapsed uncollapse 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 Uncollapse in continuous qubit measurement: K. Murch et al., Nature-2013

24 Decoherence suppression by uncollapse Protocol: ρ 11 storage period t (zero temperature) partial collapse towards ground state (strength p) π π uncollapse (measurem. strength p u ) } without uncollapse (1-p u ) κ 3 κ 4 = (1-p) κ 1 κ 2 κ 2 = κ ϕ = 1, 0.95 κ 1 = κ 3 = κ 4 = 1, 0.999, t 0.2 i / T1 κ i = e tσ / Tϕ κϕ = e measurement strength p Theory: A.K & Keane, PRA-2010 Sleeping beauty analogy First realized in optics Lee et al., Opt. Expr Also used for entanglement preservation Kim et al., Nature Phys Realization with superconducting phase qubits Zhong et al., Nature Comm T 1 = 2.5 µs natural Increases effective T 1 by 3x τ store = 3 µs improved

25 Non-decaying (persistent) Rabi oscillations left excited ground z left e g right right - Relaxes to the ground state if left alone (low-t) - Becomes fully mixed if coupled to a high-t envir. - Oscillates persistently between left and right if (weakly) measured continuously ( reason : attraction to left/right states) qubit C detector I(t) A.K., LT-1999, 2001 A.K.-Averin, 2001 Integral under peak zz 2 (Bloch sph.) ρ Reρ Imρ time A.K., 1999 Rabi frequency Ω perfect Rabi: z 2 = cos 2 =1/2 quantum: z 2 = 1 classical limit: SS peak dddd < 8/ππ 2 Ruskov, A.K., Mizel (2006)

26 Continuous monitoring of Rabi oscillations A.Palacios-Laloy, F.Mallet, F.Nguyen, P. Bertet, D. Vion, D. Esteve, and A. Korotkov, Nature Phys. (2010) n= superconducting qubit (transmon) in circuit QED setup microwave reflection from cavity driven Rabi oscillations ( gg ee ) average photon number: n = 0.23 Theory by dashed lines, very good agreement n = 1.56 Pre-amplifier noise temperature T N = 4 K quantum efficiency T + ω 1 N η = S 4S ~ 10 2

27 Violation of Leggett-Garg inequalities In time domain Rescaled to qubit z-coordinate K( τ) zt () zt ( + τ) A. Palacios-Laloy et al., 2010 K( τ ) + K( τ ) K( τ + τ ) 1 2 K( τ) K(2 τ) 1 flg ( τ ) 2 K ( τ ) K (2 τ ) f (0) = K(0) = z LG 2 z = 1.01 ± 0.15 f (17 ns) = 1.44 ± 0.12 LG Ideal f LG,max =1.5 Standard deviation σ = violation by 5σ 2 Many later experiments on Leggett-Garg ineq. violation, incl. optics and NMR M. Goggin et al., PNAS-2011 J. Dressel et al., PRL-2011 G. Walhder et al., PRL-2011 V. Athalye et al., PRL-2011 J. Groen et al., PRL-2013 A. Souza et al., NJP-2011 (s/c, DiCarlo s group) G. Knee et al., Nat. Comm.-2011

28 Quantum feedback to stabilize Rabi oscillations Bayesian Direct Simple Best but very difficult Similar to Wiseman- Imperfect but simple Milburn (1993, optics) qubit H e control stage (barrier height) environment D (feedback fidelity) (monitor quantum state and control deviation) C<<1 C=C det =1 τ a =0 detector feedback signal I(t) comparison circuit Bayesian equations C env /C det = desired evolution H fb / H = F ϕ F (feedback strength) R. Ruskov & A.K., 2002 ρ ij (t) (apply measurement signal to control with minimal processing) D (feedback fidelity) H / H = Fsin( Ωt) fb I() t I0 cosωt I /2 C=1 η =1 averaging time τ a = (2π/Ω)/ F (f db k t th) F (feedback strength) R. Ruskov & A.K., 2002 C<<1 qubit detector D (feedback fidelity) (do as in usual classical feedback) Hfb = F φm H I(t) η eff = Berkeley-2012 experiment: direct and simple control cos (Ω t), τ-average local oscil. sin (Ω t), τ-average ε/h 0 = Ω/CΩ=0.2 1 X Y rel. phase C = 0.1 τ [( I) 2 /S I ] = 1 φ m F/C (feedback strength) A.K., 2005

29 Quantum feedback of Rabi oscillations R. Vijay, C. Macklin, D. Slichter, S. Weber, K. Murch, R.Naik, A. Korotkov, and Irfan Siddiqi, Nature-2012 (quantum feedback with atoms, stabilizing photon number: C. Sayrin, S. Haroche, Nature-2011) Simple idea: Ω / Ω = F sin( θ ), sin( θ ) ~ I( t)sin( Ω t) R R ERR I( t) ~ cos( Ω t θ ) + noise R ERR ERR R arb. units Feedback OFF phase-sensitive paramp arb. units Feedback ON quantum state tomography Rabi freq. 3 MHz, paramp BW 10 MHz, cavity LW 8 MHz, env. deph MHz

30 Quantum trajectories of transmon qubit K. Murch, S. Weber, C. Macklin, Partial measurement: expt. vs. Bayesian theory phase-sensitive paramp η = 0.49 non-trivial causality & I. Siddiqi, Nature-2013 Coupling 0.52 MHz Cavity LW 10.8 MHz Paramp BW 20 MHz Individual quantum trajectories: experiment vs. Bayesian theory _t =1.8µs n=0.4 η = 0.49 S = 3.15 VV mm = 1 tt VV tt ddddd _ n=0.4 dashed: theory, solid: tomography along meridian along equator Good agreement with simple Bayesian theory (dashed)

31 Quantum trajectories with Rabi drive S. Weber, Chantasri, Dressel, Jordan, Murch, and I. Siddiqi, Nature-2014 η tot = 0.4 Ensembleaveraged dashed: theory, solid: tomography Individual Good agreement with simple Bayesian theory

32 Phase-preserving continuous measurement M. Hatridge, Shankar, Mirrahimi, Schackert, Geerlings, Brecht, Sliwa, Abdo, Frunzio, Girvin, Schoelkopf, and M. Devoret, Science-2013 phase-preserving paramp η = 0.2 Protocol: 1) Start with 0>+ 1> 2) Measure with controlled strength 3) Tomography of resulting state Experimental findings: Result of I-quadrature measurement determines state shift along meridian of the Bloch sphere Q-quadrature meas. result determines shift along parallel (within equator) Agrees well with the theory

33 Suppression of measurement-induced dephasing by feedback (undoing motion along equator) G. de Lange, Riste, Tiggelman, Eichler, Tornberg, Johansson, Wallraff, Schouten, and L. DiCarlo, PRL-2014 η = 0.5 Phase-sensitive amplifier, measure non-informational quadrature (back-action is along parallels) Idea: collect measurement signal (with weight function) to find back-action; then undo refocusing (feedback) increases qubit coherence 2 ρρ 01 from 0.40 to 0.56

34 Entanglement by measurement (theory) entangled qubit 1 qubit 2 ρ (t) I(t) R.Ruskov & A.K., 2003 V ga V I(t) V gb detector I(t) same current for 01 and 10 entangles gradually Ha Hb DQDa QPC DQDb H a qubit a SET qubit b Hb (probabilistically, even with Rabi oscillations) Similar proposal in optics J. Kerckhoff, L. Bouten, A. Silberfarb, and H. Mabuchi, 2009

35 Entanglement by measurement (expt.) D. Riste, Dukalski, Watson, de Lange, Tiggelman, Blanter, Lehnert, Schouten, and L. DiCarlo, Nature-2013 Two superconducting qubits in the same resonator, indistinguishable 01 and 10 Max. concurrence 0.77 Trick: 00 and 11 are only slightly distinguishable Max. deterministic concurrence 0.34 Race against decoherence (η is not very important)

36 Measurement-induced entanglement of remote qubits N. Roch, Schwartz, Motzoi, Macklin, Vijay, Eddins, Korotkov, Whaley, Sarovar, I. Siddiqi, PRL qubits separated by 1.3 m of cable η loss = 0.81 η meas = 0.4 dots: experiment dashed: simple theory solid: full theory same output signal for 01 and 10 entanglement First demonstration of remote entanglement of superconducting qubits Remote entanglement is much more difficult than local (η loss is very important) Simple theory is close to full quantum trajectory, fits well experimental data.

37 Simultaneous measurement of non-commuting observables of a qubit Measurement of three complementary observables for a qubit Evolution: For continuous measurement, nothing forbids simultaneous measurement of non-commuting observables Very simple quantum Bayesian description: just add terms for evolution dr dt = γ r aut { ( )(1 r) [ r [ r ut ( )]]} purity state purification η = 1 η = 0.5 1/ η = 1+ 2γτ η = time (t /τmeas ) meas Ruskov, A.K., Molmer, PRL-2010 monitoring fidelity diffusion over Bloch sphere simple monitoring η = 1 η = 0.5 red: exponential window η = 0.1 blue: rectangular averaging time (τ/τmeas) Until recently it was unclear how to realize experimentally

38 Simultaneous measurement of σσ xx and σσ zz Actually, any σσ zz cos φφ + σσ xx sin φφ S. Hacohen-Gourgy, L. Martin, E. Flurin, V. Ramasesh, B. Whaley, and I. Siddiqi, Nature-2016 Measurement in rotating frame of fast Rabi oscillations (40 MHz) Double-sideband rf wave modulation with the same frequency Two resonator modes for two channels quantum trajectory theory for simulations Ω Rabi = Ω SB = 2ππ 40 MHz κκ 2ππ = 4.3 and 7.2 MHz Γ 1 1 = Γ 2 1 = 1.3 µs ΓΓ κκ ΩΩ RRRRRRRR

39 ωω rr ωω rr ± Ω RR αα tt κκ rel. phase φφ qubit Rabi Ω RR Physical qubit (Rabi Ω RR ) zz ph tt = rr 0 cos(ω RR tt + φφ 0 ) xx ph tt = rr 0 sin(ω RR tt + φφ 0 ) yy ph tt = yy 0 This modulates resonator frequency ωω rr tt = ωω rr bb + χχrr 0 cos(ω RR tt + φφ 0 ) Drive with modulated amplitude AA tt = εε sin(ω RR tt + φφ) Then evolution of field αα(tt) is αα = iiiirr 0 cos Ω RR tt + φφ 0 Simple physical picture κκ Ω RR αα iiii sin Ω RR tt + φφ κκ 2 αα Fast oscillations (neglect κκ) Δαα tt = ii εε cos Ω Ω RR tt + φφ RR Insert, then slow evolution is αα ss = χχχχ rr 2Ω 0 cos φφ 0 φφ κκ RR 2 αα ss Thus, slow evolution is determined by effective qubit (in rotating frame), zz = rr 0 cos φφ 0, xx = rr 0 sin φφ 0, yy = yy 0, measured along axis φφ (basis 1 φφ, 0 φφ ) rr 0 cos φφ 0 φφ = Tr[σσ φφ ρρ] σσ φφ = σσ zz cos φφ + σσ xx sin φφ Stationary state αα st,1 = αα st,0 = χχχχ Ω RR κκ From this point, usual Bayesian theory Now solve this differential equation J. Atalaya, S. Hacohen-Gourgy, L. Martin, I. Siddiqi, and A.K., arxiv:

40 Correlators in simultaneous measurement of non-commuting qubit observables J. Atalaya, S. Hacohen-Gourgy, L. Martin, I. Siddiqi, and A.K., arxiv: KK iiii ττ = II jj tt + ττ II ii tt Collapse recipe : replace continuous measurement with projective at tt and tt + ττ, use ensemble-averaged evolution in between KK zzzz ττ = Γ zz + cos 2φφ Γ φφ Γ + Γ ee Γ ττ Γ zz + cos 2φφ Γ φφ Γ + Γ ee Γ +ττ KK zzzz ττ = Γ zz + Γ φφ cos φφ + 2 Ω RR sin φφ ee Γ ττ ee Γ +ττ cos φφ + Γ + Γ 2 ee Γ ττ + ee Γ +ττ Γ ± = 1 2 Γ zz + Γ φφ ± Γ zz 2 + Γ φφ 2 + 2Γ zz Γ φφ cos(2φφ) 4 Ω RR 2 1/ TT Γ zz, Γ φφ : measurement-induced decoherence rates, Ω RR : residual Rabi frequency 2TT 2

41 Comparison with experiment Cross-correlators for 11 values of φφ between 0 and ππ Maximally non-commuting: φφ = ππ 2 200,000 experimental traces Very good agreement Self-correlators δδδδ = κκ φφ κκ zz 2Ω RR (correction to angle)

42 Parameter estimation via correlators Rabi frequency mismatch: Ω RR = Ω RR Ω sideband KK zzzz ττ KK φφφφ (ττ) = Ω RR sin φφ Γ + Γ ee Γ +ττ ee Γ ττ Fitting: Ω R = Ω RR Ω sideband 2ππ 12 khz Very sensitive technique (Ω RR /2ππ = 40 MHz) J. Atalaya, S. Hacohen-Gourgy, L. Martin, I. Siddiqi, and A.K., arxiv:

43 Generalization to N-time correlators Many detectors KK ll1 ll 2 ll NN tt 1, tt 2 tt NN = II LLNN tt NN II ll2 tt 2 II(tt 1 ) J. Atalaya, S. Hacohen-Gourgy, L. Martin, I. Siddiqi, and A.K., PRA-2018 Surprising factorization: KK ll1 ll NN tt 1, tt NN = KK ll1..ll NN 2 tt 1, tt NN 2 KK llnn 1 ll NN (tt NN 1, tt NN ) NN = 3 NN = 4 good agreement with experiment

44 Arrow of time for continuous measurement Unitary evolution is time-reversible. Is continuous quantum measurement time-reversible? J. Dressel. A. Chantasri, A. Jordan, and A. Korotkov, PRL-2017 If yes, can we distinguish forward and backward evolutions? Classical mechanics Dynamics is time-reversible. However, for more than a few degrees of freedom, one time direction is much more probable than the other. Posing of the problem: a game We are given a movie, showing quantum evolution ψψ tt of a qubit due to continuous measurement and Hamiltonian, together with soundtrack, representing noisy measurement record. We need to tell if the movie is played forward of backward.

45 Hamiltonian: HH = ħωσσ yy /2 Reversing qubit evolution Measurement output: rr tt = zz tt + ττ ξξ(tt), measurement (collapse) time ττ, white noise ξξ tt ξξ 0 = δδ tt Quantum Bayesian equations (Stratonovich form, ηη = 1) xx = Ωzz xxxxxx ττ, Time-reversal symmetry: yy = yyyy rr ττ, zz = Ωxx + (1 zz 2 ) rr ττ tt tt, Ω Ω, rr rr (so, need to flip Rabi direction and measurement record) This quantum movie, played backwards, is fully legitimate (soundtrack is flipped) Is there a way to distinguish forward from backward?

46 Emergence of an arrow of time Use classical Bayes rule to distinguish forward from backward movie PP FF rr tt = PP FF [rr(tt)] PP FF rr tt + PP BB [rr(tt)] = RR 1 + RR, RR = PP FF[rr(tt)] PP BB [rr(tt)] Since the measurement record ( soundtrack ) is flipped, the particular noise realization becomes less probable (usually) rr tt = zz tt + rr tt = zz tt + ττ ξξ(tt) ττ ξξ BB (tt) ξξ BB tt = ξξ tt 2zz(tt) ττ ξξ BB (tt) is less probable than ξξ tt ln RR = 2 ττ 0 TT rr tt zz tt dddd Relative log-likelihood, distinguishing time running forward or backward For a long movie time TT, almost certainly ln RR > 0, so we will know the direction of time. For a short TT, we will often make a mistake in guessing the time direction.

47 Probability distribution for ln RR Numerical results ln RR = 2 ττ 0 RR = PP FF[rr(tt)] PP BB [rr(tt)] TT rr tt zz tt dddd 2ππ Ω = 0.5ττ xx tt = 0 = 1 Statistical arrow of time emerges at timescale of measurement time ττ (seemingly backward-in-time trajectories are still possible at TT > ττ) Asymptotic behavior (long T) For a long trajectory, probability of guessing the direction of time incorrectly is PP eeeeee Erf RR 3TT 2ττ ± ττ ππππ 4 2TT ττ TT ττ exp 9 TT 16 ττ (decreases exponentially with the ratio TT ττ)

48 Conclusions It is easy to see what is inside collapse: simple Bayesian framework works for many solid-state setups Measurement back-action necessarily has a spooky part (informational, without a physical mechanism); it may also have a unitary part (with a physical mechanism) Quantum Bayesian theory is similar to Quantum Trajectory theory, though looks quite different; also equivalent to POVM Many experiments with superconducting qubits have demonstrated what is inside collapse (most of our proposals already realized) Possibly useful (especially quantum feedback) Simultaneous measurement of non-commuting observables has become possible experimentally Continuous measurement of a qubit is time-reversible (with flipped record), but a statistical arrow of time emerges

49 Thank you!