Quantum measurement and control of solid-state qubits and nanoresonators

Transcription

1 Quantum measurement and control of solid-state qubits and nanoresonators QUEST 4, Santa Fe, August 4 Outline: Introduction (Bayesian approach) Simple quantum feedback of a solid-state qubit (Korotkov, cond-mat/44696) Quadratic quantum measurements (Mao, Averin, Ruskov, Korotkov, PRL 93, 5683, 4) QND squeezing of a nanoresonator (Ruskov, Schwab, Korotkov, cond-mat/4646) Support:

2 Examples of solid-state qubits and detectors qubit H e detector I(t) I(t) Double-quantum-qot and quantum point contact (QPC) Cooper-pair box and single-electron transistor (SET) Two SQUIDs H = H QB + H DET + H INT H QB = (ε/)(c + c -c + c )+H(c + c +c + c ) ε - asymmetry, H tunneling Ω = (4H +ε ) / frequency of quantum coherent (Rabi) oscillations Two levels of average detector current: I for qubit state, I for Response: I = I -I Detector noise: white, spectral density S I

3 H e What happens to a qubit state during measurement? I(t) For simplicity (for a moment) H=ε=, infinite barrier (frozen qubit), evolution due to measurement only Orthodox answer Conventional (decoherence) answer (Leggett, Zurek) exp( Γt) exp( Γt) > or >, depending on the result no measurement result! ensemble averaged Orthodox and decoherence answers contradict each other! applicable for: Orthodox Conventional (ensemble) Bayesian Single quantum systems yes no yes Continuous measurements no yes yes Bayesian formalism describes gradual collapse of single quantum systems Noisy detector output I(t) should be taken into account

4 H e I(t) Bayesian formalism for a single qubit ˆ ε HQB = ( cc cc ) + Hcc ( + cc ) Ò ÆI, ÒÆI I=I -I, I =(I +I )/, S I detector noise { dρ / dt = dρ / dt = HIm ρ + ρρ ( I / SI )[ I( t) I] dρ / dt = iερ + ih( ρ ρ ) + ρ ( ρ ρ )( I / S )[ I( t) I ] γρ γ A.K., 998 η fi % I =Γ ( I) /4 SI, Γ ensemble decoherence = γ / Γ = ( I ) / 4SIΓ detector ideality (ef ciency), η For simulations: I() t I ( ρ ρ) I / + ξ(), t Sξ = SI Averaging over ξ(t) master equation Ideal detector (η=) does not decohere a single qubit (pure state remains pure), then random evolution of the qubit wavefunction can be monitored Similar formalisms developed earlier. Key words: Imprecise, weak, selective, or conditional measurements, POVM, Quantum trajectories, Quantum jumps, Restricted path integral, etc. Names: E.B. Davies, K. Kraus, A.S. Holevo, C.W. Gardiner, H.J. Carmichael, C.M. Caves, M.B. Plenio, P.L. Knight, M.B. Mensky, D.F. Walls, N. Gisin, I.C. Percival, G.J. Milburn, H.M. Wiseman, R. Onofrio, S. Habib, A. Doherty, etc. (very incomplete list)

5 P "Quantum Bayes theorem (ideal detector assumed) > H > e I(t) Measurement (during time τ): D / _ I actual I I H = ε = (frozen qubit) After the measurement during time τ, the probabilities can be updated using the standard Bayes formula: Quantum Bayes formulas: _ I Initial state: τ ρ() ρ() ρ() ρ() I I() t dt τ P( I, τ ) = ρ() P ( I, τ ) + ρ() P ( I, τ ) P i ( I, τ ) = exp[ ( I I ) / ], i D π D D= S I I I S I I / τ, i, τ I / i P( B ) ( ) ( ) i P A B P B i i A = P( Bk) P( A Bk) k ρ() exp[ ( I I) / D] ( τ) = ρ + ρ ρ () exp[ ( I I ) / D] () exp[ ( I I ) / D] ρ ( τ) ρ () =, ρ( τ ) = ρ( τ ) [ ρ ( τ) ρ ( τ)] [ ρ () ρ ()] / /

6 classical noise affecting ε qubit (ε, H) Nonideal detectors with input-output noise correlation signal quantum backaction noise classical noise affecting ε ξ (t) = Aξ (t) ideal detector ξ 3 (t) fully correlated I d (t) S classical current S ξ (t) + detector I(t) S +S A.K., AS+ θ S K =, SI = S + S S d d I ρ = ρ = HIm ρ + ρρ [ I( t) I] dt dt SI d I ρ = i ερ + ih( ρ ρ ) + ρ ( ρ ρ ) [ I( t) I dt ] + ik[ I( t) I ] γρ SI I K correlation between output and backaction noises Fundamental limits for ensemble decoherence Γ = γ + ( I) /4S I, γ Γ ( I) /4S I Γ = γ + ( I) /4S I + K S I /4, γ Γ ( I) /4S I + K S I /4 Translated into energy sensitivity: (Є I Є BA ) / / or (Є I Є BA Є I,BA ) / /

7 Ideality of realistic solid-state detectors (ideal detector does not cause single qubit decoherence). Quantum point contact Theoretically, ideal quantum detector, η= A.K., 998 (Gurvitz, 997; Aleiner et al., 997) I(t) Experimentally, η > 8% (using Buks et al., 998). SET-transistor I(t) Very non-ideal in usual operation regime, η Shnirman-Schőn, 998; A.K.,, Devoret-Schoelkopf, However, reaches ideality, η = if: - in deep cotunneling regime (Averin,, vanden Brink, ) - S-SET, using supercurrent (Zorin, 996) - S-SET, double-jqp peak (Clerk et al., )??? S-SET, usual JQP (Johansson et al.), onset of QP branch (?) - resonant-tunneling SET, low bias (Averin, ) 3. SQUID V(t) Can reach ideality, η = (Danilov-Likharev-Zorin, 983; Averin, ) 4. FET?? HEMT?? ballistic FET/HEMT??

8 Bayesian formalism for N entangled qubits qb qb qb qb N ρ (t) Up to N levels of current d ρij = ρij [ I ()( t Ii + I j ρkkik ) ( Ii + I j ρkkik )] dt SI k k i ε [ ˆ ij Ii + Ij + Hqb, ρ] ij + i ρij + ikij[ I( t) ] ρij γijρij (Stratonovich form) I() t = ρ () t I + ξ() t Averaging over ξ(t) î master equation Stratonovich: i Ito: ii i detector I(t) A.K., PRA 65, 534 (); PRB 67, 3548 (3) df ( t) f ( t + t/ ) f ( t t / ) lim (easy derivatives and dt t t physical meaning) df () t f ( t+ t) f () t lim (easy averaging over noise) dt t t

9 Experimental predictions and proposals based on the Bayesian formalism Direct experiments on Bayesian evolution (998) Measured spectral density of Rabi oscillations (999,, ) Bell-type correlation experiment () Quantum feedback control of a qubit () Entanglement by measurement () Measurement and entanglement by a quadratic detector (4) Simple quantum feedback via quadratures (4) QND squeezing of a nanoresonator (4)

10 S I (ω)/s S I (ω)/s qubit Measured spectrum of qubit coherent oscillations (or spin precession) α C=3 3.3 detector C = I HS ( ) / I I(t) ε /H= ω/ω α =. η = classical limit ω /Ω S I What is the spectral density S I (ω) of detector current? Assume classical output, ev» Ω ε = Γ = ( ω ), η ( I) / 4 = S + S Ω ( I ) Γ ( ω Ω ) +Γ ω Spectral peak can be seen, but peak-to-pedestal ratio 4η 4 (result can be obtained using various methods, not only Bayesian method) Weak coupling, α = C/8 «S I ( ω ) ηs ε / H = S + + ω Ω Γ ( /4 H ) 4 ηs( + ε / H ) + + [( ω Ω ) Γ ( H / Ω )] A.K., LT 99 Averin-A.K., A.K., Averin, Goan-Milburn, Makhlin et al., Balatsky-Martin, Ruskov-A.K., Mozyrsky et al., Balatsky et al., Bulaevskii et al., Shnirman et al., Bulaevskii-Ortiz, 3 Shnirman et al., 3 Contrary: Stace-Barrett, 3 (PRL 4)

11 Quantum feedback control of a solid-state qubit desired evolution Ruskov & A.K., control stage (barrier height) feedback signal comparison circuit H qb = Hσ X qubit H e C<< detector I(t) Bayesian equations ρ ij (t) environment Goal: maintain desired phase of coherent (Rabi) oscillations in spite of environmental dephasing (keep qubit fresh ) Idea: monitor the Rabi phase φ by continuous measurement and apply feedback control of the qubit barrier height, H FB /H = -F φ To monitor phase φ we plug detector output I(t) into Bayesian equations Quantum feedback in quantum optics is discussed since 993 (Wiseman-Milburn), recently first successful experiments in Mabuchi s group (, 4).

12 K z (τ) Performance of quantum feedback (no extra environment) Qubit correlation function C=, η=, F=,.5, τ Ω /π cosωt C FHτ / Kz( τ ) = exp ( e ) 6F (for weak coupling and good fidelity) Detector current correlation function ( I) cosωt FHτ / KI ( τ ) = ( + e ) 4 C FH / S exp ( e τ ) I + δτ ( ) 6F D (synchronization degree) Fidelity (synchronization degree) C= η = d e = "direct" τ a / T=/ F (feedback factor) I C = ( I) / S H coupling - τ a available bandwidth F feedback strength D= Trρρ desir /3 For ideal detector and wide bandwidth, fidelity can be arbitrary close to % D = exp(-c/3f) Ruskov & Korotkov, PRB 66, 44(R) ()

13 Suppression of environment-induced decoherence by quantum feedback D (synchronization degree) C env /C det =..5 C=C det = τ a = F (feedback factor) D max (synchronization) D max C C env det C env /C det (relative coupling to environment) Big experimental problem: necessity of very fast (>>Ω, GHz-range) real-time solution of the Bayesian equations; therefore wide bandwidth Some help: direct ( naïve ) feedback H / H = F {[ I( t) I ]/ I cos( Ωt)}sin( Ωt) fb However, still wide bandwidth (>> Ω) required D (synchronization degree) C= η = d e = /3 "direct" τ a / T=/3 F (feedback factor)

14 Simple quantum feedback of a solid-state qubit H qb = Hσ X qubit CÜ detector H =H [ F φ m (t)] I(t) control cos(ω t), τ-average sin(ω t), τ-average X Y phase φ m (A.K., cond-mat/44696) Idea: use two quadrature components of the detector current I(t) to monitor approximately the phase of qubit oscillations (a very natural way for usual classical feedback!) t X( t) = [ I( t') I ] cos( Ωt') exp[ ( t t') / τ ] dt t Y( t) = [ I( t') I ] sin( Ωt') exp[ ( t t') / τ ] dt We want to maintain coherent (Rabi) oscillations for arbitrary long time, ρ -ρ =cos(ωt), ρ =isin(ωt)/ φ m = arctan( Y / X) (similar formulas for a tank circuit instead of mixing with local oscillator) Advantage: simplicity and relatively narrow bandwidth (/ τ ~ Γ Ω) Anticipated problem: without feedback the spectral peak-to-pedestal ratio <4, therefore not much information in quadratures (surprisingly, situation is much better than anticipated!) d

15 φ rms (phase inaccuracy) Accuracy of phase monitoring via quadratures π/3 / no noise C<< /Γ d =4S I /( I) C dimensionless coupling (no feedback yet) τ [( I) /S I ] uncorrelated noise C<< C =.,.3, 3 3 (averaging time) p ( φ) τ [( I) /S I ] = C =. Noise improves the monitoring accuracy! (purely quantum effect, reality follows observations ) dφ / dt = [ I( t) I]sin( Ω t+ φ)( I / S I ) dφ / dt = [ I( t) I ]sin( Ω t+ φ )/( X + Y ) m m / (non-gaussian distributions) weak coupling Cá φ =φ -φ 4 m 8 φ Best approximation X +Y =(S I / I) (/5)(4 / -).6 (actual phase shift, ideal detector) (observed phase shift) Noise enters the actual and observed phase evolution in a similar way

16 (fidelity) D, <X>(4/τ I) C =. η = Simple quantum feedback F/C. τ[( I) /S I ]= classical feedback fidelity for different averaging τ (feedback strength) Fidelity F up to ~95% achievable (D~9%) Natural, practically classical feedback setup Averaging τ~/γ>>/ω (narrow bandwidth!) Detector efficiency (ideality) η. still OK Robust to asymmetry ε and frequency shift Ω Very simple verification just positive in-phase quadrature X η eff =.5. (weak coupling C).5 ε/h =. Ω/CΩ=. C =. τ [( I) /S I ] =. nonideal detectors F/C D F F Tr ρ( t) ρ des() t D X (4/ τ I) X in-phase quadrature of the detector current Simple experiment?!

17 Quantum feedback in optics Recent experiment: Science 34, 7 (4) First detailed theory: H.M. Wiseman and G. J. Milburn, Phys. Rev. Lett. 7, 548 (993) No experimental attempts of quantum feedback in solid-state yet (even theory is still considered controversial) Experiments soon?

18 Summary on simple quantum feedback of a solid-state qubit Very straightforward, practically classical feedback idea (monitoring the phase of oscillations via quadratures) works well for the qubit coherent oscillations Price for simplicity is a less-then-ideal operation (fidelity is limited by ~95%) Feedback operation is much better than expected Relatively simple experiment (simple setup, narrow bandwidth, inefficient detectors OK, simple verification)

19 Quadratic Quantum Measurements Mao, Averin, Ruskov, Korotkov; Phys. Rev. Lett. 93, 5683 (4) V ga V I(t) V gb V(f) V, I quadratic H a Hb qubit a SET qubit b I bias Setup similar to Ruskov-Korotkov, PRB 67, 435(R) (3), but a nonlinear (instead of a linear) detector is considered φ, q Linear detector I( ) I( )=I( ) Noninear detector I( ) I( )=I( ) Quadratic detector I( )=I( ) I( ) I( ) I( )= I( ) Quadratic detection is useful for quantum error correction (Averin-Fazio, )

20 Bayesian formalism for a nonlinear detector j j H = HQBs + HDET + [({ t σz}) ξ + t ({ σz}) ξ ] t( x) j=, δσ z + δσ z λσσ z z = t + + Assumed: ) weak tunneling in the detector, ) large detector voltage (fast detector dynamics, and 3) weak response. The model describes an ideal detector (no extra noises). Recipe: Coupled detector-qubits evolution and frequent collapses of the number n of electrons passed through the detector Two-qubit evolution (Ito form): d ρ = ih [, ρ] + [ It ( ) I ][ ( I dt S + I I ) iϕ ] ρ γ ρ kl QBs kl k l kl kl kl kl * kl = (/ )( Γ + +Γ-)[( tk tl ) + kl t ], kl = arg( tktl ) γ ϕ ϕ δ = quadratic detector ρ jj j, k ( -) k, ( -) j + + I = I I = Γ Γ t S = Γ +Γ t (The formula happens to be the same as for linear detector) j

21 S I (ω)/s S I (ω)/s S I (ω)/s Linear detector 8 analytical 6 numerical 4 ω /Ω 3 Nonlinear detector 6 4 Quadratic detector 6 4 ω /Ω 3 ω /Ω 3 =, =, 3=, 4= S I Two-qubit detection (oscillatory subspace) ( ω) = S + η 8 Ω ( I ) Γ 3 ( ω Ω ) +Γ ω Γ= ( I) /4 S, I = I I = I I Spectral peak at Ω, peak/noise = (3/3)η (Ω is the Rabi frequency) Extra spectral peaks at Ω and S I ( ω ) 4 Ω ( I ) Γ = S + ( ω 4 Ω ) +Γ ω ( I = I I, I = I, I = I ) (Ruskov-A.K., ) (analytical formula for weak coupling case) Peak only at Ω, peak/noise = 4η Mao, Averin, Ruskov, A.K., 4

22 Two-qubit quadratic detection: scenarios and switching B ρ (t) Three scenarios: (distinguishable by average current) B 34 B ε Ωt 6E+4 8E+4 3) Slightly asymmetric qubits, ε B B ε Γ 34 = Γ/ Ω ) collapse into - Ú = Ú B, current I Æ, flat spectrum ) collapse into - Ú = Ú B, current I ÆÆ, flat spectrum 3) collapse into remaining subspace 34Ú B, current (I Æ +I ÆÆ )/, spectral peak at Ω, peak/pedestal = 4η. B Switching between states due to imperfections ) Slightly different Rabi frequencies, Ω=Ω -Ω Γ B B = Γ B B = ( Ω) / Γ, Γ= η ( I) /4S ( I ) Γ SI ( ω) = S + ( Ω ) + ω /( ) Γ Ω ) Slightly nonquadratic detector, I I 4 Γ B B = [( I I ) / I] Γ / 34 4 Mao, Averin, Ruskov, Korotkov, 4 4 Ω ( I ) Γ SI ( ω) = S + 3 ( ω 4 Ω ) +Γ ω 4 8( I ) + 7 Γ( I I ) + [4 ( I) / 3 Γ( I I ) ] 4 ω 4

23 S I /S Effect of qubit-qubit interaction ω/ω Ω= H ( εσ + σ σσ j j QBs = j z j x)/+ z z j ν - interaction between two qubits First spectral peak splits (first order in ν), second peak shifts (second order in ν) ν ω - =[ +(ν/)] / - ν/ ω + =[ +(ν/)] / + ν/ ω =[ +(ν/)] / =ω - + ω + Summary on quadratic quantum measurements Bayesian formalism is the same as for linear detectors Detector nonlinearity leads to the second peak in the spectrum (at Ω), in purely quadratic case there is no peak at Ω (very similar to classical nonlinear and quadratic detectors) Qubits become entangled (with some probability) due to measurement, detection of entanglement is easier than for a linear detector (current instead of spectrum), imperfections lead to switching to/from entanglement

24 resonator QPC x QND squeezing of a nanoresonator V(t) m, ω I(t) Ruskov, Schwab, Korotkov, cond-mat/4646 Hˆ pˆ / m m xˆ / = + ω ˆ DET = l l l + r r r + ( l r +..) l r l, r H Eaa Eaa Maa Hc H ˆ ( ˆ INT = Mxaa l r + Hc..) lr, ω GHz, T 5 mk, quantum behavior T< ω or Tτ obs /Q< / Quite similar to Hopkins, Jacobs, Habib, Schwab, PRB 3 (continuous monitoring and quantum feedback to cool down) New feature: Braginsky s stroboscopic QND measurement using modulation of detector voltage squeezing becomes possible Potential application: ultrasensitive force measurements Other most important papers: Doherty, Jacobs, PRA 999 (formalism for Gaussian states) Mozyrsky, Martin, PRL (ensemble-averaged evolution)

25 Stroboscopic QND measurements Quantum nondemolition (QND) measurements (Braginsky-Khalili book) (a way to suppress measurement backaction and overcome standard quantum limit) Idea: to avoid measuring the magnitude conjugated to the magnitude of interest Standard quantum limit Example: measurement of x(t )-x(t ) p> xt ( ) xt ( ) / x First measurement: p(t )> / x(t ), then even for accurate second measurement inaccuracy of position difference is x(t )+(t -t ) /m x(t )> (t -t ) / / m Stroboscopic QND measurements (Braginsky et al., 978; Thorne et al., 978) oscillator Difference in our case: Idea: second measurement exactly one oscillation period later is insensitive to p (or t = nt/, T=π/ω ) continuous measurement weak coupling with detector quantum feedback to suppress heating

26 QPC x V(t) Bayesian formalism for continuous measurement of a nanoresonator m, ω I(t) ˆ ˆ ˆ H = p / m+ mωx / H Eaa Eaa Maa Hc ˆ DET = l l l + r r r + ( l r +..) l r l, r H ˆ ( ˆ INT = M xa l a r + H. c.) Current lr, ( ) ρ ρ Ix = π M + M x re V / = I + kx Detector noise Sx = S ei Recipe: frequent collapses of the number of QPC electrons l Nanoresonator evolution (Stratonovich form), same Eqn as for qubits: dρ( x, x ) i ˆ ρ( x, x ) = [ H, ρ ] + I( t)( Ix + Ix I ) ( Ix + Ix I ) dt S I = I ρ( x, x), I( t) = I + ξ( t), S = S Ito form (same as in many papers on conditional measurement of oscillators): dρ( x, x ) i ˆ k k = [ H, ρ] ( x x ) ρ( x, x ) + ( x+ x x ) ρ( x, x ) ξ( t) dt 4Sη S After that we practically follow Doherty-Jacobs (999) and Hopkins et al. (3) x x ξ

27 ρ Evolution of Gaussian states Assume Gaussian states (following Doherty-Jacobs and Hopkins-Jacobs-Habib-Schwab), then ρ(x,x ) is described by only 5 magnitudes: x, p - average position and momentum (packet center), D x, D p, D xp variances (packet width) Assume large Q-factor (then no temperature) Voltage modulation f(t)v : k = f() t k, Ix = f()( t I + kx), SI = f() t S Then coupling (measurement strength) is also modulated in time: I C = f( t) C, C = k / S mω = 4/ ω τmeas Packet center evolves randomly and needs feedback (force F) to cool down d x / dt = p / m+ ( k / S )sgn[ f( t)] D ξ ( t) d p / dt = mω x + ( k / S )sgn[ f ( t)] D ξ( t) + F( t) Packet width evolves deterministically and is QND squeezed by periodic f(t) d D / dt = (/ m) D ( k / S ) f( t) D x xp x p / ω xp ( / η) ( ) ( / ) ( ) xp xp / (/ ) p ω x ( / ) ( ) x xp d D dt = m D + k S f t k S f t D d D dt = m D m D k S f t D D x xp

28 Squeezing S D x, D x f(t) ( x / x) Squeezing by sine-modulation, V(t)=V sin(ωt) D x D x D = D x + D x no modulation with modulation C =. ω = ω t /π η = C = η =.8 η = ω / ω ω Squeezing obviously oscillates in time, maximum squeezing at maximum voltage, momentum squeezing shifted in phase by π/. S max t ( x) / Dx Analytics (weak coupling): S( ω ) = 3 η, ω =.36 ω C / η η - detector efficiency, C coupling x = ( /mω ) / ground state width D x =( x), D x = x - x Quantum feedback: F = mω γ x γ p x p Ruskov-Schwab-Korotkov (same as in Hopkins et al.; without modulation it cools the state down to the ground state) Feedback is sufficiently efficient, D x ÜD x Squeezing up to.73 at ω =ω

29 Squeezing S f(t) D x, D x Squeezing by stroboscopic (pulse) modulation pulse modulation D x ω = ω /n D x << D x D using ( x / x) x D x =( x) ω t /π /5 / /3 C =.5 δt /T =.5 η = ω / ω feedback ω t /π Momentum squeezing as well Dx Dp Dxp ω t /π Sá Efficient squeezing at ω=ω /n Ruskov-Schwab-Korotkov Tr[ ρ ( x, x )] State purification (natural QND condition)

30 Squeezing by stroboscopic modulation f(t) Squeezing S Squeezing S =( x / x) ( x / x) 3 δ t =.5T C = 8 η = ω / ω τ wait /T = C =. ω = ω δt /T Analytics (weak coupling, short pulses) Maximum squeezing Linewidth S( ω / n) = 3η ωδt ω = 3 4 ω 4 C( δt) πn 3η C dimensionless coupling with detector δt pulse duration, T =π/ω η quantum efficiency of detector (long formula for the line shape) Finite Q-factor limits the time we can afford to wait before squeezing develops, τ wait /T ~Q/π Squeezing saturates as ~exp(-n/n ) after n = 3 η / C ( ωδt) measurements Therefore, squeezing cannot exceed 4 S C Q η

31 Observability of nanoresonator squeezing Ruskov-Schwab-Korotkov Procedure: ) prepare squeezed state by stroboscopic measurement, ) switch off quantum feedback N 3) measure in the stroboscopic way x X N = N j= For instantaneous measurements (δt ) the variance of X N is DXN, = + ( x ) at N mω S NCωδt S Then distinguishable from ground state (S=) in one run for Sà (error probability ~S -/ ) j S squeezing, x ground state width Not as easy for continuous measurements because of extra heating. D X,N has a minimum at some N and then increases. However, numerically it seems min D ( x ) / S (only twice worse) N X, N Example: min N DX, N /( x) =.78 for C =., η=, δt/t =., /S=.36 Squeezed state is distinguishable in one run (with small error probability), therefore suitable for ultrasensitive force measurement beyond standard quantum limit

32 Summary on QND squeezing of a nanoresonator Periodic modulation of the detector voltage modulates measurement strength and periodically squeezes the width of the nanoresonator state ( breathing mode ) Packet center oscillates and is randomly heated by measurement; quantum feedback can cool it down (keep it near zero in both position and momentum) Sine-modulation leads to a small squeezing (<.73), stroboscopic (pulse) modulation can lead to a strong squeezing (>>) even for a weak coupling with detector Still to be done: correct account of Q-factor and temperature Potential application: force measurement beyond standard quantum limit

33 Conclusions Bayesian formalism for solid-state quantum measurements is being used to produce various experimental predictions (though still not well-accepted in solid-state community) Simple, practically classical feedback using quadratures of the detector current should work well for qubit oscillations; relatively simple experiment Measurements by nonlinear (quadratic) detectors are described by the Bayesian formalism (same formulas as for linear detector), nonlinearity leads to the spectral peak at double frequency and makes easier qubit entanglement by measurement Measurement of a nanoresonator with strength modulated in time (modulating detector voltage) can produce a squeezed state; squeezed state is measurable and potentially useful No solid-state experiments yet; hopefully, reasonably soon