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Michigan State Univ., 09/21/15 Measurement of superconducting and causality Alexander Korotkov University of California, Riverside

Outline:  Introduction (causality, POVM measurement)  Circuit QED setup for measurement of superconducting qubits (transmons, Xmons)  evolution due to measurement in circuit QED (simple and better theories)  Experiments

Alexander Korotkov University of California, Riverside Causality principle in mechanics

objects a and b observers A and B (and C) observers have “free will”; A B they can choose an action A choice made by observer A can affect light a b evolution of object b “back in time” cones

time However, this retroactive control cannot pass space “useful” information to B (no signaling) C Randomness saves causality (even C Our focus: continuous cannot predict result of A measurement) |0  collapse Ensemble-averaged evolution of object b  cannot depend on actions of observer A

|1  Alexander Korotkov University of California, Riverside 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 (1724-1804), 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 A naïve philosophy should not be a roadblock for good physics, requires a non-naïve philosophy Wavefunction is not a reality, it is only our description of reality

Alexander Korotkov University of California, Riverside Textbook (“orthodox”) measurement

Operator of measured Ar  kkk||  k Measurement result rk (eigenvalue of A) 2 with probability Prk |||kin | k  After measurement with result | in  (eigenstate of A) || ||| k  Why the change (collapse) to the eigenstate? Just common sense: “you get what you see” (gives the same result for sequential measurements of a non-evolving object)

Simple ways to spoil “orthodox” measurement  rotation before measurement (possibly random)  misreporting measurement result  rotation after measurement (possibly depending on result) “Orthodox” description assumes a “good” measurement

Alexander Korotkov University of California, Riverside Generalized (POVM) measurement Davies, Kraus, Holevo, 1970s-1980s

Physics: unitary interaction with another system (ancilla), then system ancilla “orthodox” measurement of ancilla † Mathematics: measurement Mr |  MMrr |  or   (Kraus) operators M † r |||||Mr   Tr()MMrr 2 † Probability : PMrr|||||  or PMMr Tr( r r ) Completeness : † r MMrr 1

Reduces to “orthodox” measurement when Mr are orthogonal projectors

This is how theorists think about the most general quantum measurement

Easy to introduce imperfections (incl. decoherence), e.g., via averaging over subsets of Kraus operators (information loss) 5/48 Alexander Korotkov University of California, Riverside Generalized (POVM) measurement (cont.)

Mr |  2 † |  PMrr|||||  r MMrr 1 ||Mr |  || Easy to show that local generalized measurement obeys causality (for ensemble-average state, i.e. averaged over results) A useful way of thinking † polar decomposition: MUMMr r r r unitary quantum Bayes rule † 푀푟 defines a measurement basis (which diagonalizes 푀푟 푀푟). + In this basis operator 푀푟 푀푟 acts as the quantum Bayes rule:

c P( i ) | i Bayes rule (1763, Laplace-1812) ||    ci i i r i i Norm posterior prior probability probab. likelihood so that probabilities 푐 2 follow the classical 푖 P( i ) P (result | i ) Bayes rule Again, you get what you see Pi( | result)  Norm Often the same basis for all results Alexander Korotkov University of California, Riverside Continuous quantum measurement

POVM measurement is still instantaneous (as “orthodox” collapse). In practice measurements are often continuous (noise, SNR).

gradual acquisition of information  gradual collapse

Continuous measurement can be viewed as a sequence of weak POVM measurements.

Still the same general principle: you get what you see (quantum evolution follows information obtained from measurement)

Many people contributed (different approaches): Davies, Kraus, Holevo, Mensky, Caves, Carmichael, Milburn, Wiseman, Aharonov, Gisin, Belavkin, etc. (very incomplete list)

Key words: POVM, restricted path integral, quantum trajectories, quantum filtering, quantum jumps, stochastic master equation, etc.

Alexander Korotkov University of California, Riverside First solid-state example: DQD and QPC |1 (t )  ( t ) |1    ( t ) | 2  or ij ()t H e |2 1  Itdt()  0 measured V I(t) I1 I2

1) |(t)|2 and |(t)|2 evolve as probabilities,

i.e. according to the Bayes rule (same for ii) 2) phases of (t) and (t) do not change 1/2 (no dephasing), ij /(ii jj) = const (A.K., 1999) Can (indirectly) monitor wavefunction evolution in real time However, experiments have been realized with superconducting qubits in circuit QED setup (microwave readout)

Narrowband (two signals): 퐼 푡 cos 휔푡 + 푄 푡 sin(휔푡)

Alexander Korotkov University of California, Riverside Transmon (or Xmon): at present the most widely used superconducting qubit Josephson (tunnel) junction + capacitor Cg (2e)2 V HnnE ()cos 2  g 2C gJ 푬푱 푪 [,] ni nCVeggg /2 (superconducting phase , charge 2en)

- single- box with EJ/EC~100 (~5 charge states involved)

- almost insensitive to ng - EJ often tunable (two junctions) A slightly nonlinear oscillator with two lowest levels used as 0 and 1 .

J. Koch et al., 2007 (Yale) 훿 ≡ 휔01 − 휔12 훿/휔01 ≈ 3 − 6%

휔01/2휋 ≈ 4-6 GHz 훿/2휋 ≈ 0.2-0.4 GHz

Alexander Korotkov University of California, Riverside Circuit QED setup for qubit measurement A. Blais et al., 2004 (Yale)

Schoelkopf & Girvin

A. Wallraff et al., 2004 (Yale) Idea: 1) qubit is coupled with a (microwave) (CPW or lumped); 2) qubit state 0 or |1〉 slightly changes the resonator frequency; 3) change of resonator frequency is sensed by a microwave transmission (or reflection), amplification, and mixing

1 † † H zr  a a aa z (dispersive interaction) 2 qb 10/48 Alexander Korotkov University of California, Riverside Some details and complications

† 1. Dispersive interaction 휒푎 푎휎푧 follows from Janes-Cummings (or Tavis-Cummings) interaction at large qubit-resonator detuning

휔 J-C 퐻 = 푞 휎 + 휔 푎†푎 + 푔(푎† 0 〈1| + 푎 1 〈0|) 2 푧 푟 2 2 푔 휔푞 − 휔푟 휒 ≈ only if |휔푞 − 휔푟| ≫ 푔, 푛 ≪ 푛푐푟푖푡 = 2 휔푞 − 휔푟 4푔 푛 is number of photons, 푛 = 〈푎+푎〉

In experiments usually 푛 ≪ 푛푐푟푖푡 or 푛 < 푛푐푟푖푡 or 푛 < 4푛푐푟푖푡 (increase of 푛 improves measurement, but something goes wrong at big 푛 )

2. For a transmon at least 3 levels should be considered to find χ 2 푔 훿푞 휒 ≈ − 2 훿푞 ≡ 휔01 − 휔12 휔푞 − 휔푟 (then still OK to use two-level approximation in eigenbasis)

Alexander Korotkov University of California, Riverside Some details and complications (cont.) 3. Significant qubit energy relaxation due to coupling with decaying resonator (Purcell effect) 푔2 ΓP = 휅 2 휔푞 − 휔푟 휅 is decay rate for resonator The same result from classical EE analysis (Esteve et al., 1986)

At present the best way to avoid Purcell decay is to use Purcell filter Reed et al., 2010 (Yale), Jeffrey et al., 2014 (UCSB) 41G2 eff () f1 [2(  f  ) /  f ] Purcell decay  1 [2(  ) /  ] suppression q  r ff rq1 [2(  ff ) /  ] Sete et al., 2015 Classical analysis is easier than quantum

Alexander Korotkov University of California, Riverside Circuit QED measurement Our focus: qubit evolution due to measurement resonator mixer   microwave d  generator r amplifier output (two qubit quadratures) (transmon) homodyne meas.

1 † † Qubit ensemble Ha a  zr aa z 2 qb dephasing rate itd † (e ah cH..)  8 2n  qubit state changes resonator frequency;  number of photons affects qubit frequency assuming 휔푟 − 휔푑 ± 휒 ≪ 휅 (ac Stark shift), (easy to derive by tracing over  fluctuations lead to dephasing emitted microwave field) Blais et al., 2004 −푖휀 훼± = Gambetta et al., 2006, 2008 휅 2 + 푖(휔푟 − 휔푑 ± 휒) We are interested in non-averaged qubit evolution Alexander Korotkov University of California, Riverside Physical roles of two quadratures

resonator mixer   microwave d  generator r paramp output (two qubit quantum signal quadratures) (transmon) (2 quadratures)

Two quadratures: carries information about qubit A(t) cos dt + B(t) sin dt ( “quantum” back-action)

cos()dt Assume “bad cavity” regime |e (): |g sin( t ) 휅 ≫ Γ = 8휒2 푛 휅 d carries information about fluctuating photon number in the resonator ( “classical” back-action) With parametric amplifier we can choose which quadrature to amplify (measure)

Alexander Korotkov University of California, Riverside Phase-sensitive and phase-preserving parametric amplifiers (paramps)

Paramps are traditionally discussed in terms of noise temperature 

  0 for phase-sensitive (degenerate, homodyne) paramp    for phase-preserving (non-degenerate, heterodyne) paramp 2 (adds “half a quantum” Haus, Mullen (1962), Giffard (1976), of noise) Caves (1982), Devyatov et al. (1986)

A way to understand

Game: Charlie prepares “coherent state” of an oscillator, 푥푐 푡 = 퐴 cos 휔푡 + 퐵 sin 휔푡, and gives it to David. David’s goal is to find A (or both A and B).

휎푔푟 A can be measured with accuracy 휎푔푟 (“orthodox” or stroboscopic QND at times 휋푛/휔) 푥푐 푡 Both A and B can be measured with accuracy 2휎푔푟 each (adds “half a quantum” into each quadrature)

15/48 Alexander Korotkov University of California, Riverside A phase-sensitive superconducting paramp

Circulators, directional couplers, and hybrids

Theory based on Dykman, Krivoglaz (1971-1984, 1980)

Vijay et al., 2011 (UCB)

- A (classical) nonlinear oscillator: frequency changes with amplitude - A strong pump is added to the signal at the same frequency - Signal in-phase with pump increases amplitude, changes frequency - Frequency change leads to the phase change at the output Other modes of operation: double-pump at 휔 ± 훿, parametric pump at 2휔 Important: one quadrature is amplified, the other quadrature is de-amplified Same device operates as phase-preserving paramp if pumped at shifted frequency Alexander Korotkov University of California, Riverside Continuous measurement of a qubit: simple quantum Bayesian approach Korotkov, arXiv:1111.4061 resonator mixer   microwave d  generator r paramp output (two qubit quadratures) (transmon)

Assume: †† carries information Ha aa a zrz/2 (dispersive) qb about qubit (z)  max(,)  R (“bad cavity”, Markovian, almost no entanglement) cos()dt Also assume (for simplicity): |e  out (everything collected) |g sin()dt d  r (center of resonance) assume everything most ideal carries information about fluctuating Equivalent to (simplified) quantum trajectory photon number in the resonator theory by Gambetta et al., PRA-2008 Alexander Korotkov University of California, Riverside Amplify “informational” quadrature

If qubit evolves only due to measurement, then the I(t) |e diagonal elements of its must evolve as probabilities (i.e., via classical Bayes rule). |g 2 휌 휌 ggggg( )(0) exp[()/ 2] --IID qubit 푒푒 푒푔 = DS /2 state 휌 휌 2 I (noise) 푔푒 푔푔 eeee( )(0) exp[()/--IID 2] e For quantum-limited 1  phase-sensitive paramp IItdt  () measured  0 I 2 I I ()II  g e S  eg I 32 2n Now average over measurement results 퐼 8 2n | (  ) |(  )    (  )(0) (0) exp() egeeee gggg  But Γ = 8휒2 푛 휅. Therefore, no additional dephasing is possible.

gg()()   ee  Number of photons in the resonator does ge(  )=  ge (0) gg(0) ee (0) not fluctuate (we do not measure it) (in rotating frame) Alexander Korotkov University of California, Riverside Now amplify orthogonal quadrature

No information about the qubits state

|e Q(t) ee(  )  ee (0),  gg (  )  gg (0)

|g −1 휏 From 푄 = 휏 0 푄 푡 푑푡 we can infer fluctuation of photon number in the resonator

Number of emitted photons: 푁 = 푛휅휏 ± 푛휅휏. Therefore, 훿푁 푛휅휏 = 훿푄 휎푄. Each photon shifts qubit frequency by 2휒 (ac Stark shift), rotates qubit phase by 2휒 × (2/휅).

()IIeg max ge(  )=  ge (0) exp[i Q] DS /2 2D

Now the number of photons in the resonator fluctuates, but we can monitor how it fluctuates

Alexander Korotkov University of California, Riverside Phase-sensitive paramp, amplify arbitrary phase 

resonator mixer wave d  gen. r paramp |e qubit |g

get some information (~cos2) about qubit state and some information (~sin2) about photon fluctuations

2 1  ggggg( )(0) exp[ () / 2-- ]I ID IItdt () DS /2 =  0 I (  )(0) exp[-- ()I / 2 ID ]2 eeee e I IIIge  cos K  sin ggee(  )(  )  S gege(  )(0)exp()= iKI I (0)(0) (I cos )2 S I 2 8 2 n ggee =K 2 I = = unitary 4SS 4 4  Bayes II (rotating frame) Same as for QPC, but  controls trade-off Can monitor wavefunction! between “quantum” & “classical” back-actions A.K., arXiv:1111.4016 (we choose if photon number fluctuates or not) 15/48 Alexander Korotkov University of California, Riverside Choosing qubit evolution retroactively

resonator I or Q mixer wave d gen. r paramp qubit circulator here

|e  We can retroactively choose the qubit evolution  to be either along meridian or along parallel or in between (delayed choice) |g 

Does not violate causality because ensemble-averaged evolution is not affected (cannot send “useful” information)

A.K., arXiv:1111.4016

Alexander Korotkov University of California, Riverside Phase-preserving paramp Now information in both I(t) and Q(t) I(t) cos() t resonator mixer d wave d gen. r paramp |e qubit |g sin()dt Q(t) Ways to derive: - informational (with twice larger noise) - phase-preserving with rotating  (then average) - split the signal, use two paramps for I and Q

2 1  1  SI ggggg( )(0) exp[ () / 2-- ]I ID II tdt () QQ tdt () D  =  0  0 2 (  )(0) exp[-- ()I / 2ID ]2 eeee e I I II K  ()()    ge gg ee 2 2SI (  )=  (0) exp(iKQ  ) 2 2 2 ge ge (0) (0) I I8 n gg ee == Bayes unitary 88SSII Separated information Equal contributions to ensemble dephasing in I and Q channels from “quantum” & “classical” back-actions Again, can monitor wavefunction A.K., arXiv:1111.4016 2-channel feedback, if needed (easy to undo Q) Alexander Korotkov University of California, Riverside Imperfect quantum efficiency

 collectionamplifier

amplified S     quantum-limited collection  amplifier total Sactual

Experimentally measurable parameters: qubit ensemble dephasing Γ, signal Δ퐼 = (퐼푒 − 퐼푔)max, and noise 푆

Imperfect efficiency is equivalent to qubit dephasing: (I) 2 phase-sensitive:    (1) 4S ggee(  )(  )   gege(  )(0)exp()= iKI e ggee(0)(0) ( I)2 phase-preserving:     (1 ) 2S gg()()   ee   ge(  )=  ge (0) exp(iKQ  ) e gg(0) ee (0)

Alexander Korotkov University of California, Riverside Evolving qubit

Easy to add qubit evolution:

 Time derivative of the above discussed (Bayesian) equations  Add terms for the qubit evolution (Rabi osc., decoherence, etc.)

Need to be careful with definition of the derivative:

df( tf )(/ ttf 2)(/ tt 2)    (Stratonovich)  limt 0 dtt  usual calculus

df()()() t f t  t  f t  lim (Ito) dtt 0  t simple averaging

Quantum trajectory theory uses Ito definition, I usually use Stratonovich definition

Alexander Korotkov University of California, Riverside A better theory A.K., in preparation Remove the “bad cavity” assumption (now significant qubit-cavity entanglement is OK)

Easy for a non-evolving qubit; then the theory can be based on coherent states for the resonator (equiv. to “polaron” transformation approach) Just an elementary quantum mechanics and common sense resonator mixer wave d  gen. r paramp qubit

Idea: - Consider evolution for the qubit in the state |0〉, then in |1〉 - Combine as superposition - Simple “orthodox” model for homodyne measurement of field

Also equivalent to quantum trajectory theory by Gambetta et al., PRA-2008 25/48 Alexander Korotkov University of California, Riverside Preliminaries (optical coherent states)

|푟 훼〉 Passing through a beamsplitter |훼〉 |푡 훼〉 no entanglement, classical Driven resonator with leakage  |0〉

remains pure in spite of dissipation (explanation via beamsplitter or jump – no jump Lindblad)

Alexander Korotkov University of California, Riverside Measurement and “history tail”

resonator mixer |훼(푡)〉 wave |. . (푡 − Δ푡)〉 |. . (푡 − 2Δ푡)〉 |. . (푡 − 3Δ푡)〉 gen. r amp qubit

Assume a non-evolving qubit in state 0 , then resonator frequency 휔푟 − 휒

훼 0 = . . . 휑 0 = . . . Similar for qubit in state 1 , then frequency 휔푟 + 휒

Now non-evolving superposition of 0 and 1 (as in “many worlds”)

Measure pieces of the “history tail” in “orthodox” way (all coherent states)

Alexander Korotkov University of California, Riverside Homodyne measurement of “history tail” pieces

|훼0(푡)〉 |. . 〉 |. . 〉 |. . 〉 |훼0 푡 − 푚Δ푡 휅Δ푡 〉 |. . 〉

|. . 〉 |. . 〉 |훼1 푡 − 푚Δ푡 휅Δ푡 〉 |. . 〉 |. . 〉 |훼1(푡)〉 measure Simple model of homodyne measurement  Add a large coherent state (pump)  Measure number of photons n  “Orthodox” collapse onto obtained random n

This changes superposition coefficients 푐0 and 푐1:

(Gaussian approximation, 휎 is noise variance, 훼푝 is added pump)

Alexander Korotkov University of California, Riverside Resulting evolution (phase-sensitive case)

 Practically the same as in simple quantum Bayesian approach  Now applied to entangled qubit-resonator system  Allows arbitrary , transient evolution  Qubit evolves only due to measurement (no Rabi)  Equivalent to “polaron” approach in quantum trajectories, but very simple derivation

(similar for phase-preserving case, just I for diagonal and Q for off-diagonal) Alexander Korotkov University of California, Riverside How to include qubit evolution properly?

 Similar approach of the “history tail”, but cannot use coherent states, use density matrices in Fock space  Much more difficult computationally  Equivalent to “full” quantum trajectory theory, but different numerical procedure with finite time steps

30/48 Alexander Korotkov University of California, Riverside Quantum trajectory equations

퐽 푡 is homodyne (phase-sensitive) measurement result J. Gambetta et al., PRA-2008 Similar to Wiseman, Milburn (1993) Alexander Korotkov University of California, Riverside . Experiments on partial and 10

continuous measurement of expts superconducting qubits

total # of of # total 1 2006 2008 2010 2012 2014 1. N. Katz et al. (UCSB), Science 312, 1498 (2006); partial collapse 2. N. Katz et al. (UCSB), PRL 101, 200401 (2008); uncollapse 3. A. Palacios-Laloy et al. (Saclay), Nature Phys. 6, 442 (2010); continuous Rabi + weak Leggett-Garg 4. R. Vijay et al. (Berkeley), Nature 490, 77 (2012); quantum feedback of Rabi oscillations 5. M. Hatridge et al. (Yale), Science 339, 178 (2013); partial meas. in cQED (phase-preserving) 6. K. Murch et al. (Berkeley), Nature 502, 211 (2013); quantum trajectories (phase-sensitive) 7. Campagne-Ibarcq et al. (Paris), PRX 3, 021008 (2013); stroboscopic meas. and feedback 8. D. Riste et al. (Delft), Nature 502, 350 (2013); entanglement by measurement 9. J. Groen et al. (Delft), PRL 111, 090506 (2013); partial meas., weak values, LG (via ancilla) 10. J. Zhong et al. (Zhejiang U., UCSB), Nature Comm. 5, 3135 (2014); T1 increase (3x) by uncollapse 11. N. Roch et al. (Berkeley), PRL 112, 170501 (2014); entanglement of remote qubit by meas. 12. S. Weber et al. (Berkeley), Nature 511, 570 (2014); trajectories with Rabi, most likely path 13. G. De Lange et al. (Delft), PRL 112, 080501 (2014); dephasing suppression by feedback 14. Campagne-Ibarcq et al. (Paris), PRL 112, 180402 (2014); interference between past and future 15. D. Tan et al. (Wash.U.), PRL 114, 090403 (2015); prediction/retrodicton 16. N. Foroozani et a. (Wash. U), arXiv:1508.01185; state-signal correlations 17. T. White et al. (UCSB), arXiv:1504.02707; Bell-Leggett-Garg with weak meas. (via ancilla) 18. Y. Liu et al. (Yale), arXiv:1509.00860; entanglement by measurement and by dissipation Alexander Korotkov University of California, Riverside Early experiments on partial collapse with superconducting phase qubits

Partial collapse N. Katz et al., Science-2006 (UCSB)

|1  |,out if tunneled  |0 ==| 0|1(  )  t | 0|1  e-t/2  , if not tunneled  22-t  |||| e

Reversal of partial collapse (uncollapse) N. Katz et al., PRL-2008 (UCSB) | 0|1|  0|1 eeettt/2/2/2 | 0  |1|  0  |1    NormNorm

0 (still  partial 1 unknown) 0 (partially (unknown) meas. collapsed) 2 Theory: A.K. & Jordan, PRL-2006 uncollapse (information erasure) Alexander Korotkov University of California, Riverside Decoherence suppression by uncollapse

1.0 Theory: A.K & Protocol:   0.8 Keane, PRA-2010 (1-pu) 34 = (1-p) 12 11 storage period t 0.6  = 0.3 } without 2 (zero temperature) uncollapse 0.4 = 1, 0.95  =  =  = 1, 0.999, partial collapse uncollapse 1 3 4 0.2 tTi / 1 0.99. 0.9 i  e towards ground (measurem. tT /    e state (strength p) strength pu) 0.0 0.0 0.2 0.4 0.6 0.8 1.0 measurement strength p “Sleeping beauty” analogy First realized in optics Lee et al., Opt. Expr.-2011 Also used for entanglement preservation Kim et al., Nature Phys.-2012

Realization with superconducting phase qubits Zhong et al., Nature Comm.-2014

T = 2.5 s 1  natural

Increases effective T1 by 3x store = 3 s improved Alexander Korotkov University of California, Riverside Non-decaying (persistent) Rabi oscillations excited - Relaxes to the if left alone (low-T) - Becomes fully mixed if coupled to a high-T envir. left right - Oscillates persistently between left and right if (weakly) measured continuously ground (“reason”: attraction to left/right states) z |left C qubit detector I(t) A.K., LT-1999, 2001 |e   |g A.K.-Averin, 2001 |right Integral under peak  〈푧2〉 (Bloch sph.)

1.0 perfect Rabi: 11 z2=cos2=1/2 0.5 2 Re12 quantum: z  = 1 0.0 Im12 -0.5 classical limit: 0 5 10 15 20 25 30 time 2 A.K., 1999 푆peak푑푓 < 8/휋 Rabi frequency  Ruskov, A.K., Mizel (2006) 35/48 Alexander Korotkov University of California, Riverside 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=0.23  superconducting qubit (transmon) 0.78 in circuit QED setup 1.56  microwave reflection from cavity 3.9  driven Rabi oscillations (|g> |e>) 7.8 15.6

average photon Pre-amplifier noise number: n  1.56 temperature TN= 4 K n  0.23 1  0.03 2T 1  N  S quantum   ~ 102 Theory by dashed lines, efficiency 4S very good agreement Alexander Korotkov University of California, Riverside Violation of Leggett-Garg inequalities A. Palacios-Laloy et al., 2010 In time domain Rescaled to qubit z-coordinate Kztzt()()() 

KKKKK()1 ()  2  (  1   2 )1   2()   (2)1   2 fKzLG (0)(0)  fLG ( ) 2 K (  ) K (2  ) z2  1.010.15

fLG (17ns)1.440.12

Ideal fLG,max=1.5 Standard deviation  = 0.065  violation by 5

Many later experiments on Leggett-Garg ineq. violation, incl. optics and NMR M. Goggin et al., PNAS-2011 V. Athalye et al., PRL-2011 J. Groen et al., PRL-2013 J. Dressel et al., PRL-2011 A. Souza et al., NJP-2011 (s/c, DiCarlo’s group) G. Walhder et al., PRL-2011 G. Knee et al., Nat. Comm.-2011 Alexander Korotkov University of California, Riverside Quantum feedback to stabilize Rabi oscillations Bayesian “Direct” “Simple” Best but very difficult Similar to Wiseman- Imperfect but simple (monitor Milburn (1993) (do as in usual classical and control deviation) (apply measurement signal to feedback)

desired evolution control with minimal processing) Hfb feedback F m signal comparison control stage H

(barrier height) circuit HHFtfb /sin() control ij(t) qubit C<<1 detector ItI() X I(t) Bayesian 0 C<

equations  e I /2 I(t) m detector local oscil. 1.0 Y )  sin ( t), τ-average

y environment t

i rel. phase HHF /  l 0.8 fb e

d

i

f

1.0

) 1.00

k 0.6 e 1 c  = e eff C = 0.1 r averaging time

a

g b  = (2/)/10 2 e a 0.8 d 0.4  [(I) /S ] = 1

d 0.95 I

e

n C=1 e 0.5 0

f

o

( i

t 0.2 =1 0.6

C /C = 0 0.1 0.5

a env det 0.5

z

0.90 D

i

n /H = 1 (feedback fidelity) (feedback 0

o C=C =1 0.0

r det 0.4 0

h D 0.0 0.2 0.4 0.6 0.8 0.1 c 0.85 a=0 0.2 .2 n 0 F (feedback strength) C= y /

s F (feedback strength) 0.2

(

(feedback fidelity) (feedback 0.1

(feedback fidelity) (feedback

D 0.80 D D 0 1 2 3 4 5 6 7 8 9 10 R. Ruskov & A.K., 2002 0.0 F (feedbackF (feedback strength) factor) 0.0 0.2 0.4 0.6 0.8 F/C (feedback strength) R. Ruskov & A.K., 2002 Berkeley-2012 experiment: A.K., 2005 “direct” and “simple” Alexander Korotkov University of California, Riverside 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: Itt( ) ~ cos()noiseRERR

RRERR/sin(),  F  sin()ERRR ~()sin()Itt  arb. arb. units Feedback OFF

phase-sensitive paramp arb. arb. units Feedback ON quantum state tomography Rabi freq. 3 MHz, paramp BW 10 MHz, cavity LW 8 MHz, env. deph. 0.05 MHz Alexander Korotkov University of California, Riverside Single quantum trajectories of transmon qubit K. Murch, S. Weber, C. Macklin, & I. Siddiqi, Nature-2013 phase-sensitive Coupling 0.52 MHz paramp  = 0.49 Cavity LW 10.8 MHz non-trivial causality Paramp BW 20 MHz Partial measurement: Individual quantum trajectories: expt. vs. Bayesian theory experiment vs. Bayesian theory

_t =1.8s  = 0.49 _ n=0.4 S = 3.15 1 푉 = 푉 푡′ 푑푡′ n=0.4 푚 푡

dashed: theory, solid: tomography along meridian along equator

Good agreement with simple Bayesian theory (dashed) 40/48 Alexander Korotkov University of California, Riverside Quantum trajectories with Rabi drive S. Weber, Chantasri, Dressel, Jordan, Murch, and I. Siddiqi, Nature-2014

tot = 0.4

Ensemble- averaged

dashed: theory, solid: tomography Individual

Good agreement with simple Bayesian theory Alexander Korotkov University of California, Riverside Most likely path with Rabi and post-selection S. Weber et al. Nature-2014 Only |휓in〉 is fixed Both |휓in〉 and |휓fin〉 are fixed p(z,t)

p(x,t)

Good agreement with theory (dashed) Alexander Korotkov University of California, Riverside Phase-preserving continuous measurement M. Hatridge, Shankar, Mirrahimi, Schackert, Geerlings, Brecht, Sliwa, Abdo, Frunzio, Girvin, Schoelkopf, and M. Devoret, Science-2013

phase-preserving paramp  = 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 simple (Bayesian) theory

Alexander Korotkov University of California, Riverside 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

 = 0.5

Phase-sensitive amplifier, measure non-informational quadrature (back-action is along parallels)

Echo sequence to analyze dephasing

Idea: collect measurement signal (with weight function) to find back-action; then undo

“refocusing” (feedback) increases qubit 2|휌01| from 0.40 to 0.56

Alexander Korotkov University of California, Riverside Entanglement by measurement (theory)

entangled R. Ruskov & A.K., 2003 I(t) qubit 1 qubit 2 I(t)  (t) Vga V Vgb

detector Ha Hb I(t) H same current for |01> and |10> a Hb  entangles gradually DQDa QPC DQDb qubit a SET qubit b ||||||01011001     (probabilistically, even 222 with Rabi oscillations)

Similar proposal in optics J. Kerckhoff, L. Bouten, A. Silberfarb, and H. Mabuchi, 2009

45/48 Alexander Korotkov University of California, Riverside 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 ( is not very important)

Alexander Korotkov University of California, Riverside Measurement-induced entanglement of remote qubits N. Roch, Schwartz, Motzoi, Macklin, Vijay, Eddins, Korotkov, Whaley, Sarovar, I. Siddiqi, PRL- 2014

same output signal for 01 and 10 qubits separated  entanglement by 1.3 m of cable loss = 0.81 meas = 0.4  First demonstration of remote entanglement of superconducting qubits  Remote entanglement is dots: experiment much more difficult than local dashed: simple theory solid: full theory (loss is very important)  Simple theory is close to full quantum trajectory, fits well experimental data.

Alexander Korotkov University of California, Riverside Conclusions

 A very simple quantum Bayesian theory works well for continuous cQED measurement of superconducting qubits in the “bad cavity” regime (large )  A better theory (for arbitrary ) is still easy to understand  Most of experimental proposals have been realized (incl. monitoring of trajectories, quantum feedback, and entanglement by measurement), though quantum efficiency is still low, 0.5  Causality principle in quantum mechanics applies only to ensemble-averaged states (individual trajectories can be affected retroactively)

Alexander Korotkov University of California, Riverside