Michigan State Univ., 09/21/15 Measurement of superconducting qubits and causality Alexander Korotkov University of California, Riverside Outline: Introduction (causality, POVM measurement) Circuit QED setup for measurement of superconducting qubits (transmons, Xmons) Qubit evolution due to measurement in circuit QED (simple and better theories) Experiments Alexander Korotkov University of California, Riverside Causality principle in quantum 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 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 Niels Bohr 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, quantum mechanics 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 observable Ark|| k k k Measurement result rk (eigenvalue of A) 2 with probability Prk | k | in | | 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 quantum information 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.) Easy to show that local generalized measurement obeys causality (for ensemble-average state, i.e. averaged over results) M | A useful way of thinking | r ||M |† || polar decomposition: MUMMr rr r r 2 PMrrunitary|| |quantum || Bayes rule † † Operator 푀푟 defines a measurement basis (whichMM diagonalizes 1 푀푟 푀푟). + r rr 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 I() t dt 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 H( n n )2 E cos g 2C gJ 푬푱 푪 [,] ni ng C g V g /2 e (superconducting phase , charge 2en) - single-electron 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) resonator (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 H zr a a aa z 2 qb dephasing rate itd † (e a h. c .) H 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 (weak measurement): |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,