ICTP, Trieste, 05/04/2019 Continuous measurement of solid-state qubits Alexander Korotkov Google, Venice, CA on leave from University of California, Riverside Outline: Short introduction (QM philosophy) Quantum Bayesian theory for continuous measurement of a qubit Short review of first experiments Correlators in simultaneous measurement of non-commuting observables of a qubit Arrow of time in continuous measurement of a qubit Alexander Korotkov Google, UCR
“Orthodox” (Copenhagen) quantum mechanics Schrödinger equation + collapse postulate 1 ) Fundamentally random measurement result 𝑠 2 𝑞 𝑠 = 𝜔 𝜔 𝑠 (out of allowed set of eigenvalues). Probability: |𝜔 𝑠 〉 2) State after measurement corresponds to result: Instantaneous, single quantum system (not ensemble) Contradicts Schröd. Eq., but comes from common sense Needs “observer”, reality follows observer’s knowledge Why so strange (unobjective)? - “Shut up and calculate” - May be QM founders were stupid? - Use proper philosophy? Alexander Korotkov Google, UCR
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 Google, UCR
Causality principle in quantum mechanics objects a and b observers A and B (and C) o bservers have “free will”; they can choose an action A B A choice made by observer A can affect a evolution of object b “back in time” b light time cones However, this retroactive control cannot pass “useful” information to B (no signaling) space C Randomness saves causality (even C cannot predict result of A measurement) Our focus: continuous | 0 collapse Ensemble-averaged evolution of object b cannot depend on actions of observer A | 1 Alexander Korotkov Google, UCR
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, Diosi, Carmichael, Milburn, Wiseman, Aharonov, Vaidman, Molmer, Gisin, Percival, Belavkin, … (very incomplete list) Key words : POVM, restricted path integral, quantum trajectories, quantum filtering, quantum jumps, stochastic master equation, etc. solid-state qubit We consider: detector classical output Quantum Bayesian approach Alexander Korotkov Google, UCR
ҧ ҧ ҧ ҧ Quantum Bayesian formalism for qubit meas. Qubit evolution due to measurement (informational back-action) |𝟐〉 qubit 𝜔 𝑢 = 𝛽 𝑢 0 + 𝛾 𝑢 1 or 𝜍 𝑗𝑘 (𝑢) |𝟏〉 (double Qdot) 2 and 𝛾 𝑢 2 evolve as probabilities, 1) 𝛽 𝑢 i.e. according to the Bayes rule (same for 𝜍 𝑗𝑗 ) I ( t ) 2) phases of 𝛽 𝑢 and 𝛾 𝑢 do not change V Τ 𝜍 𝑗𝑘 𝜍 𝑗𝑗 𝜍 𝑘𝑘 = const (no dephasing!), detector (A.K., 1998) (quantum point contact) 𝑄( ҧ 𝐽|0) 𝐽 m 𝑄( ҧ 𝐽|1) 𝑢 𝐽 𝑢 ′ 𝑒𝑢′ 0 𝐽 m = 𝐽 0 𝐽 1 measured 𝑢 Bayes rule (1763, Laplace-1812): prior 𝐽 0 + 𝜍 11 0 𝑄( ҧ posterior 𝑄 𝐽 = 𝜍 00 0 𝑄 𝐽|1) probab. likelihood probability So simple because: 𝑄 𝐵 𝑗 res = 𝑄 𝐵 𝑗 𝑄(res|𝐵 𝑗 ) 1) no entanglement at large QPC voltage norm 2) QPC is ideal detector 3) no other evolution of qubit ( 𝐼 qb = 0 ) Alexander Korotkov Google, UCR
ҧ ҧ ҧ ҧ ҧ ҧ ҧ ҧ Further steps in quantum Bayesian formalism |1〉 𝑄( ҧ 𝑄( ҧ 𝐽 m 𝐽|0) 𝐽|1) |0〉 𝛽 𝑢 0 + 𝛾 𝑢 1 𝑢 𝐽 𝑢 ′ 𝑒𝑢′ I ( t ) 0 𝐽 m = 𝐽 0 𝐽 1 measured 𝜍 𝑗𝑘 𝑢 𝑢 1. Informational back-action (“spooky”, no mechanism), × likelihood 𝑄 𝐽 m 0 𝛽 0 0 + 𝑄 𝐽 m 1 𝛾 0 1 𝜔 𝑢 = norm 2. Add unitary (phase) back-action, physical mechanisms for QPC and cQED 𝐽 m − 𝐽 0 + 𝐽 1 𝑄 𝐽 m 0 exp 𝑗𝐿 𝛽 0 0 + 𝑄 𝐽 m 1 𝛾(0) 1 2 𝜔 𝑢 = norm 𝛿 = Γ − Δ𝐽 2 − 𝐿 2 𝑇 𝐽 3. Add detector non-ideality (equivalent to dephasing) 4𝑇 𝐽 4 𝐽 m − 𝐽 0 +𝐽 1 = 𝑓 𝑗𝐿( ҧ ) 𝜍 01 0 𝜍 01 𝑢 𝜍 𝑗𝑗 𝑢 = 𝑄 𝐽 m 𝑗 𝜍 𝑗𝑗 0 2 exp(−𝛿𝑢) , norm 𝜍 00 𝑢 𝜍 11 𝑢 𝜍 00 0 𝜍 11 0 Alexander Korotkov Google, UCR
Further steps in quantum Bayesian formalism 4. Take derivative over time (if differential equation is desired) Simple, but be careful about definition of derivative Τ 𝑒𝑔 𝑢 = 𝑔 𝑢 + 𝑒𝑢 2 − 𝑔(𝑢 − 𝑒𝑢/2) Stratonovich form 𝑒𝑢 𝑒𝑢 preserves usual calculus 𝑒𝑔 𝑢 = 𝑔 𝑢 + 𝑒𝑢 − 𝑔(𝑢) requires special calculus, Ito form 𝑒𝑢 𝑒𝑢 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., 1998 — 2001) Alexander Korotkov Google, UCR
ҧ ሶ ሶ Generalization: measurement of operator 𝑩 “Informational” quantum Bayesian evolution in differential (Ito) form: (𝐵 2 𝜍 + 𝜍𝐵 2 ) 2 Τ 𝜍 = 𝐵𝜍𝐵 − + 𝐵𝜍 + 𝜍𝐵 − 2𝜍Tr (𝐵𝜍) 𝜊(𝑢) 2𝜃𝑇 2𝑇 𝐽 𝑢 = Tr 𝐵𝜍 + 𝑇 2 𝜊(𝑢) Τ noisy detector output 𝑇 : spectral density of the output noise 𝐽 m 𝜊 𝑢 𝜊 𝑢 ′ = 𝜀(𝑢 − 𝑢 ′ ) normalized white noise 𝐽 1 𝐽 2 𝐽 3 𝐽 𝑙 𝜃 : quantum efficiency With additional unitary (Hamiltonian) back-action 𝐶 and additional evolution 𝜍 = ℒ 𝜍 + 𝐵𝜍 + 𝜍𝐵 − 2𝜍Tr (𝐵𝜍) 1 𝜊 𝑢 − 𝑗 𝐶, 𝜍 𝜊 𝑢 2𝑇 2𝑇 ℒ[𝜍] : ensemble-averaged (Lindblad) evolution The same as in the Quantum Trajectory theory (Wiseman, Milburn, …) Nowadays “quantum trajectory“ often means Q.Bayesian real-time monitoring Alexander Korotkov Google, UCR
Quantum trajectory theory H. J. Carmichael, 1993 optics H. M. Wiseman and G. J. Milburn, 1993 solid-state, H.-S. Goan and G. J. Milburn, 2001 quantum point contact H.-S. Goan, G. J. Milburn, H. M. Wiseman, and H. B. Sun, 2001 J. Gambetta, A. Blais, M. Boissonneault, A. A. Houck, circuit QED D. I. Schuster, and S. M. Girvin, 2008 Relation between Quantum Trajectory and Quantum Bayesian formalisms Essentially the same thing, but look different Quantum trajectory theory uses mathematical language (superoperators), quantum Bayesian theory uses simple physical approach (undergraduate-level) Computationally, Bayesian theory is usually better (more than first-order) Another meaning of “quantum trajectories“: real -time monitoring of evolution (often done by quantum Bayesian theory) Alexander Korotkov Google, UCR
Quantum measurement in POVM formalism Davies, Kraus, Holevo, etc. system ancilla † 𝑁 𝑠 𝜍𝑁 𝑠 𝑁 𝑠 𝜔 Measurement (Kraus) operator 𝜍 → 𝜔 → or † 𝑁 𝑠 𝜍) 𝑁 𝑠 (any linear operator in H.S.) : | 𝑁 𝑠 𝜔 | Tr(𝑁 𝑠 † 𝑁 𝑠 𝜍) 𝑁 𝑠 𝜔 2 𝑄 𝑠 = Tr(𝑁 𝑠 𝑄 𝑠 = or Probability : † 𝑁 𝑠 = 1 σ 𝑠 𝑁 𝑠 Completeness : (People often prefer linear evolution and non-normalized states) Relation between POVM and quantum Bayesian formalism † 𝑁 𝑠 𝑁 𝑠 = 𝑉 𝑠 𝑁 𝑠 polar decomposition: unitary Bayes ( steps 1 and 2 above) Alexander Korotkov Google, UCR
ҧ ҧ ҧ ҧ ҧ ҧ Quantum Bayesian theory for circuit QED setup d mixer resonator 𝜆 phase-sensitive r microwave amplifier generator output (two quadratures) qubit homodyne meas. (transmon) A. Blais et al., PRA 2004 A. Wallraff et al., Nature 2004 Two quadratures: |1 J. Gambetta et al., PRA 2008 1) information on qubit state informational back-action |0 𝑄( ҧ 𝑄( ҧ 𝐽 m |0) 𝐽 m |1) 2) information on fluct. photon number 𝐽 0 𝐽 1 unitary (phase) back-action 2 2𝐸] 𝑄 𝐽 m = 𝜍 00 0 𝑄 𝐽 m 0 + 𝜍 11 0 𝑄 𝐽 m 1 𝜍 11 (𝜐) 𝜍 00 (𝜐) = 𝜍 11 0 exp[− 𝐽 m − 𝐽 1 Τ 𝜐 𝐽 𝑢 𝑒𝑢 𝐽 m = 𝜐 −1 2 2𝐸] 𝐸 = 𝑇 𝐽 /2𝜐 Τ 𝜍 00 0 exp[− 𝐽 m − 𝐽 0 0 Τ 𝐽 0 − 𝐽 1 = Δ𝐽 cos 𝜒 𝐿 = Δ𝐽 sin 𝜒 𝑇 𝐽 𝜍 00 𝜐 𝜍 11 𝜐 = 8𝜓 2 ത Γ = Δ𝐽 cos 𝜒 2 4 = Δ𝐽 2 + 𝐿 2 𝑇 𝐽 𝑜 exp 𝑗𝐿 ҧ 𝜍 01 𝜐 = 𝜍 01 0 𝐽 m 𝜐 𝜍 00 0 𝜍 11 0 4𝑇 𝐽 4𝑇 𝐽 𝜆 Amplified phase controls trade-off between Bayes unitary informational and phase back-actions (we A.K., arXiv:1111.4016 choose if photon number fluctuates or not) Alexander Korotkov Google, UCR
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