DOCSLIB.ORG

Partial Collapse and Uncollapse of a Wavefunction: Theory and Experiments (What Is “Inside” Collapse) Alexander Korotkov EE Dept

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Partial Collapse and Uncollapse of a Wavefunction: Theory and Experiments (What Is “Inside” Collapse) Alexander Korotkov EE Dept UCR, 5/06/10 Partial collapse and uncollapse of a wavefunction: theory and experiments (what is “inside” collapse) Alexander Korotkov EE dept. (& Physics), University of California, Riverside Outline: • Bayesian formalism for quantum measurement • Persistent Rabi oscillations (+ expt.) • Wavefunction uncollapse (+ expts.) Acknowledgements: Funding: Theory: R. Ruskov, A. Jordan, K. Keane Expt.: UCSB (J. Martinis, N. Katz et al.), Saclay (D. Esteve, P. Bertet et al.) Alexander Korotkov University of California, Riverside Quantum mechanics is weird… Niels Bohr: “If you are not confused by quantum physics then you haven’t really understood it” Richard Feynman: “I think I can safely say that nobody understands quantum mechanics” Weirdest part is quantum measurement Alexander Korotkov University of California, Riverside Quantum mechanics = Schrödinger equation (evolution) + collapse postulate (measurement) 2 1) Probability of measurement result pr =||〈ψψr 〉 | 2) Wavefunction after measurement = ψ r • State collapse follows from common sense • Does not follow from Schrödinger Eq. (contradicts) What is “inside” collapse? What if collapse is stopped half-way? Alexander Korotkov University of California, Riverside What is the evolution due to measurement? (What is “inside” collapse?) • controversial for last 80 years, many wrong answers, many correct answers • solid-state systems are more natural to answer this question Various approaches to non-projective (weak, continuous, partial, generalized, etc.) quantum measurements Names: Davies, Kraus, Holevo, Mensky, Caves, Knight, Plenio, Walls, Carmichael, Milburn, Wiseman, Gisin, Percival, Belavkin, etc. (very incomplete list) Key words: POVM, restricted path integral, quantum trajectories, quantum filtering, quantum jumps, stochastic master equation, etc. Our limited scope: solid-state qubit (simplest system, experimental setups) detector I(t), noise S Alexander Korotkov University of California, Riverside “Typical” setup: double-quantum-dot (DQD) qubit + quantum point contact (QPC) detector Gurvitz, 1997 |1Ò |1Ò H = H + H + H H |2Ò |2Ò QB DET INT e ε |2Ò |1Ò HH=+σ σ QB 2 ZX I(t) ΔI It()=+ I zt () +ξ () t I(t) 0 2 const + signal + noise Two levels of average detector current: I1 for qubit state |1〉, I2 for |2〉 Response: ΔI= I1 –I2 Detector noise: white, spectral density SI For low-transparency QPC †† †† HDET =++∑∑∑lrl Eaall l+ Eaarrr ,r Taa()rl aa l r †††† SeI= 2 HINT =Δ∑lr, T()() cc11 − cc 22 aarrll + aa I 5/38 Alexander Korotkov University of California, Riverside Bayesian formalism for DQD-QPC system Qubit evolution due to measurement (quantum back-action): HQB =0 ρ ()t |1Ò ψ ()tt= αβ ()|1〉+ ()|2 t 〉 or ij H e 1) |α(t)|2 and |β(t)|2 evolve as probabilities, |2Ò i.e. according to the Bayes rule (same for ρii) 2) phases of α(t) and β(t) do not change I(t) 1/2 (no dephasing!), ρij /(ρii ρjj) = const (A.K., 1998) 1 τ Bayes rule (1763, Laplace-1812): Itdt() τ ∫0 posterior prior measured probab. likelihood probability I1 I2 PA()(res|)ii P A PA(|res)i = So simple because: ∑k P(APkk ) (res | A ) 1) no entaglement at large QPC voltage 2) QPC happens to be an ideal detector 3) no Hamiltonian evolution of the qubit Alexander Korotkov University of California, Riverside Bayesian formalism for a single qubit Vg V H e • Time derivative of the quantum Bayes rule I(t) • Add unitary evolution of the qubit I(t) 2e • Add decoherence γ (if any) •• 2ΔI ρρ11=- 22 =-2ImHI ρρρ12 + 11 22 []It()- 0 SI • ΔI ρ 12 =+iiHερ12 ()()[] ρ11 - ρ 22+ ρ 12 ρ 11 - ρ 22 It() --I0γ ρ 12 SI 2 (A.K., 1998) γ =-ΓΔ()/4,ISI Γ−ensemble decoherence γ = 0 for QPC detector Averaging over result I(t) leads to conventional master equation with Γ Evolution of qubit wavefunction can be monitored if γ=0 (quantum-limited) Natural generalizations: • add classical back-action • entangled qubits Alexander Korotkov University of California, Riverside Assumptions needed for the Bayesian formalism: • Detector voltage is much larger than the qubit energies involved eV >> ÑΩ, eV >> ÑΓ, Ñ/eV << (1/Ω,1/Γ), Ω=(4H2+ε2)1/2 (no coherence in the detector, classical output, Markovian approximation) • Simpler if weak response, |ΔI | << I0, (coupling C ~ Γ/Ω is arbitrary) Derivations: 1) “logical”: via correspondence principle and comparison with decoherence approach (A.K., 1998) 2) “microscopic”: Schr. eq. + collapse of the detector (A.K., 2000) ρ n()t ij nt()k classical qubit detector pointer information quantum frequent n – number of electrons interaction quantum collapse passed through detector 3) from “quantum trajectory” formalism developed for quantum optics (Goan-Milburn, 2001; also: Wiseman, Sun, Oxtoby, etc.) 4) from POVM formalism (Jordan-A.K., 2006) 5) from Keldysh formalism (Wei-Nazarov, 2007) Alexander Korotkov University of California, Riverside Why not just use Schrödinger equation for the whole system? qubit detector information Impossible in principle! Technical reason: Outgoing information (measurement result) makes it an open system Philosophical reason: Random measurement result, but deterministic Schrödinger equation Einstein: God does not play dice Heisenberg: unavoidable quantum-classical boundary Alexander Korotkov University of California, Riverside Quantum limit for ensemble decoherence 2 “measurement time” (S/N=1) Γ = (ΔI) /4SI + γ 2 τ m =Δ2/()SI I ensemble single-qubit decoherence rate decoherence 1 Γτ m ≥ ~ information flow [bit/s] 2 A.K., 1998, 2000 S. Pilgram et al., 2002 2 γ ≥ 0 ⇒ Γ ≥ (ΔI) /4SI A. Clerk et al., 2002 D. Averin, 2000,2003 γ ()/4ΔIS2 η =-1 = I detector ideality (quantum efficiency) ΓΓ η ≤ 100% Translated into energy sensitivity: (ε ε )1/2 ≥ =/2 Danilov, Likharev, O BA Zorin, 1983 εO, εBA: sensitivities [J/Hz] limited by output noise and back-action Known since 1980s (Caves, Clarke, Likharev, Zorin, Vorontsov, Khalili, etc.) 2 1/2 2 2 (εO εBA - εO,BA ) ≥ =/2 ⇔ Γ ≥ (ΔI) /4SI + K SI/4 Quantum limits for measurement are due to quantum (informational) back-action 10/38 Alexander Korotkov University of California, Riverside POVM vs. Bayesian formalism General quantum measurement (POVM formalism) (Nielsen-Chuang, p. 85,100): † Measurement (Kraus) operator M ψ MMρ r ρ → rr M (any linear operator in H.S.) : ψ → or † r ||Mrψ || Tr(MMrrρ ) 2 † Probability : PMrr=||ψ || or PMMrrr= Tr(ρ ) † (People often prefer linear evolution Completeness : ∑r MMrr= 1 and non-normalized states) • POVM is essentially a projective measurement in an extended Hilbert space • Easy to derive: interaction with ancilla + projective measurement of ancilla • For extra decoherence: incoherent sum over subsets of results † Relation between POVM and decomposition MUMMrrrr= quantum Bayesian formalism: unitary Bayes Mathematically POVM and quantum Bayes are almost equivalent We focus not on the mathematical structures, but on particular setups and experimental consequences Alexander Korotkov University of California, Riverside Can we verify the Bayesian formalism experimentally? Direct way: partial control projective prepare measur. (rotation) measur. A.K.,1998 However, difficult: bandwidth, control, efficiency (expt. realized only for supercond. phase qubits) Tricks are needed for real experiments (proposals 1999-2010) Alexander Korotkov University of California, Riverside Non-decaying (persistent) Rabi oscillations excited - Relaxes to the ground state if left alone (low-T) left right - Becomes fully mixed if coupled to a high-T (non-equilibrium) environment ground - Oscillates persistently between left and right z if (weakly) measured continuously (measure left-right) to verify: stop& check z 1.0 |left〉 ρ11 0.5 |e〉 ••• |g〉 Reρ11 |right〉 0.0 Imρ11 -0.5 0 5 10 15 20 25 30 time A.K., PRB-1999 Direct experiment is difficult Alexander Korotkov University of California, Riverside Indirect experiment: spectrum of persistent Rabi oscillations A.K., LT’1999 C A.K.-Averin, 2000 qubit detector ΔI I(t) It()=+ I0 zt () +ξ () t z is Bloch 2 coordinate 12 (const + signal + noise) C=13 10 S (ω) Ω =2H amplifier noise fl higher pedestal, I 1 0 8 10 2 poor quantum efficiency, )/S CIHS=Δ()/ I ηÜ1 ω 6 0 ( but the peak is the same!!! I 012ω/Ω S 4 integral under the peak ‹ variance ‚z2Ú 3 2 1 How to distinguish experimentally 0 0.3 0.00.51.01.52.0 persistent from non-persistent? Easy! ω/Ω Ω - Rabi frequency perfect Rabi oscillations: ·z2Ò=·cos2Ò=1/2 peak-to-pedestal ratio = 4η≤4 imperfect (non-persistent): ·z2ÒÜ1/2 ΩΔ22()I Γ quantum (Bayesian) result: ·z2Ò =1 (!!!) SS()ω =+ I 0 ()ω 22222−Ω +Γ ω (demonstrated in Saclay expt.) Alexander Korotkov University of California, Riverside z |left〉 How to understand ·z2Ò =1? |e〉 ••• |g〉 qubit |right〉 ΔI It()=+ I0 zt () +ξ () t I(t) 2 detector First way (mathematical) We actually measure operator: z →σz (What does it mean? 2 2 Difficult to say…) z →σz =1 Second way (Bayesian) ΔΔII2 SS()ω =+ S()ωω + S () I ξξ 42zz ξ z quantum back-action changes z Equal contributions (for weak in accordance with the noise ξ coupling and η=1) “what you see is what you get”: observation becomes reality Can we explain it in a more reasonable way (without spooks/ghosts)? +1 z(t)? No (under assumptions of macrorealism; -1 Leggett-Garg, 1985) or some other z(t)? 15/38 Alexander Korotkov University of California, Riverside Leggett-Garg-type inequalities for continuous measurement of a qubit Ruskov-A.K.-Mizel, PRL-2006 qubit detector I(t) Jordan-A.K.-Büttiker, PRL-2006 Assumptions of macrorealism Leggett-Garg,1985 6 SI (ω) (similar to Leggett-Garg’85): 0 Kij = ·Qi QjÒ 4 0 )/S if Q = ±1, then 4S It()= I+Δ( I /2)() zt +ξ () t ω ≤ 0 ( 2 1+K +K +K ≥0 I |()|1,zt≤〈 ξ ()( t zt +〉τ )= 0 12 23 13 S K12+K23+K34 -K14 £2 0 012ω/Ω Then for correlation function quantum result violation KItIt()τ = 〈+〉 ()(τ ) 3 3 KKK()ττττ+−+≤Δ () ( ) ( I /2)2 (/2)ΔI 2 × 1212 2 2 and for area under narrow spectral peak π 2 [()Sf−≤ Sdf ] (8/)(/2)π 22 Δ I (/2)ΔI 2 × ∫ I 0 8 η is not important! Experimentally measurable violation (Saclay experiment) Alexander Korotkov University of California, Riverside Experiment with supercond.
Load more

Recommended publications
  • Finally Making Sense of the Double-Slit Experiment
    Finally making sense of the double-slit experiment Yakir Aharonova,b,c,1, Eliahu Cohend,1,2, Fabrizio Colomboe, Tomer Landsbergerc,2, Irene Sabadinie, Daniele C. Struppaa,b, and Jeff Tollaksena,b aInstitute for Quantum Studies, Chapman University, Orange, CA 92866; bSchmid College of Science and Technology, Chapman University, Orange, CA 92866; cSchool of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel; dH. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom; and eDipartimento di Matematica, Politecnico di Milano, 9 20133 Milan, Italy Contributed by Yakir Aharonov, March 20, 2017 (sent for review September 26, 2016; reviewed by Pawel Mazur and Neil Turok) Feynman stated that the double-slit experiment “. has in it the modular momentum operator will arise as particularly signifi- heart of quantum mechanics. In reality, it contains the only mys- cant in explaining interference phenomena. This approach along tery” and that “nobody can give you a deeper explanation of with the use of Heisenberg’s unitary equations of motion intro- this phenomenon than I have given; that is, a description of it” duce a notion of dynamical nonlocality. Dynamical nonlocality [Feynman R, Leighton R, Sands M (1965) The Feynman Lectures should be distinguished from the more familiar kinematical non- on Physics]. We rise to the challenge with an alternative to the locality [implicit in entangled states (10) and previously ana- wave function-centered interpretations: instead of a quantum lyzed in the Heisenberg picture by Deutsch and Hayden (11)], wave passing through both slits, we have a localized particle with because dynamical nonlocality has observable effects on prob- nonlocal interactions with the other slit.
    [Show full text]
  • Weak Measurements, the Energy-Momentum Tensor and the Bohm Approach Robert Flack and Basil J
    6 Weak Measurements, the Energy-Momentum Tensor and the Bohm Approach Robert Flack and Basil J. Hiley Abstract μ In this paper we show how the weak values, x(t)|P |ψ(t0)/x(t)|ψ(t0), are related to the T 0μ(x, t) component of the energy-momentum tensor. This enables the local energy and momentum to be measured using weak measurement techniques. We also show how the Bohm energy and momentum are related to T 0μ(x, t) and there- fore it follows that these quantities can also be measured using the same methods. Thus the Bohm ‘trajectories’ can be empirically determined as was shown by Kocis et al in the case of photons. Because of the difficulties with the notion of a photon trajectory, we argue the case for determining experimentally similar trajectories for atoms where a trajectory does not cause these particular difficulties. 6.1 Introduction The notion of weak measurements introduced by Aharonov, Albert and Vaidman (1988,1990) has opened up a radically new way of exploring quantum phenomena. In contrast to the strong measurement (von Neumann 1955), which involves the collapse of the wave function, a weak measurement induces a more subtle phase change which does not involve any collapse. This phase change can then be ampli- fied and revealed in a subsequent strong measurement of a complementary oper- ator that does not commute with the operator being measured. This amplification explains why it is possible for the result of a weak spin measurement of a spin-1/2 atom to be magnified by a factor of 100 (Aharonov et al.
    [Show full text]
  • Weak Measurement and Feedback in Superconducting Quantum Circuits
    1 Weak Measurement and Feedback in Superconducting Quantum Circuits K. W. Murch, R. Vijay, and I. Siddiqi 1 Department of Physics, Washington University, St. Louis, MO, USA [email protected] 2 Tata Institute of Fundamental Research, Mumbai, India [email protected] 3 Quantum Nanoelectronics Laboratory, Department of Physics, University of California, Berkeley, CA, USA [email protected] Abstract. We describe the implementation of weak quantum measurements in su- perconducting qubits, focusing specifically on transmon type devices in the circuit quantum electrodynamics architecture. To access this regime, the readout cavity is probed with on average a single microwave photon. Such low-level signals are detected using near quantum-noise-limited superconducting parametric amplifiers. Weak measurements yield partial information about the quantum state, and corre- spondingly do not completely project the qubit into an eigenstate. As such, we use the measurement record to either sequentially reconstruct the quantum state at a given time, yielding a quantum trajectory, or to close a direct quantum feedback loop, stabilizing Rabi oscillations indefinitely. Measurement-based feedback routines are commonplace in modern elec- tronics, including the thermostats regulating the temperature in our homes to sophisticated motion stabilization hardware needed for the autonomous oper- ation of aircraft. The basic elements present in such classical feedback loops include a sensor element which provides information to a controller, which in turn steers the system of interest toward a desired target. In this paradigm, the act of sensing itself does not a priori perturb the system in a significant way. Moreover, extracting more or less information during the measurement process does not factor into the control algorithm.
    [Show full text]
  • Direct Measurement of the Density Matrix of a Quantum System
    Selected for a Viewpoint in Physics week ending PRL 117, 120401 (2016) PHYSICAL REVIEW LETTERS 16 SEPTEMBER 2016 Direct Measurement of the Density Matrix of a Quantum System G. S. Thekkadath, L. Giner, Y. Chalich, M. J. Horton, J. Banker, and J. S. Lundeen Department of Physics and Max Planck Centre for Extreme and Quantum Photonics, University of Ottawa, 25 Templeton Street, Ottawa, Ontario K1N 6N5, Canada (Received 26 April 2016; revised manuscript received 14 June 2016; published 12 September 2016) One drawback of conventional quantum state tomography is that it does not readily provide access to single density matrix elements since it requires a global reconstruction. Here, we experimentally demonstrate a scheme that can be used to directly measure individual density matrix elements of general quantum states. The scheme relies on measuring a sequence of three observables, each complementary to the last. The first two measurements are made weak to minimize the disturbance they cause to the state, while the final measurement is strong. We perform this joint measurement on polarized photons in pure and mixed states to directly measure their density matrix. The weak measurements are achieved using two walk-off crystals, each inducing a polarization-dependent spatial shift that couples the spatial and polarization degrees of freedom of the photons. This direct measurement method provides an operational meaning to the density matrix and promises to be especially useful for large dimensional states. DOI: 10.1103/PhysRevLett.117.120401 Shortly after the inception of the quantum state, Pauli of the reconstruction algorithm renders the task increasingly questioned its measurability, and in particular, whether or difficult for large dimensional systems.
    [Show full text]
  • Weak Measurements for Quantum Characterization and Control Jonathan A
    University of New Mexico UNM Digital Repository Physics & Astronomy ETDs Electronic Theses and Dissertations Summer 7-27-2018 Weak measurements for quantum characterization and control Jonathan A. Gross University of New Mexico Follow this and additional works at: https://digitalrepository.unm.edu/phyc_etds Part of the Atomic, Molecular and Optical Physics Commons, and the Quantum Physics Commons Recommended Citation Gross, Jonathan A.. "Weak measurements for quantum characterization and control." (2018). https://digitalrepository.unm.edu/ phyc_etds/204 This Dissertation is brought to you for free and open access by the Electronic Theses and Dissertations at UNM Digital Repository. It has been accepted for inclusion in Physics & Astronomy ETDs by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected]. Weak measurements for quantum characterization and control by Jonathan A. Gross B.M., Piano Performance, University of Arizona, 2011 B.S., Computer Engineering, University of Arizona, 2011 M.S., Physics, University of New Mexico, 2015 DISSERTATION Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Physics The University of New Mexico Albuquerque, New Mexico December, 2018 c 2018, Jonathan A. Gross iii Dedication The posting of witnesses by a third and other path altogether might also be called in evidence as appearing to beggar chance, yet the judge, who had put his horse forward until he was abreast of the speculants, said that in this was expressed the very nature of the witness and that his proximity was no third thing but rather the prime, for what could be said to occur unobserved? |Cormac McCarthy, Blood Meridian: Or the Evening Redness in the West And he sayde vnto me: my grace is sufficient for the.
    [Show full text]
  • Weak Measurement, Projective Measurement and Quantum-To-Classical Transitions in Free Electron-Photon Interactions
    Weak measurement, projective measurement and quantum-to-classical transitions in free electron-photon interactions Yiming Pan1†, Eliahu Cohen2†, Ebrahim Karimi3, Avraham Gover4, Ido Kaminer5 and Yakir Aharonov6,7 1. Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel 2. Faculty of Engineering and the Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat Gan 5290002, Israel 3. Department of Physics, University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada 4. Department of Electrical Engineering Physical Electronics, Tel Aviv University, Ramat Aviv 6997801, Israel 5. Department of Electrical Engineering, Technion – Israel Institute of Technology, Haifa 3200003, Israel 6. School of Physics and Astronomy, Tel Aviv University, Ramat Aviv 6997801, Israel 7. Institute for Quantum Studies, Chapman University, Orange, CA 92866, USA †Correspondence and requests for materials should be addressed to E.C. ([email protected]) and Y.P. ([email protected]). 1 Abstract How does the quantum-to-classical transition of measurement occur? This question is vital for both foundations and applications of quantum mechanics. We developed a new measurement- based framework for characterizing the classical and quantum free electron-photon interactions. We first analyze the transition from projective measurement to weak measurement in generic light- matter interactions, and show that any classical electron-laser-beam interaction can be represented as an outcome of a weak measurement. In particular, the appearance of classical point-particle acceleration is an example of an amplified weak value resulting from weak measurement. A universal decay factor exp(-G2 2), quantifies the measurement regimes and their transition from quantum to classical, where G corresponds to the ratio between the electron wavepacket size and the optical wavelength.
    [Show full text]
  • How Quantum Is Quantum Counterfactual Communication? Arxiv
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Explore Bristol Research Hance, J. R. , Ladyman, J., & Rarity, J. (Accepted/In press). How Quantum is Quantum Counterfactual Communication? arXiv. Peer reviewed version Link to publication record in Explore Bristol Research PDF-document University of Bristol - Explore Bristol Research General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms How Quantum is Quantum Counterfactual Communication? Jonte R. Hance,1, ∗ James Ladyman,2 and John Rarity1 1Quantum Engineering Technology Laboratories, Department of Electrical and Electronic Engineering, University of Bristol, Woodland Road, Bristol, BS8 1US, UK 2Department of Philosophy, University of Bristol, Cotham House, Bristol, BS6 6JL Quantum Counterfactual Communication is the recently-proposed idea of using quantum mechan- ics to send messages between two parties, without any particles travelling between them. While this has excited massive interest, both for potential `un-hackable' communication, and insight into the foundations of quantum mechanics, it has been asked whether this phenomena is truly quantum, or could be performed classically. We examine counterfactuality, both classical and quantum, and the protocols proposed so far, and conclude it must be quantum, at least insofar as it requires particle quantisation. I. INTRODUCTION municated to counterfactually (e.g. knowing a car's en- gine works by the `check engine' light being off; knowing Quantum Counterfactual Communication is the no-one is calling you by your phone not ringing; or being combination of counterfactual circumstances (where sure your house is not on fire by hearing no alarm).
    [Show full text]
  • Understanding the Born Rule in Weak Measurements
    Understanding the Born Rule in Weak Measurements Apoorva Patel Centre for High Energy Physics, Indian Institute of Science, Bangalore 31 July 2017, Open Quantum Systems 2017, ICTS-TIFR N. Gisin, Phys. Rev. Lett. 52 (1984) 1657 A. Patel and P. Kumar, Phys Rev. A (to appear), arXiv:1509.08253 S. Kundu, T. Roy, R. Vijayaraghavan, P. Kumar and A. Patel (in progress) 31 July 2017, Open Quantum Systems 2017, A. Patel (CHEP, IISc) Weak Measurements and Born Rule / 29 Abstract Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigenstate will be observed in which experimental run. The probabilistic Born rule gives it an ensemble interpretation, predicting proportions of various outcomes over many experimental runs. Understanding gradual weak measurements requires replacing this scenario with a dynamical evolution equation for the collapse of the quantum state in individual experimental runs. We revisit the framework to model quantum measurement as a continuous nonlinear stochastic process. It combines attraction towards the measured operator eigenstates with white noise, and for a specific ratio of the two reproduces the Born rule. This fluctuation-dissipation relation implies that the quantum state collapse involves the system-apparatus interaction only, and the Born rule is a consequence of the noise contributed by the apparatus. The ensemble of the quantum trajectories is predicted by the stochastic process in terms of a single evolution parameter, and matches well with the weak measurement results for superconducting transmon qubits. 31 July 2017, Open Quantum Systems 2017, A. Patel (CHEP, IISc) Weak Measurements and Born Rule / 29 Axioms of Quantum Dynamics (1) Unitary evolution (Schr¨odinger): d d i dt |ψi = H|ψi , i dt ρ =[H,ρ] .
    [Show full text]
  • WEAK MEASUREMENT of PHOTON POLARIZATION James M. KNIGHT
    Volume 143, number 8 PHYSICS LETFERS A 29 January 1990 WEAK MEASUREMENT OF PHOTON POLARIZATION James M. KNIGHT and Lev VAIDMAN Department ofPhysics and Astronomy, University ofSouth Carolina, Columbia, SC 29208, USA Received 12 October 1989; revised manuscript received 27 November 1989; accepted for publication 28 November 1989 Communicated by J.P. Vigier An optical procedure for measuringthe recently introduced weak value ofa photon polarization variable is suggested and some of its experimental aspects are discussed. The deflection of a beam passing through a birefringent prism is considered. It is shown that if the beam also passes through preselection and postselection polarization filters, it may have a deflection which is greater than the deflection of the beam without the filters. The magnitude of the deflection is give by weak measurement theory. The concept of the weak value of a quantum me- measurement of photon polarization”. The optical chanical variable A has recently been introduced by polarization filters employed provide an especially Aharonov, Albert, and Vaidman (AAV) [1] (see also precise means of performing the preselection and ref. [2]), and discussed by other authors [3—71.This postselection which play an essential role in weak value is the statistical result of the standard meas- measurement. In addition to the interest of this ex- uring procedure carried out on a preselected and periment as an example ofweak measurement, it may postselected ensemble when the coupling of the sys- suggest practical optical applications of the under- tern with the measuring device is sufficiently weak, lying AAV interference phenomenon. a procedure called weak measurement.
    [Show full text]
  • Weak Measurement and Bohmian Conditional Wave Functions Travis Norsen
    Masthead Logo Smith ScholarWorks Physics: Faculty Publications Physics 11-2014 Weak Measurement and Bohmian Conditional Wave Functions Travis Norsen Follow this and additional works at: https://scholarworks.smith.edu/phy_facpubs Part of the Quantum Physics Commons Recommended Citation Norsen, Travis, "Weak Measurement and Bohmian Conditional Wave Functions" (2014). Physics: Faculty Publications, Smith College, Northampton, MA. https://scholarworks.smith.edu/phy_facpubs/21 This Article has been accepted for inclusion in Physics: Faculty Publications by an authorized administrator of Smith ScholarWorks. For more information, please contact [email protected] Weak Measurement and (Bohmian) Conditional Wave Functions Travis Norsen Physics Department, Smith College, Northampton, MA 01060, USA.∗ Ward Struyve Departments of Mathematics and Philosophy, Rutgers University, Hill Center, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA.† (Dated: October 8, 2013) It was recently pointed out (and demonstrated experimentally) by Lundeen et al. that the wave function of a particle (more precisely, the wave function possessed by each member of an ensemble of identically-prepared particles) can be “directly measured” using weak measurement. Here it is shown that if this same technique is applied, with appropriate post-selection, to one particle from a (perhaps entangled) multi-particle system, the result is precisely the so-called “conditional wave function” of Bohmian mechanics. Thus, a plausibly operationalist method for defining the wave function of a quantum mechanical sub-system corresponds to the natural definition of a sub- system wave function which Bohmian mechanics (uniquely) makes possible. Similarly, a weak- measurement-based procedure for directly measuring a sub-system’s density matrix should yield, under appropriate circumstances, the Bohmian “conditional density matrix” as opposed to the standard reduced density matrix.
    [Show full text]
  • Weak Measurements Is to Make the Coupling with the Measuring Device Sufficiently Weak So B Does Not Change
    WEAK MEASUREMENTS Lev Vaidman School of Physics and Astronomy Raymond and Beverly Sackler Faculty of Exact Sciences Tel-Aviv University Tel-Aviv, 69978 ISRAEL 1. PRE- AND POST-SELECTED QUANTUM SYSTEMS In 1964 Aharonov, Bergmann and Lebowitz1 considered measurements performed on a quantum system between two other measurements, results of which were given. They proposed describing the quantum system between two measurements by using two states: the usual one, evolving towards the future from the time of the first measurement, and a second state evolving backwards in time, from the time of the second measurement. If a system has been prepared at time t in a state Ψ and is 1 | 1i found at time t2 in a state Ψ2 , then at time t, t1 <t<t2, the system is described by | i t t i Hdt −i Hdt t t Ψ2 e R 2 and e R 1 Ψ1 . arXiv:hep-th/9408154v2 8 Sep 1994 h | | i For simplicity, we shall consider the free Hamiltonian to be zero; then the system at time t is described by the two states Ψ and Ψ , see Fig. 1. In order to obtain such h 2| | 1i a system we prepare an ensemble of systems in the state Ψ1 , perform measurement of desired variable using separate measuring devices for ea|chi system in the ensemble, and perform the post-selection measurement. If the outcome of the post-selection was not the desired result, we discard the system and corresponding measuring device. We look only on measuring devices corresponding to the systems post-selected in the state Ψ .
    [Show full text]
  • An Interaction-Free Quantum Measurement-Driven Engine
    An Interaction-Free Quantum Measurement-Driven Engine Cyril Elouard,1 Mordecai Waegell,2 Benjamin Huard,3 and Andrew N. Jordan1 1Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA∗ 2Institute for Quantum studies, Chapman University, Orange, CA 92866, USA 3Laboratoire de Physique, Ecole´ Normale Sup´erieure de Lyon, 46 all´eedItalie, 69364 Lyon Cedex 7, France Recently highly-efficient quantum engines were devised by exploiting the stochastic energy changes induced by quantum measurement. Here we show that such an engine can be based on an interaction- free measurement, in which the meter seemingly does not interact with the measured object. We use a modified version of the Elitzur-Vaidman bomb tester, an interferometric setup able to detect the presence of a bomb triggered by a single photon without exploding it. In our case, a quantum bomb subject to a gravitational force is initially in a superposition of being inside and outside one of the interferometer arms. We show that the bomb can be lifted without blowing up. This occurs when a photon traversing the interferometer is detected at a port that is always dark when the bomb is located outside the arm. The required potential energy is provided by the photon (which plays the role of the meter) even though it was not absorbed by the bomb. A natural interpretation is that the photon traveled through the arm which does not contain the bomb { otherwise the bomb would have exploded { but it implies the surprising conclusion that the energy exchange occurred at a distance despite a local interaction Hamiltonian.
    [Show full text]