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Quantum metrology:An information-theoretic perspectivein three two lecturesCarlton M. CavesCenter for Quantum Information and Control, University of New Mexicohttp://info.phys.unm.edu/ cavesCenter for Quantum Information and Control

Quantum metrology:An information-theoretic perspectiveLecture 1I. Introduction. What’s the problem?II. Squeezed states and optical interferometryIII. Ramsey interferometry, cat states, and spin squeezingCarlton M. CavesCenter for Quantum Information and Control, University of New Mexicohttp://info.phys.unm.edu/ cavesCenter for Quantum Information and Control

I. Introduction. What’s the problem?View from Cape HauyTasman PeninsulaTasmania

Quantum information scienceA new way of thinkingComputer scienceComputational complexitydepends on physical law.New physicsQuantum mechanics as liberator.What can be accomplished withquantum systems that can’t bedone in a classical world?Explore what can be done withquantum systems, instead ofbeing satisfied with what Naturehands us.Quantum engineeringOld physicsQuantum mechanics as nag.The uncertainty principlerestricts what can be done.

MetrologyTaking the measure of thingsThe heart of physicsNew physicsQuantum mechanics asliberator.Explore what can bedone with quantumsystems, instead ofbeing satisfied withwhat Nature hands us.Quantum engineeringOld physicsQuantummechanics as nag.The uncertaintyprinciplerestricts what canbe done.Old conflict in new guise

Measuring a classical parameterPhase shift in an (optical) interferometerReadout of anything that changes optical path lengthsMichelson-Morley experimentGravitational-wave detectionPlanck-scale, holographic uncertainties in positionsTorque on or free precession of a collection of spinsMagnetometerLectures 1 and 2Atomic clockForce on a linear systemGravitational-wave detectionAccelerometerGravity gradiometerElectrometerLecture 3Strain meter

II. Squeezedstates and opticalinterferometryOljeto WashSouthern Utah

(Absurdly) high-precision interferometryHanford, WashingtonThe LIGO Collaboration, Rep.Prog. Phys. 72, 076901 (2009).Laser Interferometer Gravitational Observatory (LIGO)4 kmLivingston, Louisiana

(Absurdly) high-precision interferometryInitial LIGOHanford, WashingtonLaser Interferometer Gravitational Observatory (LIGO)4 kmLivingston, LouisianaHigh-power, FabryPerot-cavity(multipass), powerrecycledinterferometers

(Absurdly) high-precision interferometryAdvanced LIGOHanford, WashingtonCurrently a factor of 3short of this design goal.Laser Interferometer Gravitational Observatory (LIGO)4 kmLivingston, LouisianaHigh-power, FabryPerot-cavity(multipass), powerand signal-recycled,squeezed-lightinterferometers

Mach-Zender interferometerC. M. Caves, PRD 23, 1693 (1981).

Squeezed states of light

Squeezed states of lightGroups at Australian National University, Hannover, andTokyo have achieved up to 15 dB of squeezing at audiofrequencies for use in Advanced LIGO, VIRGO, and GEO.Squeezing by a factor of about 3.5G. Breitenbach, S. Schiller, and J. Mlynek,Nature 387, 471 (1997).

Fabry-Perot Michelson interferometerMotion of the mirrors produced by a gravitationalwave induces a transition from the symmetricmode to the antisymmetric mode; the resulting tinysignal at the vacuum port is contaminated byquantum noise that entered the vacuum port.

Squeezed statesand opticalinterferometryK. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf.R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J.Weinstein, and N. Mavalvala, Nature Physics 4,472 (2008).44% improvement indisplacement sensitivity

Squeezed statesfor opticalinterferometryH. Vahlbruch, A. Khalaidovski, N. Lastzka,C. Graef, K. Danzmann, and R. Schnabel, Classicaland Quantum Gravity 27, 084027 (2010).9dB below shot noise from10 Hz to 10 kHz

Squeezed statesand opticalinterferometryGEO 600 laserinterferometerThe LIGO Scientific Collaboration,Nature Physics 7, 962 (2011).Up to 3.5dB improvement insensitivity in the shot-noiselimited frequency band

Squeezed statesand opticalinterferometry 2 dB of shot-noisereductionSqueezed light in theLIGO Hanford detectorThe LIGO Scientific Collaboration,Nat. Phot. 7, 613 (2013).

Quantum limits on optical interferometryQuantum Noise Limit (Shot-Noise Limit)Heisenberg LimitAs much powerin the squeezedlight as in themain beam

III. Ramsey interferometry, cat states,and spin squeezingTruchas from East Pecos BaldySangre de Cristo RangeNorthern New Mexico

Ramsey interferometryN independent“atoms”Frequency measurementTime measurementClock synchronization

Cat-state Ramsey interferometryJ. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, Phys. Rev. A 54, R4649 (1996).Fringe patternwith period 2π/NN cat-state atoms

Optical interferometryRamsey interferometryQuantum Noise Limit(Shot-Noise Limit)Heisenberg LimitSomething’s going on here.

Squeezed-state opticalinterferometryCat-state RamseyinterferometryEntanglement before “beamsplitter”Between arms(wave or modal entanglement)Between atoms(particle entanglement)Between photons(particle entanglement)Between arms(modal entanglement)

Squeezed-state opticalinterferometryCat-state RamseyinterferometryEntanglement after “beamsplitter”Between arms(wave or modal entanglement)Between atoms(particle entanglement)Between photons(particle entanglement)Between arms(modal entanglement)

Spin-squeezing Ramsey interferometryJ. Ma, X. Wang, C. P. Sun, and F. Nori,Phys. Rep. 509, 89‒165 (2011).Heisenberg LimitThis is really a cat state.

Spin-squeezing Ramsey interferometryWhat’s squeezed?The y spin state has N particles; the –y spin statehas single-mode squeezing. This is like the state ofthe two arms prior to the beamsplitter in an opticalinterferometer. The up and down spin states havecorrelated squeezing like that in the arms of asqueezed-state optical interferometer.What’s entangled?No entanglement of y and –y spin states.Modal entanglement of up and down spin states.Particle entanglement.

Squeezed-state opticalinterferometrySpin-squeezing RamseyinterferometryEntanglementBetween arms(wave or modal entanglement)Between atoms(particle entanglement)Between photons(particle entanglement)Between arms(modal entanglement)

Role of entanglementEntanglement is a resource for getting my paper into Nature.Don’t accept facile explanations.Ask questions.

TransitionTelling stories is what physics is about.Lecture 1 has been about understanding fringepatterns and sources of noise and designing devicesto improve phase sensitivity based on thisunderstanding. This is telling stories.Lecture 2 is about proving that the stories aren’tfooling us.Which is better, stories or proofs? You need themboth, but stories are going to get you farther.

Quantum metrology:An information-theoretic perspectiveLecture 2I. Quantum Cramér-Rao Bound (QCRB)II. Making quantum limits relevant. Loss and decoherenceIII. Beyond the Heisenberg limit. Nonlinear interferometryCarlton M. CavesCenter for Quantum Information and Control, University of New Mexicohttp://info.phys.unm.edu/ cavesCenter for Quantum Information and Control

I. Quantum Cramér-Rao Bound (QCRB)Cable BeachWestern Australia

Quantuminformation versionof interferometryQuantumnoise limitQuantumcircuitscat stateN 3HeisenberglimitFringe pattern with period 2π/N

eterestimationMeasurement

HeisenberglimitS. L. Braunstein, C. M. Caves, and G. J. Milburn, Ann. Phys. 247, 135 (1996).V. Giovannetti, S. Lloyd, and L. Maccone, PRL 96, 041401 (2006).S. Boixo, S. T. Flammia, C. M. Caves, and JM Geremia, PRL 98, 090401 (2007).Separable inputsGeneralizeduncertainty principleQuantum Cramér-Rao bound

Achieving the Heisenberg limitcatstateProof ofQCRB

Is it entanglement? It’s the entanglement,stupid.But what about?For metrology, entanglement is part of the story, but only part.We need a generalized notion of entanglement/resources thatincludes information about the physical situation, particularlythe relevant Hamiltonian.

II. Making quantum limits relevant.Loss and decoherenceBungle Bungle RangeWestern Australia

Making quantum limits relevantQuantumcircuitsThe serial resource, T, andthe parallel resource, N, areequivalent andinterchangeable,mathematically.The serial resource, T, andthe parallel resource, N, arenot equivalent and notinterchangeable, physically.Information scienceperspectivePhysics perspectivePlatform independenceDistinctions between differentphysical systems

Making quantum limits relevant.One metrology storyA. Shaji and C. M. Caves, PRA 76, 032111 (2007).

Making quantum limits relevantRule of thumb for photon losses for large NS. Knysh, V. N. Smelyanskiy and G. A. Durkin, PRA 83, 021804(R) (2011).Heisenberg limit: lessthan one photon lost.Typically, beat shot noiseby square of root offractional loss.

Quantum limit on practical optical interferometry1.2.3.Cheap photons from a laser (coherent state)Low, but nonzero losses on the detection timescaleBeamsplitter to make differential phase detection insensitive to laser fluctuationsFreedom: state input to the second input port; optimize relative to a mean-numberconstraint.Entanglement: mixing this state with coherent state at the beamsplitter.Generalizeduncertainty principleQCRBOptimum achieved bydifferenced photodetection in aMach-Zehnder configuration.M. D. Lang and C. M. Caves,PRL 111, 17360 (2013).Achieved by squeezed vacuum into the second input port

Practical optical interferometry: Photon lossesM. D. Lang , UNM PhD dissertation, 2015.B. M. Escher, R. L. de Matos Filho, and L.Davidovich, Nat. Phys. 7, 406‒411 (2011).Z. Jiang, PRA 89, 032128 (2014).Optimum achieved by differenced photodetection in a Mach-Zehnder configuration.

III. Beyond the Heisenberg limit.Nonlinear interferometryEchidna GorgeBungle Bungle RangeWestern Australia

Beyond the Heisenberg limitThe purpose of theorems inphysics is to lay out theassumptions clearly so onecan discover whichassumptions have to beviolated.

Improving the scaling with NCat state does the job.Metrologicallyrelevant k-bodycouplingS. Boixo, S. T. Flammia, C. M. Caves, andJM Geremia, PRL 98, 090401 (2007).Nonlinear Ramsey interferometry

Improving the scaling with NBoixo, A. Datta, S. T. Flammia, A.without entanglement S.Shaji,E. Bagan, and C. M. Caves,PRA 77, 012317 (2008).ProductinputProductmeasurement

Improving the scaling with N without entanglement.Two-body couplingsS. Boixo, A. Datta, S. T. Flammia, A. Shaji, E. Bagan,and C. M. Caves, PRA 77, 012317 (2008); M. J. Woolley,G. J. Milburn, and C. M. Caves, NJP 10, 125018 (2008).Loss and decoherence?

Improving the scaling with N without entanglement.Two-body couplingsSuper-Heisenberg scalingfrom nonlinear dynamics (Nenhanced rotation of a spincoherent state), without anyparticle entanglementS. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M.Caves, PRL 101, 040403 (2008); A. B. Tacla, S. Boixo, A. Datta, A.Shaji, and C. M. Caves, PRA 82, 053636 (2010).Loss and decoherence?

Improving the scaling with N without entanglement.Optical experimentM. Napolitano, M. Koschorreck, B. Dubost,N. Behbood, R. J. Sewell, and M. W. Mitchell,Nature 471, 486 (2011).

Quantum metrology:An information-theoretic perspectiveLecture 3II.I. Introduction. What’s the problem?Standard quantum limit (SQL) for force detection.The right wrong storyIII. Beating the SQL. Three strategiesCarlton M. CavesCenter for Quantum Information and Control, University of New MexicoCentre for Engineered Quantum Systems, University of Queenslandhttp://info.phys.unm.edu/ cavesCenter for Quantum Information and Control

I. Introduction. What’s the problem?Pecos WildernessSangre de Cristo RangeNorthern New Mexico

(Absurdly) high-precision interferometryfor force sensingHanford, WashingtonThe LIGO Collaboration, Rep.Prog. Phys. 72, 076901 (2009).Laser Interferometer Gravitational Observatory (LIGO)4 kmLivingston, Louisiana

(Absurdly) high-precision interferometryfor force sensingInitial LIGOHanford, WashingtonLaser Interferometer Gravitational Observatory (LIGO)4 kmLivingston, LouisianaHigh-power, FabryPerot-cavity(multipass), powerrecycledinterferometers

(Absurdly) high-precision interferometryfor force sensingAdvanced LIGOHanford, WashingtonLaser Interferometer Gravitational Observatory (LIGO)4 kmLivingston, LouisianaHigh-power, FabryPerot-cavity(multipass), powerand signal-recycled,squeezed-lightinterferometers

Opto,atomic,electro micromechanics30 μm long170 nm wide140 nm thickBeam microresonator10 μmT. Rocheleau, T. Ndukum, C. Macklin ,J. B. Hertzberg, A. A. Clerk, and K. C.Schwab, Nature 463, 72 (2010).Atomic force microscopeDielectric micromembraneJ. C. Sankey, C. Yang, B. M. Zwickl,A. M. Jayich, and J. G. E. Harris,Nature Physics 6, 707 (2010).

Opto,atomic, electro micromechanicsZipper-cavity microresonatorDrum microresonatorA. D. O’Connell et al.,Nature 464, 697 (2010).M. Eichenfield, R. Camacho, J.Chan, K. J. Vahala, and O.Painter, Nature 459, 550 (2009).Toroidal microresonatorA. Schliesser and T. J. Kippenberg,Advances in Atomic, Molecular, andOptical Physics, Vol. 58, (AcademicPress, San Diego, 2010), p. 207.

Mechanics forforce sensingT. J. Kippenberg and K. J. Vahala, Science 321,172 (2008).

Standard quantum limit (SQL)Wideband detection of force f on free mass mLIGO interferometerBack action

Standard quantum limit (SQL)Narrowband, on-resonance detection of force f onoscillator of mass m and resonant frequency ω0NanoresonatorBack action?

SQLWideband force f on free mass mOn-resonance force f on oscillator ofmass m and resonant frequency ω0It’s wrong.It’s not even the right wrong story.The right wrong story. Waveform estimation.

II. Standard quantum limit (SQL) for forcedetection. The right wrong storySan Juan River canyonsSouthern Utah

SQL for force detectionMonitorpositionBack-action forceLangevin forcemeasurement (shot) noise

InterferometricreadoutLaser—

Interferometricreadout—Vacuum input portLasermeasurement(shot) noiseBack-action noiseIf shot noise dominates,squeeze the phase quadrature.

SQL for force detectionTime domainBack-action forceLangevin forcemeasurement noiseFrequency domainBack-action forcemeasurement noiseLangevin force

Noise-power spectral densitiesZero-mean, time-stationary random process u(t)Noise-power spectral density of u

SQL for force detectionBack-action forcemeasurement noiseLangevin force

SQL for force detection

Langevin force

SQL for force detectionThe right wrong story.In an opto-mechanical setting, achieving the SQL at aparticular frequency requires squeezing at thatfrequency, and achieving the SQL over a widebandwidth requires frequency-dependent squeezing.

III. Beating the SQL. Three strategiesTruchas from East Pecos BaldySangre de Cristo RangeNorthern New Mexico

Beating the SQL. Strategy 11. Couple parameter to observable h, and monitor observable oconjugate to h.2. Arrange that h and o are conserved in the absence of theparameter interaction; o is the simplest sort of quantumnondemolition (QND) or back-action-evading (BAE) observable.3. Give o as small an uncertainty as possible, thereby giving h asbig an uncertainty as possible (back action).Strategy 1. Monitor a quadrature component.Downsides1. Detect only one quadrature of the force.2. Mainly narrowband (no convenient free-mass version).3. Need new kind of coupling to monitor oscillator.

Strategy 2.Interferometricreadout—Vacuum input portOutput noiseLaserAll the output noise comes from the(frequency-dependent) purple quadrature.Squeeze it.W. G. Unruh, in Quantum Optics, Experimental Gravitation, andMeasurement Theory, edited by P. Meystre and M. O. Scully(Plenum, 1983), p. 647; F. Ya. Khalili, PRD 81, 122002 (2010).

Beating the SQL. Strategy 2Strategy 2. Squeeze the entire output noise bycorrelating the measurement and back-action noise.

Quantum Cramér-Rao Bound (QCRB)Single-parameter estimation: Bound on the error inestimating a classical parameter that is coupled to aquantum system in terms of the inverse of the quantumFisher information.Multi-parameter estimation: Bound on the covariancematrix in estimating a set of classical parameters that arecoupled to a quantum system in terms of the inverse of aquantum Fisher-information matrix.Waveform estimation: Bound on the continuous covariancematrix for estimating a continuous waveform that iscoupled to a quantum system in terms of the inverse of acontinuous, two-time quantum Fisher-information matrix.

Waveform QCRB.Spectral uncertainty principleM. Tsang, H. M. Wiseman, and C. M. Caves,PRL 106, 090401 (2011).Prior-information termAt frequencies where there is little prior information,Minimum-uncertainty noiseNo hint of SQL—no back-action noise, onlymeasurement noise—but can the bound be achieved?

Beating the SQL. Strategy 3Strategy 3. Quantum noise cancellation (QNC)using oscillator and negative-mass oscillator.Primary oscillatorNegative-mass oscillatorMonitor collectiveposition QConjugate pairsOscillator pairsQCRB

Quantum noise cancellationM. Tsang and C. M. Caves,PRL 105,123601 (2010).Conjugate pairsOscillator pairsPaired sidebands about a carrier frequencyPaired collective spins, polarized along opposite directionsW. Wasilewski , K. Jensen, H. Krauter, J. J. Renema,M. V. Balbas, and E. S. Polzik, PRL 104, 133601 (2010).

That’s all. Thanks for your attention.Tent RocksKasha-Katuwe National MonumentNorthern New Mexico

Using quantum circuit rometer2.IIC. M. Caves and A. Shaji, Opt. Commun. 283, 695 (2010) .

Proof of QCRB. Setting

Proof of QCRB. Classical CRB

Proof of QCRB.Classical Fisher information

Proof of QCRB. Quantum mechanics

For metrology, entanglement is part of the story, but only part. We need a generalized notion of entanglement/resources that includes information about the physical situation, particularly the relevant Ham