Review Article Quantum Sensors: Improved Optical ...downloads.hindawi.com/journals/js/2016/6051286.pdfReview Article Quantum Sensors: Improved Optical Measurement via Specialized Quantum

Review ArticleQuantum Sensors Improved Optical Measurement viaSpecialized Quantum States

David S Simon12

1Department of Electrical and Computer Engineering Boston University 8 Saint Maryrsquos Street Boston MA 02215 USA2Department of Physics and Astronomy Stonehill College 320 Washington Street Easton MA 02357 USA

Correspondence should be addressed to David S Simon simondbuedu

Received 14 June 2015 Accepted 2 November 2015

Academic Editor Tao Zhu

Copyright copy 2016 David S Simon This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Classical measurement strategies in many areas are approaching their maximum resolution and sensitivity levels but these levelsoften still fall far short of the ultimate limits allowed by the laws of physics To go further strategies must be adopted that take intoaccount the quantum nature of the probe particles and that optimize their quantum states for the desired application Here wereview some of these approaches in which quantum entanglement the orbital angular momentum of single photons and quantuminterferometry are used to produce optical measurements beyond the classical limit

1 Introduction

In the search for greater sensitivity and resolution in mea-suring and sensing applications there are fundamental limitsimposed by the laws of physics In order to reach these limitsit has become clear that it is ultimately necessary to take intoaccount the fundamentally quantum mechanical nature ofboth the system beingmeasured and the devices that are usedto measure the system The goal of this review is to give anintroduction to some of the ideas needed in order to continuepushing measurement capabilities of future devices towardtheir ultimate capacity

In imaging applications it has long been common lorethat the Abbe limit prevents an optical system from resolvingfeatures smaller than the wavelength of the illuminating lightHowever it has been recognized in recent years that there areloopholes in the Abbe limit which allow the so-called super-resolution the resolution of subwavelength-sized features [12] This can be done by using (i) spatially structured lightand (ii) nonlinear interactions between the light and theobject The spatial structure means that the wavefronts havesome nontrivial set of spatial frequencies which in a sensecombine with the lightrsquos intrinsic wavelength to produce asmaller effective wavelength thus allowing the resolving ofsmaller features To detect this structure some nonlinear

interaction is utilized such as the two-photon absorptionprocesses commonly used in microscopy

Similar ldquosuper-resolutionrdquo or ldquosuper-sensitivityrdquo effectsalso occur in other measurement and sensing contextsoutside of the realm of imaging and these effects are oftenassociatedwith purely quantummechanical phenomena suchas entanglement Entanglement occurs when a compositesystem made of two or more subsystems is described by asingle quantum state for the entire system in such away that itcannot be divided into separate well-defined substates for theindividual subsystems (see the appendix for a more precisedefinition of entanglement) In this review we give a shortdescription of some of these quantum-based super-resolvingeffects in areas other than standard imaging Specificallywe will focus on the use of nonclassical optical states tomeasure quantities related to phase angular and rotationalvariables and dispersive properties We will see that in eachof these cases the use of quantum-based methods allowsmore precise measurements than is possible through purelyclassical means For example entangled pairs of photonscan carry out phase measurements that violate the so-calledstandard quantum limit (see the next section) In many ofthese cases quantum entanglement can be seen as a resourcethat may be used to carry out practical tasks a viewpointis common in the quantum computation and quantum

Hindawi Publishing CorporationJournal of SensorsVolume 2016 Article ID 6051286 13 pageshttpdxdoiorg10115520166051286

2 Journal of Sensors

information communities but is not as widely recognizedin other areas The improved resolution and sensitivity thatcan be obtained from an 119873-photon entangled state can beseen as being due to the fact that the set of photons (each ofwavelength 120582) can be treated as a single object (essentially asmall Bose-Einstein condensate) with a single joint de Brogliewavelength 120582radic119873 [3ndash8] In this way the true quantummechanical limit the Heisenberg limit can be reached

By optimizing the detection method for a given measure-ment the weaker standard quantum limit can be achievedbut to reach the stronger Heisenberg limit it is necessary tooptimize the probe state as well [9] and tailor its properties tothe desired measurement this is where entanglement gener-ally enters the picture The comparison to super-resolutionimaging also holds here the Abbe limit can be reachedby optimizing the detection system (large-aperture lenseshigh resolution detectors etc) but to exceed the limit it isnecessary to also optimize the illuminating light (the probestate) by introducing appropriate spatial structure As inthe imaging case nonlinear processes are also involved innonimaging quantum sensing applications as well nonlinearinteractions in solids (spontaneous parametric downconver-sion [10] in nonlinear crystals) are commonly used to preparethe entangled probe state and the detection process itselftypically involves coincidence counting or other forms ofmultiphoton detection such methods are essentially nonlin-ear since the signals from several photons are multiplied foreach detection event In particular two-detector coincidencecounting is often used which measures the second-order(intensity-intensity) correlations of a system as opposed tothe first-order (amplitude-amplitude) correlations measuredin classical interferometryMany purely quantummechanicaleffects that are invisible to classical interferometry showup in the second-order correlations [11 12] In additioninterference effects in such measurements have a maximumvisibility of 1radic2 asymp 71 for classical states of light [13 14]whereas entangled states can have visibilities approaching100

A review of the more strictly technological aspects ofquantum sensing such as the development of on-chip photonsources and interferometers can be found in [15] and variousother aspects of quantum metrology have previously beenreviewed in [9 16ndash18] Here we concentrate on the physicalideas and some specific applicationsThemain ingredients inall of these applications are (i) entanglement (ii) using statestailored to optimize the given measurement and (iii) coinci-dence counting or intensity-intensity correlations Althoughmost of the applications in this area are still confined tothe lab some have begun to make steps toward real-worlduse As one example some of the methods described herehave recently been used to measure the polarization modedispersion of a switching unit in a fiber optics communicationnetwork [19 20] the resulting measurement had uncertain-ties of 01 fs for group delay and 2 attoseconds for phase delaywhich is orders of magnitude smaller than the uncertaintiespresent in previous classical measurement strategies

One note on terminology super-resolution is takento mean producing interference patterns that oscillate on

smaller scales than expected classically while super-sensitiv-ity means reducing the uncertainty in a measured variablebelow its classical value The two effects often go hand-in-hand

A very brief summary of the necessary quantummechan-ical background is given in the appendix

2 Phase Estimation Entanglementand Quantum Limits

We consider phase measurements first [17 21] and ask whatfundamental limit does quantum mechanics place on ourability to measure themThe question is muddled a bit by thefollowing problem In quantummechanics physical variablesare represented by operators (see the appendix) and wewould expect the photon number operator and the phaseoperator 120601 to be canonically conjugate operators with theircommutator leading to an uncertainty relation that definesthe ultimate physical limit on measurements However it hasbeen known since at least the 1960s that it is impossible todefine a unitary operator 119890119894120601 or equivalently a Hermitianoperator 120601 that gives the correct canonical commutationrelations There is a large literature discussing this problem[22ndash34] and there are several possible resolutions to it Herewe will simply follow the most common path and make useof the non-Hermitian Susskind-Glogower (SG) operator [22]

119878 =119890119894120601 =

infin

sum

119899=0

|119899⟩ ⟨119899 + 1| = 119886minus12

= ( + 1)minus12

119886 (1)

where 119886 and |119899⟩ are respectively the photon annihilationoperator the photon number operator and the Fock state ofphoton number 119899 The eigenstates of 119878 are |120601⟩ = sum

infin

119899=0119890119894119899120601|119899⟩

The corresponding eigenvalue relation is given by 119878|120601⟩ =

119890119894120601|120601⟩ where 120601 is the phase of the state (up to some

unobservable overall reference phase) To get a Hermitian(and therefore physically measurable) operator we can add119878 to its Hermitian conjugate to form the new operator 119860 =

suminfin

119899=0(|119899⟩⟨119899 + 1| + |119899 + 1⟩⟨119899|) The expectation value of

this operator measures the rate of single-photon transitionswithin the apparatus Now that we have a phase-dependentobservable we can measure its value on some quantum stateand extract an estimate of the statersquos phase (note that any otherphase-dependent Hermitian operator could be used insteadhowever some operators give better phase estimation thanothers and are easier to measure experimentally)

Suppose the phase is measured using an interferometerwith input to the system only at a single portWe can considerseveral types of input such as a coherent state input (wherethe number of photons fluctuates about some mean value119873 according to Poisson statistics) a single Fock state (offixed photon number 119873) or 119873 separate measurements ona stream of single-photon states In each of these cases thephase sensitivity (the minimum uncertainty in the measuredphase value) is proportional to 1radic119873when119873 is largeWe cansee this most easily by looking at the case of119873measurementsof consecutive single-photon inputs

Journal of Sensors 3

Consider a photon entering port 119860 in the Mach-Zehnderinterferometer of Figure 1 It can travel along the lower pathreflecting offmirror119872

2 or it can travel along the upper path

reflecting off1198721and gaining a phase shift of120601 (wewill neglect

any phase shift due to reflection at the beam splitter sinceit can be absorbed into the definition of 120601) We may denotethe state leaving the first beam splitter along the lower pathby |119871⟩ and the state leaving along the upper path by |119880⟩ (seethe appendix for an explanation of state vector notation) Orequivalently we could label these two states by the number ofphotons in the upper branch so that |119871⟩ = |0⟩ and |119880⟩ = |1⟩Then the state arriving at the second beam splitter is |120595⟩ =

(1radic2)(|0⟩+119890119894120601|1⟩) Since the photonnumber 119899never exceeds

1 anywhere for this state the operator 119860 can be truncated to119860 = |0⟩⟨1| + |1⟩⟨0| Note that 119860 picks out interference termswhen wedged between two states ⟨120595

1|119860|1205952⟩ it links the |0⟩

component of one state to the |1⟩ component of the other Infact taking a basis of the form

|0⟩ = (

1

0)

|1⟩ = (

0

1)

(2)

this operator is simply the first Pauli spin matrix 119860 = 120590119909=

(0 1

1 0) It is therefore easily seen that 1198602 = 119868 the identity

operator Taking the expectation value (the mean value in thegiven state) we find

⟨119860⟩ = ⟨1205951003816100381610038161003816100381611986010038161003816100381610038161003816120595⟩ = cos120601

⟨1198602⟩ = ⟨120595

10038161003816100381610038161003816119860210038161003816100381610038161003816120595⟩ = 1

(3)

Therefore the uncertainty in the measurement of 119860 is

Δ119860 = radic⟨1198602⟩ minus ⟨119860⟩2

= radic(1 minus cos2120601) = 1003816100381610038161003816sin1206011003816100381610038161003816

(4)

If we repeat the experiment119873 times on an identical ensembleof states standard statistical theory tells us that the uncer-tainty should be reduced by a factor of radic119873 so that Δ119860 =

| sin120601|radic119873 Our best estimate of 120601 is then the value 120601 =

cosminus1(⟨119860⟩119873) and the uncertainty in this phase estimate isgiven by

Δ120601 =Δ119860

10038161003816100381610038161003816119889 ⟨119860⟩ 119889120601

10038161003816100381610038161003816

=

radic1198731003816100381610038161003816sin120601

1003816100381610038161003816

1198731003816100381610038161003816sin120601

1003816100381610038161003816

=1

radic119873

(5)

This is the standard quantum limit or shot noise limit Inthe case of single-photon states being measured 119873 timesthis uncertainty can be thought of as being due to photonicshot noise or ldquosorting noiserdquo the Poisson-distributed randomfluctuations of the photon number in each arm of theapparatus due to the randomchoicemade by the beam splitteras to which way to send each photon If instead of119873 single-photon states the experimenter uses a single coherent state

Input Output

C

DB

A

BS1 BS2

120593

M2

M2

BS = 5050 beam splitterM = mirror

Figure 1 A Mach-Zehnder interferometer Input at either of theinput ports (119860 or119861) has two possible ways of reaching the final beamsplitter BS

2 Along the upper path a phase shift of 120601 is added The

second beam splitter makes it impossible to determine which pathwas followed simply from looking at the output at ports 119862 and 119863As a result the amplitudes for the both possibilities must be presentleading to interference at the output ports

pulse with mean photon number ⟨⟩ = 119873 the result forthe uncertainty is the same since the random fluctuationsof photon number in the coherent state beam have the samePoisson statistics as the sorting noise

The derivation of the standard quantum limit aboveassumed that the119873 photons used to obtain the estimate wereall acting independently However it is possible to send instates where the photons are entangled (see appendix) sothat the entire 119873-photon set is described by a single jointquantum state At any horizontal position in Figure 1 let aquantum state with 119899

1photons in the top branch and 119899

2

photons in the lower branch be denoted by |1198991⟩|1198992⟩ or more

simply by |1198991 1198992⟩ Then imagine sending an entangled 119873-

photon state into the system for example a so-called NOONstate (1radic2)(|119873 0⟩+|0119873⟩) [35] A two-photonNOON statecan be easily produced by means of the Hong-Ou-Mandeleffect [36] 119873 = 3 [6] 119873 = 4 [7] and 119873 = 5 [37 38]states have been produced using parametric downconversionfollowed by postselection of states with the desired formAfter the first beam splitter and the phase shift the statereaching the second beam splitter is |120595

119873⟩ = (1radic2)(|0119873⟩ +

119890119894119873120601

|119873 0⟩) Note that because of the entanglement the phaseshifts from the 119873 photons act collectively giving a totalphase shift 119873120601 to one component of the entangled stateThis is in contrast to the single-photon and separable 119873-photon cases above where each photon privately carried itsown phase shift independently of what happened to theother photons In place of operator 119860 above an appropriatemeasurement operator to extract this total phase shift is now119861119873= |0119873⟩⟨119873 0|+|119873 0⟩⟨0119873| It is straightforward to check

that the expectation values and uncertainties are given by

⟨119861119873⟩ = ⟨120595

119873

10038161003816100381610038161003816119861119873

10038161003816100381610038161003816120595119873⟩ = cos119873120601 (6)

⟨1198612

119873⟩ = ⟨120595

119873

100381610038161003816100381610038161198612

119873

10038161003816100381610038161003816120595119873⟩ = 1 (7)

Δ119861119873= sin119873120601 (8)

Δ120601 =1

119873 (9)


In other words this entangled system can beat the stan-dard quantum limit It in fact saturates the Heisenberglimit the fundamental physical bound imposed by quantummechanics [39] The use of NOON states to carry out phasemicroscopy near the Heisenberg limit has been experimen-tally demonstrated in [40]

We see that entanglement gives a 1radic119873 advantage inphase sensitivity over unentangled 119873-photon states Theincrease in oscillation frequency signalled by the factor of119873 inside the cosine in (6) indicates super-resolution as wellAlthough this improved phase measurement ability has beendemonstrated in the lab NOON states and other types ofentangled photon states are difficult to create for large 119873 soone prominent current goal is to find ways to more easilymake such states in order to allow their use in applicationsettings outside the lab

The quality of the phase estimation is normally quantifiedby the uncertainty as obtained from some observable asin (9) In more complicated situations such as imagingthe optimal uncertainty may be obtained via informationtheoretic means through the Cramer-Rao bound [41 42]Alternatively the uncertainty may not be used at all insteadthe quality of the estimation may be measured by meansof mutual information [43 44] Bayesian analysis [45 46]is sometimes used to provide strategies for optimizing theestimation strategy

Other entangled 119873-photon states such as those of [4748] may be used Some of these states give slightly bettersensitivity or are slightly more robust to noise but overallthey give qualitatively similar results to those derived fromNOON states For completeness we should mention thatentangled states are not necessary to surpass the standardquantum limit For example squeezed states have beenshown to be capable of achieving 1119873

34 phase sensitivity[49] However up to this point only entangled states seemto be capable of fully reaching the Heisenberg bound Alsoit has been shown that through a postselection scheme [8]it is possible to produce super-resolution without the use ofentangled illumination states (although super-sensitivity wasnot demonstrated)

3 Angular and Rotational Measurements

31 Optical Orbital Angular Momentum In the last sectionit was shown that phase measurements could be made withentangled optical states that exceed the standard quantumlimit It is natural to ask if a similar improvement can bemade in measurements of other quantities In this sectionwe show that this can be done for angular displacements[50] This is done by using states for which angular rotationslead to proportional phase shifts the super-sensitivity tophase then leads to super-sensitivity to orientation anglesIn this subsection we introduce the idea of optical angularmomentum and then look at its applications in the followingsubsections

It is well known that photons carry one unit of spinangular momentum This spin manifests itself as circularpolarization left- and right-circularly polarized photons have

spin quantum numbers 119904119911

= plusmn1 along the propagationaxis while linearly polarized photons are formed from equalsuper-positions of the two spin states It is only in the past twodecades that it has been widely recognized that in addition tothis intrinsic spin angularmomentumaphoton can also carryorbital angular momentum (OAM) L about its propagationaxis [51] A number of excellent reviews of the subject existincluding [52ndash54] Here we briefly summarize the basic facts

For a photon to have nonzero OAM its state must havenontrivial spatial structure such that the wavefronts haveazimuthally dependent phases of the form 119890

119894120601(120579)= 119890119894119897120579 Here

120579 is the angle about the 119911-axis and 119897 is a constant knownas the topological charge There are several ways to imprintsuch phases onto a beam The most common are by meansof spiral phase plates (plates whose optical thickness variesazimuthally according to Δ119911 = 1198971205821205792120587(119899 minus 1) for light ofwavelength 120582) [55] computer generated holograms of forkeddiffraction gratings [56] or spatial light modulators (SLM)

Since the angular momentum operator about the 119911-axisis 119911= minus119894ℏ(120597120597120579) the resulting wave has a 119911-component

of angular momentum given by 119871119911

= 119897ℏ where ℏ isPlanckrsquos constant The fact that the wavefunction must besingle valued under rotations 120579 rarr 120579 + 2120587 forces 119897 to bequantized to integer values The 119890

119894119897120579 phase factor has theeffect of tilting the wavefronts giving them a corkscrewshape (Figure 2) The Poynting vector S = E times H must beperpendicular to the wavefront so it is at a nonzero angle tothe propagation direction S therefore rotates about the axisas the wave propagates leading to the existence of nonzeroorbital angular momentum

The question of separating the angular momentum intospin and orbital parts in a gauge-independent manner iscomplicated for the general case [57ndash62] but as long as werestrict attention to the transverse propagating radiation fieldin the paraxial region the splitting is unambiguous andOAMwill be conserved in the parametric downconversion processdiscussed below

Several different optical beam modes can carry OAMhere we focus on Laguerre-Gauss (LG) modes (otherpossibilities include higher-order Bessel or Hermite-Gaussmodes) The LG wavefunction or spatial amplitude functionis [63]

119906119897119901 (119903 119911 120579) =

119862|119897|

119901

119908 (119911)(radic2119903

119908 (119911))

|119897|

119890minus11990321199082(119903)119871|119897|

119901(

21199032

1199082 (119903))

sdot 119890minus1198941198961199032119911(2(119911

2+1199112

119877))119890minus119894120579119897+119894(2119901+|119897|+1) arctan(119911119911

119877)

(10)

with normalization 119862|119897|119901= radic2119901120587(119901 + |119897|) and beam radius

119908(119911) = 1199080radic1 + 119911119911

119877at 119911 119871120572

119901(119909) are the associated Laguerre

polynomials [64] 119911119903= 120587119908

2

0120582 is the Rayleigh range 119908

0is

the radius of the beam waist and the arctangent term is theGouy phase familiar from Gaussian laser beams The index119901 characterizes the radial part of the mode the number ofintensity nodes in the radial direction is 119901 + 1 All LG modes(for 119897 = 0) have a node at the central axis about which theoptical vortex circulates For 119901 gt 1 additional nodes appearas dark rings concentric about the central axis It is important


z

Surface of constant phase

S

120579

Figure 2 Optical wavefronts with nonzero orbital angular momen-tum have corkscrew-shaped wavefronts The Poynting vector Smust be everywhere perpendicular to the wavefront so it rotates asthe wave propagates along the 119911-axis

to note that not only macroscopic light beams but also evensingle photons can have nontrivial spatial structure leading tononzero OAM

Measurement of theOAMof a beam can be accomplishedin a number of ways including an interferometric arrange-ment that sorts different 119897 values into different outgoingspatial modes [65ndash67] and sorting by means of 119902-plates [68](devices which couple the OAM to the polarization) as wellas by use of polarizing Sagnac interferometers [69] pinholearrangements followed by Fourier-transforming lenses [70]and specialized refractive elements [71ndash73]

As was mentioned earlier the most common way toproduce entangled photons is via spontaneous parametricdownconversion (SPDC) in a 1205942 nonlinear crystal In thisprocess interactions with the crystal lattice cause a highfrequency photon from the incident beam (called the pumpbeam) to split into two lower frequency photons (called forhistorical reasons the signal and idler) The signal-idler pairis entangled in a number of different variables includingenergy momentum and polarization Our interest herethough is in the fact that they are entangled in OAM [74]We will assume for simplicity that the pump beam has noOAM in which case the signal and idler have equal andopposite values of topological charge plusmn119897 The 119901 values of thesignal and idler are unconstrained but since the detectors areusually coupled via fibers that propagate only 119901 = 0 modeswe will restrict attention to the case where the outgoing 119901

values vanishTheoutput of the crystalmay then be expandedas a super-position of signal-idler states with different OAMvalues For the case we are considering this takes the form

infin

sum

119897119904119897119894=minusinfin

119870119897119904119897119894

120575 (119897119904+ 119897119894)1003816100381610038161003816119897119904 119897119894⟩ =

infin

sum

119897119904=minusinfin

119870119897119904minus119897119904

1003816100381610038161003816119897119904 minus119897119904⟩ (11)

where 119897119904and 119897119894are the OAM of the signal and idler Explicit

expressions for the expansion coefficients119870119897119904119897119894

may be foundin [75ndash78]

32 Angular Displacement Measurement In order to illus-trate super-resolution in angular measurement consider aDove prism This is a prism (Figure 3) designed so that totalinternal reflection occurs on the bottom surface Because of

Input

Output

Figure 3 Dove prism Total internal reflection at the horizontalsurface causes images to be inverted vertically

this reflection images are inverted in the vertical direction(the direction perpendicular to the largest prism face) with-out any corresponding horizontal inversion One interestingproperty of these prisms is that if they are rotated by an angle120579 about the propagation axis the image is rotated by 2120579 Forwell-collimated light with a definite value of orbital angularmomentum 119871 = 119897ℏ the Dove prism reverses the angularmomentum direction while maintaining its absolute value119897 rarr minus119897 assuming that the beam is not too tightly focused[79]

Consider the setup shown in Figure 4 [50] The Doveprism is rotated by some unknown angle 120579 and the goal isto measure 120579 as precisely as possible As input take a pairof entangled photons with signal and idler of topologicalquantum numbers +119897 and minus119897 respectivelyThe state producedby the entangled light source will be of the form of (11) butinsertion of appropriate filters can be used to block all exceptfor the plusmn119897 terms for some fixed value of 119897

1003816100381610038161003816120595in⟩ =1

radic2(|119897⟩119904 |minus119897⟩119894 + |minus119897⟩119904 |119897⟩119894) (12)

where 119904 and 119894 denote signal and idler which we take toenter the interferometer in the upper and lower branchesrespectivelyThe effect of the components of the apparatus onthe two-photon state can be traced through to find the outputstate |120595out⟩ The rate of detection events for which value 119897arrives at one detector and value minus119897 arrives at the other is thenthe expectation value of the operator

= |+119897⟩119863119887|minus119897⟩119863

119886

sdot119863119886⟨+119897|119863

119887⟨minus119897| + |minus119897⟩119863

119887|+119897⟩119863

119886

sdot119863119886⟨minus119897|119863

119887⟨+119897|

(13)

We find the expectation values

⟨⟩ = ⟨120595out1003816100381610038161003816100381610038161003816100381610038161003816120595out⟩ = cos2 (2119897120579)

⟨2⟩ = ⟨120595out

10038161003816100381610038161003816210038161003816100381610038161003816120595out⟩ = cos2 (2119897120579)

(14)

so that the uncertainty in is

Δ = radic⟨2⟩ minus ⟨⟩2

= cos (2119897120579) sin (2119897120579)

=1

2sin (4119897120579)

(15)


M1

M1

M2

M2

120579

DP

Da

Db

NLCSignal

Pump

Idler

Entangled photonsource

CCBS1 BS2

BS = 5050 beam splitterD = detectorCC = coincidence counter

M= mirrorDP = dove prismNLC = nonlinear crystal

Figure 4 Schematic of setup for super-resolution angular dis-placementmeasurements using entangledOAMstates Spontaneousparametric downconversion in a nonlinear crystal (on the left)sends entangled photon pairs into a Mach-Zehnder interferometer(middle portion) which contains a Dove prism in one arm On theright photodetectors are connected to a coincidence circuit so thatan event is registered only when both detectors fire within a shorttime window

We may then extract an estimate of the angular displacementfrom the value of ⟨⟩ The resulting estimate will haveuncertainty

Δ120579 =Δ

10038161003816100381610038161003816120597 ⟨⟩ 120597120579

10038161003816100381610038161003816

(16)

This whole procedure can be generalized from two to 119873entangled photons In an identical manner it can be shownthat the phase sensitivity is now given by

Δ120579 =1

2119873119897 (17)

This should be compared to the case when an ensemble of119873independent photons are measured with the same apparatus

Δ120579SQL =1

2radic119873119897

(18)

We not only see once again that the entanglement allowsthe sensitivity to be reduced by a factor of radic119873 beyond thecase standard quantum limit but also further improvementscan be made to the angular sensitivity by using statesof higher angular momentum The angular resolution canalso exceed the classical limit with the argument of theoscillatory term in ⟨⟩ being proportional to119873119897120579 Such highprecision measurement of rotations may ultimately be usefulin nanotechnology applications

33 Image Reconstruction and Object Identification An addi-tional set of OAM-based approaches illustrates the ideathat single-photon states can be chosen specifically to suitthe type of measurement that will be made with themThe Laguerre-Gauss modes 119906

119897119901(119903 119911 120579) form a complete set

meaning that any two-dimensional function can be built as a

linear combination of them 119891(119903 120579) = sum119897119901119886119897119901119906119897119901(119903 119911 120579) for

some set of coefficients 119886119897119901This fact leads to the idea of digital

spiral imaging [80ndash82] This is a form of angular momentumspectroscopy in which a beam of known OAM is used toilluminate an object and then the spectrumof outgoingOAMvalues is measured after reflection or transmission from theobject This allows the evaluation of the squared coefficients|119886119897119901|2 of the objectrsquos reflection or transmission profile and

reveals a great deal of information about the object structureIn this manner the object could be identified from a knownset or its difference from some desired shape could bequantifiedThis approach has two advantages first for simpleobjects where only a small number of coefficients are largethe OAM spectrum can be determined to good accuracy witha small number of photons Second rotating the object onlyadds a constant phase factor to all the amplitudes leavingthe intensities of the outgoing components unchanged Thismeans that the spatial orientation of the object does not needto be known and that the object can still be easily identifiedeven if it is rapidly rotating [83] Measurement of the OAMspectrum also allows for rapid experimental determinationof the rotational symmetries of an unknown object [84 85]These facts could make this approach ideal for a number ofapplications such as (i) looking for sickle cells cancer cellsor other irregularly shaped cells in amoving bloodstream (ii)rapidly scanning for defects in parts on an assembly line or(iii) identifying defects in rapidly spinning objects such as arotor blade

One goal of this approach is to ultimately be able toreconstruct the image from the OAM spectrum alone Twoproblems must be addressed in order to do this The first isthat measuring the intensities of the outgoing OAM intensitycomponents is not sufficient for image reconstruction thecoefficients 119886

119897119898are complex so that their relative phases

must be determined as well This fortunately can once againbe done by means of an interferometric arrangement suchas that of Figure 1 [83ndash85] this method has been calledcorrelated spiral imaging The second problem is that currenttechnology does not easily allow terms with nonzero 119901

values to be measured efficiently at the single-photon levelSo even though the angular structure of the object can bereconstructed nontrivial radial structure is difficult to detectin this manner This will likely be remedied over time asimproved detection methods are developed

Digital and correlated spiral imaging do not requireentanglement for their basic functioning However it isclear that119873-photon entangled states would once again allowthese approaches to exceed the classical limits on resolutionand sensitivity in the same manner as the other methodsdescribed in the earlier sections In particular angular oscilla-tion frequencies would again be increased by a factor of119873 asin the previous subsection allowing the same improvementin sensitivity and resolution of small angular structures Inthis way super-resolved imaging could in principle be carriedout entirely by OAMmeasurements

34 Rotational Measurements Since measurements withOAM are being discussed one more related type of mea-surement can be mentioned even though it does not


require entanglement or low photon numbers Optical orbitalangular momentum states can also be used to measurerotation rates [86ndash90] Imagine two photons of oppositeOAM plusmn119897 reflecting off the surface of a rotating object Let 120596be the photon frequency and letΩ be the rotational frequencyof the objectThe two reflected photons will experience equaland opposite Doppler shifts120596 rarr 120596plusmn119897Ω If the two reflectedphotons are allowed to interfere the combined intensity willtherefore exhibit a beat frequency of 2119897Ω If 119897 is large thenprecise measurements can be made of even very small valuesofΩ

4 Frequency Polarization and Dispersion

Here we briefly mention several other applications taking asimilar approach to those discussed in the earlier sections

41 Frequency Measurement The Ramsey interferometerused to measure atomic transition frequencies is formallyequivalent to a Mach-Zehnder interferometer A schematicof the Ramsey interferometer is shown in Figure 5 A 1205872

pulse is used to put an incoming atom into a super-positionof the ground and excited states this is analogous to the firstbeam splitter in theMach-Zehnder case which puts a photoninto a super-position of two different paths The two energystates gain different phase shifts under free propagation for afixed time so that when they are returned to the same stateagain they will produce an interference pattern as the timebetween the two pulses is varied The relative phase shift isΔ119864119905ℏ = 120596119905 where Δ119864 and 120596 are the energy difference andemitted photon frequency of the levels ℏ is Planckrsquos constantand 119905 is the time between the two pulses As a result theinterference pattern allows determination of the frequenciesin a manner exactly analogous to the phase determination inaMach-Zehnder interferometerThe formal analogy betweenthe Mach-Zehnder and Ramsey interferometers (as well aswith quantum computer circuits) has been referred to as thequantum Rosetta stone [17 91] as it allows ideas from onearea to be translated directly into the other

Because of the exact mathematical equivalence betweenthe two interferometers it follows immediately that the use ofentangled atomic states as input to the Ramsey interferometershould allow super-sensitive measurements of frequencywith the uncertainty scaling as 1119873 as the number 119873 ofentangled atoms increases [92 93] In a similar mannerHeisenberg-limited resolution can be achieved in optical andatomic gyroscopes which are also based on Mach-Zehnderinterferometry [94]

42 Dispersion Dispersion the differential propagationspeed of different states of light must be taken into accountto very high precision in modern optical telecommunicationsystems in long-range stand-off sensing and in many otherareas Here we briefly mention several related quantumapproaches to the measurement and cancelation of disper-sion

Chromatic dispersion effects can be described by consid-ering the wavenumber 119896 to be a function of frequency 119896(120596)

1205872pulse

1205872pulse

120593

Red excited stateGreen ground state

Figure 5 Schematic of a Ramsey interferometerThe first pulse putsthe ground state input into super-positions of ground and excitedstates Free propagation causes the two terms in the super-positionto differ in phase by an amount 120601 proportional to the transitionfrequency

In a dispersive medium 119896 will be a nonlinear function of 120596causing different frequencies to propagate at different speedsThis leads to broadening of wave packets as they propagatereducing the resolution of measurements made with thosepackets Expanding 119896(120596) in a Taylor series it can be dividedinto terms that are even-order and terms of odd-order infrequency It was found in the early 1990s that the use of anentangled light source combined with coincidence countingcan cause the even order terms to cancel among other thingsthis means that the largest contribution to the broadening(the second-order group delay term) will be absent Unlikethe applications in the previous section this method doesnot require simultaneous entanglement of a large number119873 of photons it only requires two photons to be entangledat a time and so can be easily carried out with standarddownconversion sources These sources naturally producephoton pairs with frequencies 120596

0plusmn Δ120596 displaced by equal

amounts about the central frequency 1205960(1205960is half the pump

frequency) The two photons are in fact entangled in thefrequency variable it is not predetermined which photonhas the higher frequency and which has the lower so bothpossibilities have to be super-posed

1003816100381610038161003816120595⟩ =1

radic2(10038161003816100381610038161205960 + Δ120596⟩signal

10038161003816100381610038161205960 minus Δ120596⟩idler

+10038161003816100381610038161205960 minus Δ120596⟩signal

10038161003816100381610038161205960 + Δ120596⟩idler)

(19)

For even-order dispersion cancelation the essential ideais to make the photons interfere in such a way that the twoopposite-signed frequency displacements add causing theeven powers in the expansion to cancel There are two waysto do thisThe Steinberg-Kwiat-Chaio (SKC)method [95 96]puts the dispersive material into one arm of a Mach-Zehnderinterferometer sends the entangled photons into the systemat one end and then measures coincidence counts in the twooutput ports at the other end The Franson method [97] onthe other hand sends the two photons into twomaterials withopposite dispersion looking again at coincidence counts at


the output Either way the second order pulse-broadeningterms are absent

It has since been shown that the SKC method can bereproduced with classical light sources (no entanglement)[98 99] but the Franson method seems to truly requireentanglement It has also been shown [100] that by con-catenating two successive interferometers (one Hong-Ou-Mandel interferometer [36] and one Mach-Zehnder) morecomplicated manipulations of the even- and odd-order fre-quency dispersion terms can be carried out such as cancelingthe odd-order terms or causing even-order and odd-ordercancelation to occur simultaneously in different parts of aninterferogram

Optical coherence tomography (OCT) is a well-knowninterferometric method for detecting and mapping subsur-face structures in biological samples It makes use of first-order amplitude-amplitude correlations Quantum opticalcoherence tomography (QOCT) [101ndash103] is a similar methodthatmakes use of entangled photon sources and second-orderintensity-intensity correlations QOCT has greatly improvedresolution due to the dispersion cancelation described aboveThe main difficulty with the method is that it is very slowdue to the need for individual pairs of entangled photonsSeveral variations [98 99 104ndash106] have been proposedthat mimic the dispersion-canceling effect of QOCT tovarying degrees using classical light sources including anexperimental demonstration of dispersion-cancelled OCTwith a 15-fold reduction in the broadening of the interferencepeaks [107]

In addition to chromatic dispersion optical pulses canalso exhibit polarization mode dispersion (PMD) in whichcomponents of an optical pulse with different polarizationstravel at different speeds With the extensive use of recon-figurable optical add-drop multiplexer (ROADM) systemsin modern fiber optics networks it has become essentialto be able to characterize the PMD of small componentssuch as wavelength selective switches to very high levels ofprecision Quantum interferometricmethods similar to thoseused for chromatic dispersion cancelation have also beenshown to allow ultra-high precision PMD measurements[19 20 108 109] beyond those possible using interferometrywith classical states of light In particular the apparatus of [19]has been demonstrated to produce time delay measurementswith sensitivities as low as the attosecond (10minus18 s) level [20]as a result of even-order dispersion cancelation this wasthe first demonstration of a quantum metrology applicationimplemented on an industrial commercial device The appa-ratus for super-resolved PMD measurement is once againconceptually similar to the interferometers of Figure 1 expectthat now the first beam splitter sends different polarizationsinto different spatial modes the phase shift in one armis introduced by dispersion and the two polarizations aremixed at the end by diagonally oriented polarizers instead ofby a second beam splitter

43 Quantum Lithography Although not a sensing or mea-suring application one other topic should be mentionedwhich is closely related to the subjects of this paper Thatis the idea of quantum lithography [35 110ndash112] in which

entangled states such as NOON states are used to writesubwavelength structures onto a substrate This again is dueto the fact that the 119873-photon entangled state oscillates 119873times faster than the single-photon state or equivalently thatthe effective wavelength of the system is reduced from 120582 to120582radic119873 The method holds great promise in principle foruse in fabricating more compact microchips and for othernanotechnology applications However practical applicationof the idea is again currently hampered by the difficultyof creating high 119873 entangled states on demand Moreoverrecent work [113] has shown that the efficiency of the processis low for small values of119873

5 Conclusions

The methods of quantum sensing described in this review(use of entangled states choosing single-photon or mul-tiphoton states that are tailored to optimize the desiredmeasurement and use of intensity-intensity correlations) inprinciple allow great advantages over classical methods Thecurrent challenges involve overcoming the limitations thatarise in real-life practice Chief among these challenges arethe following

(i) Producing high-quality entangled states with 119873 gt 2

photons is very difficult to arrange and the efficien-cies of current methods are extremely low

(ii) For more sophisticated OAM-based applicationsoptical fibers that propagate multiple 119897 values and aneffective means of measuring the radial 119901 quantumnumber are needed

(iii) NOON states are fragile against loss and noise [48] sothey are not suitable for use in noisy real-world envi-ronments However methods have been developedto find other states with more robust entanglement[48 114ndash116] Some of these states have been shownto beat the standard quantum limit and there is hopefor ultimately finding states that strongly approach theHeisenberg limit in lossy and noisy environments

If these technical challenges can be met quantum mea-surement methods will be ready to move from the laboratoryto the real world opening a wide range of new applicationsand improved measurement capabilities

Appendix

Brief Review of Quantum Mechanics

As the size scales involved in ameasurement or the number ofparticles involved becomes small the laws of classical physicsbreak down and the formalism of quantum mechanics mustbe taken into accountThere are many detailed introductionsto quantummechanics fromvarious points of view (eg [117ndash119]) Here we give just the few minimal facts needed for thisreview

The state of a particle (or more generally of any quantumsystem) is described by a vector in a complex space knownas a Hilbert space In Dirac notation these vectors are written


as |120595⟩ |120601⟩ and so forth where 120595 and 120601 are just labels for thestates One particularly useful example is the Fock state |119899⟩which contains exactly 119899 photons The operators 119886dagger and 119886known as creation and annihilation operators raise and lowerthe number of photons in the Fock state 119886dagger|119899⟩ = radic119899 + 1|119899+1⟩

and 119886|119899⟩ = radic119899|119899minus1⟩ The number operator = 119886dagger119886 counts

the number of photons in the state |119899⟩ = 119899|119899⟩To each vector |120595⟩ a corresponding dual or conjugate

vector ⟨120595| is associated This dual vector is the Hermitianconjugate (the complex transpose) of the original vectorSo for example if |120595⟩ is described by a column vectorwith components 120595

1 1205952 then ⟨120595| is a row vector with

components 120595lowast1 120595lowast

2

1003816100381610038161003816120595⟩ = (

1205951

1205952

sdot sdot sdot

) larrrarr ⟨1205951003816100381610038161003816 = (120595

lowast

1120595lowast

2sdot sdot sdot) (A1)

Inner products are defined between bras and kets ⟨120601 | 120595⟩ =

120601lowast

11205951+ 120601lowast

21205952+ 120601lowast

31205953+ sdot sdot sdot All vectors are usually assumed to

be normalized ⟨120595 | 120595⟩ = 1 Inner products such as |120601⟩⟨120595| actas operators that project state |120595⟩ onto the direction parallelto state |120601⟩

One of the defining properties of quantum mechanicsis that it is a linear theory the dynamics are definedby the Schrodinger equation a linear second-order partialdifferential equation As a result of this linearity quantumsystems obey the super-position principle any two possiblestates of a system say |120595⟩ and |120601⟩ can be added to get anotherallowed state of the system |Φ⟩ = (1radic2)(|120595⟩ + |120601⟩) wherethe 1radic2 is included to maintain normalization If |120595⟩ and|120601⟩ are not orthogonal to each other ⟨120601 | 120595⟩ = 0 then thetwo terms in the super-positionmay interferewith each otherleading to many uniquely quantum phenomena

Suppose a pair of particles 119860 and 119861 together form acomposite system 119862 If 119860 is in state |120595⟩ and 119861 is in state|120601⟩ then the composite system is in state |Ψ⟩

119862= |120595⟩

119860|120601⟩119861

where the subscripts are used to indicate which system isin which state Such a state is called a product state or aseparable state Often however the state of the compositesystem is known while the states of the individual subsystemsare not in that case all the possibilities consistent with theavailable information have to be super-posed as for examplein the state |Φ⟩

119862= (1radic2)(|120595⟩

119860|120601⟩119861+ |120601⟩119860|120595⟩119861) The two

possibilities (119860 in state |120595⟩ with 119861 in state |120601⟩ versus 119860 instate |120601⟩ with 119861 in state |120595⟩) can be thought of as existingsimultaneously Such a state which cannot be factored intoa single well-defined state for 119860 and a state for 119861 is calledentangled An example is the case where two photons arriveat a beam splitter both can exit out one output port both canexit at the other output port or there are two ways in whichthe two photons can exit at different ports If the locations ofthe outgoing photons are not measured then there is no wayto distinguish between the possibilities and they must all beincluded the state of the full system is therefore an entangledstate formed by the super-position of all four possibilities If ameasurement is made that determines the states of one of thesubsystems then the entangled state collapses to a product

state and the state of the second subsystem is also thereforeknown For example if particle 119860 is measured and found tobe in state 120595 then the collapse |Φ⟩

119862= (1radic2)(|120595⟩

119860|120601⟩119861+

|120601⟩119860|120595⟩119861) rarr |120595⟩

119860|120601⟩119861occurs After the collapse we know

that 119861 must now be in state 120601 It is important to point outthat the subsystems were not in definite states |120595⟩

119860and

|120601⟩119861before the measurement but that their prior states as

individuals did not even exist It can be shown that thecontrary assumption of prior existence of definite states foreach subsystem before measurement leads to contradictionswith experiment as demonstrated by the violation of Bell andCHSH inequalities [13 14 120ndash122] The fact that entangledsubsystems do not exist in definite states beforemeasurementis one of the odder andmore nonintuitive aspects of quantummechanics but is extremely well verified experimentally

The most common way to produce entangled pairsof photons is via spontaneous parametric downconversion(SPDC) [10] in which nonlinear interactions in a crystalmediate the conversion of an incoming photon (the pump)into two lower energy outgoing photons (the signal and idler)Only a tiny fraction of the incoming pump photons undergodownconversion (typically on the order of one in 10

6) butSPDC is very versatile in the sense that the downconvertedphotons can be entangled in multiple different variables (fre-quency time angular momentum linear momentum andpolarization) and the properties of the entangled pairs can befinely tuned by proper choice of parameters for the nonlinearcrystal and the laser beam pump Other sources such asquantum dots atomic cascades and especially designednonlinear optical fibers also exist see [123] for a review

Operators are mathematical objects that perform actionson states these operators can include for example matricesand derivatives Quantities that can be physically measuredsuch as energy and angular momentum are eigenvalues ofHermitian operators (operators which equal their Hermitianconjugate) Two operators 119860 and 119861 which do not commute[119860 119861] equiv 119860119861 minus 119861119860 = 0 obey an uncertainty relation if 119886 and119887 are the variables associated to the two operators then thereis a minimum value to the product of their uncertaintiesΔ119886Δ119887 must exceed a minimum quantity proportional to thecommutator [119860 119861] The most famous example is betweenposition (119909) and momentum (119901) 119909 and 119901 form a conjugatepair obeying the so-called canonical commutation relations[119909 119901] = 119894ℏ where ℏ is Planckrsquos constant As a result we findthe Heisenberg uncertainty relation

Δ119909Δ119901 geℏ

2 (A2)

Energy and time obey a similar relation

Δ119864Δ119905 geℏ

2 (A3)

These uncertainty relations provide the ultimate funda-mental physical limit to all measurements and are the basis ofthe Heisenberg limit given in the main text


Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

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[5] K Edamatsu R Shimizu andT Itoh ldquoMeasurement of the pho-tonic de brogliewavelength of entangled photon pairs generatedby spontaneous parametric down-conversionrdquo Physical ReviewLetters vol 89 no 21 Article ID 213601 4 pages 2002

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[7] P Walther J-W Pan M Aspelmeyer R Ursin S Gasparoniand A Zeilinger ldquoDe Broglie wavelength of a non-local four-photon staterdquo Nature vol 429 no 6988 pp 158ndash161 2004

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[48] S D Huver C F Wildfeuer and J P Dowling ldquoEntangledFock states for robust quantum optical metrology imaging andsensingrdquo Physical Review A vol 78 no 6 Article ID 063828 5pages 2008

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[69] S Slussarenko V DrsquoAmbrosio B Piccirillo L Marrucci andE Santamato ldquoThe Polarizing Sagnac Interferometer a tool forlight orbital angular momentum sorting and spin-orbit photonprocessingrdquo Optics Express vol 18 no 26 pp 27205ndash272162010

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[73] M P J Lavery D J Robertson G C G Berkhout G DLove M J Padgett and J Courtial ldquoRefractive elements forthe measurement of the orbital angular momentum of a singlephotonrdquo Optics Express vol 20 no 3 pp 2110ndash2115 2012

[74] AMair A Vaziri GWeihs and A Zeilinger ldquoEntanglement ofthe orbital angular momentum states of photonsrdquo Nature vol412 no 6844 pp 313ndash316 2001


[75] J P Torres A Alexandrescu and L Torner ldquoQuantum spiralbandwidth of entangled two-photon statesrdquo Physical Review Avol 68 no 5 Article ID 050301 4 pages 2003

[76] X-F Ren G-P Guo B Yu J Li and G-C Guo ldquoThe orbitalangular momentum of down-converted photonsrdquo Journal ofOptics B Quantum and Semiclassical Optics vol 6 no 4 pp243ndash247 2004

[77] G A Barbossa ldquoWave function for spontaneous parametricdown-conversion with orbital angular momentumrdquo PhysicalReview A vol 80 no 6 Article ID 063833 12 pages 2009

[78] FMMiatto AMYao and SM Barnett ldquoFull characterizationof the quantum spiral bandwidth of entangled biphotonsrdquoPhysical ReviewA vol 83 no 3 Article ID 033816 9 pages 2011

[79] N Gonzalez G Molina-Terriza and J P Torres ldquoHow a Doveprism transforms the orbital angular momentum of a lightbeamrdquo Optics Express vol 14 no 20 pp 9093ndash9102 2006

[80] L Torner J P Torres and S Carrasco ldquoDigital spiral imagingrdquoOptics Express vol 13 no 3 pp 873ndash881 2005

[81] G Molina-Terriza L Rebane J P Torres L Torner and SCarrasco ldquoProbing canonical geometrical objects by digitalspiral imagingrdquo Journal of the European Optical Society vol 2Article ID 07014 6 pages 2007

[82] J P Torres A Alexandrescu and L Torner ldquoQuantum spiralbandwidth of entangled two-photon statesrdquo Physical Review Avol 68 no 5 Article ID 050301(R) 4 pages 2003

[83] C A Fitzpatrick D S Simon and A V Sergienko ldquoHigh-capacity imaging and rotationally insensitive object identi-fication with correlated orbital angular momentum statesrdquoInternational Journal of Quantum Information vol 12 no 7-8Article ID 1560013 16 pages 2015

[84] D S Simon and A V Sergienko ldquoTwo-photon spiral imagingwith correlated orbital angular momentum statesrdquo PhysicalReview A vol 85 no 4 Article ID 043825 8 pages 2012

[85] N Uribe-Patarroyo A Fraine D S Simon O Minaeva andA V Sergienko ldquoObject identification using correlated orbitalangular momentum statesrdquo Physical Review Letters vol 110 no4 Article ID 043601 5 pages 2013

[86] M V Vasnetsov J P Torres D V Petrov and L TornerldquoObservation of the orbital angular momentum spectrum of alight beamrdquo Optics Letters vol 28 no 23 pp 2285ndash2287 2003

[87] M P J Lavery F C Speirits S M Barnett and M J PadgettldquoDetection of a spinning object using lightrsquos orbital angularmomentumrdquo Science vol 341 no 6145 pp 537ndash540 2013

[88] C Rosales-Guzman N Hermosa A Belmonte and J P TorresldquoExperimental detection of transverse particle movement withstructured lightrdquo Scientific Reports vol 3 article 2815 2013

[89] M P Lavery S M Barnett F C Speirits and M J PadgettldquoObservation of the rotational Doppler shift of a white-lightorbital-angular-momentum-carrying beam backscattered froma rotating bodyrdquo Optica vol 1 no 1 pp 1ndash4 2014

[90] M Padgett ldquoA new twist on the Doppler shiftrdquo Physics Todayvol 67 no 2 pp 58ndash59 2014

[91] H Lee P Kok and J P Dowling ldquoA quantum Rosetta stone forinterferometryrdquo Journal of Modern Optics vol 49 no 14-15 pp2325ndash2338 2002

[92] J J Bollinger W M Itano D J Wineland and D J HeinzenldquoOptimal frequency measurements with maximally correlatedstatesrdquo Physical ReviewA vol 54 no 6 pp R4649ndashR4652 1996

[93] P Bouyer and M A Kasevich ldquoHeisenberg-limited spec-troscopy with degenerate Bose-Einstein gasesrdquo Physical ReviewA vol 56 no 2 pp R1083ndashR1086 1997

[94] J P Dowling ldquoCorrelated input-port matter-wave interferome-ter quantum-noise limits to the atom-laser gyroscoperdquo PhysicalReview A vol 57 no 6 pp 4736ndash4746 1998

[95] A M Steinberg P G Kwiat and R Y Chiao ldquoDispersion can-cellation in a measurement of the single-photon propagationvelocity in glassrdquo Physical Review Letters vol 68 no 16 pp2421ndash2424 1992

[96] A M Steinberg P G Kwiat and R Y Chiao ldquoDispersion can-cellation and high-resolution time measurements in a fourth-order optical interferometerrdquo Physical Review A vol 45 no 9pp 6659ndash6665 1992

[97] J D Franson ldquoNonlocal cancellation of dispersionrdquo PhysicalReview A vol 45 no 5 pp 3126ndash3132 1992

[98] B I Erkmen and J H Shapiro ldquoPhase-conjugate opticalcoherence tomographyrdquoPhysical ReviewA vol 74 no 4 ArticleID 041601 4 pages 2006

[99] K J Resch P Puvanathasan J S Lundeen M W Mitchell andK Bizheva ldquoClassical dispersion-cancellation interferometryrdquoOptics Express vol 15 no 14 pp 8797ndash8804 2007

[100] O Minaeva C Bonato B E A Saleh D S Simon and AV Sergienko ldquoOdd- and even-order dispersion cancellation inquantum interferometryrdquo Physical Review Letters vol 102 no10 Article ID 100504 4 pages 2009

[101] A F Abouraddy M B Nasr B E A Saleh A V SergienkoandMCTeich ldquoQuantum-optical coherence tomographywithdispersion cancellationrdquoPhysical ReviewA vol 65 no 5 ArticleID 053817 6 pages 2002

[102] M B Nasr B E A Saleh A V Sergienko and M CTeich ldquoDemonstration of dispersion-canceled quantum-opticalcoherence tomographyrdquo Physical Review Letters vol 91 no 8Article ID 083601 4 pages 2003

[103] M B Nasr D P Goode N Nguyen et al ldquoQuantum opticalcoherence tomography of a biological samplerdquo Optics Commu-nications vol 282 no 6 pp 1154ndash1159 2009

[104] K Banaszek A S Radunsky and I A Walmsley ldquoBlinddispersion compensation for optical coherence tomographyrdquoOptics Communications vol 269 no 1 pp 152ndash155 2007

[105] R Kaltenbaek J Lavoie D N Biggerstaff and K J ReschldquoQuantum-inspired interferometry with chirped laser pulsesrdquoNature Physics vol 4 no 11 pp 864ndash868 2008

[106] J L Gouet D Venkatraman F N C Wong and J H ShapiroldquoExperimental realization of phase-conjugate optical coherencetomographyrdquo Optics Letters vol 35 no 7 pp 1001ndash1003 2010

[107] M D Mazurek K M Schreiter R Prevedel R Kaltenbaekand K J Resch ldquoDispersion-cancelled biological imaging withquantum-inspired interferometryrdquo Scientific Reports vol 3article 1582 5 pages 2013

[108] D Branning A L Migdall and A V Sergienko ldquoSimultaneousmeasurement of group and phase delay between two photonsrdquoPhysical Review A vol 62 no 6 Article ID 063808 12 pages2000

[109] E Dauler G Jaeger A Muller A Migdall and A SergienkoldquoTests of a two-photon technique for measuring polarizationmode dispersion with subfemtosecond precisionrdquo Journal ofResearch of the National Institute of Standards and Technologyvol 104 no 1 pp 1ndash10 1999

[110] M DrsquoAngelo M V Chekhova and Y Shih ldquoTwo-photondiffraction and quantum lithographyrdquo Physical Review Lettersvol 87 no 1 Article ID 013602 4 pages 2001

[111] G Bjork L L Sanchez-Soto and J Soderholm ldquoEntangled-statelithography tailoring any pattern with a single staterdquo PhysicalReview Letters vol 86 no 20 pp 4516ndash4519 2001


[112] G Bjork L L Sanchez-Soto and J Soderholm ldquoSubwavelengthlithography over extended areasrdquo Physical Review A vol 64 no1 Article ID 138111 8 pages 2001

[113] C Kothe G Bjork S Inoue and M Bourennane ldquoOn theefficiency of quantum lithographyrdquo New Journal of Physics vol13 Article ID 043028 18 pages 2011

[114] U Dorner R Demkowicz-Dobrzanski B J Smith et alldquoOptimal quantum phase estimationrdquo Physical Review Lettersvol 102 no 4 Article ID 040403 4 pages 2009

[115] R Demkowicz-Dobrzanski U Dorner B J Smith et al ldquoQuan-tum phase estimation with lossy interferometersrdquo PhysicalReview A vol 80 no 1 Article ID 013825 2009

[116] T-W Lee S D Huver H Lee et al ldquoOptimization of quantuminterferometric metrological sensors in the presence of photonlossrdquo Physical Review A vol 80 no 6 Article ID 063803 10pages 2009

[117] R Shankar Principles of Quantum Mechanics Plenum PressNew York NY USA 2nd edition 2011

[118] A Peres Quantum Theory Concepts and Methods KluwerAcademic Publishers Dordrecht The Netherlands 1995

[119] D J Griffiths Introduction to Quantum Mechanics PearsonEducation Essex UK 2nd edition 2004

[120] J S Bell ldquoOn the problem of hidden variables in quantummechanicsrdquo Reviews of Modern Physics vol 38 no 3 pp 447ndash452 1966

[121] J S Bell ldquoOn the Einstein Podolsky Rosen paradoxrdquo Physicsvol 1 no 3 pp 195ndash200 1964

[122] A Aspect P Grangier and G Roger ldquoExperimental testsof realistic local theories via Bellrsquos theoremrdquo Physical ReviewLetters vol 47 no 7 pp 460ndash463 1981

[123] K Edamatsu ldquoEntangled photons generation observation andcharacterizationrdquo Japanese Journal of Applied Physics vol 46no 11 pp 7175ndash7187 2007

International Journal of

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Control Scienceand Engineering

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



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Advances inOptoElectronics


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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



information communities but is not as widely recognizedin other areas The improved resolution and sensitivity thatcan be obtained from an 119873-photon entangled state can beseen as being due to the fact that the set of photons (each ofwavelength 120582) can be treated as a single object (essentially asmall Bose-Einstein condensate) with a single joint de Brogliewavelength 120582radic119873 [3ndash8] In this way the true quantummechanical limit the Heisenberg limit can be reached

By optimizing the detection method for a given measure-ment the weaker standard quantum limit can be achievedbut to reach the stronger Heisenberg limit it is necessary tooptimize the probe state as well [9] and tailor its properties tothe desired measurement this is where entanglement gener-ally enters the picture The comparison to super-resolutionimaging also holds here the Abbe limit can be reachedby optimizing the detection system (large-aperture lenseshigh resolution detectors etc) but to exceed the limit it isnecessary to also optimize the illuminating light (the probestate) by introducing appropriate spatial structure As inthe imaging case nonlinear processes are also involved innonimaging quantum sensing applications as well nonlinearinteractions in solids (spontaneous parametric downconver-sion [10] in nonlinear crystals) are commonly used to preparethe entangled probe state and the detection process itselftypically involves coincidence counting or other forms ofmultiphoton detection such methods are essentially nonlin-ear since the signals from several photons are multiplied foreach detection event In particular two-detector coincidencecounting is often used which measures the second-order(intensity-intensity) correlations of a system as opposed tothe first-order (amplitude-amplitude) correlations measuredin classical interferometryMany purely quantummechanicaleffects that are invisible to classical interferometry showup in the second-order correlations [11 12] In additioninterference effects in such measurements have a maximumvisibility of 1radic2 asymp 71 for classical states of light [13 14]whereas entangled states can have visibilities approaching100

A review of the more strictly technological aspects ofquantum sensing such as the development of on-chip photonsources and interferometers can be found in [15] and variousother aspects of quantum metrology have previously beenreviewed in [9 16ndash18] Here we concentrate on the physicalideas and some specific applicationsThemain ingredients inall of these applications are (i) entanglement (ii) using statestailored to optimize the given measurement and (iii) coinci-dence counting or intensity-intensity correlations Althoughmost of the applications in this area are still confined tothe lab some have begun to make steps toward real-worlduse As one example some of the methods described herehave recently been used to measure the polarization modedispersion of a switching unit in a fiber optics communicationnetwork [19 20] the resulting measurement had uncertain-ties of 01 fs for group delay and 2 attoseconds for phase delaywhich is orders of magnitude smaller than the uncertaintiespresent in previous classical measurement strategies

One note on terminology super-resolution is takento mean producing interference patterns that oscillate on

smaller scales than expected classically while super-sensitiv-ity means reducing the uncertainty in a measured variablebelow its classical value The two effects often go hand-in-hand

A very brief summary of the necessary quantummechan-ical background is given in the appendix

2 Phase Estimation Entanglementand Quantum Limits

We consider phase measurements first [17 21] and ask whatfundamental limit does quantum mechanics place on ourability to measure themThe question is muddled a bit by thefollowing problem In quantummechanics physical variablesare represented by operators (see the appendix) and wewould expect the photon number operator and the phaseoperator 120601 to be canonically conjugate operators with theircommutator leading to an uncertainty relation that definesthe ultimate physical limit on measurements However it hasbeen known since at least the 1960s that it is impossible todefine a unitary operator 119890119894120601 or equivalently a Hermitianoperator 120601 that gives the correct canonical commutationrelations There is a large literature discussing this problem[22ndash34] and there are several possible resolutions to it Herewe will simply follow the most common path and make useof the non-Hermitian Susskind-Glogower (SG) operator [22]

119878 =119890119894120601 =

infin

sum

119899=0

|119899⟩ ⟨119899 + 1| = 119886minus12

= ( + 1)minus12

119886 (1)

where 119886 and |119899⟩ are respectively the photon annihilationoperator the photon number operator and the Fock state ofphoton number 119899 The eigenstates of 119878 are |120601⟩ = sum

infin

119899=0119890119894119899120601|119899⟩

The corresponding eigenvalue relation is given by 119878|120601⟩ =

119890119894120601|120601⟩ where 120601 is the phase of the state (up to some

unobservable overall reference phase) To get a Hermitian(and therefore physically measurable) operator we can add119878 to its Hermitian conjugate to form the new operator 119860 =

suminfin

119899=0(|119899⟩⟨119899 + 1| + |119899 + 1⟩⟨119899|) The expectation value of

this operator measures the rate of single-photon transitionswithin the apparatus Now that we have a phase-dependentobservable we can measure its value on some quantum stateand extract an estimate of the statersquos phase (note that any otherphase-dependent Hermitian operator could be used insteadhowever some operators give better phase estimation thanothers and are easier to measure experimentally)

Suppose the phase is measured using an interferometerwith input to the system only at a single portWe can considerseveral types of input such as a coherent state input (wherethe number of photons fluctuates about some mean value119873 according to Poisson statistics) a single Fock state (offixed photon number 119873) or 119873 separate measurements ona stream of single-photon states In each of these cases thephase sensitivity (the minimum uncertainty in the measuredphase value) is proportional to 1radic119873when119873 is largeWe cansee this most easily by looking at the case of119873measurementsof consecutive single-photon inputs








1|119860|1205952⟩ it links the |0⟩


|0⟩ = (

1

0)

|1⟩ = (

0

1)

(2)


(0 1



⟨119860⟩ = ⟨1205951003816100381610038161003816100381611986010038161003816100381610038161003816120595⟩ = cos120601

⟨1198602⟩ = ⟨120595

10038161003816100381610038161003816119860210038161003816100381610038161003816120595⟩ = 1

(3)


Δ119860 = radic⟨1198602⟩ minus ⟨119860⟩2


(4)




Δ120601 =Δ119860

10038161003816100381610038161003816119889 ⟨119860⟩ 119889120601

10038161003816100381610038161003816

=

radic1198731003816100381610038161003816sin120601

1003816100381610038161003816

1198731003816100381610038161003816sin120601

1003816100381610038161003816

=1

radic119873

(5)


Input Output

C

DB

A

BS1 BS2

120593

M2

M2








2




119873⟩ = (1radic2)(|0119873⟩ +

119890119894119873120601



⟨119861119873⟩ = ⟨120595

119873

10038161003816100381610038161003816119861119873

10038161003816100381610038161003816120595119873⟩ = cos119873120601 (6)

⟨1198612

119873⟩ = ⟨120595

119873

100381610038161003816100381610038161198612

119873

10038161003816100381610038161003816120595119873⟩ = 1 (7)

Δ119861119873= sin119873120601 (8)

Δ120601 =1

119873 (9)













119894120601(120579)= 119890119894119897120579 Here








119906119897119901 (119903 119911 120579) =

119862|119897|

119901

119908 (119911)(radic2119903

119908 (119911))

|119897|

119890minus11990321199082(119903)119871|119897|

119901(

21199032

1199082 (119903))

sdot 119890minus1198941198961199032119911(2(119911

2+1199112

119877))119890minus119894120579119897+119894(2119901+|119897|+1) arctan(119911119911

119877)

(10)


119908(119911) = 1199080radic1 + 119911119911

119877at 119911 119871120572


polynomials [64] 119911119903= 120587119908

2


0is



z


S

120579






infin

sum

119897119904119897119894=minusinfin

119870119897119904119897119894

120575 (119897119904+ 119897119894)1003816100381610038161003816119897119904 119897119894⟩ =

infin

sum


119870119897119904minus119897119904

1003816100381610038161003816119897119904 minus119897119904⟩ (11)





Input

Output




1003816100381610038161003816120595in⟩ =1



= |+119897⟩119863119887|minus119897⟩119863

119886

sdot119863119886⟨+119897|119863

119887⟨minus119897| + |minus119897⟩119863

119887|+119897⟩119863

119886

sdot119863119886⟨minus119897|119863

119887⟨+119897|

(13)


⟨⟩ = ⟨120595out1003816100381610038161003816100381610038161003816100381610038161003816120595out⟩ = cos2 (2119897120579)

⟨2⟩ = ⟨120595out

10038161003816100381610038161003816210038161003816100381610038161003816120595out⟩ = cos2 (2119897120579)

(14)



= cos (2119897120579) sin (2119897120579)

=1

2sin (4119897120579)

(15)


M1

M1

M2

M2

120579

DP

Da

Db

NLCSignal

Pump

Idler


CCBS1 BS2





Δ120579 =Δ

10038161003816100381610038161003816120597 ⟨⟩ 120597120579

10038161003816100381610038161003816

(16)


Δ120579 =1

2119873119897 (17)


Δ120579SQL =1

2radic119873119897

(18)
























1205872pulse

1205872pulse

120593







1003816100381610038161003816120595⟩ =1

radic2(10038161003816100381610038161205960 + Δ120596⟩signal

10038161003816100381610038161205960 minus Δ120596⟩idler

+10038161003816100381610038161205960 minus Δ120596⟩signal

10038161003816100381610038161205960 + Δ120596⟩idler)

(19)









5 Conclusions







Appendix











2

1003816100381610038161003816120595⟩ = (

1205951

1205952

sdot sdot sdot

) larrrarr ⟨1205951003816100381610038161003816 = (120595

lowast

1120595lowast



120601lowast

11205951+ 120601lowast

21205952+ 120601lowast





119862= |120595⟩

119860|120601⟩119861


119862= (1radic2)(|120595⟩

119860|120601⟩119861+ |120601⟩119860|120595⟩119861) The two



119862= (1radic2)(|120595⟩

119860|120601⟩119861+

|120601⟩119860|120595⟩119861) rarr |120595⟩



119860and






Δ119909Δ119901 geℏ

2 (A2)


Δ119864Δ119905 geℏ

2 (A3)





References

































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















1|119860|1205952⟩ it links the |0⟩


|0⟩ = (

1

0)

|1⟩ = (

0

1)

(2)


(0 1



⟨119860⟩ = ⟨1205951003816100381610038161003816100381611986010038161003816100381610038161003816120595⟩ = cos120601

⟨1198602⟩ = ⟨120595

10038161003816100381610038161003816119860210038161003816100381610038161003816120595⟩ = 1

(3)


Δ119860 = radic⟨1198602⟩ minus ⟨119860⟩2


(4)




Δ120601 =Δ119860

10038161003816100381610038161003816119889 ⟨119860⟩ 119889120601

10038161003816100381610038161003816

=

radic1198731003816100381610038161003816sin120601

1003816100381610038161003816

1198731003816100381610038161003816sin120601

1003816100381610038161003816

=1

radic119873

(5)


Input Output

C

DB

A

BS1 BS2

120593

M2

M2








2




119873⟩ = (1radic2)(|0119873⟩ +

119890119894119873120601



⟨119861119873⟩ = ⟨120595

119873

10038161003816100381610038161003816119861119873

10038161003816100381610038161003816120595119873⟩ = cos119873120601 (6)

⟨1198612

119873⟩ = ⟨120595

119873

100381610038161003816100381610038161198612

119873

10038161003816100381610038161003816120595119873⟩ = 1 (7)

Δ119861119873= sin119873120601 (8)

Δ120601 =1

119873 (9)













119894120601(120579)= 119890119894119897120579 Here








119906119897119901 (119903 119911 120579) =

119862|119897|

119901

119908 (119911)(radic2119903

119908 (119911))

|119897|

119890minus11990321199082(119903)119871|119897|

119901(

21199032

1199082 (119903))

sdot 119890minus1198941198961199032119911(2(119911

2+1199112

119877))119890minus119894120579119897+119894(2119901+|119897|+1) arctan(119911119911

119877)

(10)


119908(119911) = 1199080radic1 + 119911119911

119877at 119911 119871120572


polynomials [64] 119911119903= 120587119908

2


0is



z


S

120579






infin

sum

119897119904119897119894=minusinfin

119870119897119904119897119894

120575 (119897119904+ 119897119894)1003816100381610038161003816119897119904 119897119894⟩ =

infin

sum


119870119897119904minus119897119904

1003816100381610038161003816119897119904 minus119897119904⟩ (11)





Input

Output




1003816100381610038161003816120595in⟩ =1



= |+119897⟩119863119887|minus119897⟩119863

119886

sdot119863119886⟨+119897|119863

119887⟨minus119897| + |minus119897⟩119863

119887|+119897⟩119863

119886

sdot119863119886⟨minus119897|119863

119887⟨+119897|

(13)


⟨⟩ = ⟨120595out1003816100381610038161003816100381610038161003816100381610038161003816120595out⟩ = cos2 (2119897120579)

⟨2⟩ = ⟨120595out

10038161003816100381610038161003816210038161003816100381610038161003816120595out⟩ = cos2 (2119897120579)

(14)



= cos (2119897120579) sin (2119897120579)

=1

2sin (4119897120579)

(15)


M1

M1

M2

M2

120579

DP

Da

Db

NLCSignal

Pump

Idler


CCBS1 BS2





Δ120579 =Δ

10038161003816100381610038161003816120597 ⟨⟩ 120597120579

10038161003816100381610038161003816

(16)


Δ120579 =1

2119873119897 (17)


Δ120579SQL =1

2radic119873119897

(18)
























1205872pulse

1205872pulse

120593







1003816100381610038161003816120595⟩ =1

radic2(10038161003816100381610038161205960 + Δ120596⟩signal

10038161003816100381610038161205960 minus Δ120596⟩idler

+10038161003816100381610038161205960 minus Δ120596⟩signal

10038161003816100381610038161205960 + Δ120596⟩idler)

(19)









5 Conclusions







Appendix











2

1003816100381610038161003816120595⟩ = (

1205951

1205952

sdot sdot sdot

) larrrarr ⟨1205951003816100381610038161003816 = (120595

lowast

1120595lowast



120601lowast

11205951+ 120601lowast

21205952+ 120601lowast





119862= |120595⟩

119860|120601⟩119861


119862= (1radic2)(|120595⟩

119860|120601⟩119861+ |120601⟩119860|120595⟩119861) The two



119862= (1radic2)(|120595⟩

119860|120601⟩119861+

|120601⟩119860|120595⟩119861) rarr |120595⟩



119860and






Δ119909Δ119901 geℏ

2 (A2)


Δ119864Δ119905 geℏ

2 (A3)





References

































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





















119894120601(120579)= 119890119894119897120579 Here








119906119897119901 (119903 119911 120579) =

119862|119897|

119901

119908 (119911)(radic2119903

119908 (119911))

|119897|

119890minus11990321199082(119903)119871|119897|

119901(

21199032

1199082 (119903))

sdot 119890minus1198941198961199032119911(2(119911

2+1199112

119877))119890minus119894120579119897+119894(2119901+|119897|+1) arctan(119911119911

119877)

(10)


119908(119911) = 1199080radic1 + 119911119911

119877at 119911 119871120572


polynomials [64] 119911119903= 120587119908

2


0is



z


S

120579






infin

sum

119897119904119897119894=minusinfin

119870119897119904119897119894

120575 (119897119904+ 119897119894)1003816100381610038161003816119897119904 119897119894⟩ =

infin

sum


119870119897119904minus119897119904

1003816100381610038161003816119897119904 minus119897119904⟩ (11)





Input

Output




1003816100381610038161003816120595in⟩ =1



= |+119897⟩119863119887|minus119897⟩119863

119886

sdot119863119886⟨+119897|119863

119887⟨minus119897| + |minus119897⟩119863

119887|+119897⟩119863

119886

sdot119863119886⟨minus119897|119863

119887⟨+119897|

(13)


⟨⟩ = ⟨120595out1003816100381610038161003816100381610038161003816100381610038161003816120595out⟩ = cos2 (2119897120579)

⟨2⟩ = ⟨120595out

10038161003816100381610038161003816210038161003816100381610038161003816120595out⟩ = cos2 (2119897120579)

(14)



= cos (2119897120579) sin (2119897120579)

=1

2sin (4119897120579)

(15)


M1

M1

M2

M2

120579

DP

Da

Db

NLCSignal

Pump

Idler


CCBS1 BS2





Δ120579 =Δ

10038161003816100381610038161003816120597 ⟨⟩ 120597120579

10038161003816100381610038161003816

(16)


Δ120579 =1

2119873119897 (17)


Δ120579SQL =1

2radic119873119897

(18)
























1205872pulse

1205872pulse

120593







1003816100381610038161003816120595⟩ =1

radic2(10038161003816100381610038161205960 + Δ120596⟩signal

10038161003816100381610038161205960 minus Δ120596⟩idler

+10038161003816100381610038161205960 minus Δ120596⟩signal

10038161003816100381610038161205960 + Δ120596⟩idler)

(19)









5 Conclusions







Appendix











2

1003816100381610038161003816120595⟩ = (

1205951

1205952

sdot sdot sdot

) larrrarr ⟨1205951003816100381610038161003816 = (120595

lowast

1120595lowast



120601lowast

11205951+ 120601lowast

21205952+ 120601lowast





119862= |120595⟩

119860|120601⟩119861


119862= (1radic2)(|120595⟩

119860|120601⟩119861+ |120601⟩119860|120595⟩119861) The two



119862= (1radic2)(|120595⟩

119860|120601⟩119861+

|120601⟩119860|120595⟩119861) rarr |120595⟩



119860and






Δ119909Δ119901 geℏ

2 (A2)


Δ119864Δ119905 geℏ

2 (A3)





References

































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










z


S

120579






infin

sum

119897119904119897119894=minusinfin

119870119897119904119897119894

120575 (119897119904+ 119897119894)1003816100381610038161003816119897119904 119897119894⟩ =

infin

sum


119870119897119904minus119897119904

1003816100381610038161003816119897119904 minus119897119904⟩ (11)





Input

Output




1003816100381610038161003816120595in⟩ =1



= |+119897⟩119863119887|minus119897⟩119863

119886

sdot119863119886⟨+119897|119863

119887⟨minus119897| + |minus119897⟩119863

119887|+119897⟩119863

119886

sdot119863119886⟨minus119897|119863

119887⟨+119897|

(13)


⟨⟩ = ⟨120595out1003816100381610038161003816100381610038161003816100381610038161003816120595out⟩ = cos2 (2119897120579)

⟨2⟩ = ⟨120595out

10038161003816100381610038161003816210038161003816100381610038161003816120595out⟩ = cos2 (2119897120579)

(14)



= cos (2119897120579) sin (2119897120579)

=1

2sin (4119897120579)

(15)


M1

M1

M2

M2

120579

DP

Da

Db

NLCSignal

Pump

Idler


CCBS1 BS2





Δ120579 =Δ

10038161003816100381610038161003816120597 ⟨⟩ 120597120579

10038161003816100381610038161003816

(16)


Δ120579 =1

2119873119897 (17)


Δ120579SQL =1

2radic119873119897

(18)
























1205872pulse

1205872pulse

120593







1003816100381610038161003816120595⟩ =1

radic2(10038161003816100381610038161205960 + Δ120596⟩signal

10038161003816100381610038161205960 minus Δ120596⟩idler

+10038161003816100381610038161205960 minus Δ120596⟩signal

10038161003816100381610038161205960 + Δ120596⟩idler)

(19)









5 Conclusions







Appendix











2

1003816100381610038161003816120595⟩ = (

1205951

1205952

sdot sdot sdot

) larrrarr ⟨1205951003816100381610038161003816 = (120595

lowast

1120595lowast



120601lowast

11205951+ 120601lowast

21205952+ 120601lowast





119862= |120595⟩

119860|120601⟩119861


119862= (1radic2)(|120595⟩

119860|120601⟩119861+ |120601⟩119860|120595⟩119861) The two



119862= (1radic2)(|120595⟩

119860|120601⟩119861+

|120601⟩119860|120595⟩119861) rarr |120595⟩



119860and






Δ119909Δ119901 geℏ

2 (A2)


Δ119864Δ119905 geℏ

2 (A3)





References

































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










M1

M1

M2

M2

120579

DP

Da

Db

NLCSignal

Pump

Idler


CCBS1 BS2





Δ120579 =Δ

10038161003816100381610038161003816120597 ⟨⟩ 120597120579

10038161003816100381610038161003816

(16)


Δ120579 =1

2119873119897 (17)


Δ120579SQL =1

2radic119873119897

(18)
























1205872pulse

1205872pulse

120593







1003816100381610038161003816120595⟩ =1

radic2(10038161003816100381610038161205960 + Δ120596⟩signal

10038161003816100381610038161205960 minus Δ120596⟩idler

+10038161003816100381610038161205960 minus Δ120596⟩signal

10038161003816100381610038161205960 + Δ120596⟩idler)

(19)









5 Conclusions







Appendix











2

1003816100381610038161003816120595⟩ = (

1205951

1205952

sdot sdot sdot

) larrrarr ⟨1205951003816100381610038161003816 = (120595

lowast

1120595lowast



120601lowast

11205951+ 120601lowast

21205952+ 120601lowast





119862= |120595⟩

119860|120601⟩119861


119862= (1radic2)(|120595⟩

119860|120601⟩119861+ |120601⟩119860|120595⟩119861) The two



119862= (1radic2)(|120595⟩

119860|120601⟩119861+

|120601⟩119860|120595⟩119861) rarr |120595⟩



119860and






Δ119909Δ119901 geℏ

2 (A2)


Δ119864Δ119905 geℏ

2 (A3)





References

































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















1205872pulse

1205872pulse

120593







1003816100381610038161003816120595⟩ =1

radic2(10038161003816100381610038161205960 + Δ120596⟩signal

10038161003816100381610038161205960 minus Δ120596⟩idler

+10038161003816100381610038161205960 minus Δ120596⟩signal

10038161003816100381610038161205960 + Δ120596⟩idler)

(19)









5 Conclusions







Appendix











2

1003816100381610038161003816120595⟩ = (

1205951

1205952

sdot sdot sdot

) larrrarr ⟨1205951003816100381610038161003816 = (120595

lowast

1120595lowast



120601lowast

11205951+ 120601lowast

21205952+ 120601lowast





119862= |120595⟩

119860|120601⟩119861


119862= (1radic2)(|120595⟩

119860|120601⟩119861+ |120601⟩119860|120595⟩119861) The two



119862= (1radic2)(|120595⟩

119860|120601⟩119861+

|120601⟩119860|120595⟩119861) rarr |120595⟩



119860and






Δ119909Δ119901 geℏ

2 (A2)


Δ119864Δ119905 geℏ

2 (A3)





References

































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















5 Conclusions







Appendix











2

1003816100381610038161003816120595⟩ = (

1205951

1205952

sdot sdot sdot

) larrrarr ⟨1205951003816100381610038161003816 = (120595

lowast

1120595lowast



120601lowast

11205951+ 120601lowast

21205952+ 120601lowast





119862= |120595⟩

119860|120601⟩119861


119862= (1radic2)(|120595⟩

119860|120601⟩119861+ |120601⟩119860|120595⟩119861) The two



119862= (1radic2)(|120595⟩

119860|120601⟩119861+

|120601⟩119860|120595⟩119861) rarr |120595⟩



119860and






Δ119909Δ119901 geℏ

2 (A2)


Δ119864Δ119905 geℏ

2 (A3)





References

































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















2

1003816100381610038161003816120595⟩ = (

1205951

1205952

sdot sdot sdot

) larrrarr ⟨1205951003816100381610038161003816 = (120595

lowast

1120595lowast



120601lowast

11205951+ 120601lowast

21205952+ 120601lowast





119862= |120595⟩

119860|120601⟩119861


119862= (1radic2)(|120595⟩

119860|120601⟩119861+ |120601⟩119860|120595⟩119861) The two



119862= (1radic2)(|120595⟩

119860|120601⟩119861+

|120601⟩119860|120595⟩119861) rarr |120595⟩



119860and






Δ119909Δ119901 geℏ

2 (A2)


Δ119864Δ119905 geℏ

2 (A3)





References

































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












References

































































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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