Quantizing the Electromagnetic Field II

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Quantizing the electromagnetic field II

Hyeran Kong Table of Contents

• Two flavors of quantum variables

• Physical Consequence of field quantization - Lamb shift, Quantum beat, Casimir(-Polder) effect

• Coherent state representation

• Optical equivalence Theorem Quantization with different quantum variables

1. Discrete quantum variables

Energy ; photon counting e.g. Single photon, Fock states ..

2. Continuous quantum variables

Phase and amplitude quadratures e.g. Squeezed state, Schrodinger cat state QuantizationPlane wave expansions in cube of volume with different quantum variables

1. DiscreteHomogeneous wave equaon from Maxwell eqn. quantum variables

complex analytical signal

Energy ; photon counng e.g. Single photon, Fock states ..

2. Continuous quantum variables

Phase and amplitude e.g. Squeezed state, Schrodinger cat state Quantizationeigenstate of operator with different quantum Using the commutaon relaon variables

1. Discrete quantum variables

eigenstate of Energy ; photon counng e.g. Single photon, Fock states ..

2. Continuous quantum variables creaon operator Phase and amplitude e.g. Squeezed state, Schrodinger cat state annihilaon operator

eigenvalue is quantized Quantization with different quantum variables

1. Discrete quantum variables Fock state representation

Energy ; photon counting e.g. Single photon, Fock states ..

2. Continuous quantum variables

PERSPECTIVES Phase and amplitude Coherent state e.g. Squeezed state, Schrodinger catNegative state values and positive W(Q,P) results. The Wigner function is a way W(Q,P) to describe how “quantum” a light A pulse is. Progressing from most clas- B sical to most quantum, the Wigner functions are shown for (A) the coherent state, (B) a squeezed state, (C) the single-photon state, and (D) a Schrödinger’s-cat state. The pro- Q P Q jections or “shadows” of the Wigner P function (shown on the sides) are the measured probability distributions of the quantum continuous variables Q or P. The Wigner function is a Gauss- W(Q,P) ian function for (A) and (B), but it W(Q,P) takes negative values for the strongly C quantum states (C) and (D). These D negative features vanish very quickly in the presence of decoherence. In the experiment by Lee et al., a quantum state similar to (C) and (D) was teleported and kept the negative Q P Q values of the Wigner function. This P result demonstrates an extraordinary degree of experimental control over such fragile quantum objects. on December 27, 2011

(see the figure, panel A). For a squeezed tions ( 10). Of particular interest here are can remain at the end (16 , 17 ): The cat state state, the peak is still Gaussian but now has superpositions of coherent states with oppo- must really be “erased” somewhere in order two-fold symmetry. One variance, say along site phases, which are called Schrödinger’s to be able to “reappear” elsewhere. P, is smaller than the vacuum noise, while kittens ( 11– 13) or Schrödinger’s cats ( 14, Overall, such an achievement is certainly the other one is larger (see the ﬁ gure, panel 15), depending on their size (see the ﬁ gure, very impressive, and it goes beyond pure B). For all optics experiments done during panel D). experimental virtuosity. It shows that the the last century, W(Q, P) was some kind of Such highly quantum states are desirable controlled manipulation of quantum objects

Gaussian and therefore looked like a “real” for effi cient quantum information process- has progressed steadily and achieved objec- www.sciencemag.org (positive) probability distribution. ing tasks, such as entanglement distillation tives that seemed impossible just a few years More surprisingly, the number state, in quantum communication, or as logical ago, and that tools are now available to where the number of photons in a light pulse gates for quantum computing. These tasks tackle more ambitious goals. is well defi ned, has a W(Q, P) function that involve many operations and are quite vul- is negative at the origin (see the fi gure, panel nerable to decoherence—the degradation of References and Notes C). Negative values are allowed because entanglement by unwanted coupling to the 1. N. Lee et al., Science 332, 330 (2011). 2. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, J. F. true probabilities are obtained by integrat- environment. A natural question is how well Valley, Phys. Rev. Lett. 55, 2409 (1985). ing these distributions—the integral over P highly quantum states can be controlled in a 3. A. Furusawa et al., Science 282, 706 (1998). Downloaded from gives the true probability distribution of Q. basic processing operation. 4. C. H. Bennett et al., Phys. Rev. Lett. 70, 1895 (1993). 5. D. Bouwmeester et al., Nature 390, 575 (1997). The integral of W(Q, P) over any component The quantum teleportation of a 6. D. Boschi, S. Branca, F. De Martini, L. Hardy, S. Popescu, of the electric fi eld is the probability distri- Schrödinger’s-cat state by Lee et al. suc- Phys. Rev. Lett. 80, 1121 (1998). bution of its orthogonal component (these cessfully combined many operations. They 7. A. I. Lvovsky et al., Phys. Rev. Lett. 87, 050402 (2001). 8. A. Ourjoumtsev, R. Tualle-Brouri, P. Grangier, Phys. Rev. appear as “shadows” in the fi gure panels). generated a highly entangled state—the Lett. 96, 213601 (2006). This shadow idea is behind the method resource for teleportation—and indepen- 9. V. Parigi, A. Zavatta, M. Kim, M. Bellini, Science 317, used to obtain W(Q, P) experimentally. dently a Schrödinger’s-kitten state, the one 1890 (2007). Many projections are measured so that to be teleported. Then, after the appropri- 10. A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, P. Grangier, Nat. Phys. 5, 189 (2009). W(Q, P) can be reconstructed by a pro- ate steps including the destruction of the 11. A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, P. Grangier, cess called quantum tomography. All of initial kitten, they “recreated” this kitten in Science 312, 83 (2006). the states with negative Wigner functions another place, still with negative features 12. J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, E. S. Polzik, Phys. Rev. Lett. 97, 083604 (2006). are called “nonclassical,” because their dis- in its Wigner function. These fi nal negative 13. H. Takahashi et al., Phys. Rev. Lett. 101, 233605 (2008). crete or continuous properties (or both) are features can only be observed if the qual- 14. A. Ourjoumtsev et al., Nature 448, 784 (2007). now mixed and are purely quantum features. ity of the teleportation is high enough. This 15. T. Gerrits et al., Phys. Rev. A 82, 031802(R) (2010). Many states with negative Wigner functions quality is measured by a number, called the 16. F. Grosshans, P. Grangier, Phys. Rev. A 64, 010301(R) (2001). have recently been realized experimentally, fi delity, which must be greater than 2/3 for 17. M. Ban, Phys. Rev. A 69, 054304 (2004). including number states with one ( 7) or two the operation to be successful. This 2/3 value 18. We thank A. Ourjoumtsev for help with the fi gures. photons ( 8), photon-added states ( 9), and is the so-called no-cloning limit, which also

entangled states with negative Wigner func- ensures that no other copy of the initial state 10.1126/science.1204814 OURJOUMTSEV ALEXEI CREDIT:

314 15 APRIL 2011 VOL 332 SCIENCE www.sciencemag.org Published by AAAS Quantization with different quantum variables

• Phase ?

does not satisfy uncertainty principle

is not Hermitian

do not commute

commute !

commute, orthogonal Quantization with different quantum variables

vacuum

Input2 D5 BS

Input1 D3

vacuum

J.W. Noh, A. Fougeres, L. Mandel, Phys. Rev. Lett. 67, 1467 (1991). Quantization with different quantum variables

vacuum

Input2 In the regime, D5 BS Phase diﬀerence ill deﬁned

Input1 D3

vacuum

J.W. Noh, A. Fougeres, L. Mandel, Phys. Rev. Lett. 67, 1467 (1991). Lamb Shift

• “Experimental Result”

Energy diﬀerence between by level shi

• Vacuum Fluctuation Theory (by Welton, 1957)

W E. Lamb, JR., and R C. Rutherford, Phys. Rev. Lett. 72,241 (1947) W E. Lamb, JR. Nobel Lecture, (1955) Lamb Shift

• “ExperimentalElectron Result” orbit perturbed by vacuum fluctuation perturbation Energy diﬀerence between by level shi

Non-relativistic eqn. of motion for electron

• Vacuum Fluctuation Theory (by Welton, 1957)

~1GHz in good agreement with Lamb shift measurement

W E. Lamb, JR., and R C. Rutherford, Phys. Rev. Lett. 72,241 (1947) W E. Lamb, JR. Nobel Lecture, (1955) Quantum beat

• Semiclassical Theory (SCT)

configuration configuration

Interference term Quantum beat

• Quantum mechanics

configuration configuration

S. Haroche, J.A. Paisner, A.L. Schawlow Phys. Rev. Lett. 30, 948 (1973). Casimir effect

Casimir effect manifested over different scales.

H.B.G. Casimir, and D. Polder, Phys. Rev. Lett. 73, 360 (1947). V. Sandoghdar, C. I. Sukenik, E. A. Hinds, S. Haroche, Phys. Rev. Lett. 68, 3432 (1992).

4.2. FOCK STATES 31

Problem 4.4 The annihilation operator is deﬁned as follows:

1 aˆ = Xˆ + iPˆ ; (4.11) √2 ! " 4 Representations of the Electromagnetic Field The operatora ˆ† is called the creation operator. Show that: 4.1 Basics a) the creation operator is A quantum mechanical state is completely described1 by its density matrix Ω. The density matrix Ω can be expanded in diaˆÆerent† = basisesXˆ √ i:Pˆ ; (4.12) √2 {|−i} Ω = D √ √ for a discrete! set of basis" states (158) ij | ii j i,j b) the creationTwo and annihilationX flavors≠ operatorsØ of arequantum not Hermitian; Ø The c-number matrix Dij = √i Ω √j is a representation of the density operator. c) their commutator is variablesh | One example is the number stateØ orÆ Fock state representation of the density ma- 1. Energy Discrete quantum variablesØ photon counting trix. [ˆa, aˆ†] = 1; (4.13)

Ω = P n m (159) d) positionIn andFock-state momentum representation can benm expressednm | asih | n,m X The diagonal elements in the Fock representation give the probability to find exactly 1 34 1 A. I. Lvovsky. Nonlinear and Quantum Optics n photons in the field! Xˆ = aˆ +â† ; Pˆ = aˆ aˆ† ; (4.14) Based on the data obtained in this measurement, we can construct a histogram of the experimental √2 results, a.k.a. marginal distribution pr(iX√θ). This2 marginal− distribution is the integral projection 2. Fields (phase-amplitude) Continuousof the phase-space quantum probability variables density on the vertical plane oriented at angle θ with respect to the A trivial example is the density matrix#vertical of a axis Fock (Fig.$ 4.2): State k : # $ | + i ∞ pr(X )= W (X cos θ P sin θ,Xsin θ + P cos θ)dP. (4.27) θ − e) the Hamiltonian can be writtenΩ = k k as and Ω = ±nk±km!−∞ (160) quadratures | ih | with nm W() X,P Another example is the density matrix ofˆ a thermal state of a1 free singlepr(X modeq ) field: e.g., WignerH function= ! ωfor aˆ†aˆ + ; (4.15) + squeezedexp[ states~! a a/kbT ] 2 q Ω = ° + % & X (161) Tr exp[ ~!a a/kbT ] P { ° } f) theL. Mandel., following and E. commutatorWolf, “Optical Coherence relations and Quantum hold: FigureOptics” 4.2: The phase-space probability density and the marginal distribution.

Now let us suppose the same measurement of Xˆθ is performed on a quantum particle in state In the Fock representation the matrix is:with density matrixρ ˆ that is prepared anew after each measurement. Again, we can construct the histogram, which is related to the state according to [â, aˆ†aˆ]=â;[â†, aˆ†aˆ]= aˆ†. (4.16) ˆ Ω = exp[ ~!n/kbT ][1 exp( ~!/kbT )] pr(nXθ)=Tr(ˆn ρXθ). (162) (4.28) ° ° ° | ih −| n In the quantum domain, there can exist no phase-space probability density because, according to X n the uncertainty principle, the particle cannot have certain values of position and momentum at the n same time. However, for every quantum state there exists a phase-space quasiprobability density — = h i na functionn Wρˆ(X, P) for which Eq. (4.27) holds for all θ’s. Without derivation,(163) the expression for (1 + n )n+1 | thisih quasiprobability| density (now known as the Wigner function) is as follows: 4.2 Fock states n h i + 1 ∞ Q Q X W (X, P)= eiP Q X ρˆ X + dQ. (4.29) ρˆ 2π − 2 2 !−∞ " # # $ with # # # # + This equation is called the Wigner formula1 . # # Our next goal is to find the eigenvaluesn = Tr(a aΩ) and = [exp( eigenstates~!/kbT ) 1] of° the Hamiltonian.(164) Because of Eq. (4.15) h i Problem 4.18 Prove Eq. (4.27)° for the classical case. the latter are also eigenstates ofa ˆ†aˆ. Problem 4.19 Use a mathematical software package to calculate the Wigner function and verify that37 Eq. (4.27) holds for θ = 0 and θ = π/2 for the following states: a) vacuum state; Problem 4.5 Suppose some state n is anb) coherent eigenstate state with α = 2; of the operatorn ˆ =â†aˆ [called the (photon) number operator] with eigenvalue n.Then| ⟩ c) coherent state with α =2eiπ/6; d) the single-photon state; e) the ten-photon state; f) ( 0 + 1 )/√2; a) the statea ˆ n is also an eigenstate ofa ˆ|†⟩aˆ| ⟩with eigenvalue n 1; g) ( 0 + i 1 )/√2; | ⟩ | ⟩ | ⟩ − h) state with the density matrix ( 0 0 + 1 1 )/2; | ⟩⟨ | | ⟩⟨ | b) the statea ˆ† n is also an eigenstate ofa ˆ†aˆ with eigenvalue n + 1. | ⟩ Hint: Use Eq. (4.16).

Note 4.1 The above exercise shows that the statesa ˆ n anda ˆ† n are proportional to normalized states n 1 and n +1 , respectively. In the following,| ⟩ we ﬁnd the| ⟩ proportionality coeﬃcient. | − ⟩ | ⟩

Problem 4.6 Using n aˆ†aˆ n = n, show that ⟨ | | ⟩ a) aˆ n = √n n 1 ; (4.17) | ⟩ | − ⟩ b) aˆ† n = √n +1 n +1 ; (4.18) | ⟩ | ⟩ VOLUME 10, NUMBER 7 PHYSICAL REVIEW LETTERS 1 APRIL 1963

52, 337 {1962). '4L. Mandel, Proceedings of the Third Symposium on ~L. Mandel, J. Opt. Soc. Am. 52, 1337,1408 (1962). Quantum Electronics, Paris, February, 1963 {to be ~O. S. Heavens, Suppl. Appl. Opt. 1, 4 (1962). published) . SW. Bothe, Z. Physik 41, 345 (1927). ~~R. J. Glauber, Proceedings of the Third Symposium SK. M. Purcell, Nature 178, 1449 (1956). on Quantum Electronics, Paris, February, 1963 (to ~OL. Mandel, Proc. Phys. Soc. (London} 72, 1037 be published). (1958); 74, 233 (1959). ~SL. Mandel, J. Opt. Soc. Am. 51, 797 (1961). E. Wolf, Proc. Phys. Soc. (London} 76, 424 (1960). ~ Since this was written, E. C. G. Sudarshan tPhys. 2C. T. J. Alkemande, Physica 25, 1145 (1959}. Rev. Letters 10, 277 {1963)]has demonstrated that ~3E. Wolf, Proceedings of the Third Symposium on the semiclassical and the quantum mechanical de- Quantum Electronics, Paris, February, 1963 (to be scriptions are completely equivalent to each other as published) . long as no nonlinear effects are considered.

EQUIVALENCE OF SEMICLASSICAL AND QUANTUM MECHANICAL DESCRlPTjONS OF STATISTICAL LIGHT BEAMS E. C. G. Sudarshan Department of Physics and Astronomy, University of Rochester, Rochester, New York (Received 1 March 1963)

With the advent of the laser, attention has been tation are focused on the problem of the complete descrip- (y(m), aq(n)) = In5 tion of the electromagnetic field associated with arbitrary light beams. The classical theory of optical coherence' works almost exclusively with (q(m), a~y(n)) = (n+ 1)"'5 two-point correlations; and this theory is adequate myel + ~ for the description of the classical optical phenom- One could, however, introduce an overcomplete ena of interference and diffraction in general. set of eigenstates of the destruction operator More sophisticated experiments on intensity in- given by terferometry and photoelectric counting statistics necessitated special higher order correlations. Ire ) =— Iz) =exp(- iz I') »g(n), Most of this work' was done usingVOLUMEa classical10, NUMBERor 7 PHYSICAL REVIEW n=OLETTERSQ, 1 APRIL 1963 a semiclassical formulation of the problem. On 52, 337 {1962). '4L. Mandel, Proceedings of the Third Symposium on the other hand, statistical states ~L.of aMandel,quantizedJ. Opt. Soc. Am.satisfying52, 1337,1408 (1962). Quantum Electronics, Paris, February, 1963 {to be (electromagnetic) field have been ~O.consideredS. Heavens, re-Suppl. Appl. Opt. 1, 4 (1962). published) . ~~R. = SW. a iz) =z I (z lat =z*(zJ. Glauber,I; (z Iz)Proceedings of the Third Symposium cently, ' and a quantum mechanical definitionBothe, Z. Physikof 41, 345 (1927). z); l, SK. M. Purcell, Nature 178, 1449 (1956). on Quantum Electronics, Paris, February, 1963 (to coherence functions of arbitrary ~OL.orderMandel,presented.Proc. Phys. Soc. (London} 72, 1037 be published). It is the aim of this note to elaborate(1958);on74, this233 (1959).defi- for every complex number~SL.z.Mandel,TheseJ. Opt.statesSoc.areAm. 51, 797 (1961). E. Wolf, Proc. Phys. Soc.all(London}normalized76, 424 (1960).but not orthogonal',~ Since this wastheywritten,are E.corn-C. G. Sudarshan tPhys. nition and to demonstrate its complete2C. T. equivalenceAlkemande, Physica 25, 1145 (1959}. Rev. Letters 10, 277 {1963)]has demonstrated that J. plete in the sense that they furnish a resolution to the classical description as long~3E.OpticalasWolf,no Proceedingsnon- of the Third Symposium on equivalencethe semiclassical and the quantum mechanical de- theorem linear effects are considered. Quantum Electronics, Paris, ofFebruary,the identity1963 (to be scriptions are completely equivalent to each other as published) . long as no nonlinear effects are considered. We begin with an outline of the analytic function = representation~ of canonical creationVOLUMEand10, NUMBERdestruc-7 PHYSICALl (l/w) REVIEWjfrdrd& I re LETTERS) (re' 1 APRIL 1963 tion operators. If a and a~ satisfy the relations More generally, '4L. Mandel, Proceedings of the Third Symposium on 52, 337 {1962).EQUIVALENCE OF SEMICLASSICAL AND QUANTUM MECHANICAL DESCRlPTjONS Quantum Electronics, Paris, February, 1963 {to be = ~L. Mandel, J. Opt. Soc. Am. 52, 1337,OF 1408STATISTICAL(1962). LIGHT BEAMS [a, a t] 1, ~O. d8 published) .y S. Heavens, Suppl. Appl. Opt. 1,—4Ire'(1962).)(re' I=e g(n)gt(n). SW. Bothe, Z. Physik 41, 345 (1927).2n ~~R. J. Glauber, Proceedings of the Third Symposium every irreducible representation is equivalent to ~ E. C. G. Sudarshan~g,OPg! SK. M. Purcell,DepartmentNature 178,of Physics1449 (1956).and Astronomy, Universityon Quantumof Rochester,Electronics,Rochester,Paris, NewFebruary,York 1963 (to the Fock representation in terms ~OL.of Mandel,the statesProc. Phys. Soc. (London} 72, (Received1037 1 Marchbe published).1963) (1958); 74, 233 (1959). ~SL. Mandel, J. Opt. Soc. Am. 51, 797 (1961). g(n), satisfying ~ Soc. (London} 424 (1960). Since this was written, E. C. G. Sudarshan tPhys. E.WithWolf,the Proc.adventPhys.of the laser,We canattentionmake76, hasusebeenof the overcompleteness'tation are of the 2C. T. J. Alkemande, Physica 25, 1145 (1959}. Rev. Letters 10, 277 {1963)]has demonstrated that focused on the problem ofstatesthe completeto representdescrip- ~3E. Wolf, Proceedings of the Third Symposium on everythe densitysemiclassicalmatrix,(y(m),and aq(n))the quantum= In5 mechanical de- a'tap(n) =ng(n); (y(rn), g(n))tion =of5 the electromagnetic field associated with Quantum mnElectronics, Paris, February, 1963 (to be scriptions are completely equivalent to each other as published)arbitraryVoLvMz light10, NvMsERbeams. 7The classical PHYSICALtheory of long REVIEWas no nonlinear effectsLETTERSare considered. 1 APRIL 1963 . VoLvMz 10, NvMsER 7 n')jc(n)y~(n'),PHYSICAL REVIEW LETTERS 1 APRIL 1963 optical coherence' works almost exclusively withQ p(n, The matrix elements of a and af in this represen- n =On~ =0 (q(m), a~y(n)) = (n+ 1)"'5 two-point and this theory is adequate myel + ~ in the "diagonal"correlations;form for the description of the inclassicalthe "diagonal"optical phenom-form n+n'One could, however, introduce an overcomplete ena of interference and diffraction in general. ' "~ n+n'277 EQUIVALENCE= OF SEMICLASSICALn') AND QUANTUMset of eigenstatesMECHANICALexp[r'+i(n'-n)&jire'~)(re'of the DESCRlPTjONSdestruction' "~ operator More sophisticatedp experimentsp(n, on intensity= in- n') exp[r'+i(n'-n)&jire'~)(re' OFpSTATISTICALp(n,LIGHTgiven byBEAMS terferometry and photoelectric counting statistics ) ) E. C. G. Sudarshan necessitatedThis formspecialis particularlyhigher order interestingcorrelations. since if =— Department of PhysicsThis andformAstronomy,is particularlyUniversity ofinterestingIreRochester,)quantumIz) =exp(-Rochester,sincemechanicalizif I')New Yorkdensity»g(n), matrices. Hermitic- Most= of this work' was done using a classical or n=OQ,quantum mechanical density Hermitic- 0 (a~)"aP' be any normal= ordered(Receivedoperator1 March(i. e.1963), matrices. a semiclassical formulation0 of(a~)"aP'the problem.be any Onnormal orderedityoperatorof p implies(i. e. ,that p(z) is a "real" function in all creation operators to the left of all annihilation ity of p implies that p(z) is a "real" function in the other hand, statistical allstatescreationof a quantizedoperators to satisfyingthe left oftheallsenseannihilationthat p*(z*)= but not necessarily With the advent of the laser, attention has been tation are theP(z),sense that p*(z*)= but not necessarily operators), itsfieldexpectationhave been valueconsideredin there-statistical P(z), (electromagnetic) operators), its expectation value positivein the statisticaldef inite. = focused on the problem of the complete descrip- a iz) =z I (z lat =z*(z I; (z Iz) cently,state 'representedand a quantum bymechanicalthe densitydefinitionmatrixof in the (y(m),z); aq(n)) = In5 positive l,def inite. tion of the electromagnetic statefield representedassociated withby the density matrixTheseinconsiderationsthe Glauber-Sudarshangeneralize in a straight- function coherence"diagonal"functionsform of arbitrary order presented. These considerations generalize in a straight- arbitrary light beams. The"diagonal"classical theoryform of for forward numbermanner to an arbitrary (countable) num- It is the aim of this note to elaborate on this defi- every complex z. Theseforward•states Universalmannerare characterto an arbitrary of all (countable)states of EMnum- fields optical coherence' works= almost exclusively with all normalized buta~y(n))not orthogonal',= (n+ 1)"'5 they are corn- I ber(q(m),of degrees of freedom, finite or infinite. nition and to demonstratep Jd'z4its(z)completeiz) (z equivalence= myel• +Quasi-probability~ density in the “diagonal” basis two-point correlations; and this theory is adequatep Jd'z4 (z) iz) (zin I the that furnish ber of degrees of freedom, finite or infinite. toisthegivenclassicalby description as long as no non- plete Thesensestates theyare now representeda resolution by a sequence of for the description of the classicalis given opticalby phenom- Oneof thecould,identityhowever, introduce an Theovercompletestates are now represented by a sequence of enalinearof interferenceeffects are considered.and diffraction in general. complex numbers {z and the Fock representa- = p(aT)"a&& = fd'zy(z)(z*)"z P. set(5)of eigenstates of the destructioncomplex};operator numbers and the Fock representa- MoreWe sophisticatedbegintr{po}withtr{an experimentsoutline of the onanalyticintensity= functionp(aT)"a&&in- = fd'zy(z)(z*)"ztion basisP. is labeled {z}; tr{po} tr{ given by = (5) by a sequence of non-negative representation~ of canonical creation and destruc- l (l/w) jfrdrd& I re (re'tion basis is labeled by a sequence of non-negative terferometryThis is theandsamephotoelectricas the expectationcounting statisticsvalue of the integers and density) matrices functions tion operators. If a and a~Thissatisfyis thethe samerelations the value {n}of the integers andby density matrices functions necessitated special higher order correlations.as expectation=— {n} by function (z*)"zP a MoreIregenerally,) ofIz)two=exp(-such izsequencesI') »g(n), Any such Mostcomplexof this classicalwork' was done using a classicalforor probabi- P n=OQ, p({n},{n'}). complex= classical function (z*)"z for a probabi- of two such sequences p({n},{n'}). Any such lity distribution [a,q(z)a t] over1, the complex Onplane. The d8 state can be put intoy one-to-one correspondence a semiclassical formulationlityofdistributionthe problem. q(z) over the complex—Ire' )(re'plane.I=e The stateg(n)gt(n).can be put into one-to-one correspondence the other hand, statistical states of a quantized satisfying~ 2n with classical probability~g,OPg! distributions in se- everydemonstrationirreducible representationabove shows isthatequivalentany statisticalto with a (electromagnetic) field havedemonstrationbeen consideredabovere- shows that any statistical classical probability distributions in a se- thestateFockofrepresentationRepresentationthe quantum inmechanicalterms of the systemstates of quantumbe quence statesof complex in thevariables coherent-state= such that the basis may a iz) =z I z); (z lat =z*(z I; (z Iz) l,P({z}) cently, ' and a quantum mechanicalstate of thedefinitionquantumof mechanical system may be quence of complex variables P({z})such that the g(n),describedsatisfyingby a classical probability distribution expectation value of any normal ordered operator coherence Predictionfunctions of arbitrarydescribed oforder genuinebypresented.a classical Weprobabilityquantumcan make usedistributionof effects,the overcompleteness'expectation whichof the valuecannotof any normal be describedordered operator classically. It overis the aaimcomplexof this noteplane,to elaborateprovided onallthisoperatorsdefi- forstatesareeveryto representcomplex0({at},number{a})every isdensityz.givenThesematrix,by states are a'tap(n) =ng(n); (y(rn),over g(n))a complex= 5 plane, provided all operators are 0({at},{a})is given by nitionwrittenand toindemonstratethe normal itsorderedcompleteform.mnequivalenceIn otherall normalized but not orthogonal', they are corn- written in the normal orderedplete inform.the senseInthatotherthey furnish a resolution E. C.to G.words,the classicalSudarshanthe classicaldescription, Phys.complexas long asRev.representations'no non- Lett. 10, 277 (1963)trOa,Q p(n,a n')jc(n)y~(n'),p = d'z 0 z*, z P z The matrix elements of awords,and af inthethisclassicalrepresen- complexof the identityrepresentations'n =On~ =0 trOa, a p = d'z 0 z*, z P z R. J.linear Glaubercan beeffectsput ,arein Phys.one-to-oneconsidered. Rev.correspondence 131, 2766 with(1963). We begin with an outline canof thebe analyticput in one-to-onefunction correspondencewhere thewith"real" function p({z})is given by = where the 277"real" function is given representation~ of canonical creation and destruc- l (l/w) jfrdrd& I re ) (re' p({z}) by tion operators. If a and a~ satisfy the relations !n '!)"' p({n},{n'})(n More iE ~A. generally,p({n},{n'})(n !n '!)"' ~A. ~({})=n exp[r '+i(n '-n )0 ] &(r ) iE z= z~ exp[r '+i(n '-n )0 [a, a t] =On1, =O~({})=n(n +n ')!(2vr ) d8 y ] &(r ) z=On z~ =O—Ire'(n +n)(re'')!(2vrI=e ) g(n)gt(n). every irreducible representation is equivalent to ~ 2n ~g,OPg! theConsequentlyFock representationthe descriptionin terms of ofthestatisticalstates states g(n), satisfying Consequently the description of statisticaltions correspondingstates to a density matrix diagonal of a quantum mechanical system with an arbitraryWe can make use of the overcompleteness'tions correspondingof the to a density matrix diagonal of a quantum mechanical system within theanoccupationarbitrary numbers {n}given by the grand (countably infinite) number of degrees of freedomstates to represent every density matrix,in the occupation numbers {n}given by the grand a'tap(n) =ng(n); (y(rn),(countablyg(n)) = 5 infinite) number of degreescanonicalof freedomensemble for the blackbody radiation, is completely equivalent to the mndescription in terms canonical ensemble for the blackbody radiation, is completely equivalent to the descriptionthere areQ inp(n,othertermsn')jc(n)y~(n'),probability functions! A particu- Theof matrixela, ssicalelementsprobabilityof a and distributionsaf in this represen-in the same n =On~ =0 there are other probability functions! A particu- of ela, ssical probability distributionslar onein themaysamenot be diagonal in the occupation (countably infinite) number of complex variables. lar one may not be diagonal in the occupation (countably infinite) number of complexnumbervariables.sequence {n}and this implies, in accord- In particular, the statistical states of the quantized number sequence277 and this implies, in accord- In particular, the statistical statesanceof thewithquantizedEq. (6), that not all phase-angle{n} se- electromagnetic field may be described uniquely ance with Eq. (6), that not all phase-angle se- electromagnetic field may be describedquencesuniquely{8}have equal weight. In such a case by classical complex linear functions on the clas- quences {8}have equal weight. In such a case by classical complex linear functionsthe expectationon the clas-values of operators with unequal sical electromagnetic field. This functional will the expectation values of operators with unequal sical electromagnetic field. Thisnumberfunctionalof creationwill and destruction operators be "real" reflecting the Hermiticity of the density number of creation and destruction operators be "real" reflecting the Hermiticityneedof notthe alldensityvanish. %e note in passing, that matrix; and leads in either version to real ex- need not all vanish. %e note in passing, that matrix; and leads in either versionEq to(6)realforex-.P({z})in terms of p({n},{n'})ean be pectation values for Hermitian (real) dynamical Eq (6) for. in terms of ean be pectation values for Hermitian inverteddynamicalto yield P({z}) p({n},{n'}) variables. (real) inverted variables. to yield Several additional remarks are in order. First- Several additional remarks are in order. First- ly, since the states i{z})are eigenfunctions of ly, since the states i{z})are eigenfunctions of i)3/2 the a,nnihilation operators, the analogous states A. i)3/2 the a,nnihilation operators, the analogous states A. for a quantized field are eigenfunctions of the If we do this for the Gaussian functions, we ob- for a quantized field are eigenfunctions of the If we do this for the Gaussian functions, we ob- positive-frequency (annihilation) part of the field. tain the Bose-Einstein distribution, diagonal in positive-frequency (annihilation) part of the field. tain the Bose-Einstein distribution, The corresponding classical theory should then the occupation number sequences corresponding diagonal in The corresponding classical theory should then the occupation number sequences work with positive-frequency parts of the elassi- to the equal weightage of all phase angles. It is corresponding work with positive-frequency parts of the elassi- to the equal weightage of all ca.l field; but this is precisely what is involved worth pointing out that this result reproduces the phase angles. It is ca.l field; but this is precisely' what is involved worth pointing out in the concept of the (classical) analytic signal. Purcell-Mandel derivation' for photoelectricthat this result reproduces the in the concept of the (classical) analytic signal. ' Purcell-Mandel derivation' for Secondly, while therma, l beams are usually rep- counting statistics. The method of inverting the photoelectric Secondly, while therma, l beams are usually rep- counting statistics. The method resented by Gaussian classical probability func- expectation values to obtain the probability dis- of inverting the resented by Gaussian classical probability func- expectation values to obtain the probability dis- VOLUME 10, NUMBER 7 PHYSICAL REVIEW LETTERS 1 APRIL 1963

With the advent of the laser, attention has been tation are focused on the problem of the complete descrip- (y(m), aq(n)) = In5 tion of the electromagnetic field associated with arbitrary light beams. The classical theory of optical coherence' works almost exclusively with (q(m), a~y(n)) = (n+ 1)"'5 two-point correlations; and this theory is adequate myel + ~ for the description of the classical optical phenom- One could, however, introduce an overcomplete ena of interference and diffraction in general. set of eigenstates of the destruction operator More sophisticated experiments on intensity in- given by terferometry and photoelectric counting statistics necessitated special higher order correlations. Ire ) =— Iz) =exp(- iz I') »g(n), Most of this work' was done usingVOLUMEa classical10, NUMBERor 7 PHYSICAL REVIEW n=OLETTERSQ, 1 APRIL 1963 a semiclassical formulation of the problem. On 52, 337 {1962). '4L. Mandel, Proceedings of the Third Symposium on the other hand, statistical states ~L.of aMandel,quantizedJ. Opt. Soc. Am.satisfying52, 1337,1408 (1962). Quantum Electronics, Paris, February, 1963 {to be (electromagnetic) field have been ~O.consideredS. Heavens, re-Suppl. Appl. Opt. 1, 4 (1962). published) . ~~R. = SW. a iz) =z I (z lat =z*(zJ. Glauber,I; (z Iz)Proceedings of the Third Symposium cently, ' and a quantum mechanical definitionBothe, Z. Physikof 41, 345 (1927). z); l, SK. M. Purcell, Nature 178, 1449 (1956). on Quantum Electronics, Paris, February, 1963 (to coherence functions of arbitrary ~OL.orderMandel,presented.Proc. Phys. Soc. (London} 72, 1037 be published). It is the aim of this note to elaborate(1958);on74, this233 (1959).defi- for every complex number~SL.z.Mandel,TheseJ. Opt.statesSoc.areAm. 51, 797 (1961). E. Wolf, Proc. Phys. Soc.all(London}normalized76, 424 (1960).but not orthogonal',~ Since this wastheywritten,are E.corn-C. G. Sudarshan tPhys. nition and to demonstrate its complete2C. T. equivalenceAlkemande, Physica 25, 1145 (1959}. Rev. Letters 10, 277 {1963)]has demonstrated that J. plete in the sense that they furnish a resolution to the classical description as long~3E.OpticalasWolf,no Proceedingsnon- of the Third Symposium on equivalencethe semiclassical and the quantum mechanical de- theorem linear effects are considered. Quantum Electronics, Paris, ofFebruary,the identity1963 (to be scriptions are completely equivalent to each other as published) . long as no nonlinear effects are considered. We begin with an outline of the analytic function = representation~ of canonical creationVOLUMEand10, NUMBERdestruc-7 PHYSICALl (l/w) REVIEWjfrdrd& I re LETTERS) (re' 1 APRIL 1963 tion operators. If a and a~ satisfy the relations More generally, '4L. Mandel, Proceedings of the Third Symposium on 52, 337 {1962).EQUIVALENCE OF SEMICLASSICAL AND QUANTUM MECHANICAL DESCRlPTjONS Quantum Electronics, Paris, February, 1963 {to be = ~L. Mandel, J. Opt. Soc. Am. 52, 1337,OF 1408STATISTICAL(1962). LIGHT BEAMS [a, a t] 1, ~O. d8 published) .y S. Heavens, Suppl. Appl. Opt. 1,—4Ire'(1962).)(re' I=e g(n)gt(n). SW. Bothe, Z. Physik 41, 345 (1927).2n ~~R. J. Glauber, Proceedings of the Third Symposium every irreducible representation is equivalent to ~ E. C. G. Sudarshan~g,OPg! SK. M. Purcell,DepartmentNature 178,of Physics1449 (1956).and Astronomy, Universityon Quantum of Rochester,Electronics,An Rochester,extremelyParis, NewFebruary, Yorkimportant1963 (to aspect of these developments was the Optical Equivalence the Fock representation in terms ~OL.of Mandel,the statesProc. Phys. Soc. (London} 72, (Received1037 1 Marchbe published).Theorem,1963) first stated by George in Ref. [3] and elaborated in his essential book with John Klauder (1958); 74, 233 (1959). ~SL. Mandel, J. Opt. Soc. Am. 51, 797 (1961). g(n), satisfying ~ Soc. (London} 424 (1960). Since this was written, E. C. G. Sudarshan tPhys. E.WithWolf,the Proc.adventPhys.of the laser,We canattentionmake76, hasusebeenof the overcompleteness'tation[5].are Over the fiveof thedecades since then, the Optical Equivalence Theorem has provided the central 2C. T. J. Alkemande, Physica 25, 1145 (1959}. Rev. Letters 10, 277 {1963)]has demonstrated that focused on the problem ofstatesthe completeto representdescrip- ~3E. Wolf, Proceedings of the Third Symposium on everythe densitytoolsemiclassical inmatrix, (y(m), Quantumand aq(n))the quantum= In5 Opticsmechanical forde- distinguishing between classical and manifestly quantum (or a'tap(n) =ng(n); (y(rn), g(n))tion =of5 the electromagnetic field associated with Quantum mnElectronics, Paris, February, 1963 (to be scriptions are completely equivalent to each other as published)arbitraryVoLvMz light10, NvMsERbeams. 7The classical PHYSICALtheory of longnonclassical)REVIEWas no nonlinear effectsLETTERS regimesare considered. for the electromagnetic1 APRIL field.1963 . VoLvMz 10, NvMsER 7 n')jc(n)y~(n'),PHYSICAL REVIEW LETTERS 1 APRIL 1963 optical coherence' works almost exclusively withQ p(n, The matrix elements of a and af in this represen- n =On~ =0 (q(m),As revieweda~y(n)) = (n+ 1)"'5in the presentation by Professor Mehta, the density operator !ˆ for the two-point and this theory is adequate myel + ~ in the "diagonal"correlations;form for the description of the inclassicalthe "diagonal"optical phenom-form n+n'Oneelectromagneticcould, however, introduce field canan overcompletebe expressed in the form [3] ena of interference and diffraction in general. ' "~ n+n'277 EQUIVALENCE= OF SEMICLASSICALn') AND QUANTUMset of eigenstatesMECHANICALexp[r'+i(n'-n)&jire'~)(re'of the DESCRlPTjONSdestruction' "~ operator 2 More sophisticatedp experimentsp(n, on intensity= in- p(n, n') exp[r'+i(n'-n)&jire'~)(re'"#ˆ = dz() z z z . OFpSTATISTICAL)LIGHTgiven byBEAMS terferometry and photoelectric counting statisticsClassical domain:) A well-behaved! P-function can be E. C. G. Sudarshan necessitatedThis formspecialis particularlyhigher order interestingcorrelations. since if =— ()z Department of PhysicsThis andformAstronomy,is particularlyUniversityHere,ofinterestingIreRochester, )thequantumIz) function=exp(-Rochester,sincemechanicalizif I')!New York is,density»g(n), in modernmatrices. terms,Hermitic- called the Glauber-Sudarshan phase-space function Most= of this work' was done using a classical or n=OQ,quantum mechanical density Hermitic- 0 (a~)"aP' be any normal= ordered(Receivedoperatorinterpreted1 March(i. e.1963), as a probability density to matrices.find in a semiclassical formulation0 of(a~)"aP'the problem.be any Onnormal orderedityoperatorof p implies(i. e. ,that p(z) is a "real" function in all creation operators to the left of all annihilation[6]. The aspect of this functionity of p thatimplies Georgethat p(z)emphasizedis a "real" (andfunction formallyin established) is its universal the other hand, statistical allstatescreationof a quantizedoperators to satisfyingthe left oftheallsenseannihilationthat p*(z*)= but not necessarily With the advent of the laser, attention has been tation are theP(z),sense that p*(z*)= but not necessarily operators), itsfieldexpectationhave been valueconsideredin there- statistical character for all states of the vs. electromagnetic P(z), field, ranging from blackbody radiation, for which (electromagnetic) operators), its expectation value positivein the statisticaldef inite. = focused on the problem of the complete descrip- a iz) =z I (z lat =z*(z I; (z Iz) cently,state 'representedand a quantum bymechanicalthe densitydefinitionmatrixof in the ()z (y(m),z); aq(n)) = In5 positive l,def inite. tion of the electromagnetic statefield representedassociated withby the density! matrixis Thesea Gaussianinconsiderationsthe distribution,Glauber-Sudarshangeneralize to morein apathologicalstraight- function examples, such as a number (Fock) state |n> coherence functions of order Quantum domain: PathologicalThese considerations examplesgeneralize in ofa straight- P-functions such arbitrary"diagonal"light formbeams. arbitraryThe classical theorypresented.of "diagonal" form defi- forofevery the field,forwardcomplex fornumbermanner whichz. to Thesean arbitrary•states Universalare (countable) characternum- of all states of EM fields opticalIt is thecoherence'aim of thisworksnote toalmostelaborateexclusivelyon this with forward manner to an arbitrary (countable) num- = all normalized buta~y(n))not orthogonal',= (n+ 1)"'5 they are corn- 2 I ber(q(m),of degrees of freedom, finite or infinite. nition and to demonstratep Jd'z4its(z)completeiz) (z equivalence= as photon number-statemyel• + Quasi-probability~ ||zn density 2 in the “diagonal” basis two-point correlations; and this theory is adequateJd'z4 (z) iz) I ber of degrees of freedom, finite or infinite. p plete (zin the sense that they furnish a resolution e (2) fortoisthethegivenclassicaldescriptionby descriptionof the classicalas long opticalas no non-phenom- The states are now represented by a sequence !of is given by Oneof thecould,identityhowever, introduce an Theovercompletestates#()arezz=now represented by"a sequence. of enalinearof interferenceeffects are considered.and diffraction in general. complex numbers {z and the Fock representa-nn () = p(aT)"a&& = fd'zy(z)(z*)"z P. set(5)of eigenstates of the destructioncomplex};operator numbers nzz!*!!and the Fock representa- MoreWe sophisticatedbegintr{po}withtr{an experimentsoutline of the onanalyticintensity= functionp(aT)"a&&in- = fd'zy(z)(z*)"ztion basisP. is labeled {z}; tr{po} tr{ given by = (5) by a sequence of non-negative representation~ of canonical creation and destruc- l (l/w) jfrdrd& I re (re'tion basis is labeled by a sequence of non-negative terferometryThis is theandsamephotoelectricas the expectationcounting statisticsvalue of the integers and density) matrices functions tion operators. If a and a~Thissatisfyis thethe samerelations the value {n}of the integers andby density matrices functions necessitated special higher order correlations.as expectation=— {n} by function (z*)"zP a MoreIregenerally,) ofIz)two=exp(-such izsequencesI') »g(n), Any such Mostcomplexof this classicalwork' was done using a classicalforor probabi- P n=OQ, p({n},{n'}). complex= classical function (z*)"z for a probabi- of two such sequences p({n},{n'}). Any such lity distribution [a,q(z)a t] over1, the complex Onplane. The d8 state can be put intoy one-to-one correspondence a semiclassical formulationlityofdistributionthe problem. q(z) over the complex—Ire' )(re'plane.I=e The stateg(n)gt(n).can be put into one-to-one correspondence the other hand, statistical states of a quantized satisfying~ 2n with classical probability~g,OPg! distributions in se- everydemonstrationirreducible representationabove shows isthatequivalentany statisticalto with a (electromagnetic) field havedemonstrationbeen consideredabovere- shows that any statistical classical probability distributions in a se- thestateFockofrepresentationRepresentationthe quantum inmechanicalterms of the systemstates of quantumbe quence statesof complex in thevariables coherent-state= such that the basis may a iz) =z I z); (z lat =z*(z I; (z Iz) l,P({z}) cently, ' and a quantum mechanicalstate of thedefinitionquantumof mechanical system may be quence of complex variables P({z})such that the g(n),describedsatisfyingby a classical probability distribution expectation value of any normal ordered operator coherence Predictionfunctions of arbitrarydescribed oforder genuinebypresented.a classical Weprobabilityquantumcan make usedistributionof effects,the overcompleteness'expectation whichof the valuecannotof any normal be describedordered operator classically. It overis the aaimcomplexof this noteplane,to elaborateprovided onallthisoperatorsdefi- forstatesareeveryto representcomplex0({at},number{a})every isdensityz.givenThesematrix,by states are a'tap(n) =ng(n); (y(rn),over g(n))a complex= 5 plane, provided all operators are 0({at},{a})is given by nitionwrittenand toindemonstratethe normal itsorderedcompleteform.mnequivalenceIn otherall normalized but not orthogonal', they are corn- written in the normal orderedplete inform.the senseInthatotherthey furnish a resolution E. C.to G.words,the classicalSudarshanthe classicaldescription, Phys.complexas long asRev.representations'no non- Lett. 10, 277 (1963)trOa,Q p(n,a n')jc(n)y~(n'),p = d'z 0 z*, z P z The matrix elements of awords,and af inthethisclassicalrepresen- complexof the identityrepresentations'n =On~ =0 trOa, a p = d'z 0 z*, z P z R. J.linear Glaubercan beeffectsput ,arein Phys.one-to-oneconsidered. Rev.correspondence 131, 2766 with(1963). We begin with an outline canof thebe analyticput in one-to-onefunction correspondencewhere thewith"real" function p({z})is given by = where the 277"real" function is given representation~ of canonical creation and destruc- l (l/w) jfrdrd& I re ) (re' p({z}) by tion operators. If a and a~ satisfy the relations !n '!)"' p({n},{n'})(n More iE ~A. generally,p({n},{n'})(n !n '!)"' ~A. ~({})=n exp[r '+i(n '-n )0 ] &(r ) iE z= z~ exp[r '+i(n '-n )0 [a, a t] =On1, =O~({})=n(n +n ')!(2vr ) d8 y ] &(r ) z=On z~ =O—Ire'(n +n)(re'')!(2vrI=e ) g(n)gt(n). every irreducible representation is equivalent to ~ 2n ~g,OPg! theConsequentlyFock representationthe descriptionin terms of ofthestatisticalstates states g(n), satisfying Consequently the description of statisticaltions correspondingstates to a density matrix diagonal of a quantum mechanical system with an arbitraryWe can make use of the overcompleteness'tions correspondingof the to a density matrix diagonal of a quantum mechanical system within theanoccupationarbitrary numbers {n}given by the grand (countably infinite) number of degrees of freedomstates to represent every density matrix,in the occupation numbers {n}given by the grand a'tap(n) =ng(n); (y(rn),(countablyg(n)) = 5 infinite) number of degreescanonicalof freedomensemble for the blackbody radiation, is completely equivalent to the mndescription in terms canonical ensemble for the blackbody radiation, is completely equivalent to the descriptionthere areQ inp(n,othertermsn')jc(n)y~(n'),probability functions! A particu- Theof matrixela, ssicalelementsprobabilityof a and distributionsaf in this represen-in the same n =On~ =0 there are other probability functions! A particu- of ela, ssical probability distributionslar onein themaysamenot be diagonal in the occupation (countably infinite) number of complex variables. lar one may not be diagonal in the occupation (countably infinite) number of complexnumbervariables.sequence {n}and this implies, in accord- In particular, the statistical states of the quantized number sequence277 and this implies, in accord- In particular, the statistical statesanceof thewithquantizedEq. (6), that not all phase-angle{n} se- electromagnetic field may be described uniquely ance with Eq. (6), that not all phase-angle se- electromagnetic field may be describedquencesuniquely{8}have equal weight. In such a case by classical complex linear functions on the clas- quences {8}have equal weight. In such a case by classical complex linear functionsthe expectationon the clas-values of operators with unequal sical electromagnetic field. This functional will the expectation values of operators with unequal sical electromagnetic field. Thisnumberfunctionalof creationwill and destruction operators be "real" reflecting the Hermiticity of the density number of creation and destruction operators be "real" reflecting the Hermiticityneedof notthe alldensityvanish. %e note in passing, that matrix; and leads in either version to real ex- need not all vanish. %e note in passing, that matrix; and leads in either versionEq to(6)realforex-.P({z})in terms of p({n},{n'})ean be pectation values for Hermitian (real) dynamical Eq (6) for. in terms of ean be pectation values for Hermitian inverteddynamicalto yield P({z}) p({n},{n'}) variables. (real) inverted variables. to yield Several additional remarks are in order. First- Several additional remarks are in order. First- ly, since the states i{z})are eigenfunctions of ly, since the states i{z})are eigenfunctions of i)3/2 the a,nnihilation operators, the analogous states A. i)3/2 the a,nnihilation operators, the analogous states A. for a quantized field are eigenfunctions of the If we do this for the Gaussian functions, we ob- for a quantized field are eigenfunctions of the If we do this for the Gaussian functions, we ob- positive-frequency (annihilation) part of the field. tain the Bose-Einstein distribution, diagonal in positive-frequency (annihilation) part of the field. tain the Bose-Einstein distribution, The corresponding classical theory should then the occupation number sequences corresponding diagonal in The corresponding classical theory should then the occupation number sequences work with positive-frequency parts of the elassi- Figureto the1. Theequal Opticalweightage Equivalenceof all phase Theoremangles. Itprovidesis correspondinga definitive line of demarcation between work with positive-frequency parts of the elassi- to the equal weightage of all ca.l field; but this is precisely what is involved worth pointing out that this classicalresult reproduces and quantumthe phase worldsangles. [3, 5].It is ca.l field; but this is precisely' what is involved worth pointing out in the concept of the (classical) analytic signal. Purcell-Mandel derivation' for photoelectricthat this result reproduces the in the concept of the (classical) analytic signal. ' Purcell-Mandel derivation' for Secondly, while therma, l beams are usually rep- counting statistics. The method of inverting the photoelectric Secondly, while therma, l beams are usually rep- counting statistics. The method resented by Gaussian classical probability func-As youexpectation can see andvalues as Professorto obtain the Mehtaprobability discussed,dis- this ofis invertingnot an ordinarythe function. As Dorothy in resented by Gaussian classical probability func- the Wizard of Oz expressedexpectation so appropriately,values to obtain we’rethe probability certainly dis- not in Kansas anymore! For an

2 2 A model-independent line between classical and manifestly quantum domains

Manifestly Quantum Domain Nonclassical Light Classical Domain

Number states Blackbody radiation Squeezed states Radio waves Cell phone emissions EPR states P()α < 0P()α ≥ 0 “Twin” beams Laser light … …

provides the basis for a No classical description of with classical statistical theory of the EM field positive probability distribution

for the field amplitudes =probability distribution for field amplitudes

Glauber-Sudarshan Phase-Space Function P (α ) ραααˆ = ∫ dzP2 () Coherent State Representation

Fock state representation

normalize factor

Probability follows Poisson distribution

annihilation without changing state ! Coherent State Representation

• Displacement operator

displaced vacuum state

unitaryQ operator orthogonal coherent state to another coherent state displaced

phase P Coherent State Representation

• Uncertainty Relation for coherent state

definite complex amplitude coherent state

squeezed state

“quantum” and “classical” Q Q squeezed state Minimum uncertainty ? coherent state Unitary operator

phase phase P P

Optical equivalence theorem

Glauber-Sudarshan P-representation for “all” EM field

phase space

Non-orthogonality detection

Completeness What is inside ?

Observable

Optical equivalence theorem Optical equivalence theorem

Photon number fluctuation – Mandel Q parameter

at, (classical probability density)

2nd order coherence

for all classical EM fields

anti-bunching Quasi-probability density

Can’t be explained by classical statistics 34 A. I. Lvovsky. Nonlinear and Quantum Optics

Based on the data obtained in this measurement, we can construct a histogram of the experimental results, a.k.a. marginal distribution pr(Xθ). This marginal distribution is the integral projection of the phase-space probability density on the vertical plane oriented at angle θ with respect to the vertical axis (Fig. 4.2):

+ ∞ pr(X )= W (X cos θ P sin θ,Xsin θ + P cos θ)dP. (4.27) θ − !−∞

W() X,P Glauber-Sudarshan pr(Xq ) P-representation q Quasi-probability density function X P from with coherent state Figure 4.2: The phase-space probability density and the marginal distribution.

Now let us suppose the same measurement of Xˆθ is performed on a quantum particle in state with density matrixρ ˆ that is prepared anew after each measurement. Again, we can construct the histogram, which is related to the state according to By Fourier integral, pr(Xθ)=Tr(ˆρXˆθ). (4.28) characteristic function In the quantum domain, there can exist no phase-space probability density because, according to More generally,the uncertainty there principle, are other the kinds particle of cannot phase-space have certain functions values of position and momentum at the same time. However, for every quantum state there exists a phase-space quasiprobability density — a function W (X, P) for which Eq. (4.27) holds for all θ’s. Without derivation, the expression for Wigner representationρˆ this quasiprobability density (now known as the Wigner function) is as follows:

+ 1 ∞ Q Q W (X, P)= eiP Q X ρˆ X + dQ. (4.29) ρˆ 2π − 2 2 !−∞ " # # $ # # # # This equation is called the Wigner formula. # # Problem 4.18 Prove Eq. (4.27) for the classical case. Problem 4.19 Use a mathematical software package to calculate the Wigner function and verify that Eq. (4.27) holds for θ = 0 and θ = π/2 for the following states: a) vacuum state; b) coherent state with α = 2; c) coherent state with α =2eiπ/6; d) the single-photon state; e) the ten-photon state; f) ( 0 + 1 )/√2; | ⟩ | ⟩ g) ( 0 + i 1 )/√2; | ⟩ | ⟩ h) state with the density matrix ( 0 0 + 1 1 )/2; | ⟩⟨ | | ⟩⟨ | Examples of Phase-space funcon

Coherent states

Using optical equivalence theorem,

PERSPECTIVES Coherent state Negative values and positive W(Q,P) results. The Wigner function is a way W(Q,P) to describe how “quantum” a light A As a alternative representation,pulse is. Progressing from most clas- B sical to most quantum, the Wigner functions are shown for (A) the coherent state, (B) a squeezed state, (C) the single-photon state, and (D) a Schrödinger’s-cat state. The pro- Q P Q jections or “shadows” of the Wigner P function (shown on the sides) are the measured probability distributions of the quantum continuous variables Q or P. The Wigner function is a Gauss- W(Q,P) ian function for (A) and (B), but it W(Q,P) takes negative values for the strongly C quantum states (C) and (D). These D negative features vanish very quickly in the presence of decoherence. In the experiment by Lee et al., a quantum state similar to (C) and (D) was teleported and kept the negative Q P Q values of the Wigner function. This P result demonstrates an extraordinary degree of experimental control over such fragile quantum objects. on December 27, 2011

entangled states with negative Wigner func- ensures that no other copy of the initial state 10.1126/science.1204814 OURJOUMTSEV ALEXEI CREDIT:

314 15 APRIL 2011 VOL 332 SCIENCE www.sciencemag.org Published by AAAS Examples of Phase-space funcon • Thermal light

mean photon number

• Fock state PERSPECTIVES Negative values and positive W(Q,P) results. The Wigner function is a way W(Q,P) to describe how “quantum” a light A pulse is. Progressing from most clas- B sical to most quantum, the Wigner functions are shown for (A) the coherent state, (B) a squeezed state, (C) the single-photon state, and (D) sharper than delta function a Schrödinger’s-cat state. The pro- Q P Q jections or “shadows” of the Wigner P function (shown on the sides) are the measured probability distributions of the quantum continuous variables Q Single photon or P. The Wigner function is a Gauss- W(Q,P) ian function for (A) and (B), but it W(Q,P) takes negative values for the strongly C quantum states (C) and (D). These D negative features vanish very quickly in the presence of decoherence. In the experiment by Lee et al., a quantum state similar to (C) and (D) was teleported and kept the negative Q P Q values of the Wigner function. This P result demonstrates an extraordinary degree of experimental control over such fragile quantum objects. on December 27, 2011

entangled states with negative Wigner func- ensures that no other copy of the initial state 10.1126/science.1204814 OURJOUMTSEV ALEXEI CREDIT:

314 15 APRIL 2011 VOL 332 SCIENCE www.sciencemag.org Published by AAAS