Manipulation of Photonic Orbital Angular Momentum for

29 Pages·2014·0.93 MB·English

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40
Manipulation of Photonic Orbital Angular
Momentum for Quantum Information Processing
Eleonora Nagali and Fabio Sciarrino
Dipartimento di Fisica, Sapienza Universitá di Roma
Italy
1.Introduction
Morethanacenturyago,pioneeringworkscarriedoutbyPoynting(1909)andotherphysicists
have provided the evidence of the validity of Marxwell equations for an electromagnetic
field, showing how a beam of light carries energy and momentum, both in the linear and
angular components. In particular the angular momentum of light, related to the generator
of rotations in quantum mechanics, has been typically associated with its polarization, and
more specifically with its circular polarization components. An optical beam traveling in the
positive direction of the z axis that is circularly polarized, carries a z-component angular
momentum content σ = h¯ per photon, which is positive if the circular polarization is
±
left-handed and negative if it is right-handed. This angular momentum content is not just
a formal property, but a very concrete one that can have significant mechanical effects as for
exampleanobservableinducedrotationbyabsorptiontoamaterialparticleBeth(1936);Friese
etal.(1998)
Atthesametime,calculationsofangularmomentumforafreeelectromagneticbeamgaverise
toasecondcontributionnotrelatedtophotonspin,towhichwasattributedthenameoforbital
angularmomentum(OAM).Unlikethespinorialangularmomentum(SAM),consideredasthe
intrinsic part of angular momentum since it does not depend on the specific reference frame,
theorbitalcomponentisassociatedtothetransversespatialstructureofthewavefront. More
precisely, this angular momentum appears when the beam wavefront acquires a helicoidal
structure, or equivalently, its field spatial dependence contains a helical phase factor having
the form eimϕ, where ϕ is the azimuthal phase of the position vector r around the beam axis
z and m is any integer, positive or negative, providing the direction and the "velocity" of
the phase spiraling along the beam direction. It can be shown that in this case the optical
beam carries an angular momentum along its axis z equal to mh¯ per photon, in addition to
the polarization one σ. When m is nonzero, the helical phase factor imposes the existence
of an optical vortex at the center of the beam where the light intensity vanishes. Although
the fundamental concept of orbital angular momentum associated with a light field was
already known since the early forties, the research on the orbital angular momentum of
light has begun only in 1992, with the appearance of a seminal publication carried out by
Allen et al. (1992). In this work, Allen et al. demonstrated experimentally that a particular
set of solutions of Helmoltz equation in paraxial approximation, the Laguerre-Gauss modes
(LG), carry a fixed amount of orbital angular momentum. Moreover, such beam could be
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generated experimentally in the laboratory by manipulating gaussian beams emerging from
alasercavity. Interestingly,eventhoughitwaslargelydiffusedtoindicatethe"single-photon
contribution" to the whole value of OAM carried by the beam, the first experimental test on
the OAM as an individual property of single photons has been carried out only in 2001 by
Mairetal.(2001)
For all the reasons listed above, the orbital angular momentum is considered a recently
discovered photonic degree of freedom. In general, only the global angular momentum
is associated to an observable quantity, however in the paraxial approximation, both SAM
and OAM can be manipulated and measured separately. Indeed the OAM of light can
be exchanged with matter, thus opening new perspectives in several fields of classical and
quantum physics as well as in biology. In contrast to SAM, which couples only with the
materiallocalanisotropy(birefringence),OAMcouplesmainlywithmaterialinhomogeneities
characterized by a rotational asymmetry around the beam axis. This coupling may be
considered a negative feature when OAM is considered for communication purposes, as
it makes it very sensitive to turbulence or other sources of noise Paterson (2005), but it
becomes an useful property when OAM is adopted as a tool for sensitively probing the
properties of a given medium as considered in several recent works (see Molina-Terriza,
Rebane, Torres, Torner & Carrasco (2007); Torner et al. (2005)). The use of OAM for probing
can lead to microscopic imaging with a spatial resolution that is higher than the Rayleigh
limit Tamburini et al. (2006) and, when OAM fields are used in combination with suitable
fluorescence methods (e.g., the stimulated emission depletion), they enable new methods of
far-field microscopy with theoretically unlimited resolution Harke et al. (2006); Hell (2009).
Optimal spatially structured light beams have also been considered as tools to cage/uncage
specific molecules for accurate and rapid biological imaging Shoham et al. (2005). Some of
these approaches may have relevant applications in the imaging of biological tissues, e.g. for
diagnostic or research purposes. Finally the characteristic doughnut profile of the intensity
patternofaLGmodeallowsanefficientionandatomtrappingwithlowscatteringandhence
heatingoftheatom,usefulforatomopticsandBECspurposesAndersenetal.(2006).
Beyond all these applications, it has been recognized that the orbital angular momentum
has a great potential for quantum photonics, in particular regarding quantum information
protocols implemented through quantum optics techniques. Quantum information (QI)
is based on the combination of classical information theory and quantum mechanics. In
the last few decades, the development of this new field has opened far-reaching prospects
both for fundamental physics, such as the capability of a full coherent control of quantum
systems, as well as in technological applications, most significantly in the communication
field. In particular, quantum optics has enabled the implementation of a variety of quantum
informationprotocols. ThefundamentalunitofinformationinQItheoryisatwo-levelsystem,
the quantum bit or qubit. Exploiting the features of quantum states, however, it has been
proventhatqubitsallowsthetransferofmoreinformationthantheoneencodedinaclassical
boolean alphabet and, at the same time, the quantumness of qubit systems ensure high level
of security in communication processing. In this context, the information encoding based
on two-dimensional system can be experimentally implemented by exploiting degrees of
freedom of single photons as, for example, the polarization. Up to now, several quantum
information protocols have been successfully implemented, thanks to a notable control on
the polarization degree of freedom achieved through different efficient devices. However
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Manipulation of Photonic Orbital Angular Momentum for Quantum Information Processing 77
ManipulationofPhotonicOrbitalAngularMomentumforQuantumInformationProcessing 3
there may be a significant advantage in introducing the use of higher dimensional systems
for encoding and manipulating the quantum information. Such d-level quantum systems, or
qudits,provideanaturalextensionofqubitsthathasbeenshowntobesuitableforprospective
applicationssuchasquantumcryptographyandcomputationCerfetal.(2002);Lanyonetal.
(2009). Inthisframeworktheorbitalangularmomentum,definedinaninfinitelydimensional
Hilbert space, provides a natural choice for implementing qudits encoded in a single photon
state (see Franke-Arnold et al. (2008); Molina-Terriza, Torres & Torner (2007)). This can be
animportantpracticaladvantage,asitallowsincreasingtheinformationcontentperphoton,
and this, in turn, may cut down substantially the noise and losses arising from the imperfect
generation and detection efficiency, by reducing the total number of photons needed in a
givenprocess. Moreover, thecombineduseofdifferentdegreesoffreedomofaphoton, such
as OAM and spin, enables the implementation of entirely new quantum tasks, as shown by
Aolita&Walborn(2007);Barreiroetal.(2005;2008)
Since the seminal paper of 1992 a large effort has been spent to develop optical tools able to
manipulateandcontrolefficaciouslytheorbitalangularmomentumdegreeoffreedom. Upto
now, by observing the transfer of OAM to matter, some devices have been adopted in order
togenerate/analyseLGmodes(computergeneratedholograms,spatiallightmodulators),or
manipulate the OAM analogously to what is commonly carried out through waveplates on
polarization (cylindrical lenses mode converters). Despite these successes, the optical tools
for controlling the OAM quantum states remain rather of limited use: a wider and more
practical control of the OAM resource somehow analogous to that currently possible for the
polarizationdegreeoffreedomisbythewayunderprogress.
Here we present a brief introduction on orbital angular momentum in quantum optics,
describing the main devices adopted in order to achieve an efficient manipulation.
Furthermore, we describe some experiments that have been carried out by adopting an
opticaldevice,theq-plate,abletocouplethespinorialandorbitalcontributionsoftheangular
momentumoflightbyexploitingthepropertiesofliquidcrystals.
2.The orbital angular momentum of light
Light beams carry energy and momentum, the latter both in its linear and angular
components. Thus, when an electromagnetic field interacts with matter, an exchange of
energyandmomentumoccurs,manifestedininterestingmechanicaleffectsorinchangingof
thebeamproperties. Herewewillfocusonlightangularmomentum,composedbyaspinorial
andorbitalcomponent. Themanipulationofspinorialangularmomentum(SAM),commonly
associatedtothepolarizationoflight,islargelydiffusedinallthedifferentfieldofphysics. On
the other hand, the orbital angular momentum (OAM) of light did not go through the same
diffusion for many years, since after seminal works of Poynting (1909) and Beth (1936), only
in 1992 Allen et al. (1992) demonstrated that particular solutions of the Helmoltz equation in
the paraxial regime exhibit an azimuthal phase structure typical of beams possessing OAM.
Such beams, as for example the Laguerre-Gauss beams, can be experimentally manipulated
andthusofferavaluableresourceinquantuminformation,wherethepossibilitytoexploitthe
infinite-dimensionalityofOAMopensinterestingperspectives. Herewepresenttheoretically
themainelementsthatcharacterize,bothinclassicalandquantumregimes,theorbitalangular
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momentum of light, deriving the solutions of Helmoltz equation and presenting the main
ones,namelytheHermite-GaussandLaguerre-Gaussbeams.
3.Classical orbital angular momentum of light
(cid:2)
Inclassicalphysicsthetotalangularmomentum J ofanelectromagneticfieldreads:
(cid:2)J = ǫ d(cid:2)r(cid:2)r ((cid:2)E (cid:2)B)
0
× ×
(cid:2)
(cid:2) (cid:2)
where E and B are the electric and magnetic fields, respectively. The angular momentum is
a conserved quantity for a free field, due to the invariance under arbitrary rotations of free
Marxwell equations Enk & Nienhuis (1994). Moreover, as the electric field can be separated
(cid:2) (cid:2) (cid:2)
in longitudinal and transverse component E = E + E , it is possible to single out the
|| ⊥
contribution from the transverse part to the angular momentum related to the radiation field
(cid:2)
J whosecomponentsreads:
rad
(cid:2) (cid:2) (cid:2)
J = L+S (1)
rad
(cid:2) (cid:2)
where S is the spin angular momentum and L represent the orbital angular momentum
contribution. Thespinangularmomentumdoesnotdependonthespecificcoordinatesystem
andhenceitisoftencalledastheintrinsicangularmomentum. Moreover,sinceforamassless
particle, as a photon, it is not possible to define a rest frame and hence a spin vector, it is
possible only to define the projection of the spin along the propagation direction zˆ, whose
values can be σ = s with s both integer or half-integer values Enk & Nienhuis (1994). For
±
the photon, having s = 1, the helicity σ can takes only values σ = 1 (in h¯ units). As a form
±
ofangularmomentum,spinangularmomentumisphysicallyassociatedtotherotationofthe
particle around its own axis, and for this reason physically is associated to the polarization
of single photons. In particular, right-circular polarization is associated to σ = h¯ and
−
left-circular polarization to σ = +h¯. On the other hand as shown in eq.(1) light beams carry
alsoorbitalangularmomentum,consideredastheextrinsiccomponentofangularmomentum
since it depends on the chosen coordinates system. It has been demonstrated that OAM is
associated to the azimuthal phase structure of light beams, as will be described in section 3.2
(Allen et al. (1992)). Differently from SAM, OAM causes a rotation of the particle around the
beam axis. The experimental evidence of the angular momentum and its components can be
achieved only by studying the interaction between light and matter, where it is possible to
observe a transfer of angular momentum. In particular, differences between SAM and OAM
canbeappreciatedintheparaxialregime.
3.1The paraxial approximation
In classical optics a beam that propagates along the zˆ direction is described by the Helmoltz
equation:
2U((cid:2)r)+k2U((cid:2)r) = 0 (2)
∇
where k is the wavevector k = ωc 1. For beams that are characterized by small divergence
−
angle respect to the propagation direction, eq.(2) can be investigated within the paraxial
approximation. Such approximation provides a proper description for laser beams, whose
beam variation along the transverse plane (xˆyˆ) are slow compared to the ones along the
directionofpropagation. InthiscasethebeamcanberepresentedasU((cid:2)r) = A((cid:2)r)e ikz,where
−
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ManipulationofPhotonicOrbitalAngularMomentumforQuantumInformationProcessing 5
the exponential component is the plane wave contribution and A((cid:2)r) C is the complex
∈
amplitude whose variations are slow along the beam propagation and thus the following
relationholds:
∂2A((cid:2)r) ∂2A((cid:2)r) ∂2A((cid:2)r)
<<
, (3)
∂z2 ∂2 ∂y2
Replacing in eq.(2) the expression of U((cid:2)r) and considering the paraxial approximation in
eq.(3),weobtaintheparaxialHelmoltzequation:
∂2A((cid:2)r) ∂2A((cid:2)r) ∂A((cid:2)r)
+ 2ik = 0 (4)
∂x2 ∂y2 − ∂z
This equation admits different families of solutions depending on the coordinates system
adopted. For example adopting Cartesiancoordinates, the solutions are the Gaussian beams,
widelyusedinopticstodescribelaserbeamspropagation. Howeveriftheamplitude A((cid:2)r) is
modulated by another function slowly varying on direction zˆ, other valid solutions of eq.(4)
can be obtained. Among them, a common one is represented by the Hermite-Gauss (HG)
modes, where the Gaussian envelope is multiplied by Hermite polynomials. Interestingly,
if instead of a beam described by U(x,y,z) the Helmoltz equation is solved in cylindrical
coordinates that is, the function U(r,ϕ,z) is adopted, we obtain another important family of
solutions, known as Laguerre-Gauss (LG) modes. As it will be shown in the next section, LG
modesarecharacterizedbyadependenceofthephasetermtotheazimuthalangleandhence
carry a well-determinate value of orbital angular momentum. Finally, it is worth notice that
otherclassesofsolutioncanbefoundbyadoptingpolarcoordinates(Hypergeometricbeams
HGKarimietal.(2007)),orellipticcoordinates(Ince-Gaussianbeams),ormodifyingproperly
theenvelopes(BesselandMathieubeams).
3.2Laguerre-Gauss modes
When the paraxial equation is solved in cylindrical coordinates (x = rcosϕ,y = rsinϕ,z =
z) the solution is represented by a function family known as Laguerre-Gauss (LG) modes,
where the gaussian envelope is multiplied by Laguerre polynomials Lm(τ). The amplitude
p
distributionofthesemodesreads:
m
C r√2 2r2 r2 i kr2z
uLp,Gm(r,ϕ,z) = 1+ zz22 (cid:4)w(z)(cid:5) Lmp (cid:6)w2(z)(cid:7)e−w2(z)e− 2(z2+z2R)e−imϕeiς(z) (5)
R
(cid:3)
where C is a normalization constant, z is the Rayleigh range, w(z) = w 1+ z2 is the
R 0 z2
(cid:8) R
beam size along zˆ starting from the beam waist w , and ς(z) = (2p+m+1)arct(cid:9)an z(cid:10) is the
0 z
R
Gouyphaseterm. Here(p,m)aretheradialandazimuthalindex,respectively,anddetermine
theorderofthemodeN = 2p+ m . Inparticular(p,m)areintegernumberswith p 0while
| | ≥
mcanassumebothpositiveornegativevalues. Thelowestorderofthesemodes p = m = 0is
theGaussianbeam(TEM mode).
00
Let us focus on the phase term e imϕ. This term leads to a phase front that varies with the
−
azimuthal angle along the propagation direction: points on the phase front diametrically
opposite are indeed dephased by a factor π. This means that the phase front follows an
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helicoidal pattern along zˆ so that in the center of the transverse plane the contact between
all different phases causes a phase singularity. The torque shown along the directional axis
thus leads to an optical vortex, that is LG beams are characterized by a singularity in the
center where the field intensity vanishes. For each optical vortex it is possible to associate a
number, known as topological charge, that refers to the number of twist over a distance of the
wavelength λ. Suchnumber q isalwaysaninteger,andcanassumebothpositiveornegative
values, depending on the direction of the twist. The larger is the number of torsions, greater
is the twisting velocity along the beam direction. It is exactly this twisting that leads to a
non-vanishingcontributionoforbitalangularmomentum
Fig.(1) reports some examples of LG modes intensities and phases for different values of
indices p,m. For p = 0 the intensity in the transverse plane is distributed among an annular
patternwithazero-intensityspotinthecenter,sothatLGmodesareknownalsoas"doughnut
modes". In the next section we will show that Laguerre-Gauss modes constitute a complete
orthogonal basis of orbital angular momentum operator L eigenvectors, however here we
z
present a notable result of Allen et al. (1992) where for the first time was demonstrated
that beam characterized by a helical behavior of the phase front, as for LG modes, carry a
well-definedamountofOAMperphotonfixedbythevalueoftheazimuthalmodeindexm
LetusfirstconsideravectorpotentialwritteninCartesiancoordinatesas:
A(cid:2)(x,y,z) = u(x,y,z)eikz (6)
(cid:2) (cid:2)
Adopting the Lorentz gauge, the relation between magnetic B and electric E fields and
potential vector are known, so that is possible to evaluate the mean value of the Poynting
(cid:2)
vector Pthatdeterminestheenergycarriedbythebeam:
p = ǫ < (cid:2)E (cid:2)B >= iωǫ0(u u u u)+wkǫ u 2
0 ∗ ∗ 0
× 2 ∇ − ∇ | |
This result can be extended to the case of a vector potential expressed in cylindrical
coordinates, that is where the beam is characterized by the azimuthal angular dependence
e imϕ. Now we are interested in the estimation of the angular momentum carried by the
−
beamalongthezˆ direction. Theangularmomentumreads:
(cid:2)j =(cid:2)r (cid:2)p
×
where the Poynting vector provides three component(cid:2)p = (p ,p ,p ). Thus the component
r ϕ z
of angular momentum along the propagation direction of the beam j can be calculated as
z
j = rp where:
z ϕ
wm u 2
p = ǫ | | (7)
ϕ 0
r
Justbyreplacingthisresultinthe j expression,weget:
z
j = ǫ wm u 2 (8)
z 0
| |
The value of angular momentum carried by each photon can be obtained considering the
ratio between angular momentum density and the energy one, and then integrating over all
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Manipulation of Photonic Orbital Angular Momentum for Quantum Information Processing 81
ManipulationofPhotonicOrbitalAngularMomentumforQuantumInformationProcessing 7
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Fig.1. ExamplesofLGmodes. Top: phaseprofilesfordifferentLGmodesandBottom:
intensityprofilesofthesamemodes.
thetransverseplane:
J rdrdϕ((cid:2)r < (cid:2)E (cid:2)B >) m
z z
= × × = (9)
W c rdrdϕ < (cid:2)E (cid:2)B > ω
(cid:11) (cid:11) z
×
Equation (9) shows that each photon(cid:11)t(cid:11)hat constitutes the beam carries an amount of angular
momentum equal to mh¯ Allen et al. (1992; 1999). It is possible to demonstrate that this value
refers to the orbital component of angular momentum and is related to the azimuthal phase
factor. Indeed if we consider a circularly polarized beam, i.e. a beam with no vanishing
contributionofspinangularmomentum,itsvectorpotentialis:
A(cid:2) = (αxˆ+βyˆ)u(x,y,z)eikz
and going through the same procedure carried before, including the transformation to
cylindricalcoordinates,theratiobetweenangularmomentumandenergydensityleadsto:
J rdrdϕ((cid:2)r < (cid:2)E (cid:2)B >) m σ
z z
= × × = + (10)
W c rdrdϕ < (cid:2)E (cid:2)B > ω ω
(cid:11) (cid:11) z
×
(cid:11) (cid:11)
We observe from eq.(10) that each photon contributes to the total angular momentum J
z
with two components. The σ/ω one belongs to the spinorial contribution, and is due to the
polarization of the beam, since σ = 1 depending on the right or left circular polarization.
±
Analogously, we observe a further term purely related to the phase dependence e imϕ
−
that does not belongs to the polarization. Such contribution is thus related to the orbital
angular momentum, showing that, depending on the twisting velocity of the phase along
thepropagationdirection,eachphotonofthebeamcarriesmh¯ OAM.
3.3Hermite-Gauss modes
The transverse field distribution of laser cavities is typically well described in terms of
Hermite-Gauss(HG)modes. Theseoneformacompleteorthogonalsetthatsolvetheparaxial
equationinCartesiancoordinatesandtheiramplitudeisgiven,apartforaphaseterm,bythe
product of a gaussian function and Hermite polynomials of indexes (n˜,m˜) Walborn et al.
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Fig.2. IntensityprofilesfordifferentHGmodes. Ascanbeobserved,thezeroorderofHG
coincideswiththeoneofLGmodesthatis,thegaussianmode.
(2005):
2 1 √2x √2y x2+y2 ik(x2+y2)z
unH˜,mG˜ = (cid:8)2(n˜+m˜)πn˜!m˜! w(z)Hn˜ (cid:4)w(z)(cid:5)Hm˜ (cid:4)w(z)(cid:5)e− w2(z) e− 2(z2+z2R)e−i(n˜+m˜+1)ς(z) (11)
where ς(z) = actan z and the last exponential term is the Gouy phase. This solution of
z
r
the paraxial Helmoltz equation represents modes structurally stables, so that their intensity
remain the same along the propagation Allen et al. (1999). The mode indexes n˜,m˜ define the
shapeofthebeamprofilealongthexˆandyˆdirection,respectively(Saleh&Teich(1991)). Some
examples of intensity distribution of such modes are reported in Fig.(2). We observe that the
intensityI = uHG 2hasn˜ nodesonthehorizontaldirection,andm˜ alongthevertical,while
n˜,m˜ | n˜,m˜|
forn˜ = m˜ = 0theshapeistheoneofaGaussianbeam(TEM mode). Theorderofthemode
00
is defined as N = n˜ +m˜. As can be observed from eq.(11), in HG modes each component of
thetransverseplanetravelinphase,asforaplanewave. Indeedthismeansthatsurfacesthat
includeallthepointswithsamephaseareplanesseparatedbyadistanceequalto λ. Forthis
reason there is a nil component of linear momentum along the axial direction, which means
that such modes do not carry a well defined amount of OAM, as deducible by the lack of the
phase term eimϕ. Hermite-Gauss modes form a complete basis set so that any arbitrary field
distributioncanbewrittenasasuperpositionofHGmodeswithindexes n˜,m˜. Howeveralso
Laguerre-Gauss modes form a complete basis set, hence a HG mode can be rewritten as a
linearsuperpositionofLGmodesandviceversathroughtherelations:
N
uLG (x,y,z) = ∑ ikb(m,p,k)uHG (x,y,z) (12)
m,p N k,k
k=0 −
where
(N k!k!) 21 1 dk
b(m,p,k) = − [(1 t)m(1+t)p] = 0 (13)
t
2Nm!p! k! dtk −
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Manipulation of Photonic Orbital Angular Momentum for Quantum Information Processing 83
ManipulationofPhotonicOrbitalAngularMomentumforQuantumInformationProcessing 9
DifferentindexesthatcharacterizeHGandLGarerelatedonesettotheotheras:
m = n˜ m˜ (14)
−
p = min(n˜,m˜) (15)
Coefficientsgivenineq.(13)areofcrucialimportanceastheydefinethematrixelementneeded
for the conversion between HG and LG modes (or viceversa): O’Neil & Courtial (2000).
Moreover, the correspondence between HG and LG modes is fixed by the mode order N,
definedas:
N = n˜ +m˜ = 2p+ m
| |
Invertingrelationsgivenineq.(16)itispossibletoestimatetheindexesofHGmodesinterms
ofthemodeorder N andtheazimuthalindexm:
N+m
n˜ =
2
N m
m˜ = −
2
All these relations can be exploited to generate Laguerre-Gauss modes starting from
Hermite-Gauss modes emitted by lasers. Experimentally such mode converters based on
eq.(12)areimplementedthroughcylindricallenses,asexplainedinsection4.3.
3.4The orbital angular momentum in quantum mechanics
In1992Allenetal. demonstratedthatLaguerre-Gaussmodescarryawelldefinedamountof
orbital angular momentum through classical mechanics approach, as shown in section 3.2.
On the other hand, it is possible to demonstrate this property also following a quantum
mechanical approach, showing that LG modes are eigenvectors of the orbital angular
momentumoperator L (Bransden&Joachain(2003);Sakurai(1994)).
z
Theclassicalorbitalangularmomentum(cid:2)L =(cid:2)r (cid:2)pfindsitscounterpartinquantummechanics
×
usingthedefinitionofoperatormomentum pˆ = ih¯ ,sothat
− ∇
Lˆ = ih¯(r )
− ×∇
Thusthecartesiancomponentsoftheorbitalangularmomentumoperatorreads:
∂ ∂
L = ih¯ y z
x
− ∂z − ∂y
(cid:6) (cid:7)
∂ ∂
L = ih¯ z x (16)
y
− ∂x − ∂z
(cid:6) (cid:7)
∂ ∂
L = ih¯ x y
z
− ∂y − ∂x
(cid:6) (cid:7)
(17)
wherethefollowingcommutationrulesapply:
[L ,L ] = ih¯ǫ L (18)
i j ijk k
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