Photon reflection by a quantum mirror: a wave function approach

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by  Raul Corrêa

Photon reflection by a quantum mirror: a wave function approach
Raul Corrˆea∗ and Pablo L. Saldanha
Departamento de F´ısica, Universidade Federal de Minas Gerais,
Caixa Postal 701, 30161-970, Belo Horizonte, MG, Brazil
Wederivefromfirstprinciplesthemomentumexchangebetweenaphotonandaquantummirror
upon reflection, by considering the boundary conditions imposed by the mirror surface on the
photonwaveequation. Weshowthatthesystemgenerallyendsupinanentangledstate,unlessthe
mirror position uncertainty is much smaller than the photon wavelength, when the mirror behaves
classically. Our treatment leads us directly to the conclusion that the photon momentum has the
known value (cid:126)k. This implies that when the mirror is immersed in a dielectric medium the photon
radiation pressure is proportional to the medium refractive index n. Our work thus contributes
to the longstanding Abraham–Minkowski debate about the momentum of light in a medium. We
interpret the result by associating the Minkowski momentum (which is proportional to n) with the
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canonical momentum of light, which appears naturally in quantum formulations.
1
0
2 PACSnumbers: 42.50.Ct,03.65.Ta,03.65.Ud

n
a The advances in nanofabrication techniques have al- thiseffectbasedontheassociationoftheMinkowskimo-
J
lowed extensive research on mechanical systems whose mentum (which is proportional to n) with the canon-

5 masses are small enough so that they can significantly ical momentum of light [16], contributing to the long-
1
exchange momentum with electromagnetic fields – the standingAbraham-Minkowskidebateaboutthemomen-

so-called optomechanical systems. An important class of tum of light in material media [17, 18].
]
h those systems are the optomechanical cavities, in which We consider one photon in a paraxial beam state with
p the mechanical system is described by a harmonic oscil- arbitrarypolarizationthatreachesaperfectlyconducting
-
t lator that couples with the light [1]. In some cases the plane surface (the mirror) and interacts with it during a
n cavity is composed by mirrors, and since the mid 1990s finite time. The mirror surface is considered to be larger
a
u there were proposals of studying quantum properties of than the photon beam diameter and its description is
q such mirrors, for instance quantum fluctuations [2] and quantized in the z direction, which corresponds to the
[ the construction of superposition states [3–7]. Also, re- direction orthogonal to the surface plane. We consider
cently there have been many theoretical discussions on the mirror initially in an arbitrary quantum state. We
1
entanglement between vibration modes of mirrors in a want to know the state of the system after the photon is
v
6 quantumregimeandlight[8–10],whileithasbeenexper- completely reflected, while considering that the photon-
9 imentallyreportedforothermechanicalsystems[11,12]. mirror interaction occurs in a time scale much smaller
7 Thequantummechanicaltreatmentforacavityquan- than the one by the quantum mirror free evolution –
3 tum mirror interacting with photons can be based on in such case the mirror wave function can be considered
0 a Hamiltonian formulation [13], with a second quanti- stationary during the reflection process. Our strategy
.
1 zation approach for light. But here we take a different to solve the problem is to make use of the linearity of
0 path, by using a photon wave function approach to treat both Schr¨odinger’s and Maxwell’s equations, to find the
6 the reflection of a single photon by a quantum mirror. knownsolutionofamonochromaticclassicalelectromag-
1 Our treatment is constructed from first principles and neticfieldbeingreflectedbyafixedinfiniteplanemirror,
:
v can be used to describe photons interacting with quan- and to construct the arbitrary solution from the super-
i tum mirrors in a cavity as well as a single photon reflec- position of those. In order to do this, we are going to
X
tion by a quantum mirror. The mirror imposes bound- describeboththemirrorandthephotonwithwavefunc-
r
a aryconditionsforthephotonwavefunctionatitssurface tions.
andthisnaturallyleadstothephotonradiationpressure
The mirror wave function is the traditional quantum
on the mirror, which is associated to the photon phase
mechanical wave function whose time evolution is de-
changeuponreflection. Wediscussthemirror-photonen-
scribed by the Schr¨odinger equation. For a classical per-
tanglement and its dependence on the relation between
fectly fixed mirror, its quantum state can be approxi-
the photon momentum and the mirror momentum un-
mated as a Dirac delta wave function in the position
certainty. When the mirror is immersed in a medium
space, having a momentum uncertainty that tends to in-
with refractive index n, we show that the radiation pres-
finity. Therefore no matter how much momentum it ex-
sure is proportional to n, which agrees with experiments
changeswithaphoton,itswavefunctioncanonlyacquire
performed with classical mirrors [14, 15]. We analyze
a global phase. Since an arbitrary quantum state for the
mirror can be decomposed in the position eigenfunctions
and since the Schr¨odinger equation is linear, if we know
whattheinteractiondoestoeverypositioneigenfunction,
∗ raulcs@fisica.ufmg.br we know what it does to an arbitrary mirror state.
2
We use the Bialynicki-Birula–Sipe wave function de-
scriptionforthephoton[19–22]. Thisphotonwavefunc-
tion is a complex vector function of the spatial and time
coordinatesthatcompletelydescribesthequantumstate
of a photon. It can be decomposed in the eigenstates of
the helicity operator σˆ in the following way:
Ψ(r,t)=Ψ (r,t)+Ψ (r,t), (1)
+ −
where
(cid:114) (cid:114)
(cid:15) 1
Ψ (r,t)= 0E (r,t)±i B (r,t). (2)
± 2 ± 2µ ±
0
ε represents the electric permittivity and µ the mag-
0 0
netic permeability of free space. We have σˆΨ = ±Ψ
± ±
and the condition ∇·Ψ = 0 is imposed. The helicity FIG. 1. An electromagnetic field with wave vector k is re-
eigenstates are associated to photons with circular po- flected by a perfect mirror, resulting in a reflected wave with
larizations and the photon electric and magnetic fields wave vector k(cid:48) =k−2(k·ˆzm)ˆzm.
(cid:112)
are given by E = Re[ 2/ε Ψ(r,t)] = E + E and
√ 0 + −
B = Im[ 2µ σˆΨ(r,t)] = B + B . By introducing
0 + −
the term J(r,t) accounting for the induced current in approximately zero at its surface. Hence, by solving
the medium due to the presence of the photon field, Maxwell’s equations with this condition, we find that if
Maxwell’s equations for the electromagnetic field in a there is a field E±(r,t)=uˆk±E0kei(k·r−ωt) in the zm <
mediumcanberecoveredwiththeuseofthephotonwave z0 region, called the incident field, then there must be
equation [22] a field E(cid:48)∓(r,t)=−uˆk(cid:48)∓E0kei(k(cid:48)·r−ωt)e2i(k·ˆzm)z0, called
the reflected field, in the same region. This guarantees
∂Ψ(r,t) J(r,t) that, at the interface, the component of the electric field
i =cσˆ∇×Ψ(r,t)−i √ . (3)
∂t 2(cid:15)0 parallel to it is zero. Here k(cid:48) =k−2(k·ˆzm)ˆzm, where
ˆz is the unit vector perpendicular to the surface of re-
m
This Maxwell wave equation determines the photon evo- flection and pointing inward the conductor, and uˆ are
k±
lution,justliketheSchr¨odingerequationdoesforaquan- the circular polarization unit vectors. Given the refer-
tum massive particle. We note that to apply the sec- ence frames (x,y,z) and (x(cid:48),y(cid:48),z(cid:48)) indicated in Fig. 1,
√ √
ond quantization procedure to the electromagnetic field we define uˆ =(xˆ±iyˆ)/ 2 and uˆ =(xˆ(cid:48)±iyˆ(cid:48))/ 2.
k± k(cid:48)±
in the presence of matter is an extremely difficult task
Hence, in the same spatial configuration, the boundary
[23–26]. The boundary conditions imposed by the in-
conditions demand that if the incident photon in this
terface between different media and the interaction of
space is described by the wave function
the electromagnetic field with dispersive and absorptive
mediamakesthequantizationprocesstobeverycompli- Ψ (r,t)=uˆ Aei(k·r−ωt), (4)
± k±
cated. In this sense, the use of the Maxwell wave equa-
tion greatly simplifies the treatment in relation to the in the region z ≤ z , then in that same region there
m 0
secondquantizationmethodwhenthereisnoabsorption must be a reflected part of the the wave function given
oremissionofphotonsintheproblemtobetreated. This by
is the case in the present problem of the photon reflec-
tion by a quantum mirror. Since the photon equation is Ψ(cid:48)∓(r,t)=−uˆk(cid:48)∓Aei(k(cid:48)·r−ωt)e2i(k·ˆzm)z0, (5)
equivalent to the Maxwell’s equations, a boundary con-
ditions problem for a photon interacting with different with k(cid:48) =k−2(k·ˆz )ˆz . From now on, we represent
m m
media has the same solution as the one for a classical the state of a photon with wave vector k and helicity
electromagnetic field. It is worth mentioning that the ± in the Dirac notation as |k±(cid:105). The reflection implies
photonwavefunctionformalismcanbeusefulevenwhen that for every |k±(cid:105) component of the field, there must
there is the absorption and generation of photons in a be another component |k(cid:48)∓(cid:105) with the same amplitude
scatteringprocess,asinthegenerationofentangledtwin andaphasedifference−e2i(k·ˆzm)z0 inordertosatisfythe
photons with parametric down conversion [22, 27]. boundary conditions. Of course, the wave function must
Let z =z be the plane of the mirror interface, with be zero for z >z .
m 0 m 0
the region z > z being a perfect conductor, as shown Up to now we have been dealing with plane waves,
m 0
in Fig. 1. It can be shown that the electromagnetic field which extend themselves with the same amplitude
inside the conductor falls to zero rapidly and there is no through all times and all space, but in our problem the
field propagation inside the conductor [28]. When the interaction takes place during a finite time and in a re-
penetration depth is much smaller than the field wave- stricted region of the mirror. So in order to talk about
length, the field component parallel to the interface is before and after the reflection of the photon and to con-
3
sider that the mirror surface is larger than the beam di- 2(cid:126)(k·ˆz ) to it. This is the exact necessary amount to
m
ameter, we make use of the superposition principle and conserve momentum, since the reflection simply inverts
allow the state of the photon to be a superposition of everyphotonwavevectorcomponentinthez direction.
m
differentwavevectors,thereforeconfiningitinspaceand It is interesting to note that we arrived at this result of
time. For an incident photon in a beam state the momentum transfer from the photon to the mirror
simply by imposing boundary conditions on the photon
(cid:90)
|ψ(cid:105)= ψ(k)(c |k+(cid:105)+c |k−(cid:105)) d3k, (6) reflection. Nospecificationofthephotonmomentumwas
k+ k−
made. In other words, we can conclude that the pho-
ton momentum is given by the expression (cid:126)k simply by
with|c |2+|c |2 =1foreverykand(cid:82) |ψ(k)|2d3k =1,
k+ k− computing the momentum transfer to the mirror upon
ourtreatmentimpliesthat,apartfromaglobalphase,the
reflection and imposing momentum conservation.
reflected photon state must be
We can analyze some classical limits of the quantum
(cid:90) state of Eqs. (9) and (10). In the case when the mir-
|ψ(cid:48)(cid:105)= ψ(k)(c |k(cid:48)+(cid:105)+c |k(cid:48)−(cid:105))e2i(k·ˆzm)z0 d3k. rorpositionwavefunctionapproximatesadeltafunction,
k− k+
Eq. (9)reducestoEq. (7)forthereflectedphoton,with
(7)
the mirror state unaltered by the photon reflection. In
the view of Eq. (10), this approximation is valid when
We are finally ready to include the wave function for
the mirror momentum uncertainty is much larger than
the z position of the quantum mirror in the description.
the momentum gained by the reflection of each k com-
Eqs. (6)and(7)correspondtothesituationoffixedmir-
ponent of the photon state. Since ∆p ∆z ∼ (cid:126), this is
ror at the position z =z , that is, its state is described m m
m 0
equivalenttothemirrorpositionuncertaintybeingmuch
by the wave function (cid:104)z |z (cid:105) = δ(z −z ). Hence, for
m 0 m 0
smaller than the wavelengths that compose the photon
a mirror in an arbitrary state |φ(cid:105) with wave function
state. For an optical photon with average wavelength
φ(z ) = (cid:104)z |φ(cid:105), the composite state of the system be-
m m
λ ∼ 500nm, it means that the mirror should have a po-
fore the interaction is
sition uncertainty at least around ∆x ∼ 10−7m for sig-
(cid:90)
nificant entanglement effects to appear. It is important
|Ψ(cid:105)= ψ(k)(c |k+(cid:105)+c |k−(cid:105)) d3k
k+ k− to note that the momentum transfer from the photon
(cid:90) to the quantum mirror can be increased by a factor of
⊗ φ(zm)|zm(cid:105) dzm, (8) Q if the quantum mirror is one of the mirrors of a cav-
ity with a quality factor Q. This is because the pho-
which leads us to the state after the reflection ton is reflected on average Q times by the quantum mir-
ror before leaving the cavity. Entanglement effects may
(cid:90)(cid:90)
|Ψ(cid:48)(cid:105)= ψ(k)φ(z )e2i(k·ˆzm)zm arise in that way with, for instance, ∆x∼10−13m, with
m
Q ∼ 106. If the mirror is in the ground state of a quan-
×(c |k(cid:48)+(cid:105)+c |k(cid:48)−(cid:105))|z (cid:105) d3k dz . (9) tum harmonic oscillator, the relation between its mass
k− k+ m m
m , its resonance frequency ω and its position uncer-
0 0
The state described in Eq. (9) has a different phase fac- taintyis∆x=(cid:112)(cid:126)/2m ω [29]. Inthatsense,m ω gets
0 0 0 0
tor for each ket |k±(cid:105)|zm(cid:105) of the composite system state. smaller as ∆x grows larger, hence entanglement effects
This phase depends on the eigenvalues k and zm, which arise whenever m0ω0 ∼ 10−8kg/s or smaller. A look
means that this is a non-separable, or entangled, state. at table II of [1] shows us that the suspended mirrors
Soingeneraltherearenon-classicalcorrelationsbetween with smallest m ω have it of order 10−6kg/s along with
0 0
the photon and the quantum mirror after the photon re- Q∼106 [30], which is a bit far from the regime needed.
flection. So it is still not possible to effectively entangle spatial
We can also write the state of Eq. (9) in the lin- modes of a photon and a mirror upon reflection.
ear momentum basis for the mirror {|pm(cid:105)}, given that Another disentangled state limit occurs if the photon
(cid:104)pm|zm(cid:105)=(2π(cid:126))−12e−ipmzm/(cid:126). We are led to propagates as a nearly monochromatic beam along the
directionk (butnon-monochromaticenoughsothatthe
(cid:90)(cid:90) 0
|Ψ(cid:48)(cid:105)= ψ(k) φ˜(p −2(cid:126)(k·ˆz )) interaction is still much faster than any evolution due to
m m
thefreemirrorHamiltonian). Inaroughapproximation,
×(c |k(cid:48)+(cid:105)+c |k(cid:48)−(cid:105))|p (cid:105) d3k dp , (10) the final state of Eq. (10) is then almost disentangled
k− k+ m m
and the mirror momentum wave function is displaced by
where 2(cid:126)(k·ˆz ):
m
(cid:90) (cid:90)
φ˜(pm)=(cid:104)pm|φ(cid:105)=(2π(cid:126))−12 φ(zm)e−ipmzm/(cid:126)dzm. |Ψ(cid:48)(cid:105)≈ ψ(k) (ck0−|k(cid:48)+(cid:105)+ck0+|k(cid:48)−(cid:105)) d3k
(11) (cid:90)
⊗ φ˜(p −2(cid:126)(k ·ˆz ))|p (cid:105) dp . (12)
m 0 m m m
From the above equations it is clear that every compo-
nent|k±(cid:105)pushesthemirrorbytransferringamomentum Intermediate regimes account for mirror position un-
4
certainty of the order of the average wavelength of non- the Minkowski momentum for the photon, which is di-
monochromatic light, and those generally result in a rectly proportional to n. This behavior was observed
non-separable state, as explicit in Eq. (10). Following in the experiments with classical light being reflected by
the discussions of the above paragraphs, such regimes classical mirrors immersed in dielectric media [14, 15],
can in principle be achieved by engineering cavities with and we present a fully quantum justification here. The
larger quality factors tuned to smaller light wavelengths. answertowhyistheMinkowskimomentumthatappears
But since a high quality factor is associated with highly in this case lies on the fact that quantum mechanics is a
monochromatic light allowed in the cavity, present tech- Hamiltoniantheory,basedoncanonicalrelationsbetween
nologyseemstobeinadeadlocktotrytoprobethiskind position and momentum. The phase acquired upon re-
of entanglement. It is important to note, though, that flectionbythephotononEq. (7), whichisdependenton
if a cavity with the quantum mirror is in one arm of an the mirror position, is shared by both mirror and pho-
interferometer, asproposedin[5], entanglementbetween ton on Eq. (9). The canonical commutation relations
thephotonandthemirrorcouldbegeneratedduetothe in quantum mechanics define translation operators with
superposition of the single photon propagating on each the same form as these phase factors [29], hence turning
arm of the interferometer. The quantum superposition those phase factors into momentum kicks, made explicit
ofthepathinwhichthephotoninteractswiththemirror on each component of Eq. (10). It is natural then that
and transfer momentum to it with the path in which the our system will reveal the canonical momentum of the
photon does not interact and does not transfer momen- photon,whichcorrespondstotheMinkowskimomentum.
tumtothemirrormayresultinanentangledstate. But In summary, we have treated a single-photon reflec-
an experimental realization of this proposal has not yet tion by a quantum mirror using the photon wave func-
been accomplished. tion formalism. This allowed us to treat the problem
using boundary conditions on the photon wave equation
Now we address the historical Abraham–Minkowski instead of using the second quantization formalism for
debate, which concerns how the linear momentum car- light. By computing the momentum transferred from
ried by light behaves when it propagates through a di- the photon to the mirror, we concluded that a photon
electric medium [17, 18]. The two apparently contrary with wavevector k must have momentum (cid:126)k in order to
views, due to Max Abraham and Hermann Minkowski, achieve momentum conservation in the system, as ex-
respectively indentify the momentum of the electromag- pected. We also showed that in the case that the pho-
netic field either inversely or directly proportional to the ton is not monochromatic and its average wavelength
refractive index of the medium. But it is important to is of the order of the mirror position uncertainty, en-
note that when both the electromagnetic and material tanglement between them might appear with the reflec-
energy-momentum tensors are taken into account, the tion process. Finally we addressed a contribution to the
experimental predictions of Abraham’s and Minkowski’s Abraham-Minkowski debate by showing, with a quan-
formulationsareequivalent[17,31,32]. RecentlyBarnett tum treatment from first principles, that the momentum
showed how the Abraham and Minkowski momenta can transferred from a photon to a mirror immersed in a di-
be associated to the kinetic and canonical momentum of electric medium upon reflection is proportional to the
the field respectively [16, 18]. It is clear that, according medium refractive index. This result associates the pho-
to equation (10), the momentum gained by every com- ton momentum with the Minkowski momentum. This is
ponent |p (cid:105) of the mirror is proportional to the wave natural given that the Minkowski momentum is associ-
m
vector component k. If uˆ is the unitary vector along ated with the canonical momentum of light, which is the
k
the direction of k, then we can write k = n(ω/c)uˆ , momentum that should appear in a quantum treatment.
k
where n is the refractive index of the medium in which This work was supported by the Brazilian agencies
the photon is propagating. Clearly, this corresponds to CNPq and CAPES.
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