Download A Hamilton operator for quantum optics in gravitational fields

10 July 1995
PHYSICS
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Physics Letters A 203 (1995) 12-17
A Hamilton operator for quantum optics in gravitational fields
Claus Liimmerzahll
Lnboratoire de Gravitation et Cosmologie Relativiste, UniversitP Pierre et Marie Curie. CNRS/URA 679, 75252 Paris Cedex 05, France
Fakultiitfir Physik der Universitiit Konstanz, D-78434 Constance, Germany
Received 20
June
1994; revised manuscript received 24 April 1995; accepted for publication 5 May 1995
Communicated by P.R. Holland
Abstract
Performing a relativistic approximation of the Klein-Gordon equation the interaction of matter with light in the presence
of a gravitational field is treated. One main result is that the electric dipole coupling remains unchanged provided the electric
field as well as the dipole are operationally defined by measured quantities. In contrast, the magnetic dipole coupling will
be modified by the gravitational field.
1. Introduction
The interaction of matter fields with the electromagnetic field is essential for describing the effect of radiation on atoms. An interaction of this kind is the dipole
interaction -d . E, which is basic for doing quantum
optics. An important application of this dipole coupling is atomic beam interferometry
[ I]. Especially
the BordC atom beam interferometer as proposed in
Ref. [2] (see also Ref. [3 J) and realised by Riehle
et al. [4], and the light pulse atom beam interferometer of Kasevich and Chu [ 51 take advantage of this
interaction, which causes a momentum transfer from
the laser beam to the atoms. These devices have been
used to do interferometry in non-inertial frames and
gravitational fields. Observed effects are the Sagnac
effect [ 41 and the effect of acceleration [ 51. Thereby
a great accuracy is achieved.
On the level of the minimally coupled Schriidinger
equation the dipole interaction and the interaction with
the magnetic field can be introduced (see, e.g., Refs.
[ 6,7] ) by performing a unitary transformation. In lowest order of approximation one gets the dipole coupling Hdipole = -d. ET, which describes the coupling
of the atomic dipole with the radiation field. Here ET =
-( l/c)&A is the transversal electric field representing the radiation field in the Coulomb gauge and d :=
e (x - x,) is the dipole operator where xa is the position of the atom or of the center of mass of the matter
field in question. A static dipole term arises by expanding the electrostatic potential 4 around the center
ofmassn,ofthematterfield:4(x)
=+(q,)-d.E
with the electric field E := -V#J.
The great accuracy achieved by the experiments indicates that also relativistic contributions in the gravitational field may be measurable by these devices. This
possibility and theoretical consistency make it necessary to consider the interaction of matter with radiation also in the case of non-inertial motion and in the
presence of gravitational fields in first relativistic approximation. Therefore the question we are going to
answer is: does the gravitational field modify the interaction of matter with the radiation field? Does the
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C. Liimmerzahl/Physics Letters A 203 (1995) 12-17
gravitational field act on the atoms in quantum optical
experiments not only in a direct manner but perhaps
also via the interaction with the electromagnetic field?
In the following we treat this problem in a quite
general manner: We make a relativistic expansion of
the Klein-Gordon
equation in an arbitrary electromagnetic and gravitational field, whereby gravity is described in a post-Newtonian
approximation which is
consistent with the approximation scheme employed.
On the level of the Klein-Gordon
equation the interaction with the Maxwell and the gravitational field is
introduced by means of the minimal coupling procedure. The resulting Hamilton operator describes the
time evolution of the matter field in first relativistic
approximation,
The coupling to the electromagnetic
field strengths is worked out using a unitary transformation.
Our answer to the question stated above will be:
if the dipole interaction is described by means of the
electric field E and of the dipole operator d whereby
both are defined as quantities which are actually meusured by an observer in the gravitational field, then the
dipole interaction will remain unchanged. This answer
means that in terms of operationally defined quantities
the form of this interaction remains the same. Contrary
to that, the magnetic dipole coupling will be modified
by gravity.
Only BordC and co-workers [ 8,9] treated this question by rewriting the non-relativistic expression for the
dipole coupling in a relativistic way using the tetrad
formalism. Very recently, Marzlin [ lo] used classical
methods to confirm our result concerning the dipole
coupling.
The Maxwell field strength is given by FGy =
a,A, - &A,. We define the electric field Ei = & =
&A0 - ( 1/c) 8tAi and the magnetic field Bi = EijkFjk
where
is the Levi-Civita symbol in three dimensions, El23 = 1. Latin indices belong to a space-like
hypersurface, i = 1,2,3. Entities which are defined
in t = const three-hypersurfaces
are represented by
vectors: xi +-+ X, Ai c-t A, etc. A0 is identified with
the scalar potential -4 so that E = -VC#I - i&A.
Gravity is given by a simple version of the
parametrized post-Newtonian
approximation.
In this
approximation the metric is given by [ 111
eijk
approximation
0 = g+?DpDypKG
- m2c2qoKG,
(1)
with D,T” = 8,T” + {b}T”
- (ie/hc)A,T’
where
e is the charge of the field T” and A, is the Maxwell
potential. {&} := $gyP(aPg,, + &g,, - JPglM) denotes the Christoffel symbol. We take this field equation as our model to describe the relativistic quantum
motion of an atom. We do not consider spin effects.
(2)
gOi =
07
(3)
gij=
(1+2?$)6,,
(4)
with the Newtonian
potential
(5)
where po is the rest mass density of the matter generating the gravitational field. Einstein’s general relativity is characterised by y = p = 1. Note that the Newtonian potential is uniquely fixed by boundary conditions. There is no gauge freedom to add e.g. a constant
to this potential.
We insert the above metric into ( 1) and make according to Kiefer and Singh [ 131 for the wave function pko the ansatz
+C-2S2(X)
(P, v = 0,. . . ,3),
;
[
of the matter field
We take as matter field the Klein-Gordon
equation
minimally coupled to gravity and to the Maxwell field
( >I,
2
l-2;+2p
gm=-
SPKG(X) =exp
2. Relativistic
13
~[c2So(n)
(.
+...I
,
+$(x)
(6)
>
which is inserted into ( 1 ), too. Then we compare
equal powers of the expansion parameter c2 whereby
we treat (e/c)Ai
to be of order 1 (terms of this
kind have to appear in the non-relativistic Schrodinger
equation).
The lowest order term VSo = 0 implies that SO is a
function of t only. The next order gives (6’&/&)2 m2 = 0 which has the solutions Se = fmt. For our purposes we choose the solution SO = --mt. The substitution cpi := exp[ (i/n) Si ] transforms the next order
14
C. Liimmerzahl/Physics
Letters A 203 (1995) 12-17
to the Schriidinger equation for ~1 minimally coupled
to the Maxwell field and coupled to the Newtonian
potential
First we present the
covariant form which
explicitly its Hermitian
the derivatives by the
the three-hypersurface,
where p := -iFi[V - (ie/flc)A].
The next order
is an equation for St and S2. Replacing St by ~1
by means of the above substitution and introducing
p := cpt exp(i&//Ic*)
we get a Schrijdinger equation
im,qD = Hq with relativistic correction terms. The
Hamiltonian is given by
Aco” :=&(a,-
H=g-L
8m3c2
Hamiltonian (8) in a manifest
makes it possible to identify
parts. For doing so we replace
Laplace-Beltrami
operator in
$Ai)
x [fig+
- ;A+].
(11)
This operator fulfills J[ +* A,,,qo - (A,&)
x d3x = 0 and gives to the order c-*
* cplm
-eAo-ml/-(y+f)$$p*
ACov~:= (1 -Zy$)(V-
iA)*p
_(f_P)~_~_&yVU.p
(V-
.,+J.
(8)
We introduce a scalar product by taking the relativistic approximation
of the conserved quantity const =
Insertion
;A)q.
(12)
of this covariant operator results in
HP = - ;A,,,,,
- eAoqo - rn&
SgoY(&&~ko
~~oDp~DKG)(-gOO)-“*d3V
where d3V = fid3x
(t3)g := detgi,) and integration is over a x0 = t = const hypersurface. Using
$&o = eXp[ -( i/k)mc2f] 4p We get
const =
S[
cp*q+ $-$
(&)*4p
+ co*(H~)
(9)
Therefore we define as scalar product
(‘b 140)=
S(‘4.p
- &,*V*p)fid3X.
(10)
The total probability defined by (‘p 1(p) is conserved.
In addition, since -V2 is a positive operator, the total
probability is positive definite. The Hamiltonian (8)
is Hermitian with respect to the scalar product ( 10)
if and only if d,U = 0 and &A = 0, that is, if the
Newtonian potential and the vector potential A are
stationary.
Among the many ways to reformulate the Hamilton operator we present two: A manifest covariant formulation and a representation with respect to a “flat”
scalar product.
Note that in this form the parameter y is totally absorbed in the Laplace-Beltrami
operator. Rewritten in
this manifest covariant form it is obvious that, provided we neglect the last two terms, H is manifest
Hermitian. The next-to-last term contains a Darwin
term V - E.
Next we represent our Hamiltonian (8) with respect
to a “flat” scalar product: We make a transformation
9 -+ ppf so that the scalar product (p ) $I) acquires
the usual non-relativistic
Schrodinger form (50 11,0)=
s pf t,bfd3n. The corresponding transformation of the
wave functions and the Hamilton operator is given by
(0 ---f Pf=
(1+-gJ4((~‘gP4p,(14)
H -+ Hf= (1 + -$)“4((3,g)‘/4
x H(l+-&)-“4
( c3)g) -1/4 + im,( C3)g)-1/4
(15)
C. Liimmerzahl/Physics Letters A 203 (1995) 12-l 7
P4
=- P2 ---eAo-mU
2m
8m3c2
+2y+1
M ;A
+ &B(x,,
ua,u-’
Ii2V2U
+ 3y---
= -id’
&‘(x,,
t) + &“a
-d x B(x,,t).
(16)
4c2m ’
t) . L,
(19)
- (i -p)m$
+inVU.p)
-(-Up2
- ;A]=U
;U[V
1.5
The Hamiltonian
Hf is Hermitian with respect to
the Schrodinger scalar product. All time-dependent
hermiticity-violating
terms canceled due to the timedependent transformation
(14>, (15). One advantage
of this representation
is that we can keep the usual
interpretation
of the various terms of the Hamiltonian Hr and of the operators appearing in ( 16). Note
that the electric Darwin term vanished while the
gravitational Darwin term is still present. This is in
agreement with a result of Bjorken and Drell [ 141
who showed that in the Klein-Gordon
theory electric
Darwin terms effectively appear first at the order c-~.
(20)
Here we expanded all expressions to first order in the
distance x - xa. We introduced the angular momentum operator L := (x - x,) x (-i&V)
and the dipole
operator d := e(x - x,) where both are defined with
respect to the center-of-mass. va = dx,/dt
is the velocity of the atom. We neglected terms quadratic in
the magnetic field (it has been shown in Ref. 171 that
such terms lead to negligible effects only). As result
we get from ( 16) the transformed Hamiltonian ( { , }
denotes the anti-commutator)
+-;A-
ri4
-A2
8m3c2
- eAo - mU
+(2y+l)&UA+VUV)
3. Discussion
of the coupling to the Maxwell field
Finally, we want to present the coupling of the electromagnetic field in ( 16) in terms of the field strengths
E and B. To begin with, we note that because we work
in a representation using the usual Schrodinger scalar
product we can define the coordinate position operator
as x. The expectation value of this position operator is
then given by xii = s &xpr d3x and can be identified
with the position of the atom.
According to Ref. [ 71 we make a unitary transformation with
(
I/=exp
-i
I
x
s
0
(x--x,).A(x,+A(x-xxa),t)dA
The transformed
iFiU&U-I.
Hamiltonian
is Hi = U-‘Hp!/ -
We use (ET = -( l/c)k)
“-‘(V-EA)U=V+i(x-r,)
x B(x,,t),
(18)
- d-
ET(X,, t) +
$9x B(x,,
t)
(21)
This is the main result of our calculations. This Hamiltonian is Hermitian and describes the interaction of
a scalar matter field with the electromagnetic
field in
the presence of a gravitational field which is given
to first relativistic order. The coupling to the electromagnetic field is expressed in terms of the magnetic
and transversal electric field strength and the scalar
potential. Thereby the interaction of the matter field
with the magnetic field is explicitly modified by the
gravitational potential U while the interaction with the
transversal electric field and the scalar potential A0 is
not gravitationally modified. In deriving this result we
have chosen no special gauge.
The first two terms in (21) describe the kinetic energy to first relativistic approximation. The next two
terms are the non-relativistic
coupling to the scalar
electric and Newtonian potential. Lines two and three
are relativistic corrections to the coupling of the matter
16
C. Liimmerzahl/
Physics Letters A 203 (1995) 12-17
tield to gravity. Line four is the magnetic-dipole coupling together with relativistic and gravitationally induced modifications. The last line is the usual dipolecoupling term with the radiation field, and of a term
which gives the energy of a moving electric dipole in
a magnetic field which may be called a Rontgen term
and which has been discussed in Ref. [ 15 1.
Finally we want to identify the various terms in
the Hamiltonian with measured quantities. The electromagnetic field may operationally
be defined by
means of the generally covariant Lorentz-force equation L~“D,u~ = (e/m)g~“F_&’
where L+ is the fourvelocity of the charged particle. The observed acceleration ai := h>u”D,up is obtained by projecting this
equation on a space-like tetrad h; which we choose to
be collinear with the coordinate lines: hk N 6:. Normalisation hlhi,g@” = rfJ where $j is the Minkowski
metric, implies hh = &a;.
If the particle is momentarily at rest in our coordinate system then we
have L+ N 6:. Normalisation
of the four-velocity
- -I
implies P = (l/J-goo)S~.
We get
,Q,‘LJgU
a’ = (L/TT*)( l/&)viJFjo(
l/a)
= (e/m)E’
where we defined the actually measured electric field
E’ := ( I/,,&) F;o( l/&).
This means that the
measured electric field is defined through the measured
force it exerts on a charged particle. If we introduce
the proper distance ri := Js7;x-’ = [ I + y( U/c2) ] xi
along the jth coordinate axis and the corresponding
dipole vector d’ := eri then we can replace in the
Hamiltonian
ter is the symmetrized operator-valued
form of the
corresponding classical expression. Consequently, we
expect some genuine quantum corrections to the usual
magnetic dipole coupling due to the presence of a
gravitational field which may result e.g. in a gravitationally induced modification of the splitting of the
energy levels in the Zeeman effect.
If one describes the dynamics
of the matter
field with respect to the proper time of the atom,
then the Hamiltonian
has to be multiplied
with
I /J-gm(
xa, t) (compare Ref. [ 161). Consequently,
the multiplication
with this function cancels the
factor \/-goo(x,,t)
in (22), (23) and in the magnetic parts. If, on the other hand, the dynamics is
described with respect to the proper time of an observer whose worldline does not coincide with the
worldline of the atom, then the electric as well as the
magnetic dipole coupling will be multiplied by the
usual red-shift parameter Jgoo(&, ~)/&%0(~0~ t) =
1 - (l/c2)[U(~,,t)
- U(xo,t)],
where xc is the
position of the observer.
We can now state: If all entities are introduced as
measured quantities then on our level of approximation there is no gravitational modijcation
or relativistic correction of the dipole coupling terms in the
Hamiltonian describing the evolution of the matter
field with respect to the proper time of the atom. This
concerns both the dipole couphng to the static electric field as well as to the radiation field. However, the
magnetic dipole coupling terms are modified by the
gravitational field.
(22)
Similarly
-eAo
x -eAo(x,,
4. Conclusion
t) - d - EL(x,,
t) da.
(23)
In the same way as for the electric field strength one
can show that the measured magnetic field strength is
given by
(24)
Comparison
of this expression with the magnetic
dipole interaction term in (2 I ) shows that the lat-
We have shown how in the presence of gravitational
fields the coupling of a matter field to the electromagnetic field is modified. Thereby all fields are treated
to first relativistic order. One main result is that the
dipole coupling of the matter field to the radiation field
is not modified by gravity to this order. This means especially that for treating the influence of laser beams
on atoms, which is essentially in atomic beam interferometry, the interaction term does not have to be
modified. The modifications of the magnetic dipole
coupling may lead to gravitationally induced modifications of the Zeeman splitting.
C. Liimmerzahl/Physics
Acknowledgement
I thank Professor J. Audretsch and Professor Ch.J.
BordC for discussions, Professor R. Kerner for the hospitality in the LGCR at UPMC, Paris, and the Deutsche
Forschungsgemeinschaft
for financial support.
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