Sequential measurement of conjugate variables as an alternative quantum state tomography

0.13 MB·English

by  Antonio Di Lorenzo

Sequential measurement of conjugate variables as an alternative quantum state
tomography
Antonio Di Lorenzo
Instituto de F´ısica, Universidade Federal de Uberlaˆndia,
38400-902 Uberlaˆndia, Minas Gerais, Brazil
It is shown how it is possible to reconstruct the initial state of a one-dimensional system by
measuring sequentially two conjugate variables. The procedure relies on the quasi-characteristic
function,theFourier-transform oftheWignerquasi-probability. Thepropercharacteristic function
obtained by Fourier-transforming the experimentally accessible joint probability of observing “po-
sition” then “momentum” (or vice versa) can be expressed as a product of the quasi-characteristic
function of the two detectors and that, unknown, of the quantum system. This allows state re-
3 construction through the sequence: data collection, Fourier-transform, algebraic operation, inverse
1 Fourier-transform. The strength of the measurement should be intermediate for the procedure to
0 work.
2

n
Introduction. Quantum state tomography, i.e., the re- surements of conjugate variables. In order for the pro-
a
J construction of the unknown state of a quantum system cedure to work, the strength of the measurement is not
4 is a fundamental problem. Its formulation can be traced a fundamental issue, provided it is not too strong nor
back to Pauli [1], who asked whether a measurement of too weak. The equations at the basis of this proposal

] positiononanensembleofspinlesssystemspreparedina were reported, without derivation, in Ref. [17], and they
h
pure state, complemented by a measurement of momen- are an exact result, not a perturbative expansion in the
p
- tum on a distinct ensemble, would allow to reconstruct couplings. Furthermore, the procedure proposed herein
t
n the wave function. This question has been answered in requires a fixed setup. This is to be contrasted to ho-
a thenegative[2,3]. Purestatesarerepresentedbyawave- modyne quantum state tomography, where a different
u
function,andtheyareexceptional,inthesensethatthey quadrature operator x is measured for various values
φ
q
form a zero-measure subset of all possible states (which of φ. The efficiency of the two procedures is otherwise
[
reflects in the great care one has to take in preparing a comparable, as in the sequential measurements one has

3 purestate). Themostgeneralstateismixedandusually to evaluate a joint probability of two variables, while in
v
described by a density matrix, which is then the object the homodyne detection scheme a one-parameter family
8
to be reconstructed. However, other equivalent descrip- of single variable probabilities is estimated.
3
2 tions of a mixedstate haverevealedto be moreuseful or Review of the measurement. A linearmeasurementre-
1 significantthan the density matrix: the Wigner function lies on a specific bilinear coupling between a quantum
5. [4], the Husimi Q function [5], the Glauber-Sudarshan system and a probe. The interaction is assumed bilinear
0 P function [6, 7], the recently reintroduced Dirac func- in the observable Xˆ belonging to the system and in an
2 tion [8], and the almost forgotten Fourier-transform of observable Φˆ belonging to the probe:
1
Wignerfunction,whichwerefertoastheMoyalM func-
: H =−λ~g(t−t )ΦˆXˆ, (1)
v tion [9]. Furthermore, parametric families including all int 0
Xi oftheabovefunctionshavealsobeenintroduced[10,11]. where the function g(t) is strongly peaked around t = 0
Despite the quantum state being essential in describ- and has unit integral. If the spectrum of Xˆ is bounded,
r
a ing a system, successful quantum state reconstruction, thenλmayincludeascalesuchthattheeigenvaluesofXˆ,
dubbed quantum state tomography because of an anal- indicated by X, are dimensionless and less than one[18].
ogy with the germane procedure of medicaltomography, A detector with a continuous unbounded output is con-
is relatively recent [12]. The procedure relies on a pro- sidered, so that Φˆ has a conjugate operator Jˆ, satisfying
posal by Vogel and Risken [13]. For a recent review [Φˆ,Jˆ]=i. Notice that λX andJ are homogeneous. The
of continuous-variable quantum state tomography, the variable J represents eigenvalues of Jˆand is the readout
reader may refer to Ref. [14]. ofthedetector,carryinginformationaboutthesystem. If
Recently, a remarkable experiment [15] showed how theprobeisinitiallyinawelldefinedstatewithvanishing
it is possible to determine the unknown pure state of variance in the readout variable J, i.e. its density ma-
′
a one-dimensional quantum system by making a weak trix is ρ(J,J ) = δJ,J′δ(J), then the measurement is an
measurementofthe xvariable followedbya strongmea- idealstrongone(we areindicating by δ the Kronecker
a,b
surement of the conjugate variable p. A method, based delta and by δ(a) the Dirac delta). When this latter re-
on the Dirac function, allowing to lift the restriction to quirement is relaxed the measurement is a linear ideal
pure states was proposed recently [16]. Here, we make (non-strong) one. More precisely, let X be the typical
m
an alternative proposal, allowing the reconstruction of spacing between the eigenvalues of Xˆ; the measurement
the Moyal M function by a quick sequence of two mea- is weak when the coupling constant satisfies λX ≪ ∆,
m
2
with ∆2 the initialvarianceof the readout. One candis- cumulants of the system, and only the second cumulant,
tinguishtworegimes: theweakincoherentmeasurement, which is the variance, includes a contribution from the
when κ ≪ λX ≪ ∆, with κ the coherence scale of the detector,while the firstone, the average,hasa contribu-
m
detector (in the readout basis), and the weak coherent tion from the probe only if this is biased, introducing a
measurement, when λX ≪ κ. The former case bears systematic error.
m
little interest: because of the large variance, the readout Moyal quasi-characteristic function. The results will
of the detector in each individual trial is not necessar- be specialized to a one-dimensional system, so that X
ily λX and can lie well outside the spectrum, but after represents its coordinate, and P = ~K its momentum.
averaging over many trials, this effect washes out, even A useful transformof the density matrix was introduced
if one post-selects the system [19, 20]. The latter case inRef.[9]: theMoyalquasi-characteristicfunctionwhich
was shown to produce a large averageoutput after post- is but the Fourier transform of W(K,X), the Wigner
selection [21, 22], and to allow a joint measurement of quasi-probability function [4, 9], and is defined by
non-commuting observables with optimal noise in both
outputs [17, 23]. Itis interestingto notethata measure- M(x,k)= dXdKeikX+ixKW(K,X)
mentofacontinuousvariable(X =0)isalwaysaweak
m Z
coherent measurement,and thus may show quantum co- = dXeikXρ(X+x,X−x)
herence effects when followed by a post-selection. In the 2 2
Z
following, no specific assumption about λ are made, in k k
= dKeiKxρˇ(K− ,K+ ). (6)
orderto havegeneralresults. For simplicity aninstanta- 2 2
Z
neous interaction g(t)=δ(t) is considered.
From the definition Eq. (6) one realizes that M(x,0) is
Inlackofanysufficientreasontobelieveotherwise,the
the characteristic function for the probability Πˇ (K) =
probe and the system are assumed to be initially uncor- f
hK|ρˆ|Ki, and M(0,k) the characteristic function for the
related, so that their state immediately before the inter-
action is Rˆ− =ρˆS ⊗ρˆi. We shall indicate by ρi(J,J′)= probability Πf(X) = hX|ρˆ|Xi. The generalization to a
′ ′ ′ higher dimension is straightforward. The Moyal quasi-
hJ|ρˆ|J i [ρ (X,X ) = hX|ρˆ |X i] the elements of the
i S S
characteristic function uniquely determines the density
probe (system) density matrix in the J (X) basis and
′ ′ ′ ′ matrix of a system, and vice versa. For composite sys-
by ρˇ(Φ,Φ) = hΦ|ρˆ|Φi [ρˇ (K,K ) = hK|ρˆ |K i] the
i i S S
tems,themarginalquasi-characteristicfunctionofasub-
elements in the Φ (K) basis. The system-probe state
sytem is obtained by putting the coordinates of the re-
immediately after the interaction is
maining subsystems to zero in the total function. Fur-
′ ′ ′ ′ ′
R (X,X ;J,J )=ρ (X,X )ρ (J−λX,J −λX ), (2) thermore, the Moyal function can be expressed as the
S,f S i
average of the non-Hermitian Weyl operator
where the equality exp[−iJ Φˆ]|Ji = |J −J i was used.
0 0
After tracing outthe system, the final state of the probe M(x,k)=Tr ρˆeixKˆ+ikXˆ , (7)
following the interaction is
n o
where we used in the second line of Eq. (6) |X−x/2i=
ρ (J,J′)= dµ(X)ρ (X,X)ρ (J−λX,J′−λX), (3) exp[iKˆx/2]|Xi,exploitedthecyclicpropertyofthetrace,
f S i
Z and applied the Campbell-Baker-Hausdorffformula.
with µ(X) the spectral function describing the distribu- Theusefulnessofthistransformshows,e.g.,whencon-
tion of eigenvalues (for a discrete spectrum, it is a com- sideringthe measurementillustratedin the previoussec-
bination of Dirac δ-s, for a continuous one generally it tion. The joint Moyal quasi-characteristic function for
is dµ(X)=dX). The probability distribution Π of the system and probe in terms of the initial ones is simply
f
readout J is then
M (x,k;φ,j)=M (x,k+λφ)M (φ,j−λx). (8)
S,f S i
Πf(J)=ρf(J,J)= dµ(X)ρS(X,X)Πi(J−λX), (4) Results. A system interacting in rapid sequence with
Z two probes, one coupling to X, the other to K, is con-
with Πi its initial distribution, and the corresponding sidered, so that the interaction term is
characteristic function is
H =−~ λ δ(t+ε)Φˆ Xˆ +λ δ(t−ε)Φˆ Kˆ . (9)
int X X K K
Z (φ)≡ dJeiφJΠ (j)=Z (λφ)Z (φ). (5)
f f S i h i
Z For ε → 0− a measurement of K is followed by a mea-
Equation (5) reveals that the contribution of the detec- surementofX,and vice versa for ε→0+. Forε=0 the
tor to the cumulants (logarithmic derivatives of Z ) are measurement is a joint measurement `a la Arthurs and
f
simply additive. In particular, if the detector is initially Kelly [23], that we shall not study in detail here, and
preparedinaGaussianstate,thecumulantsoftheoutput for which we refer the reader to our previous paper [17].
starting from the third and higher reflect faithfully the The initial state is assumed to be Rˆ− =ρˆS ⊗ρˆi, with ρˆi
3
the density matrix of the two probes immediately before can be done as far as M (φ,α ) 6= 0. (A partial recon-
i ε
the first interaction. The possibility that the probes are struction of the state of the system may be satisfactory,
initiallyinacorrelatedstateisaccountedfor. Thequasi- however,if, e.g., the function M is known ona dense set
characteristicfunctionforthesystemandthetwoprobes, over R2 or everywhere but on a zero-measure set. See
after somestraightforwardcalculations[24], turns outto Ref. [25] for a detailed study.) The density matrix and
be the Wigner function are given respectively by
x x dk e−iXkZ (Vs)
MS,f(x,k;φ,j)=Mi[φ,j+Λ(2α0s+αεΛφ)] ρ X+ ,X− = f , (14)
2 2 2πM (Vs,Λα s)
×MS(s+Λφ), (10) (cid:16) k k(cid:17) Z dx e−iiKxZ (Vεs)
f
ρˇ K− ,K+ = , (15)
2 2 2πM (Vs,Λα s)
where for conciseness the two probes’s coordinates were (cid:18) (cid:19) Z i ε
arranged in two column vectors φ = (φK,φX), j = W(K,X)= dkdx e−i(Kx+Xk)Zf(Vs), (16)
(jK,jX), s = (x,k) represents the symplectic coordi- (2π)2 Mi(Vs,Λαεs)
Z
nates,weintroducedthe 2×2matrixΛ=diag(λK,λX), with V =Λ−1 =diag(λ−1,λ−1).
K X
ε∈{+,−,0},and
For an ideal strong measurement, the ini-
′
tial pointer density matrix would be ρ (J,J ) =
i
α+ = 01 00 , α− = 00 −01 , α0 = α++2 α−. (11) δδJK,JδK′ δJX,,JsX′oδ(tJhKat)δt(hJeX)p,rocaenddurehwenocueld Mnoit(φw,jo)rk fo=r
(cid:18) (cid:19) (cid:18) (cid:19) jK,0 jX,0
ε = +, φ 6= 0, nor for ε = −, φ 6= 0, nor for
K X
The case ε = + corresponds to a measurement of X ε = 0, φK 6= 0 or φX 6= 0. It is therefore desirable not
followed by one of K, ε = − to the opposite order, and to work in the strong regime, but at the same time one
ε=0tothejointmeasurement. Then,the(proper)char- does not need to keep the measurement in the weak
acteristicfunction, i.e. the Fouriertransformofthe joint regime. For instance, the probes could be prepared in
probability of observing J = (J ,J ) as the output, is the mixed gaussian state
K X
obtained by tracing out the system (k = 0,x = 0), and
exp − [J¯2/2∆2+j2/2κ2]
by putting j = 0, so that the experimentally accessible ρ (J,J′)= a a a a a , (17)
i
quantity (cid:8) P 2π∆K∆X (cid:9)
withJ¯ =(J +J′)/2,j =(J −J′),andκ representing
a a a a a a a
Zf(φ)= dJKdJXeiJ·φΠf(JK,JX) (12) the coherence scale (that satisfies κa = 1/∆˜a, where ∆˜a
Z isthe spreadofthe conjugatevariableΦa,sothatbythe
uncertainty principle κ ≤2∆ ). Then
isexpressedintermsoftheproductofthetargetM and a a
S
the known M
i
M (φ,j)=exp − φ2∆2/2+j2/2κ2 . (18)
i a a a a
( )
Z (φ)=M (Λφ)M (φ,Λα Λφ). (13) a
f S i ε X(cid:2) (cid:3)
Theidealstrongcaseisobtainedforκ →0,∆ →0. In
a a
Equations (10) and (13) are the main results of this pa- the ideal weak coherent regime, κ → ∞, ∆ → ∞, the
a a
per. Since performinga jointmeasurementmay be more state is M (φ,j) → δ δ (which is but the strong
i φK,0 φX,0
difficult than a sequential one, we concentrate on the regime for when the Φ-variables are used as a readout),
cases ε = ±, but in principle the joint measurement of hence Eq. (13) can not determine M (λ φ ,λ φ ) for
S K K X X
positionandmomentumallowsaswellthereconstruction φ ,φ 6= 0. Thus the procedure suggested here works
K X
of the initial state. with intermediate measurement strength.
If the probes are initially in a known state, then the Inparticular,whenthedetectorispreparedinthestate
unknown initial state of the system can be evinced after given by Eq. (18), the Wigner function in terms of the
dividingthelefthandsideofEq.(13)byM (φ,α ). This characteristic function of Eq. (12) is
i ε
2 2
dkdx 1 ∆ (1+ε)λ
W(K,X)= Z (Vs)exp −i(Kx+Xk)+ K + X x2
(2π)2 f 2 λ 2κ
Z ( "(cid:18) K (cid:19) (cid:18) X (cid:19) #
2 2
1 ∆ (1−ε)λ
+ X + K k2 . (19)
2 λ 2κ
"(cid:18) X (cid:19) (cid:18) K (cid:19) # )
4
After substituting Eq. (12) into Eq. (19), one should be mined. The finite sampling introduces a statistical error
careful not to exchange recklessly the order of integra- in Z (φ), that can be estimated according to standard
f
tion, or an artificial divergence appears. Furthermore, statistical analysis as δZ2 = [1 − |Z (φ)|2]/N. There
f
for ε = 0 and κ = 2∆ , ∆ ∆ = λ λ /4, Eqs. (19) is also a numerical error introduced by the integration,
a a X K X K
and (12) give a relation between W(K,X) and Π (J) η . Notice how these uncertainties are present also in
f num
that is the formula relating the Wigner and the Q func- QHT. In conclusion, in QST, the relative error can be
tion,sothat,asiswellknown[26],thejointmeasurement estimated as
of positionand momentum providesdirectly the Q func-
δ|M (s)|2 1−|Z (Vs)|2 δ|M (Vs,Λα s)|2
tion, provided the detectors are properly prepared. By S ∼ f + i ε
contrast, the general procedure proposed here for mea- |MS(s)|2 N|Zf(Vs)|2 |Mi(Vs,Λαεs)|2
suringtheMoyalM functionallowsmuchmoreflexibility +η2 (Vs). (20)
num
in terms of detectors preparation.
Finally, an alternative application of Eq. (13) could Conclusions. Aprocedurefordetermininganunknown
consist in obtaining the quasi-characteristic function of quantumstatewasproposed. Twomeasurementsofcon-
the system for two fixed values of φ ,φ and then vary jugate variables are made in quick sequence, the joint
K X
the couplingstrengths λK,λX by keeping λKλX fixedin probability is estimated from the collected data, it is
order to reconstruct the state. Fourier-transformed to give the characteristic function,
Estimates. Let us summarize the steps needed to pro- then divided by the quasi-characteristic function of the
ceed to the quantum state tomographywith the method probes appearing in the right hand side of Eq. (13).
proposed here, which we may call quantum sequential This yields the Moyal quasi-characteristic function of
tomography (QST), while comparing them to the anal- the system. The density matrix is obtained by Fourier-
ogous steps done in quantum homodyne tomography transforming the latter function. On the other hand,
(QHT). it is sufficient to determine the Moyal function only in
In QST, first, a joint probability Π (J ,J ) is mea- a neighborhood of x = 0,k = 0 in order to estimate
f K X
sured by observing the J variables of the probe. Exper- the cumulants. Furthermore, the method proposed has
imentally, one should divide the J ,J plane in a suffi- the advantage of requiring one fixed setup and does not
K X
cientnumberofsmallbins,eachhavinganarea∆J ∆J require a sharp measurement of either position or mo-
X K
determined by the precision of the measurement. The mentum, rather it thrives over the unsharpness of the
measurement is repeated a large number of times N, so measurement.
thattheresultinghistogramapproximatesthetrueprob- Inperspective,itwouldbeinterestingtoextendthere-
ability within the precision of the probes. Analogously, sultstofinite-dimensionalHilbertspaces,forwhichthere
inQHT,theconditionalprobabilitypr(x |φ)ismeasured is a wide interest (see, e.g., the recent Ref. [27]), espe-
φ
fordifferentvaluesofφ∈[0,π]. Thebinningisinthex,φ cially in the light of a recent generalization [28] of the
strip, and it is determined by the precision with which concept of conjugate variables.
xφ canbemeasuredandφcontrolled. Then,inQST,the I acknowledge stimulating discussions with J. Lun-
MoyalfunctionMS(x,k)isobtainedby makingadouble deen. This work was performed as part of the Brazil-
integral, precisely a Fourier transform of the observed ian Instituto Nacional de Ciˆencia e Tecnologia para a
probability, which can be done efficiently fast with the Informac¸a˜o Quaˆntica (INCT–IQ) and it was supported
Fast-FourierTransformalgorithm,andbydividingfinally by Funda¸ca˜o de Amparo `a Pesquisa do Estado de Minas
by the knownstate of the probes,which is computation- Gerais through Process No. APQ-02804-10.
ally trivial. In QHT, the Wigner function is obtained by
making an inverse Radon transform of pr(x |φ), which
φ
consists as well in a double integral. Finally, the density
matrixcanbeobtainedbymakingasingleFouriertrans-
[1] W. Pauli, Die allgemeinen Prinzipien der Wellen-
formofthe Moyalcharacteristicfunctionin QST,andof
mechanik (Dover,Princeton,1933)part1Reprinted: En-
the Wigner function in QHT. A further Fourier trans-
cyclopedia of Physics, vol. V, p. 17, Springer, Berlin
form is needed to obtain the Wigner function in QST
1958,in: Geiger, Scheel (Eds.), Handbuch der Physik,
and the Moyal function in QHT. Thus we can say that Vol. XXIV(1933).
QSTismoreefficientofQHTforthedeterminationofthe [2] M. G. Raymer,Cont. Phys.38, 343 (1997).
Moyal function, at least as performant as QHT for the [3] J. Corbett, Rep. Math. Phys.57, 53 (2006).
densitymatrixandlessefficientforobtainingtheWigner [4] E. Wigner, Phys. Rev.40, 749 (1932).
[5] K. Husimi, Proc. Phys.-Math. Soc. Jpn. 22, 264 (1940).
function.
[6] R. J. Glauber, Phys.Rev. Lett.10, 84 (1963).
Finally, we estimate the relative uncertainty in the
Moyal function, M (s) = Z (Vs)M−1(Vs,Λα s). As [7] E. C. G. Sudarshan, Phys.Rev.Lett. 10, 277 (1963).
S f i ε [8] S. Chaturvedi, E. Ercolessi, G. Marmo, G. Morandi,
Mi is fixed, it carries an uncertainty δMi(φ,j) that N.Mukunda, andR.Simon,J. Phys. A 39, 1405 (2006).
depends on how the state of the probes was deter- [9] J.E.Moyal,Math. Proc. Cambridge Phil. Soc. 45, 99 (1949).
5
[10] L. Cohen, J. Math. Phys. 7, 781 (1966). [19] A. Di Lorenzo and J. C. Egues,
[11] K. E. Cahill and R. J. Glauber, Phys. Rev.A 77, 042108 (2008).
Phys.Rev.177, 1882 (1969). [20] Y.-W. Cho, H.-T. Lim, Y.-S. Ra, and Y.-H. Kim,
[12] D.T.Smithey,M.Beck,M.G.Raymer, andA.Faridani, New J. Phys.12, 023036 (2010).
Phys.Rev.Lett. 70, 1244 (1993). [21] Y. Aharonov, D. Z. Albert, and L. Vaidman,
[13] K.Vogel and H.Risken, Phys.Rev.A 40, 2847 (1989). Phys. Rev.Lett. 60, 1351 (1988).
[14] A. I. Lvovsky and M. G. Raymer, [22] I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan,
Rev.Mod. Phys. 81, 299 (2009). Phys. Rev.D 40, 2112 (1989).
[15] J.S.Lundeen,B.Sutherland,A.Patel,C.Stewart, and [23] E. Arthurs and J. L. Kelly Jr., Bell System Tech. J. 44,
C. Bamber, Nature474, 188 (2011). 725 (1965).
[16] J. S. Lundeen and C. Bamber, [24] See SupplementalMaterial
Phys.Rev.Lett. 108, 070402 (2012). [25] J. Kiukas, P. Lahti, J. Schultz, and R. F. Werner,
[17] A.Di Lorenzo, Phys. Rev.A 83, 042104 (2011). J. Math. Phys. 53, 102103 (2012).
[18] Unhattedvariablesrefertoeigenvaluesofthecorrespond- [26] S. L. Braunstein, C. M. Caves, and G. J. Milburn,
ing operators. Capital letters refer to the direct space, Phys. Rev.A 43, 1153 (1991).
whilelower-caselettersrefertotheconjugatespace.E.g., [27] A.KalevandP.A.Mello,J. Phys. A 45, 235301 (2012).
k is theconjugate of X, etc. [28] C. Carmeli, T. Heinosaari, and A. Toigo,
J. Phys.A 44, 285304 (2011).
SUPPLEMENTAL MATERIAL:
DERIVATION OF THE MAIN RESULT.
The case of a measurement of momentum followed by one of position is considered for definiteness. The coupling
constants are temporarily absorbed in the rescaled variables Φ → λ Φ , J → J /λ , a = K,X. The joint density
a a a a a a
matrix after the interaction is
RˆS,f =eiΦˆXXˆeiΦˆKKˆRˆ−e−iΦˆKKˆe−iΦˆXXˆ, (21)
with Rˆ− = ρˆS ⊗ρˆi the initial joint density matrix. The most convenient representation is in terms of |Φi for the
probes and |Xifor the system
R (Φ,X,Φ′,X′)=eiΦXXρ (X +Φ ,X′+Φ′ )ρ (Φ,Φ′)e−iΦ′XX′, (22)
S,f S K K i
where exp[−iaKˆ]|Xi=|X +ai was used. Multiplication by exp[ikX¯] and by exp[ij·Φ¯], with X¯ =(X +X′)/2 and
Φ¯ =(Φ+Φ′)/2, followed by integration over X¯ and Φ¯ yields
φ φ x+φ x+φ
M (x,k;φ,j)= dΦ¯dX¯ei[xΦ¯X+j·Φ¯+(k+φX)X¯]ρ (Φ¯ + ,Φ¯ − )ρ (X¯ +Φ¯ + K,X¯ +Φ¯ − K)
S,f i S K K
2 2 2 2
Z
φ φ x+φ x+φ
= dΦ¯dY ei[xΦ¯X+j·Φ¯+(k+φX)(Y−Φ¯K)]ρ (Φ¯ + ,Φ¯ − )ρ (Y+ K,Y− K)
i S
2 2 2 2
Z
φ φ
= dΦ¯ei[xΦ¯X+j·Φ¯−(k+φX)Φ¯K]ρ (Φ¯ + ,Φ¯ − )M (x+φ ,k+φ )
i S K X
2 2
Z
= M (φ,j+γ)M (x+φ ,k+φ ) (23)
i S K X
withγ =(−k−φ ,x). The identity Φ X−Φ′ X′ =X¯φ +Φ¯ xwasused,andthe changeofvariableY =X¯+Φ¯
X X X X X K
was made.