On the Observation of Vacuum Birefringence

0.16 MB·English

by  T. Heinzl

On the Observation of Vacuum Birefringence
Thomas Heinzl∗
School of Mathematics and Statistics, University of Plymouth
Drake Circus, Plymouth PL4 8AA, UK
Ben Liesfeld, Kay-Uwe Amthor, Heinrich Schwoerer, and Roland Sauerbrey†
Institut fu¨r Optik und Quantenelektronik,
Friedrich-Schiller-Universit¨at Jena
Max-Wien-Platz 1, 07743 Jena, Germany
Andreas Wipf‡
Theoretisch-Physikalisches Institut,
Friedrich-Schiller-Universit¨at Jena
6 Max-Wien-Platz 1, 07743 Jena, Germany
0 (Dated: February 2, 2008)
0
2 We suggest an experiment to observe vacuum birefringence induced by intense laser fields. A
high-intensity laser pulse is focused to ultra-relativistic intensity and polarizes the vacuum which
n
then acts like a birefringent medium. The latter is probed by a linearly polarized x-ray pulse. We
a
calculate the resulting ellipticity signal within strong-field QED assuming Gaussian beams. The
J
laser technology required for detecting thesignal will be available within thenext threeyears.
0
1
PACSnumbers: 12.20.-m,42.50.Xa,42.60.-v

1
v
The interactions of light and matter are described by are way out of experimental reach for the time being
6
quantum electrodynamics (QED), at present the best- – unless one can utilize huge relativistic gamma factors
7
0 establishedtheoryinphysics. TheQEDLagrangiancou- produced by large scale particle accelerators [5, 6].
1 ples photons to charged Dirac particles in a gauge in- Alternatively,therearealsodispersive effectsthatmay
0 variant way. At photon energies small compared to the be considered. These include many of the phenomena
6 electron mass, ω ≪ me, electrons (and positrons) will studied in nonlinear optics as well as “birefringence of
0
generically not be produced as real particles. Neverthe- the vacuum” first addressed by Klein and Nigam [7] in
/
h less, as already stated by Heisenberg and Euler, “...even 1964,soonfollowedbymoresystematicstudies[8,9,10].
p in situations where the [photon] energy is not sufficient Inessence,thepolarizedQEDvacuumactslikeabirefrin-
p- formatterproduction,itsvirtualpossibilitywillresultin gent medium (e.g. a calcite crystal) with two indices of
e a‘polarizationofthevacuum’andhenceinanalteration refractiondepending onthe polarizationofthe incoming
h ofMaxwell’sequations”[1]. These authorswerethe first light. Inastaticmagneticfieldof5Talightpolarization
v: to explicitly derive the nonlinear terms induced by QED rotation has recently been observed [11]. The measured
i for small photon energies but arbitrary intensities (see signal differs from the QED expectations and may be
X also [2]). caused by a new coupling of photons to an hitherto un-
r The most spectacular process resulting from these observed pseudoscalar.
a
modifications presumably is pair production in a con-
Detection of the tiny dispersive effects is an enormous
stant electric field. This is an absorptive process as pho-
challenge. In this paper we point out that severalorders
tonsdisappearbydisintegrationintomatterpairs. Itcan
ofmagnitudeinfield strengthmaybe gainedbyemploy-
occurforfieldstrengthslargerthanthecriticalonegiven
inghigh-powerlasers. Distorting the vacuumwith lasers
by [3, 4]
has been suggested long ago [10] but was not considered
experimentally for lack of sufficient laser power. How-
m2
E ≡ e ≃1.3×1018V/m. (1) ever, recent progress in both laser technology and x-ray
c
e detection has lead to novel experimental capabilities. It
is therefore due time to specifically address the feasibil-
Inthis electricfieldanelectrongainsanenergym upon
e ity of a strong-fieldlaser experiment to measure vacuum
travelling a distance equal to its Compton wavelength,
birefringence. In the light of the results [11] such experi-
λ = 1/m . The associated intensity is I = E2 ≃4.4×
e e c c ments are also necessary in order to test whether strong
1029 W/cm2 such that both field strength and intensity
electromagneticfields providewindowsintonew physics.
We intend to utilize the high-repetition rate petawatt
class laser system POLARIS which is currently under
construction at the Jena high-intensity laser facility and
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] which will be fully operational in 2007 [12]. POLARIS
‡Electronicaddress: [email protected] consistsofadiode-pumpedlasersystembasedonchirped
2
Laser pulse through a region where a (strong) background field is
50/50 Beamsplitter present. The resulting corrections to pure Maxwell the-
x ory to leading order in the probe field a may be ex-
µ
z
pressed in terms of an effective action [10]
Off-axis
Off-axis parabolic
parabolic mirror
mirror δS ≡ 1 d4xd4ya (x)Πµν(x,y;A)a (y), (2)
Xpu-rlasey laser 2Z µ ν
where A denotes the background field and Πµν the po-
µ
Polarizer Analyzer larizationtensor. Tolowestorderina loop(or~)expan-
sion the former is given by the Feynman diagram
FIG.1: Proposedexperimentalsetupforthedemonstrationof
Π =
vacuum birefringence: A high-intensity laser pulse is focused µν
(3)
by an F/2.5 off-axis parabolic mirror. A hole is drilled into
the parabolic mirror in alignment with the z-axis (axes as
withthe heavylinesdenotingthedressedpropagatorde-
indicated) in such a way that an x-ray pulse can propagate
pending on the background field A,
alongthez-axisthroughthefocalregionofthehigh-intensity
laser pulse. Using a polarizer-analyzer pair the ellipticity of
the x-ray pulse may be detected. Shown in grey: Extension
S [A]≡ = + + +...
of the setup for the generation of counter propagating laser F
pulsesandahigh-intensitystandingwavewhichmaybeused (4)
for pair creation. Hence, S [A] is an infinite series of diagrams where the
F
nth term correspondsto the absorption and/or emission
of n−1 backgroundphotons (representedby the dashed
pulse amplification (CPA) which will be operating at external lines) by the “bare” electron.
Λ = 1032nm (Ω = 1.2 eV) with a repetition rate of The dressed propagator is known exactly only for a
0.1Hz. A pulse duration of about 140fs and a pulse en- few special background configurations (see [16] for an
ergyof150Jinprinciple allowsto generateintensities in overview). Typically, one obtains rather unwieldy inte-
the focalregionofI =1022W/cm2. This correspondsto gral representations which have to be analyzed numeri-
a substantial electric field E ≃2×1014V/m, still about cally. In our case, however, we can exploit the fact that
four orders of magnitude below Ec. weareworkingintheregimeofbothlowenergyandsmall
The proposed experimental setup is shown in Fig. 1. intensities leading to two small parameters [17], namely
A high-intensity laser pulse is focused by an off-axis
parabolic mirror. A linearly polarized laser-generated ν2 ≡ ω2/m2 ≃4×10−6 , (5)
e
ultra-short x-ray pulse is aligned collinearly with the fo- ǫ2 ≡ E2/E2 =I/I ≃2×10−8 . (6)
c c
cused optical laser pulse. After passing through the fo-
cus the laserinduced vacuumbirefringencewill leadto a Low intensity, ǫ2 ≪1,means thatwecanworkto lowest
smallellipticity ofthe x-raypulse which willbe detected nontrivial order in the external field i.e. O(ǫ2). In terms
by a high contrast x-ray polarimeter [13]. The whole of Feynman diagrams (3) then reduces to
setup is located in an ultra-high vacuum chamber and is
entirely computer controlled.
Shown in grey in Fig. 1 is an extension of the setup
Π = ≃ +
whichenablesustoaccuratelyoverlaptwocounterprop- µν
agatinghigh-intensitylaserpulses. Accuratecontrolover
(7)
spatial and temporal overlap was convincingly demon-
stratedcarryingoutanautocorrelationofthelaserpulses Low energy, ν ≪1,implies thatwemaysafely expand
at full intensity [14] and generating Thomson backscat- Πµν in derivatives or, after Fourier transformation, in
tered x-rays from laser-accelerated electrons [15]. This powers of the probe 4-momentum k = ω(1,nk) where
counter propagating scheme, a table-top “photon col- k2 = 1 and n ≥ 1 is the index of refraction. Thus, the
lider”, may also be employed for pair creation from the derivative expansion is in powers of ω2 or, equivalently,
vacuum. For the x-ray probe pulse we have chosen an of ν2. Again we restrict our analysis to leading order
x-ray source of photon energy ω ≃ 1 keV, since the which turns out to be ν2. The first vacuum polarization
birefringence signal is proportional to ω2 (see below) . diagram in (7) is O(ν4) while the second is O(ǫ2ν2) so
Our long-termplans areto replacethe presentsourceby we may safely neglect the former. The low-energy limit
an x-ray free electron laser (XFEL) or by a laser-based oftheremainingdiagramisobtainedfromthecelebrated
Thomsonbackscatteringsource[15]bothofwhichdeliver Heisenberg-Euler Lagrangian [1] which to leading order
ultrashort and highly polarized x-rays. in ǫ2 is given by
Refraction is a dispersive process based on modified
propagation properties of the probe photons travelling δL(S,P)= 21γ−S2+ 21γ+P2 . (8)
3
The basic building blocks in (8) are the scalar and pseu- eigenvalues of Πµν do not depend on the invariants S
doscalar invariants [4, 9] and P. Hence, under the assertion (14) constant fields
behaveas crossed fields (E andB orthogonalandof the
S ≡ −41FµνFµν = 21(E2−B2), (9) same magnitude) for which strictly S = P = 0. In
P ≡ −1F F˜µν =E·B , (10) addition, one has b2 =˜b2 and b·˜b=0 so that (12) turns
4 µν
into the spectral representation
whereF denotestheelectromagneticfield-strengthten-
µν
sor(comprisingbothbackgroundandprobephotonfield) Πµν =γ−bµbν +γ+˜bµ˜bν . (15)
andF˜ itsdual. Thenonlinearcouplingsin(8)aregiven
µν
by We read off that the (nontrivial) eigenvectors are given
by (13) correspondingto eigenvalues γ b2(k). Note that
α 1 ±
γ+ ≡7ρ, γ− ≡4ρ, ρ≡ 45πE2 , (11) b2 is the only nonvanishing invariant which can be built
c from crossed fields.
with α=1/137 being the fine-structure constant. Adopting a plane wave ansatz for the probe field aµ
To proceed we split the fields into an intense (laser) yields a homogeneous wave equation which in momen-
background and a weak probe field according to the re- tum space becomes linear algebraic. It has nontrivial
placement F →F +f with upper (lower) case let- solutions only if a secular equation holds which deter-
µν µν µν
ters for electromagnetic quantities henceforth referring mines the dispersion relations for k2. With the eigenval-
to the background (probe). In the following we regard uesgivenabovetherearetwo ofthem, k2−γ±b2(k)=0.
the plane wave probe field f as a weak disturbance on Inserting k = ω(1,nk), we finally obtain two solutions
µν
top of the strong background field F which we take for the index of refraction,
µν
as an electromagnetic wave of frequency Ω. It can be a
plane or standing wave or more realistic variants thereof n± =1+ 21γ±Q2 . (16)
like Gaussianbeams (see discussion below). In any case,
for the actual experiment we will have the hierarchy of The nonnegativequantity Q2 is an energydensity which
frequencies Ω≪ω ≪m in agreement with (5). in 3-vector notation becomes
e
Theleading-ordercontributiontothepolarizationten-
sor is found by performing the split F → F +f in the Q2 =E2+B2−2S·k−(E·k)2−(B·k)2 , (17)
Heisenberg-Euler action, δS = d4xδL, and writing it
with S = E ×B being the Poynting vector. The in-
in the form (2). This yields a pRolarization tensor
equality (14) holds as long as Q2 6= 0. The indices of
Πµν =−γ−k2S µν +γ−bµbν +γ+˜bµ˜bν , (12) refraction become maximal if probe and backgroundare
P counter propagating (‘head-on collision’), k = −S/|S|,
where µν = gµν −kµkν/k2 is the standard projection whereupon
orthogoPnaltokandS denotesthebackground invariant.
In addition we have introduced the new 4-vectors [9], Q2 =E2+B2+2|S|≡4I , (18)
bµ ≡Fµνk , ˜bµ ≡F˜µνk . (13)
ν ν withI denoting the backgroundintensity. Note that one
Note that we have b·k =0=˜b·k and hence Πµνk =0 gains a factor of four as compared to a purely electric
ν
or purely magnetic background. Plugging (18) into (16)
as required by gauge invariance. It is useful to diagonal-
the indices of refractionbecome n =1+2γ I or,upon
ize Πµν and rewrite it in terms of a spectral decompo- ± ±
inserting γ ,
sition. In full generality this is a bit awkward, but for ±
our purposes matters can be simplified. The eigenvalues
14 α 14 I
of Πµν in principle depend on the four invariantsk2, S, n =1+ ρI =1+ . (19)
±
P and b2. From (12) we note that there is no P de- (cid:26) 8 (cid:27) 45π (cid:26) 8 (cid:27)Ic
pendence and that only the combination k2S appears.
Tothebestofourknowledge,thesevalueshavefirstbeen
Let us count powers of ǫ and ν to determine the relative
obtained in [8]. They imply birefringence with a relative
magnitudesofthe invariants. Ifwe writethe index ofre-
phase shift between the two rays proportional to △n ≡
fractionasn=1+∆weexpect∆=O(ǫ2)the deviation
of n from unity being due to the external fields. Hence n+−n−,
k2 is no longer zero but rather k2 = O(ǫ2ν2) implying
d 4αd I 4αd
k2S = O(ǫ4ν2). For generic geometrical settings (see △φ=2π △n= = ǫ2 . (20)
λ 15 λI 15 λ
below) the invariant b2 = O(ǫ2ν2). The upshot of this c
power counting exercise is the important inequality
A realistic laser field will lead to an intensity distri-
|k2S|≪|b2|, (14) bution along the z-axis (choosing k = ez). If z0 mea-
sures the typical extension of the distribution we may
by means of which we may neglect k2S. This justifies set s ≡ z/z0 and write the intensity as I(s) = I0g(s)
the statementin[16]thatto leadingorderinǫandν the with peak intensity I0 and a dimensionless distribution
4
quantity to be determined is δ2 ≃ (1∆φ)2. In Table I
TABLEI: Numericalvaluesforthephaseshift(21)andellip- 2
expectedellipticity valuesforgivenexperimentalparam-
ticity signal δ2. First line: present specifications of the Jena
eters are listed.
laserfacility. Secondline: optimalscenariowithXFELprobe
and large Rayleigh length. The peak intensity is taken to be
I0=1022 W/cm2. These results clearly show the challenging nature but
alsothefeasibilityoftheproposedexperiment. Presently,
ω / keV λ / nm z0 / µm △φ / rad δ2 a petawatt class laser facility such as POLARIS is ex-
1.0 1.2 10 1.2×10−6 3.4×10−13 pectedtoreachabout1022W/cm2atunprecedentedrep-
15 0.08 25 4.4×10−5 4.8×10−10 etition rates of ∼ 0.1Hz [12]. The values of δ2 obtained
for such lasers (Tab. I) are at the limit of the accuracy
that can now be obtained with high-contrast x-ray po-
larimetersusingmultipleBraggreflectionsfromchannel-
function g(s). The phase shift (20) is then replaced by
cut perfect crystals [13, 19, 20]. These instruments are
the expression [18] in principle capable of a sensitivity of δ2 ≃ 10−11 [20].
Since the expected signal is proportional to both I2 and
△φ= 4αz0I0κ, (21) λ−2 it may be greatly enhanced by increasing the laser
15 λ I
c intensity or choosing a smaller probe pulse wave length.
where the correction factor κ is the integral For example, with the proposed ELI laser facility reach-
ing 1025W/cm2 [21] a sensitivity of the polarimeter of
s0 only 10−7...10−4 is required which is within presently
κ=κ(s0)≡Z dsg(s)=O(1). (22) demonstrated values of sensitivity [13]. The required x-
−s0
ray probe pulse may be generated either with an XFEL
Here, s0 denotes the half-width of the intensity distribu- synchronizedto a petawatt laser or by the use of Thom-
tion in units of z0. In general it is a reasonable approxi- son scattered laser photons from monochromatic laser
mationto lets0 →∞. Forasingle Gaussianbeam,z0 is accelerated electron beams [15, 22].
theRayleighlengthandtheintensityI1 followsaLorentz
curve, hence g1(s) = 1/(1+ s2) implying κ1(∞) = π. Itseemsworthwhiletopointoutthatalthoughastand-
Identifying d = 2z0 this differs from (20) by a factor ingwaveforthebackground(whichmaybecreatedinthe
of π/2 = O(1). For two counter propagating Gaussian “photoncollider”setupasshowninFig.1)doesnotlead
beams (‘standing wave’) obtained from splitting a beam to an increase in integrated intensity and hence of the
ofintensity I1 one gains a factorof two inpeak intensity birefringence signal, it does yield double peak intensity.
but the distribution gets thinned out due to the usual This is important for the observation of effects sensitive
cos2 modulation,whichcancelsthegaininintensitylead- to localized intensity like Cherenkov radiation and pair
ing to the same correction factor κ2 =π =κ1. production.
A linearly polarized electromagnetic wave undergoing
vacuumbirefringencewith apolarizationvectororiented This work was supported by the DFG Project TR 18.
under an angle of 45◦ with respect to both background The authors gratefully acknowledge stimulating discus-
fieldsEandBwillberenderedellipticallypolarizedwith sionswithH.Gies,A.Khvedelidze,M.Lavelle,V.Malka,
ellipticity δ (ratio of the field vectors). In the experi- D.McMullan,G.Mourou,A.Nazarkin,A.Ringwaldand
ment, intensities will be measured and the experimental O. Schr¨oder.
[1] W. Heisenberg and H.Euler, Z. Phys. 98, 714 (1936). [12] J. Hein et al., Appl.Phys. B79, 419 (2004).
[2] V.Weisskopf, K.Dan.Vidensk.Selsk. Mat. Fys.Medd., [13] M. Hart et al., Rev.Sci. Instrum. 62, 2540 (1991).
14, 6 (1936), reprinted in Quantum Electrodynamics, [14] B. Liesfeld et al., Appl. Phys.Lett., 86, 161107 (2005).
J. Schwinger, ed., Dover, New York1958. [15] H.Schwoereretal.,Phys.Rev.Lett.(2005),acceptedfor
[3] F. Sauter,Z. Phys. 69, 742 (1931). publication.
[4] J. Schwinger, Phys. Rev. 82, 664 (1951). [16] W. Dittrich and H. Gies, Probing the quantum vacuum,
[5] C. Bula et al., Phys. Rev.Lett. 76, 3116 (1996). vol.166ofSpringer Tracts Mod. Phys.(Springer,Berlin,
[6] D.Burke et al., Phys. Rev.Lett. 79, 1629 (1997). 2000).
[7] J. Klein and B. Nigam, Phys. Rev. 135, B1279 (1964). [17] I. Affleck and L. Kruglyak, Phys. Rev. Lett. 59, 1065
[8] R.BaierandP.Breitenlohner,ActaPhys.Austriaca25, (1987).
212 (1967); NuovoCim. B 47, 117 (1967). [18] K. Koch, diploma thesis, Jena (2004), in German.
[9] Z.Bial ynicka-BirulaandI.Bial ynicki-Birula, Phys.Rev. [19] Y. Hasegawa et al., Acta Cryst. A 55, 955 (1999).
D2,2341 (1970). [20] E. Alp, W. Sturhahn, and T. Toellner, Hyperfine Inter-
[10] E. Brezin and C. Itzykson,Phys.Rev. D3,618 (1970). actions 125, 45 (2000).
[11] E. Zavattini et. al., PVLAS collaboration (2005), [21] G.MourouandV.Malka,privatecommunication(2005).
hep-ex/0507107. [22] J. Faureet al., Nature431 (7008), 541 (2004).