From Single to Multiple-Photon Decoherence in an Atom Interferometer

4 Pages·0.17 MB·

by  David A. Kokorowski

From Single to Multiple-Photon Decoherence in an Atom Interferometer
David A. Kokorowski,Alexander D. Cronin, Tony D. Roberts, and David E. Pritchard
Massachusetts Institute of Technology, Cambridge, MA 02139
(February 1, 2008)
coherenceofatomswithinanatominterferometerdueto
We measure the decoherence of a spatially separated
spontaneous scattering of photons. In the many photon
atomic superposition due to spontaneous photon scattering.
limit, this represents a simple case of the generalmodels
We observe a qualitative change in decoherence versus sepa-
above; we observe coherence loss consistent with Eq. (2)
rationasthenumberofscatteredphotonsincreases, andver-
andareabletoderivethedecayconstantfromfirstprinci-
ifyquantitativelythedecoherencerateconstantinthemany-
ples. The few photon limit is of a qualitatively different
photon limit. Our results illustrate an evolution of decoher-
1 character, and we have followed the smooth transition
0 ence consistent with general models developed for a broad
between these two regimes.
0 class of decoherencephenomenon.
The atom interferometer [10] is realized by pass-
2
03.65.Bz,03.75.Dg ing a collimated, supersonic beam of Na atoms
n
(velocity 3000 m/s using a He carrier gas) through
a ≈
three diffraction gratings arranged in the Mach-Zehnder
J
Decoherence is the result of entanglement between a geometry (Fig. 1). Prior to the first grating, the
8
quantum system and an unobserved environment, and atoms are collimated and optically pumped into the
1
manifestsasthereductionofcoherentsuperpositionsinto 3S1 F =2,mf =+2 groundstate. Twopaths through
2| i
2 incoherentmixtures. Thisreductionoccursmorequickly the interferometer, separated by up to 20µm, overlap at
v as the number of particles comprising a quantum sys- the position of the third grating, forming a spatial in-
4 temincreases,establishingdecoherenceasafundamental terference pattern. This pattern is masked by the third
4
limit to large-scale quantum computation [1] and com- grating and the total transmitted flux is detected using
0
munication [2]. Progress in these fields therefore relies a 50µm hot wire. The interference pattern is measured
9
0 upon understanding and correcting for decoherence ef- as an oscillating atomic flux versus grating position. Be-
0 fects. On a macroscopic scale, decoherence is unavoid- cause the contrast of the interference pattern is propor-
0 able and explains the emergence of classical behavior in tionaltothecoherencebetweenthe twopaths,reduction
/ a world governedby quantum mechanical laws. in contrast is direct evidence of coherence loss.
h
Theoretical treatments of decoherence provide a de-
p
- scription for the evolution of a system’s density matrix
t undertheinfluenceofaspecificenvironment. Forspatial
n
a decoherence, various environments including a thermal
Detector
u bath of harmonic oscillators[3], a scalar field [4], and an
q isotropic distribution of scatterers [5,6] have been stud- x^ d
:
v ied. In the high-temperature or many scatterer limit, Atom z
i thesemodelsallyieldadiffusion-likemasterequationfor
X Beam(cid:9)
the system’s spatial density matrix, ρ(x,x′):
r
a
dρ i
= [H,ρ] D2 x x′ 2ρ, (1) Gratings
dt −¯h − | − | Scattering
where H is the Hamiltonian for the isolated system and Laser
D, the diffusion constant, depends on the details of the FIG.1. Aschematicofourapparatus: aMach-Zehnderin-
system-environment coupling. Assuming negligible in- terferometer comprised of three, evenly spaced, transmission
ternal dynamics, this equation predicts an exponential gratings. Within the interferometer, sodium atoms continu-
reduction in coherence with time and with separation ouslyabsorbandspontaneouslyemitphotonsfromavariable
squared [7]: intensity laser beam. Decoherence due to spontaneous emis-
sion results in reduced contrast interference fringes.
ρ(x,x′,t)=e−D2|x−x′|2tρ(x,x′,0). (2)
The effective decohering environment consists of pho-
Similardecoherencebehaviorarisesandhasbeenstudied tons from a laser beam directed along the xˆ axis which
in the context of anatom interacting with a high-Q cav- intersects both interfering paths. The circularly polar-
ity [8], and trapped ions interacting with a fluctuating izedlaserlightistunedtothe 3S1 2,+2 3P3 3,+3
electric field [9]. 2| i→ 2| i
transition with wavelength λ=2π/k =590 nm. Be-
0
To investigate the distinct case of decoherence due to
cause the atoms are dipole forbidden from decaying to
scattering processes, we have studied the loss of spatial
1
any state other than 3S1 2,+2 , they can continuously Inourexperiment,thetotalnumberofphotonsscattered
2| i
scatterphotonswithoutfallingoutofresonance(thenat- byanindividualatomisintrinsicallyuncertain,butisde-
ural linewidth is 200 photon recoils wide). scribed by the distribution P(n) which can be measured
At the intersec∼tion of the atomic beam and scattering or calculated. The sum in Eq. (6) is a trace over this
laser, each atom’s transverse wavefunction is peaked at additional degree of freedom of the environment.
two positions which we label x and x+d. If a photon,
initiallyinmomentumstate k0 ,scattersfromthisatom, 1.0
| i
the two become entangled:
ψ = x + x+d k scat. |total0.8
(cid:12)(cid:12) (cid:11)i (cid:16)(cid:12)x|(cid:11)i⊗(cid:12)φ|x(cid:11)+i(cid:12)(cid:17)x+⊗d|(cid:11)0⊗i−e→ik0d(cid:12)φx+d(cid:11), (3) bntrast = | 0.6
where φ is(cid:12)thew(cid:12)avefunc(cid:12)tionofaphoto(cid:12)nspontaneously Co 0.4
emitted| fxriompositionxandthe factoreik0d accountsfor ve
ati
thedifferenceinspatialphaseoftheinitialphotonatthe el 0.2
R
two positions. Generalizing the entangled wavefunction
in Eq. 3 to a density matrix and tracing over a basis of 0.0
scattered photon states, the net effect of scattering on 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
the atom’s spatial density matrix is: Path Separation, d (units of l )
FIG.2. The total decoherence function, |βtotal|, measured
ρ(x,x+d)scat.ρ(x,x+d)β(d), (4) asthenormalized contrast after spontaneousphoton scatter-
−→ ing. The solid lineis thesingle photondecoherence function.
whereβ(d)isknownasthedecoherencefunctionandhas Alsodisplayedarethebestfitsfromwhichwedeterminen¯=
the properties β(d) 1 andβ(0)=1. The decoherence 0.9 (△),1.4 (⋄), 1.8 (◦), 2.6 (▽), and 8.2 (+).
| |≤
functionthusdefinedisequaltothe innerproductofthe
two final photon states, which are identical apart from Figure2showsmeasurementsofthedecoherencefunc-
an overall translation: tion for laser intensities corresponding to an average
number of scattered photons, n¯, ranging from 1 to
β(d)=eik0d φx φx+d =eik0d φx e−ikˆxd φx 8. Ateachintensity,areferencecontrastandpha∼sewas
h | i h | | i ∼
measured, with the scattering laser positioned such that
= d∆kP(∆k)e−i∆kd, (5)
Z theinterferingpathswerecompletelyoverlapped(d=0).
We then adjusted the longitudinal position of the scat-
where the operator kˆx is the generator of photon trans- tering laser, z, to select specific path separations in the
lationsalongthexˆ axis. The resultingdecoherencefunc- range0<d<1.4λ atwhich to measurethe decoherence
tionistheFouriertransformofaprobabilitydistribution function(see Fig.1). For eachpathseparation,the ratio
P(∆k),with∆k =kx k0beingthechangeinmomentum of the measured atom interference contrast to the ref-
−
of the photon along the xˆ axis. erence contrast yields the magnitude of the decoherence
Previous experiments [11,12] have measured the deco- function, β (d). The difference between the mea-
total
| |
herence function for an atom which spontaneously scat- sured atom interference phase and the reference phase
ters a single photon. The theoretical prediction which yields the phase of the decoherence function.
these experiments confirm is displayed as the solid line We fit the data using Eq. (6) and taking
in Fig. 2. Beneath an overall decay in coherence with P(n) exp 1(n n¯)2/σ2 . This form was chosen as
≃ −2 − n
distance, periodic coherence revivals are observed. This agoodappr(cid:0)oximationtoMo(cid:1)nte-CarloWavefunctioncal-
shape follows directly from the Fourier transform of the culationsofP(n)forourlaserparameters. Fromthebest
dipoleradiationpatternforspontaneousemission. Ithas fit curves displayed in Fig. 2, values were extracted for
alsobeenexplainedintermsoftheabilityofasinglepho- n¯ and σ whichwere consistentwith, and moreaccurate
n
ton to provide which-path information [12]: the contrast than, independent measurements of P(n) based on the
dropsto zerowhen the path separationis approximately deflection and broadening of the atomic beam with the
equal to the resolving power of an ideal Heisenberg mi- scattering laser blocked versus unblocked.
croscope d λ/2, with revivals resulting from path am- In the regime d λ, a single scattered photon suf-
≈ ≫
biguity due to diffraction structure in the image. fices to completely destroythe coherencebetween paths.
Ifseveralphotons arescattered,andif successivescat- Thus, the non-zero asymptotic value (for n¯ =0.9 in Fig.
tering events are independent, the total decoherence 2) of the decoherence function at large path separation
function includes one factor of β for each scattered pho- is equal to the fraction of atoms which scatter zero pho-
ton: tons(i.e. decoherenceisproportionaltotheatom-photon
∞ interaction cross-section). This phenomenon is a sim-
β (d)= P(n)βn(d). (6) ple example of saturation of decoherence [6,13]: the loss
total
X
n=0
2
of coherence becomes independent of path separation at 1
a characteristic length scale of the environment. A re-
7
cent experiment by Cheng and Raymer [14], involving |al 6
loss of optical coherence due to a disordered collection tot 5
4
oowf pn:olycsotnytrreansetmloiscsrowspasheorbess,erhvaesdfetoatusaretusrsaitmeilwarhetno othuer bst = | 3
a
path separation reachedroughly the diameter of the mi- r 2
nt
crospheres,andtheasymptoticcontrastwasproportional o
C
to the microsphere-light scattering cross section. e 0.1
As the averagenumber of scatteredphotons increases, v
theoverallamountofdecoherenceincreases,andthecon- elati 67
trastrevivalsdisappear. Thisbehaviorcanbeformalized R 5
4
as the Fourier transform of the total momentum distri-
3
bution of all scattered photons:
0 2 4 6 8 10 12 14
βn(d)= d∆KP(∆K)ei∆Kd, (7) n(cid:13)
Z
FIG.3. Loss of interfering contrast as a function of mean
number of photons spontaneously scattered by atoms within
where∆K = n ∆k . Asn ,thecentrallimitthe-
i=1 i →∞ the interferometer. Each curve represents a different path
orem predictsPthat P(∆K) will tend towards a Gaussian
separation: d/λ=0.06(△),0.13(⋄), and 0.16(◦).
with mean nk and variance nσ2 (where σ = 2k is the
0 k k 5 0
rmstransversemomentumofanemittedphoton). Inthe
The data follow a nearly exponential decay with n¯.
case of spontaneous emission, P(∆K) is approximately
The upward trend at large n¯ is a result of the finite size
Gaussianforn>3andthedecoherencefunctionreduces
ofourhot-wire: thetraceoverfinalphotonstates(Eq.8)
to:
must be restricted to those states which allow the atom
toreachthe detector. AsaresultκinEq.(9)isreplaced
βn(d)=Z d∆K he−12(∆K−nk0)2/nσk2i ei∆Kd (8) with κ′ where 1/κ′2 =1/κ2+1/κ2d and κd =3.3(1)k0 is
our detector’s effective momentum acceptance.
=e−21nσk2d2e−ink0d. In the previous single-photonexperiment of Chapman
etal.[12]lostcoherencewassimilarly“recovered”bypo-
InsertingthisexpressionintoEq.(6)andtakingd/λ 1,
≪ sitioning a hot-wire detector to count only atoms which
we find:
had scattered photons into a small range of momentum
lim βtotal(d)=e−12κ2d2e−in¯k0d, (9) states. This scheme required [15] that the atomic beam
n¯→∞ width, σx, be greater than the path separation, d, so
that the two interfering paths partially overlapped at
where
the pointofscattering,andascatteredphotoncouldnot
κ2 =n¯σ2+σ2k2 (10) have provided complete which-path information, even if
k n 0
d λ. The condition σ < d need not be satisfied to
x
≫
is the varianceofthe totalmomentumtransferredto the demonstrate the features of decoherence in the current
atom from the scattered photons. The first term in Eq. experiment, however. Even when it is in principle possi-
(10) comes from the trace over modes available to the ble to recoversome coherence by measuring the environ-
spontaneously emitted photon, while the second is re- ment, if no suchattempt is made then the predicted loss
lated to the uncertainty in number of absorbed photons of contrast is independent of σx.
combined with the fixed phase k d imparted by each. In the many photon limit, the decoherence function
0
If σ = √n¯ (i.e. Poissonian statistics), Eq. (9) pre- we have derived agrees with the solution to the master
n
dicts an exponential decay in contrast with number of equation presented in the introduction. Comparing Eqs.
scattered photons (κ2 n¯). If in addition the scattering (2) and (9), taking into account the time varying inten-
∝
rate, Γ, is constant, then n¯ = Γt and the decoherence sity profile, I(t), of the scattering light as experienced
has exactly the exponential form derived from a master by atoms in the beam, we identify: κ2 = D2τ where
equation like Eq. (1). τ is the amount of time needed to scatter n¯ photons
τ
We have measured this exponential reduction of spa- (n¯ = Γ(I(t))dt). Because the atom-photon scatter-
0
tial coherence by varying the average number of scat- ing intReraction is well defined, and our decohering envi-
teredphotons,leavingthe pathseparationfixed(Fig.3). ronment well controlled, we can accurately calculate the
Theory curves (solid lines) are based on Eq. (9) with σ constant κ (equivalently D) for any laser parameters.
n
determinedfromthe broadeningofthe atomicbeamdue Displayed in Figure 4 are data which demonstrate
tothemomentumofthescatteredphotons. Theproduct Gaussian reduction in contrast as a function of path
of the two remaining free parameters, n¯d, was obtained separation for two different laser intensities. As be-
from the measured phase of the decoherence function. fore, we independently determined n¯ and σn for each
3
intensity, and from these values along with κ we cal- limitexhibitfeaturesofdecoherencewhicharequitegen-
d
culate κ′ = 2.5(1)k for the higher laser intensity and eral. We have observed the exponential coherence loss
0
κ′ =1.8(1)k for the lower. Fitting the contrast data to with time and path separation squared characteristic of
0
Eq. (9) yields κ′ = 2.39(5)k and κ′ = 1.71(5)k , within this generalbehavior,and we havefor the first time pre-
0 0
error of the calculated values. dicted and experimentally verified the decoherence rate
constant κ. The particular model we have explored is
1.0 not only the most relevant for macroscopic systems but
also applies generally to situations in which decoherence
|otal 0.8 arisesslowlythoughaseriesofindependent,mildlydeco-
t
heringinteractions,asituationofinterestfordecoherence
b= |
st 0.6 avoidance or correction protocols.
a ThisworkwassupportedbyArmyResearchOfficecon-
r
nt tracts DAAG55-98-1-0429 and DAA55-97-1-0236, Office
o
C 0.4 of Naval Research contract N00014-96-1-0432, and Na-
e
v tional Science Foundation grant PHY-9877041.
ati 0.2
el
R
0.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30
Path Separation, d (units of l )
FIG.4. Lossofcontrastinthemany-photonregime. Over- [1] W. G. Unruh,Phys. Rev.A 51, 992 (1995).
layed are theory curves generated from Eq. (9) using pa-
[2] H.-J. Briegel et al., in The Physics of Quantum Infor-
rameters (•) n¯ = 4.8(2), σn = 1.8(1) and (◦) n¯ = 8.1(3), mation, edited by D. Bouwmeester, A. Ekert, and A.
σn = 3.5(1) determined from independent beam deflection Zeilinger (Springer-Verlag, Berlin, 2000), pp. 281–293.
measurements. [3] A. O. Caldeira and A. J. Leggett, Physica A 121, 587
(1983).
Our systemexhibits what havebeen referredto as the [4] W. G. Unruh and W. H. Zurek, Phys. Rev. D 40, 1071
“naive”[13]generalizationsofdecoherencephenomenon:
(1989).
exponentiallossofcontrastwithpathseparationsquared [5] E. Joos and H. D. Zeh, Z. Phys. B-Condens. Mat. 59,
andwithnumberofscatteredparticles. Thesimilarityof 223 (1985).
Eq. (1) to a diffusion equation [16], invites identification [6] M. R. Gallis and G. N. Fleming, Phys. Rev. A 42, 38
ofthis type ofdecoherencewithphasediffusionora ran- (1990).
dom phase walk. To make the identification explicit, we [7] For one of the earliest derivations of a solution of this
use the identity φ = e−ikˆxd φ to rewrite Eq. (3) form, see Walls, D.F. and Milburn, G.J., Phys. Rev. A
x+d x
as: | (cid:11) | (cid:11) 79, 2403 (1985).
[8] M. Bruneet al., Phys.Rev.Lett. 77, 4887 (1996).
ψ scat. x φ + x+d eik0de−ikˆxd φ = [9] C. J. Myatt et al.,Nature 403, 269 (2000).
(cid:12) (cid:11)i −→(cid:12) (cid:11)⊗(cid:12) x(cid:11) (cid:12) (cid:11)⊗ (cid:12) x(cid:11) [10] J.Schmiedmayeretal.,inAtomInterferometry,Vol.Ad-
(cid:12) d~k x(cid:12) +e−(cid:12)i(kx−k0(cid:12))d x+d ~k ~k φ (cid:12) (11) vances in Atomic and Molecular Physics of Advances
Z (cid:16)| i | i(cid:17)⊗| ih | xi in Atomic, Molecular and Optical Physics, edited by P.
Berman (Academic Press, San Diego, 1997), pp.2–83.
In this expression for the entangled atom-photon wave- [11] T. Pfau et al., Phys.Rev.Lett. 73, 1223 (1994).
function, a photon state ~k corresponds to an atomic [12] M. S.Chapman et al.,Phys.Rev.Lett.75,3783 (1995).
| i
superpositionstatewiththephasebetweenthetwocom- [13] J. R. Anglin, J. P. Paz, and W. H. Zurek, Phys. Rev. A
ponents shifted by an amount ∆φ = (kx k0)d. Corre- 55, 4041 (1997).
−
lating interference data with measurements of eachscat- [14] C. C. Cheng and M. G. Raymer, Phys. Rev. Lett. 82,
teredphotonmomentum(effectivelyarandomlysampled 4807 (1999).
element of the distribution P(k)) would allow complete [15] H. M. Wiseman et al.,Phys. Rev.A 56, 55 (1997).
recovery of lost contrast. In the absence of such post- [16] W. H. Zurek,Phys. Today 44, 36 (1991).
processing, however, the phase of each atom’s interfer- [17] A. Stern, Y. Aharonov, and Y. Imry, Phys. Rev. A 41,
ence fringes will vary randomly, andtheir sum, the mea- 3436 (1990).
sured interference pattern, will have reduced contrast.
The phase diffusion and (previously discussed) which-
path pictures are equally valid when the experimenter
does not measure the scattered photons [17].
In conclusion, we have studied the decoherence of a
spatialsuperpositionduetophotonscattering. Ourdata
confirm theoretical predictions, and in the many-photon
4