Scattering distributions in the presence of measurement backaction

0.48 MB·English

by  James S. Douglas

Scattering distributions in the presence of measurement backaction
James S. Douglas
Institut de Ciencies Fotoniques, 08860 Castelldefels, Barcelona, Spain
Keith Burnett
University of Sheffield, Western Bank, Sheffield S10 2TN, United Kingdom
Scattering probe particles from a quantum system can provide experimental access to informa-
tion about the system’s state. However, measurement backaction and momentum transfer during
scatteringchangesthestateofthesystem,potentiallydestroyingthestateofthesystemwewishto
probe. Hereweinvestigatehowtoprobethesystem’sinitialstateeveninthepresenceofbackaction
andmomentumtransfer. Weshowthatsummingthescatteringdistributionsofanensembleofmea-
3
surementsrevealstheinitialstatescatteringpatternevenwheneachoftheindividualmeasurements
1
completely destroystheinitialstate. Thisprocedureiseffectiveprovidedthescatteringtakesplace
0
on a timescale that is short compared with thefree evolution of thesystem.
2

n PACSnumbers: 03.65.Ta37.10.Jk03.65.Nk
a
J
9 I. INTRODUCTION caused by consecutive scattering and measurement, and
1 see how quantum backaction changes the scattering dis-
tributionformultipleparticlesystems. Aswehaveprevi-
] Scattering of particles from matter is a fundamental ouslyshown,scatteringleadstolocalizationofthesystem
h method used to reveal a system’s properties. Ruther-
particles in position space and, in some regimes, super-
p
ford’s revolutionary discovery of the atomic nucleus in-
- positions of localized particle configurations [11]. Here
t volved scattering alpha particles from gold foil [1], while
n we show that the average scattering pattern for an en-
scattering of x-ray photons has led to numerous discov-
a sembleofexperimentsgivesthepatternpredictedforthe
eriesaboutcrystalline structure,suchasthe structureof
u initial state even if this initial state is completely de-
q DNA [2]. The range of phenomena investigated by scat-
stroyed in each individual scattering experiment. This
[ teringisbroad,whereforexampleneutronscatteringhas
is possible when the scattering measurements all occur
been used to probe the nature of exotic forms of matter
1 on a timescale shortcomparedwith the free evolutionof
such as superfluid helium and high temperature super-
v the system. Experimental access to the initial state of
0 conductors[3],andscatteringofopticalphotonshasbeen the system is hence possible even for systems that are
8 used to probe correlations within Bose-Einstein conden-
easilyperturbed. Thisisanimportantresultthatweex-
5 sates [4]. In all these cases probe particles scatter from
pectcanbeutilized,forexample,inthefieldofquantum
4 the matter because of density or charge fluctuations in
simulation,whereexperimentsusingcoldatomsareused
.
1 the system or by creating or destroying excitations in
to simulate many-body physics [12, 13]. The outcome
0 thesystem. Measuringthescatteringpatternthenyields
of each cold atom experiment can be measured by scat-
3 valuableinformationaboutthesystem’sspatialstructure
tering light or matter from the system [14–20] and then
1 and excitations, and furthermore interference between
: the state of the system can be reproduced by combining
v different scattering channels reveals information about
results from multiple runs of the experiment.
i correlations within the system.
X
Typically in experiments and theoretical discussions
r
a involving scattering, the effect of the probe on the sys- II. CONSECUTIVE SCATTERING FROM
tem is assumed to be weak and the scattering pattern MANYBODY STATES
is calculated in the Born approximation from the ini-
tial system state [5]. However during each scattering
The model of consecutive scattering that we use was
event energy and momentum may be transferred from
presented in Ref. [11] and we reproduce the relevant re-
theprobetothesystem. Furthermoretheactofmeasur-
sults here for completeness. We consider coherent scat-
ingtheprobe’sscatteringpatternmayproducequantum
tering of a probe beam from a system of identical par-
backactiononthestateofthesystem[6,7]. Theseeffects
ticles, where the probe particles are assumed to be in
cancombinetosignificantlyalterthestateofthesystem,
approximately plain wave form with initial wave-vector
indeedfortwoparticles,scatteringoflightleadstolocal- k . Through scattering the probe can acquire a new
i
ization of relative position while the particles’ momen- wave-vector k , which is then associated with a mo-
f
tum states become entangled [8–10]. This complicates mentum transfer to the system of ~k = ~(k k ).
i f
the use of scattering to probe the state of a system, as −
For a weak probe-system interaction the probability of
afterasmallnumberofscatteringeventsthesystemmay
multiple scattering is low and we can then assume this
be in a completely different state from which it started.
momentum transfer occurs in a single scattering event.
Here we model the evolution of manybody states For an N-particle state ψ = dRψ(R)R , where
| i | i
R
2
R = r ,...,r gives the coordinates of the N par- Together consecutive scattering and non-scattering
1 N
{ }
ticles, the scattering interaction results in a combined events lead to a dynamic evolution of the many-body
probe-system state of state. Neglecting, for the moment, the free evolution of
the system between each scattering event, the net result
ψ ki dRdkfS(k,R)ψ(R)R kf (1) of the scattering evolution is that each scattering mea-
| i⊗| i−→ | i⊗| i
Z surement provides a partial measurement of the relative
where positions of the system particles. In the limit of an in-
S(k,R)= √g4π Nj=1eik·rj for kf 6=ki (2) fiofnittheensuymstbeemr opfarstciactletesrianrge edveetnertms itnheedreelxaatcivtley,paotsitleioansst
( PA(R) for kf =ki in principle, and the resulting state is an eigenstate of
relative position, such as the completely localized state
This state is superposition of all possible single scatter-
R . The probability of the state ending in R is given
ing events along with the possibility that the probe did
notscatterandremainsin k [11]. Thefactorg isgiven |byithe initial state wavefunction ψ(R)2 an|dithe scat-
| ii tering pattern then observed is S|(k,R|)2. If we then
by the strength of the probe-system interaction and for
| |
simplicityweassumeg isindependentofk ,whichisthe do multiple repetitions of the experiment and image the
f
scattering pattern for each, we will get an ensemble of
casewhenenergyconservationrestrictsscatteringsothat
k k [5]. This occurs when the probe particles are scattering patterings corresponding to various R with
| f|∼| i| probability ψ(R)2. Summing the scattering|paitterns
much lighter than the system particles, as in the case of
| |
overthe ensemble then givesthe same resultasthe scat-
light scattering,or can occur as we see below in the case
tering pattern for the initial state (without backaction)
of cold atoms in optical lattices where the system par-
ticles are confined to energy bands. We further assume dR S(k,R)ψ(R)2.
| |
the probe-system interaction is isotropic, as is the case More generally for a finite number of
R
for s-wave scattering in cold atoms. The treatment can scattering events, m, the state becomes
be generalized to take into account anisotropic scatter- dRS(k1,R)...S(km,R)ψ(R)R . Then in
| i
ing,suchasthedipolepatterninlightscattering,butthis an ensemble of measurements with m events
R
does not affect the qualitative behavior discussed below. each, any scattering event we choose, say
Measurement of the final probe wave-vector, for ex- j, has a probability distribution given by
ample by imaging in the far field, causes a backac- dk1...dkj 1dkj+1...dkm S(k1,R)...S(km,R)ψ(R)2 =
tion on the system state collapsing this superposition dRS(kj,R−)ψ(R)2, that |is the initial state pattern.|
R | |
ψ dRS(k,R)ψ(R)R . If a probe particle is Combining the scattering patterns from an ensemble of
o|biser−v→ed to be scattered w|ithiwave-vector k then the mR easurements then gives an estimate of the initial state
f
newsystemR stateisasuperpositionofeachsystemparti- scatteringpatternwitha samplingerrorthatscaleswith
cle having receivedmomentum ~k from the probe, given 1/√n where n is the number of experimental runs. This
thenewstateabroaderrelativemomentumdistribution. occurs despite the fact that scattering leads the initial
The superposition occurs as detecting the probe’s final state to be destroyed in each individual run.
wave-vector reveals no information about which particle
scattered it. As the system’s state broadens in relative
momentum space it localizes in relative position space III. SCATTERING AND FREE EVOLUTION IN
[8]. However, because the scattering probability distri- THE BOSE-HUBBARD MODEL
butioniscompletelydeterminedbytherelativepositions
of pairs of system particles, the backaction of the mea- Theanalysisaboveassumesthesystemdoesnotevolve
surement on the state will then preserve superpositions while the scattering measurements are taking place. To
of states that have the same set of relative position vec- examinehowthe system’sfree evolutionaffects the mea-
tors. Measuringthescatteringdistributioncanthenlead surement we now examine a specific example that dis-
to spatial superpositions of the system particles as we plays the key characteristics of the dynamic scatter-
describe in Ref. [11]. ing process. We consider scattering from a system of
Due tothesmallvalueofg itisoftenthe casethatthe Bosons on a one-dimensional lattice described by the
probe particles do no scatter from the system. Naively Bose-Hubbard model. This can be achieved in experi-
one would assume that this does not affect the system, ment by matter-wave scattering from a one-dimensional
but the measurement of a non-scattering event also re- lattice of bosonic atoms in an optical lattice [16, 17, 21].
veals information about the state and leads to measure- We take the lattice to have M sites and to be oriented
ment backaction [8]. This is because some system con- alongthey-axis,whiletheinitialwave-vectoroftheprobe
figurations scatter less than others and observing non- isinthex-direction. Forsimplicityweonlyconsiderscat-
scatteringprojectsthestatetowardtheseconfigurations. tering within the xy-plane, where the full three dimen-
The non-scattering amplitude A(R) is fixed by the con- sional scattering is a straightforwardgeneralization.
dition that probe particles must either be scattered or At low temperature the atoms all reside in the lowest
non-scattered, that is 1 = A(R)2+ k=0dk|S(k,R)|2 ≡ band of the lattice and the energy of the probe can be
dkS(k,R)2. 6 arrangedso that excitationofthe atoms in the lattice to
| | R
R
3
higher bands is negligible [16]. The state of the system scattering and non-scattering. A pseudo-random num-
can then be described in terms of lowest band Wannier ber is then used to determine if scattering occurs and in
functionsw(r r ),wherer isthepositionofthejthlat- which direction. If it does then the projection in Eq. (4)
j j
−
tice site [22]. When this condition is not met we expect is applied, if it does not the non-scattering projection in
to get similar results by taking into account the contri- Eq. (7) is applied. In either case the many-body state is
butions from higher bands [18]. We assume that the lat- then normalizedand evolvesaccording to Bose-Hubbard
tice potential is strong enough that the overlap between Hamiltonian before becoming the input state and the
neighboringWannierfunctionsisnegligible. Thestateof process repeats. For simplicity we assume the scatter-
the system can be expressed in terms of a number basis ing events are evenly spaced by time dt.
|nui ≡ |{nuj,j = 1,...,M}i, where nuj is the number of As an example of the dynamic process we look at the
atomsatsite j anduuniquely identifieseachbasisstate. simplecaseofathreesitelattice containingthreeatoms.
After the mth scattering event we expand the state as In this case there are only ten basis states making it
|Ψmi= uψum|nui. Scattering from this state then oc- straightforward to track the development of the many-
curs at angle θ to the x-axis with probability body state. In Figure 1 we show realizations of the dy-
P
namic scattering process for the three site lattice in the
2
g2 superfluid regime of the Bose-Hubbard model. For this
Pm(θ)= 2π (cid:12)I(θ)ψum eirj·k(θ)nuj(cid:12) . (3) example we have set the coupling constant g = 0.1 and
u (cid:12) j (cid:12) as a result the vast majority of detection events result
X(cid:12) X (cid:12)
(cid:12) (cid:12) from non-scattering, shown as θ = 0 in Figure 1(a). In
where I(θ) = dreik((cid:12)(cid:12)θ)·r w(r)2 and k(θ) =(cid:12)(cid:12) k0(1 Figure 1(c) we show the development of the many-body
| | −
cosθ, sinθ). state for the case of no system evolution between each
− R
Followingadetectionatθ the many-bodystate is pro- scattering event, dt=0. In this example the many-body
jected into the new state stateprogressestowardasuperpositionofthestates 201
| i
and 102 , two states which produce the same scattering
|Ψm+1i= √1 ψum eirj·k(θ)nuj|nui, (4) patte|rn.iAs discussed above, continued scattering from
N Xu Xj this end superposition does not change the constituent
basis vectors but does change the phase of the superpo-
where normalizesthe state. We note thatthe number
N sition. In Figure 1(b) we see that detection at non-zero
basis states are eigenstates of this projection, and the
anglequicklyreducestheoverlapofthemany-bodystate
scattering process will preserve any state that begins in
with the original ground state. We see that the overlap
abasisstate. Moreover,someofthe basisstatesproduce
makes quantum jumps when a scattering event occurs
the same scattering pattern as the relative positions of
and gradually evolves due to non-scattering events.
pairs of atoms in the lattice are the same, for example
Figure 1(d) and (e) demonstrate the effect of system
in the N = M = 3 case, 201 and 102 result in the
| i | i evolution between each scattering event. For the exam-
same light scattering. Superposition of these states are ple evolution in Figure 1(d) with dt=0.001J/~ the sys-
partially preserved by the projection, in that the weight
tem progresses toward a superposition like that in Fig-
ofeachstate inthe superpositionremainsthe same after
ure 1(b), however after some time the system evolution
scattering but the phase relationship is changed.
causes the system to jump to another set of states. In
Alternatively detection events where the probe is not Figure1(e)withdt=0.01J/~theevolutioniscompletely
scattered occur with probability
differentwiththesystemexploringarangeofbasisstates
over the simulation. This is detrimental to determining
PNS = ψmA 2 (5)
m | u u| theinitialstatescatteringpatternand,aswewillseebe-
u
X low,todosowemustlimitthetimeoverwhichscattering
and occurs to less than the natural system evolution time.
2
g2 π
Au =vu1− 2π dθ(cid:12)I(θ) eirj·k(θ)nuj(cid:12) . (6) IV. SCATTERING DISTRIBUTIONS FROM
uu Z−π (cid:12)(cid:12) Xj (cid:12)(cid:12) ENSEMBLE MEASUREMENTS
t (cid:12) (cid:12)
Detecting a non-scattering ev(cid:12)ent projects the st(cid:12)ate into
(cid:12) (cid:12) By repeatedsimulationofthe dynamic scatteringpro-
the new state
cess with dt = 0, we find that the simulations all set-
1 tle into a final state after a small number of scatter-
Ψ = ψmA n , (7)
| m+1i √ u u| ui ing events, as in Figure 1(c). The end states are al-
N′ Xu ways superpositions of eigenstates of the scattering pro-
whichfavorsstateswithalowerprobabilityofscattering. jection that produce the same scattering pattern. We
The dynamic scattering process can now be simulated also see numerically that the proportion of times a sim-
usingaquantumjumpprocedure[23,24]. Takingtheini- ulation ends in a particular state is determined by the
tial state, we calculate the probability distributions for initialstate. Forexampleforinitialstate Ψ wegetthe
0
| i
4
1
(a)
(a)
200
π
/ 0 U/J =0.1
θ
U/J =10
150
−1
0 200 400 600 800 1000
)
m (θ 100
d
N
1
(b)
2 50
i|
Ψ0 0.5
|m
Ψ
h 0
| −1 −0.5 0 0.5 1
0 θ/π
0 200 400 600 800 1000
m
(b)
1500
|3 0 0> (c)
|0 3 0> U/J =0.1
Basisvector|||||||0210021 0122100 3001212>>>>>>> θ) 1000 U/J =10
(
|1 1 1> d
N
200 400 600 800 1000
m 500
|3 0 0> (d)
|0 3 0>
vector||||0210 0122 3001>>>> 0−1 −0.5 0 0.5 1
asis||02 10 21>> θ/π
B|1 0 2>
|1 1 1>
200 400 600 800 1000
m FIG. 2. Comparison of scattering distribution summed over
multiplesimulationsforalatticeof9sitescontaining9atoms
|3 0 0> (e) for two different ratios of the Bose-Hubbard parameters U
|0 3 0>
vector||||0210 0122 3001>>>> asuinlmadtuiJolan.tsi(opwnosiitnhotfs1)100N000u0m0dbedteeertceotcfioteinovneenvetevdnetensttesecaetciaohcn,hswaNnhdde(rθe()bt)ahf1et0ear0n0(gasu)ilm1a0r-
asis||02 10 21>> rangeisdividedinto600binsforeventcounting. (lines)Scat-
B|1 0 2>
|1 1 1> tering distributions predicted from the initial ground states.
200 400 600 800 1000 Parameters used are g=0.1 and k0 =π/a.
m
end state in Figure 1(c) with probability 201Ψ 2 +
0
0 0.1 0.2 0.3 0.4 0.5 102Ψ 2. |h | i|
0
|c(um)|2 |h Bec|ausi|ethefinalstateproportionsarethesameasthe
initial basis state probabilities, we can obtain the initial
FIG.1. Developmentofanatomicstatecausedby1000scat- statescatteringdistributionifweaverageoveranensem-
teringeventsforalatticewiththreesitesandthreeatoms. (a) ble of scattering experiments. This occurs even though
Detectedevents. (b)Overlapofthemany-bodystatewiththe the initial state is completely changed in the scattering
initialgroundstate|Ψ0i. (c)-(e)Modulussquaredofthebasis process. For m scattering events per run of the experi-
coefficientsofthestate|Ψmifor(c)dt=0(b)dt=0.001J/~ ment andn runs, eachindividual runwill push the state
and (c) dt = 0.01J/~. Parameters used are U/J = 0.05,
towardsa particular set ofbasis vectors with a probabil-
g = 0.1 and k0 = π/a, where a is the lattice site separation
ity determined by the original state. Repetitions of the
andU andJ aretheparametersoftheBose-Hubbardmodel.
experimentthenleadtoastatisticaldeterminationofthe
initialstatescatteringdistributionasweprobabilistically
sample the scattering distributions for the basis states
thatmakeuptheinitialstate. Togetthebestestimateof
the initial state pattern with limited recourses, if we are
restricted only by the total number of scattering events
5
mn then the optimal method would be to take m=1 in thesystemtowardseigenstatesofthescatteringmeasure-
whichcaseback-actiondoesnotplayapart. Inmanyin- ment. Each individual experiment then yields a scatter-
teresting experiments however,doing an experiment run ingpatterncorrespondingtoaparticulareigenstatewith
is more restrictive than scattering. Then the error in es- probabilities determined by the initial state. Summing
timating the distribution by sampling n runs scales with
1/√n. Furthermore, when n 1 increasing the number
≫ (a)1500
of scattering events per run increases accuracy with the
U/J =0.1
same scaling in m and the accuracy of determining the
U/J =10
photon distribution scales with 1/√mn. 1000
)
θ
In Figure 2 we show how summing over multiple sim- (d
N
ulations reproduces the initial ground state pattern for 500
a lattice with 9 sites and 9 atoms. Despite there being
6420uniquescatteringpatternsfromtheassociatedbasis 0
vectors, sampling only 10 is sufficient to distinguish the −1 −0.5 0 0.5 1
θ/π
two different ground states of the Bose-Hubbard model
(a)
1500
we have show here. In Figure 2(a) each of the 10 sim-
U/J =0.1
ulations includes 10000 events of which 10% (5%) were
U/J =10
scattering events for U/J = 0.1(10). In Figure 2(b) we ) 1000
θ
show the distribution from summing 1000 simulations (d
N
with 1000 events each, in which case the error in the 500
scattering pattern compared with the exact initial state
pattern is reduced by a factor of approximately √10 in 0
−1 −0.5 0 0.5 1
accordance with the 1/√mn scaling.
θ/π
Theaboveresultsappliedwithnosystemevolutionbe- (c)
1500
tweenscatteringevents. AswesawinFigure1(e)system
U/J =0.1
evolutioncausesthestatetoeventuallyprogressthrough
U/J =10
1000
the basis states rather than just settling into one group. )
θ
One might wonder whether the basis states might then (d
N
besampledinproportiontotheinitialstateandallowus 500
to get the ground state scattering pattern without hav-
ing to re-run the experiment multiple times. This is not 0
−1 −0.5 0 0.5 1
the case and in fact for simulations long compared to
θ/π
the system’s natural evolution time the basis states end
up being sampled equally and the scattering pattern is
independent of the initial state as shown in Figure 3(a).
Thereareintermediateregimeshoweverwherescattering FIG.3. Comparison ofscattering distributionswith different
time dt between scattering events for a lattice of 3 sites con-
still typicallyresults ina single groupofbasis states and
taining 3 atoms. (points) Number of event detections Nd(θ)
the ground state can be recovered by summing over ex-
for (a) one simulation of 1000000 events with dt = 1J/~,
periments. Figure3(b)showstheresultofsummingover
(b-c) 1000 simulations of 1000 detection events each for (b)
1000 simulations where each simulation lasted at total dt = 0.001J/~, (c) dt = 0.01J/~, where the angular range
time of one natural unit of system time. In this case the isdivided into600 binsfor eventcounting. (lines) Scattering
summed scattering distribution closely matches the ini- distributionspredictedfromtheinitialgroundstates. Param-
tial state pattern. For the same parameters but a total eters used are g=0.3 and k0 =π/a.
time of 10 units, Figure 3(c), we see a significant depar-
ture from the initial state pattern and the patterns from
the two states begin to merge. these scattering patterns then gives an estimate of the
initial state scattering pattern. This allows the initial
statetobeprobedeventhoughitiscompletelydestroyed
V. CONCLUSION
in each run of the experiment. We have further shown
that when the scattering happens over a time that is
In conclusion, we have shown how scattering in the larger than the natural evolution of system this method
presence of measurementbackactionscattering leads the no longeryields the initial state pattern, andthus to op-
initial state of the system to be completely destroyed. timize measurement of the initial state we must scatter
When the systemdoesnothavetime to evolvethis leads all particles within the natural evolution time or less.
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