Testing unconventional decoherence models with atoms in optical lattices

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by  Jiří Minář

Testing unconventional decoherence models with atoms in optical lattices
Jiˇr´ı Min´aˇr,1 Pavel Sekatski,2 Robin Stevenson,1 and Nicolas Sangouard3
1School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom
2Institut for Theoretische Physik, Universitat of Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria
3Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
VariousmodelshavebeenproposedinwhichtheSchro¨dingerequationismodifiedtoaccountfor
a decay of spatial coherences of massive objects. While optomechanical systems and matter-wave
interferometrywithlargeclustersarepromisingcandidatestotestthesemodels,wehereshowthat
usingavailabletechniquesforatomsinopticallattices,someofthesemodelscanbeefficientlytested.
Inparticular,wecompareunconventionaldecoherenceduetoquantumgravityasintroducedbyEllis
and co-workers [Phys. Lett. B 221, 113 (1989)] and conventional decoherence due to scattering of
the lattice photons and conclude that optimal performances are achieved with a few atoms in the
realisticcasewhereproductatomicstatesareprepared. Adetailedanalysisshowsthatasingleatom
delocalized on a scale of 10 cm for about a second can be used to test efficiently the hypothetical
quantum gravity induced decoherence.
PACSnumbers: 03.65.Yz,42.50.Xa,37.10.Jk,67.85.d
6
1
0 Introduction –Byvirtueofthelinearityofquantumthe- the capability of cold atoms in optical lattices to test
2 ory, a system like an atom can be in a superposition of collapse models. The analysis is performed by taking
two different positions, as shown e.g. in atom interfer- intoaccountthedominantsourceofdecoherence,namely
n
ometry[1–3]. Thissuperpositionprincipleissupposedto the photon scattering from the trap lasers. We com-
a
J holdformoremassivesystemsasquantumtheorymakes pare this standard decoherence to the unconventional

0 no distinction between small and large systems. To bet- decoherence model first introduced in Ref. [10] and
2 ter account for what we observe at macroscopic scales, further elaborated in Refs. [11, 12], suggesting that
post-quantummodelshavebeenproposedinwhichquan- the coupling to the topologically non trivial spacetime

] tumtheoryissupplementedwithexplicitcollapsemecha- configurations admitted by the underlying theory of
h
p nisms [4]. Various systems are being investigated to test quantum gravity, termed as wormholes, leads to the
- these hypothetical models. Impressive experiments are decay of spatial coherences of a test massive system. We
t
n beingperformed,forexampleinmatter-waveinterferom- show that the effects of these decoherence mechanisms
a etrywithfreefallingatomsinwhiche.g. thewavepacket are qualitatively different: while the former operates
u ofasingleRbatomgetsseparatedby1.4centimetersfor locally, the latter induces a collective noise on the state
q
2.3 seconds [5]. Since the mass in these experiments is of the atoms. Focusing on the experimentally relevant
[
limited to that of a single atom, matter-wave interfer- case where the atoms are prepared in product states,

1 ometry with larger and larger molecules and clusters is we find that optimal performances are achieved with
v being developed [6, 7]. Ref. [8] is an example of ongoing small atom numbers. A detailed analysis shows that
1 experiment using a molecule made with more than 800 a single rubidium atom delocalized over ∼ 10 cm can
8
atomswithamolecularweightexceedingahundredtimes be used to test the hypothetical decoherence due to
3
theoneofasingleRubidiumatom. Optomechanicalsys- quantumgravity[11]. Whilewetakethisunconventional
5
0 tems where heavy nano and micro mechanical oscilla- decoherence model as an example, the proposed system
. tors are driven through the radiation pressure or dielec- might be used to probe other collapse models, such
1
tric nanoparticles levitating in the focus of intense laser as the Diosi-Penrose model [13, 14], the GRW model
0
fieldsarealsoatthecoreofintenseresearchprograms[9]. [15–17]oradecoherenceduetochameleonfieldsrelevant
6
1 While all these techniques are promising to test the su- for cosmology, where it is actually favorable to operate
: perposition principle with massive systems in incoming on small masses [18].
v
years, it is natural to ask whether well-controlled and
i
X easy-to-use systems could be used to test these wave-
Principle – We consider atoms trapped in a state depen-
r function collapse models. dent optical lattice, i.e. two ground states |g(cid:105), |s(cid:105) are
a
While the ready-to-use toolbox to manipulate indi- trapped independently such that an atom prepared in a
vidual atoms trapped in optical lattices makes them superposition |g(cid:105)+|s(cid:105) can be spatially delocalized into
very attractive, it is usually suggested that the mass of |g,x (cid:105)+|s,x (cid:105), see Fig. 1a,b. To measure the decay of
g s
a single atom is too small and the possibility to control spatial coherences, various techniques can be envisioned
millions of atoms does not overcome the problem as depending e.g. on the atom number. In the case of
they cannot be prepared in GHZ-like states in practice a single atom, the atom is spatially relocalized and
(which would mimic the superposition of the center the coherence between the internal states is measured.
of mass of the heavier composite systems). Here we To record the decay of spatial coherences in time, the
challenge these preconceptions by analyzing in detail atoms are kept in the state |g,x (cid:105) + |s,x (cid:105) for longer
g s
2
times before relocalization and measurement of the |g(cid:105)
– |s(cid:105) coherence. Comparing the timescales of both the
standarddecoherenceresultingformthescatteringofthe
photons from the trap and unconventional decoherence
models with the observed coherence decay time might
make it possible to confirm or rule out predictions from
unconventional decoherence mechanisms.
State dependent manipulation of atomic motion – We
here show how to coherently manipulate the position of
atoms in state-dependent atomic lattices. For concrete-
ness, we focus on a far-off resonant dipole trapping of
87Rb [19, 20]. The relevant level structure is shown in
Fig. 1c. State dependent manipulation of atomic posi-
tion can be obtained if |g(cid:105) and |s(cid:105) are chosen such that
each state can be trapped with different polarizations.
The transition |g(cid:105) – |s(cid:105) also needs to be addressed co-
herentlyforinternalstatepreparationandmeasurement.
For example, one can identify |g(cid:105) ≡ |F =2,m =−2(cid:105)
F
and |s(cid:105)≡|F =1,m =−1(cid:105). The detunings of the trap-
F
pinglaserswithrespecttotheP manifoldare∆ ,∆
1/2 − π
for the σ− and π polarizations respectively. We should FIG. 1. (Color online) a) Schematic of the trapping poten-
note that the ground states couple to higher lying man- tials: Atoms in state |s(cid:105) (blue circles) are separated by a
ifolds, namely the P manifold. However it is easy to distance d(t) with respect to the atoms in state |g(cid:105) (red cir-
3/2
be in the regime, where the dominant contribution to cles). b) Time evolution of the separation d between the |g(cid:105)
the trapping potential comes only from the coupling to and |s(cid:105) states. c) Level scheme of 87Rb. Atoms in |g(cid:105) state
the P manifold. For example if ∆ are of order 100 are traped by the π polarized trapping lasers (red arrows),
1/2 −,π while the |s(cid:105) states are trapped by both the π and σ− po-
GHz, the coupling to the P manifold becomes negligi-
3/2
larized (blue arrow) laser beams. Green arrows denote the
blecomparedtothecouplingtotheP manifoldasP
1/2 3/2 microwave driving.
and P differ by 7 THz (see Fig. 1c). In the following
1/2
we work in a regime where ∆ ≈ ∆ ≡ ∆ and conse-
− π
quently for the lattice laser wavevectors k ≈ k = k.
π − Decoherence due to photon scattering: local dephasing –
Considering a one-dimensional geometry, where x is the
While the internal state coherence can be degraded by
lattice axis, the trapping potentials for the two ground
manytechnicalissuesincludinglatticedepthormagnetic
states are given by [21, p. 199]
field gradient fluctuations (see [23–26] for an extensive
study of decoherence mechanisms) we focus on the dom-
inant (and unavoidable) source of decoherence, namely
V (x,t)=V0cos2(kx) (1a)
g π the scattering of lattice photons.
V (x,t)=V0cos2(kx)+V0cos2(kx+ϕ(t)), (1b) Letusconsiderfirstasingleatominaspatialsuperpo-
s π −
sition with separation d between the superposed states.
where V0 = (cid:126)Ω2/(4∆), j = π,−, and ϕ(t) is a time For short spatial separation d (cid:28) λ as compared to the
j j
dependent offset of the σ− lattice which in practice wavelength of the trapping light, the rate of decoherence
can be achieved e.g. by varying the frequencies of the due to the scattering of the electromagnetic radiation
counter-propagating lattice beams [22]. This leads to a grows quadratically with the distance [27]. For larger
time dependent separation d(t) = ϕ(t)/k of the |g(cid:105),|s(cid:105) distances, which are relevant for testing unconventional
states, see Fig. 1a. Note that we have absorbed possible decoherence, the decoherence rate saturates as predicted
multiplicities coming from the coupling of the |g(cid:105),|s(cid:105) in[28]andverifiedexperimentallye.g. in[29,30]. Inpar-
states to several levels into the Rabi frequencies. ticular, for lasers far detuned from resonance, it is given
by [21, p.180]
Decoherence – In order to evaluate the decay of coher-
ences in time, we compute the overlap of the initial and Γ = 1Γ0 (cid:18)Ωπ(cid:19)2 (2a)
later time density matrices O = Tr(ρ(0)ρ(t)). While g 4 2 ∆
other figures of merit might be chosen, the overlap is a 1Γ Ω2 +Ω2
quantity that is easy to calculate and analytical expres- Γs = 4 20 π∆2 −, (2b)
sions are given in the following sections. Moreover, in
the case of a single atom, it yields directly the coherence forthe|g(cid:105),|s(cid:105)staterespectively,Γ beingtheatomicfree
0
decay rate. space decay rate. Now, consider the case with N atoms
where d (cid:29) λ and where the position of each atom is
3
fullyresolvedthroughthescatteredphotons. Themaster Γ∝d2 providedk d(cid:28)1(k isthewavevectorasso-
QG QG
equation governing the decoherence of N atoms is given ciated to the wormholes). In this regime one can expand
byasumofindependentscatteringprocesses(weneglect the localization function to second order
the coherent evolution as we are interested solely in the
1
decay of the coherences) Φ(d)≈Φ(0)+ ∂2 Φ(d)| d2, (9)
2 d d=0
ρ˙ =(cid:88)N (cid:88) Γα(cid:0)2|α(cid:105)(cid:104)α| ρ|α(cid:105)(cid:104)α| −{|α(cid:105)(cid:104)α| ,ρ}(cid:1). (3) wherethelineartermisabsentsinceΦisanevenfunction
2 i i i [4]. Notethattheexpansion(9)iswelljustifiedas1/kQG
i=1α∈{g,s} corresponds to long wavelengths (up to 104 m [11]).
Moreover, we focus on the case where Na (cid:28)d(t), i.e.
This equation can be rewritten as a local dephasing pro- latt
the atom number times the lattice spacing is typically
cess
negligible with respect to the state separation. Those
(cid:32) N (cid:33) two considerations allow one to rewrite the master equa-
ρ˙ = Γsc (cid:88)σiρσi −ρ , (4) tion (7) as
2 z z
i=1 (cid:18) (cid:19)
Γ 1 1
ρ˙ = QG S ρS − S2ρ− ρS2 , (10)
where Γsc = Γg + Γs and σzi = |s(cid:105)(cid:104)s|k − |g(cid:105)(cid:104)g|k is the 2 z z 2 z 2 z
usual Pauli matrix in the {|s(cid:105),|g(cid:105)} basis. Importantly,
the decoherence due to photon scattering is independent where S =(cid:80)N σi is the collective spin, i.e. the deco-
z i=1 z
of the separation d(t). We can now compute the overlap herence due to quantum gravity corresponds to a collec-
oftheinitialandlatertimedensitymatrices. Considering tive z-spin noise, and the decay rate reads [11] (see also
the initial atomic state to be a product state Appendix A)
(cid:18)|g(cid:105)+|s(cid:105)(cid:19)⊗N (cm )4
|ψ (cid:105)= √ , (5) Γ = 0 m2 d(t)2 ≡γ d(t)2, (11)
0 2 QG ((cid:126)mPl)3 at QG
it reads (see Appendix A) where c is the vacuum speed of light, m0 the nucleon
mass, m the Planck mass and m is the mass of a sin-
Pl at
(cid:18)1+e−(Γg+Γs)t(cid:19)N gle trapped atom. Most importantly, it scales quadrati-
Osc(t)=Tr(ρ(0)ρ(t))= 2 . (6) cally with the state separation, i.e. can be enhanced by
increasingd(t). Consideringtheinitialproductstate(5),
one can evaluate the overlap as (see Appendix A)
Unonmcoennvoelongtiiocanlalyl,detchoehesrpeantciea:l dgelocboahlerdeenpcheascianng –bePdhee-- O (t)=(cid:90) dΛcos2N(cid:0)Λ(cid:1) e−2γΛ(2t) , (12)
scribed by the master equation [4] QG 2 (cid:112)2πγ(t)
(cid:90) where γ(t) = 2γ (cid:82)td(t(cid:48))2dt(cid:48) is the variance of the
ρ˙ =− dx(cid:48)dxΓ(x,x(cid:48))|x(cid:105)(cid:104)x|ρ|x(cid:48)(cid:105)(cid:104)x(cid:48)| (7) QG 0
gaussian distribution.
where |x(cid:105) = |x1,...,xN(cid:105) is the position basis for the N Scalingofcoherenceswithatomnumber–Itisinteresting
particles. The localization rate Γ(x,x(cid:48)) can be written to compare how the two overlaps O ,O scale with
sc QG
as respect to the number of trapped atoms N. While O
sc
decreases exponentially with N (6), the N dependence
N
Γ(x,x(cid:48))= γ0 (cid:88) µ µ (cid:16)Φ(x −x )+Φ(x(cid:48)−x(cid:48))−2Φ(x −x(cid:48))(cid:17),is more complicated for quantum gravity (12). However,
2 i j i j i j i j in the asymptotic limit of large N (see Appendix A),
i,j=1
(8) OQG →(cid:113)2N1γ(t)ϑ3(0,e−(22γπ()t2)),i.e. exhibitsadecaywith
where the spatial dependence described by Φ (the so-
N1/2 (ϑ is the Jacobi theta function). Consequently,
called localization function) and the constants γ and µ 3
0 i we conclude that for product states of the form (5), it is
are given by the underlying microscopic theory.
preferable to use small atom numbers.
Following the treatment given in Refs. [11, 31, 32],
Note, that the situation is completely differ-
we now take the example of quantum gravity induced
ent if the initial state is the N atom GHZ state
collapse model. It has been hypothesized by Ellis and
|GHZ(cid:105) = √1 (|g(cid:105)⊗N +|s(cid:105)⊗N). In this case, the decay of
co-workers that spatial superpositions should decay due 2
to the interaction of the system (modeled by a matter the coherence term |g(cid:105)(cid:104)s|⊗N scales as e−N2 for quantum
field) with wormholes. On a formal level, the calcula- gravityande−N forthephotonscattering(seeAppendix
tion carried out in Ref. [11] uses the scattering matrix A). Hence, in this particular case, it is easier to test
approach (i.e. the same approach as in [27]) leading to unconventional decoherence with large atom numbers.
the scaling of the decoherence rate with the separation Although desirable, the creation of GHZ states with
4
large N, is known to be a difficult task in practice and as a function of Ω for various detunings ∆. It can be
−
for the remainder of the article we focus on the more seenthatforlargedetuningsandmoderateRabifrequen-
realistic case of a single atom. cies r (cid:29) 1. In order to estimate the ratio r that can
be achieved in practice, we show in Appendix B, that r
Feasibility study – We now give a detailed analysis of scales as
an experiment using a single atom. First, the over-
laps (6) and (12) both take the simple form Olabel = r ∝dmaxk∆τ2,
1(exp(−Γ τ)+1), where ”label=sc,QG”. Here
2 label i.e. one requires large d , ∆ and τ. First we note that
max
γ (cid:90) τ ∆ is bounded from above to make the coupling to the
ΓQG = QτG dt d(t)2 ≡γQGd2eff, (13) P3/2 manifoldnegligible(ascomparedtothecouplingto
0 the P manifold), i.e. to ensure a polarization selective
1/2
where τ is the time it takes to spatially separate and atomic transport. We take ∆ = 2π ·1 THz. We then
relocalize the two states |g(cid:105), |s(cid:105). We have introduced estimate the Rabi frequency Ω− yielding the optimal r
an effective distance deff which allows us to write the from (15). The optimal value of Ω− depends on the du-
expressionfortheunconventionaldecayinasimpleform rationoftheexperimentτ. Inordertogetaquantitative
Γ ∝d2 for any atomic motion d(t). idea, it is useful to estimate the timescale of the uncon-
QG eff
It follows from (13) that large spatial separations fa- ventional decoherence. deff = 10 cm yields ΓQG = 1 Hz
vor large Γ . To get large separations by manipulating (pointAinFig. 2b)andasanexample, wechooseτ =1
QG
the |s(cid:105) state independently, we consider a regime where s. The optimal Ω− maximizing r is given by the point
V0 (cid:28)V0 whichcanbeachievedalreadyformoderatera- A in Fig. 2a (Ω− ≈ 108 Hz, r ≈ 800). Further increas-
π −
tios between Ω− and Ωπ (for example Ω−/Ωπ ≈3 yields ing Ω− (for all other parameters fixed) increases Γsc and
V0/V0 ≈10). Furthermore, in order to have large sepa- reduces r. On the other hand, if Ω− is decreased, the
− π
rations in a short time, an acceleration ramp is required. onlypossibilitytoreachdmax inatimeτ/2istodecrease
Finding the exact atomic motion which maximizes the ∆, which again increases Γsc and decreases r. The de-
effectoftheunconventionaldecoherenceisamultidimen- crease of r for deviations of Ω− from the optimal value
sional optimization problem, as the motion depends on is represented by the shaded region in Fig. 2a.
various parameters such as the Rabi frequency of the To complete our discussion, specific values of Ω− and
trappinglaser,thedetuning,theabsolutetimescaleofthe ∆imposeamaximaltemperatureoftheatoms,suchthat
experiment,thetemperatureoftheatomsorthemaximal theyremaintrapped[33]. Thedot-dashedlinesinFig.2a
separation allowed by the experimental setup. To sim- show different trap temperatures, which are given by
plifythediscussion,weconsiderasmoothatomicmotion √
described by T = (cid:126)32 √2k √Ω− , (17)
tr kB mat ∆
(cid:20) (cid:21)
d T 2π|t−T|
d(t)=d + max sin −|t−T| , (14)
max T 2π T where kB is the Boltzmann constant. One can see, that
the shaded region corresponds to T (cid:38) 100 nK, i.e.
tr
where T ≡τ/2 and t∈[0,τ], i.e. d is reached in half to the temperatures achievable in today’s cold atomic
max
of the atomic round trip τ (see Appendix B for details). experiments [5].
√
This leads to d = d / 2. To ensure that the atom
eff max
followsthetrappingpotential,thetrappingforceneedsto Summary and Outlook – To conclude, we have shown
be larger than the dynamical force [22], −∂ V > m a, that unconventional decoherence models can be ef-
x s at
where a is the acceleration of the atom. Evaluating the ficiently tested with cold atoms in superpositions of
trappingforce atits maximum yieldsa constraint onthe different spatial positions. When using optical lattices
maximal acceleration for a given Rabi frequency and de- where the dominant source of decoherence is the photon
tuning scattering, the standard and quantum gravity based
decoherence mechanisms operate differently: While the
(cid:126)k Ω2 2πd former acts as a local noise, the latter corresponds to
− ≈a = max, (15)
m 4∆ max T2 a collective dephasing. Consequently, when dealing
at
with product states, we have shown that it is easier
where we have approximated V (x,t) ≈ V0cos2(kx + to observe the unconventional decoherence with small
s −
ϕ(t)) in Eq. (1b) and used the second derivative of Eq. atom numbers. We have performed a detailed feasibility
(14) to obtain the right hand side. Inspired by the ex- study showing unambiguously that a single atom in a
perimental results presented in Ref. [22], where atomic superposition of two positions separated by ∼ 10 cm
transport was realized over up to 20 cm, we choose a for ∼ 1 s can be used to test quantum gravity induced
more conservative value d =10 cm. Fig. 2a shows the collapse. This proposal might be implemented using
eff
ratio various platforms where atomic transport has been
successfully demonstrated, including hollow core fibers
r ≡Γ /Γ (16) [34,35],opticaltweezers[36],atomicchips[37–40]orfree
QG sc
5
space optical lattices [22, 41–45]. As a first outlook, we
emphasizethatourresultsallowonetoboundthemodel
of Ellis and co-workers [11] in any scenario where atoms
are spatially delocalized over sufficiently large distances.
It is interesting to quantify the bounds provided e.g. by
atom interferometers with large momentum transfers,
yielding large spatial separations [5, 46]. In particular,
we are currently evaluating [47] what are the contraints
on the model of Ref. [11] imposed by the experimental
results published very recently in Ref. [48]. Another
interesting perspective is to investigate the potential
of many body entangled states, such as spin squeezed
states [49, 50] or BEC solitons in guided interferometers
[51], to benefit from a favourable scaling of the ratio
betweenstandardandunconventionaldecoherencerates,
which we leave for future work.
Acknowledgments – J.M. would like to thank T.
Lahaye, L. Hackermu¨ller, P. Kru¨ger and I. Lesanovsky
for useful discussions. N.S. thanks P. Treutlein and
D. Meschede for stimulating discussions. J.M. was
supported by the grant EU-FET HAIRS 612862. P.S.
and N.S were supported by the Swiss National Sci-
ence Foundation grant number P2GEP2 151964 and
PP00P2 150579 respectively.
FIG. 2. (Color online) a) Ratio r between the decoherence
ratesinducedbyquantumgravityandphotonscatteringasa
function of the trapping Rabi frequency Ω for various trap-
−
pingdetunings∆/(2π)={10,100,1000}GHz(blue,green,red
lines respectively). The dashed black line is a guide for the
eye corresponding to r = 1. The dot-dashed lines represent
differenttraptemperatures. PointAcorrespondstotheopti-
mized ratio r≈800, see text for details. b) The decoherence
rateduetoquantumgravity(11)asafunctionofthedistance
d for 87Rb. Point A corresponds to the distance d =0.1
eff eff
m and Γ =1 Hz respectively.
QG
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[29] L. Hackermu¨ller et al., Nature 427, 711 (2004). [49] A. Kuzmich, L. Mandel, and N. P. Bigelow, Phys. Rev.
[30] D. A. Kokorowski, A. D. Cronin, T. D. Roberts, and Lett. 85, 1594 (2000).
D. E. Pritchard, Phys. Rev. Lett. 86, 2191 (2001). [50] J. Est`eve et al., Nature 455, 1216 (2008).
[31] O. Romero-Isart, Phys. Rev. A 84, 052121 (2011). [51] C. Weiss and Y. Castin, Phys. Rev. Lett. 102, 010403
[32] B. Pepper et al., NJP 14, 115025 (2012). (2009).
[33] The equation (17) refers to the trapping of the |s(cid:105) state [52] W. S. Bakr et al., Nature 462, 74 (2009).
along the lattice axis. In practice the limiting tempera- [53] J. F. Sherson et al., Nature 467, 68 (2010).
ture will be given also by the trapping force of the |g(cid:105)
Appendix A: Analysis of Decoherence mechanisms
1. Conventional decoherence due to photon scattering
a. Master equation. Here we introduce the master equation governing the decoherence due to the scattering
of the optical trapping photons. It is a well known fact, that in the far off resonant regime where operate optical
lattices, the induced excited state population can be typically neglected (in fact is proportional to (Ω/∆)2, where
Ω is the Rabi frequency of the trapping laser and ∆ its detuning from the excited state, which is easy to show
e.g. from the Bloch equations). This is a desired feature, since one typically wants to avoid the heating due to the
photon absorption/emission by the trapped atoms. As a consequence, one can consider only the decay of coherences
(which are of order Ω/∆) while neglecting the change in the state population. Next we consider a situation, where
the individual sites of the optical lattice can be spatially fully resolved by measuring the scattered photons. This
is indeed the feature which led to the success of and is exploited in the quantum gas microscopes [52, 53]. Under
these assumptions, the decoherence of the system reduces to the product of individual decoherence events, where
a measurement of a scattered photon results in a projection of the atom to a given spin state of the ground state
manifold. The corresponding master equation reads
N
(cid:88)(cid:16) (cid:0) 1 (cid:1) (cid:0) 1 (cid:1)(cid:17)
ρ˙ =2 Γ |g(cid:105)(cid:104)g| ρ|g(cid:105)(cid:104)g| − {|g(cid:105)(cid:104)g| ,ρ} +Γ |s(cid:105)(cid:104)s| ρ|s(cid:105)(cid:104)s| − {|s(cid:105)(cid:104)s| ,ρ} , (A1)
g i i 2 i s i i 2 i
i=1
where Γ , α = g,s, are the scattering rates (2) for the atoms in |g(cid:105), |s(cid:105) state and {·,·} is the anticommutator.
α
Straigtforward algebra allows to rewrite this expression in terms of the operators σi =|g(cid:105)(cid:104)g| −|s(cid:105)(cid:104)s| yielding
z i i
N
ρ˙ =(cid:88)Γg+Γs(cid:0)σiρσi −ρ(cid:1) (A2)
2 z z
i=1
b. State evolution. It is straightforward to solve the master equation (A2). The dephasing process acts locally
resulting in the decay of the off-diagonal terms. Formally this process can be described by the completely positive
trace preserving (CPTP) map Eloc(ρ(0)) = ρ(t), where Eloc = (cid:78)N εi is given simply by the product of the N
t t i=1 t
7
individual dephasing processes
(cid:18)1+e−(Γg+Γs)t(cid:19) (cid:18)1−e−(Γg+Γs)t(cid:19)
εi(ρ)= ρ+ σiρσi. (A3)
t 2 2 z z
(cid:16) (cid:17)⊗N
For the initial product state (5), |ψ0(cid:105)=|+(cid:105)⊗N = |g(cid:105)√+|s(cid:105) , the overlap with a state at a later time is given by
2
(cid:89)N (cid:18)1+e−(Γg+Γs)t(cid:19)N
O =Tr(|ψ (cid:105)(cid:104)ψ |ρ(t))= (cid:104)+|εi(|+(cid:105)(cid:104)+|)|+(cid:105)= (A4)
sc 0 0 t 2
i=1
2. Unconventional decoherence due to quantum gravity
c. Master equation. Asdescribedinthemaintext, thespatialdecoherencecanbedescribedphenomenologically
by the master equation [4]
(cid:90)
ρ˙ =− dx(cid:48)dxΓ(x,x(cid:48))|x(cid:105)(cid:104)x|ρ|x(cid:48)(cid:105)(cid:104)x(cid:48)| (A5)
where |x(cid:105) = |x ,...,x (cid:105) is the position basis for the N particles. We start by explicitly computing the localization
1 N
rate (8)
N
Γ(x,x(cid:48))= γ0 (cid:88) µ µ (cid:16)Φ(x −x )+Φ(x(cid:48) −x(cid:48))−2Φ(x −x(cid:48))(cid:17). (A6)
2 i j i j i j i j
i,j=1
The localization function Φ can be expanded for small separations d compared to the localization length L, which
is well justified when considering the model of [11], where L ∼ 104 m and we consider separations of d ∼ 10 cm.
With this assumption Φ(d) ≈ Φ(0)+ 1Φ(cid:48)(cid:48)d2, where Φ(cid:48)(cid:48) = ∂2 Φ(d)| and we have used the fact, that Φ has to be
2 0 0 d d=0
symmetric, Φ(d)=Φ(−d), which imposes the cancellation of the linear term in the expansion.
Next, note that in the situation where the atoms are tightly trapped in the lattice, the spin state of each atom
|(cid:96)(cid:105) is fully entangled with its position x ((cid:96)), such that the decoherence rate only depends on the global atomic state
i i
through Γ(l,l(cid:48)), where l = ((cid:96) ,...,(cid:96) ). Denoting the spin states |0(cid:105) ≡ |g(cid:105) and |1(cid:105) ≡ |s(cid:105), for lattice spacing a
1 N latt
and separation d(t) the position of each atom is given by x ((cid:96)) = ia +(cid:96)d(t). In the regime of interest of large
i latt
separation Na (cid:28)d(t), the separation of two atoms within the same lattice can be neglected and x ((cid:96))≈(cid:96)d(t). In
latt i
this approximation and putting µ =µ for all i as we consider identical particles
i
Γ(l,l(cid:48))=γ0µ2Φ(cid:48)0(cid:48)d(t)2 (cid:88)N (cid:16)((cid:96) −(cid:96) )2+((cid:96)(cid:48) −(cid:96)(cid:48))2−2((cid:96) −(cid:96)(cid:48))2(cid:17)
4 i j i j i j
i,j=1
=γ0µ2Φ(cid:48)0(cid:48)d(t)2(cid:16)4((cid:88)(cid:96) )((cid:88)(cid:96)(cid:48))−2((cid:88)(cid:96) )2−2((cid:88)(cid:96)(cid:48))2(cid:17)
4 i i i i
i i i i
γ µ2Φ(cid:48)(cid:48)d(t)2(cid:16) 1 1 (cid:17)
= 0 0 S (l)S (l(cid:48))− S (l)2− S (l(cid:48))2 (A7)
4 z z 2 z 2 z
where S (l)=(cid:80) (2(cid:96) −1) is the total z-spin of the state |l(cid:105), S |l(cid:105)=S (l)|l(cid:105), with S =(cid:80) σi =(cid:80) S (l)|l(cid:105)(cid:104)l|. This
z i i z z z i z l z
allows to rewrite the master equation (A5) governing the localization in terms of total spin
γ c2Φ(cid:48)(cid:48)d(t)2(cid:16) 1 (cid:17)
ρ˙ = 0 0 S ρS − {S2,ρ} . (A8)
4 z z 2 z
Theprefactorin(A8)canbelinkedtotheparametersof[11]bycombiningtheresultsof[11]withthepresentanalysis.
This was done e.g. in [31] with the result γ0µ2Φ2(cid:48)0(cid:48)d(t)2 =ΓQG = (((cid:126)cmmP0l))43m2atd(t)2.
8
d. State evolution. In order to solve the master equation(A8) we realize that the collective spin noise process
corresponds to the diffusion of the polar angle of the total spin. After an elementary time step, the density matrix
can be written as an average over a random variable λ
(cid:18) (cid:19)
Γ 1 1
ρ(t+dt)=ρ(t)+dt QG S ρ(t)S − S2ρ(t)− ρ(t)S2 =(cid:104)eiλSzρ e−iλSz(cid:105) (A9)
2 z z 2 z 2 z t λ
where (cid:104)·(cid:105) denotes an ensemble average over λ with the moments (cid:104)λ(cid:105) = 0, (cid:104)λ2(cid:105) = Γ dt/2 and (cid:104)λ(n>2)(cid:105) = O(dt2).
QG
This can be easily verified by substituting the expansion of the operator exp(iλS )=1+iλS − 1λ2S2+O(λ3) into
z z 2 z
(A9). After n= t time steps the state of the system is
dt
ρ(t)=Etcol(ρ(0))=(cid:104)ei((cid:80)ni=1λi)Szρ(t)e−i((cid:80)ni=1λi)Sz(cid:105)λ1,...,λn =(cid:90) dΛeiΛSzρ(0)e−iΛSze−(cid:112)Λ22/π(γ2γ(t(t))), (A10)
where we have introduced the CPTP map Ecol corresponding to the collective noise.
t
In the last equality we have used the fact that for dt → 0 the sum of independent random variables Λ = (cid:80)n λ
i=1 i
is a normally distributed random variable with the variance γ(t)=2γ (cid:82)td(t(cid:48))2dt(cid:48).
QG 0
The overlap of the state at time t with the initial state (5) can now be easily computed as follows. First we write
the expression for the overlap
OQG =Tr(|ψ0(cid:105)(cid:104)ψ0|ρ(t))=(cid:90) dΛ|(cid:104)ψ0|eiΛσz|ψ0(cid:105)|2Ne−(cid:112)Λ22/π(γ2γ(t(t))) =(cid:90) dΛcos2N(cid:0)Λ2(cid:1)e−(cid:112)Λ22/π(γ2γ(t(t))). (A11)
The evaluation of the integral can be performed numerically. One can proceed further in the large N limit - to get
the assymptotic expansion for O we remark that
QG
(cid:114) ∞
Λ π (cid:88)
cos2N( )→ δ(Λ+2πk) for N →∞, (A12)
2 N
k=−∞
where δ is the Kronecker delta and we have used
(cid:90) π2 cos2N(cid:18)Λ(cid:19) dΛ= π(cid:0)2NN(cid:1) →(cid:114)π . (A13)
2 22N N
−π
2
(cid:18) (cid:19)
Substituting(A12)backto(A11)yieldsaninfinitesum,whichcanbeevaluatedas(cid:80)∞k=−∞e−(2π2γk)2 =ϑ3 0,e−(22πγ)2 ,
where ϑ is the Jacobi ϑ function. This leads to the result
3
(cid:115)
(cid:18) (cid:19)
OQG → 2N1γ(t)ϑ3 0,e−(22γπ()t2) for N →∞. (A14)
3. GHZ states
It is interesting to compare the decay of the overlap of the product state (which we considered for the simplicity of
its experimental preparation) with the decay for some highly entangled state. A benchmark example in metrology is
√
the GHZ state ρ =|GHZ(cid:105)(cid:104)GHZ|, |GHZ(cid:105)=1/ 2(|g(cid:105)⊗N +|s(cid:105)⊗N). One gets for the local and collective dephasing
GHZ
respectively
Eloc(cid:0)ρ (cid:1)= 1(cid:16)|g(cid:105)(cid:104)g|⊗N +|s(cid:105)(cid:104)s|⊗N +(cid:0)|g(cid:105)(cid:104)s|⊗N +h.c.(cid:1)e−N(Γg+Γs)t(cid:17) (A15)
t GHZ 2
Ecol(cid:0)ρ (cid:1)= 1(cid:16)|g(cid:105)(cid:104)g|⊗N +|s(cid:105)(cid:104)s|⊗N +(cid:0)|g(cid:105)(cid:104)s|⊗N +h.c.(cid:1)e−2N2γ(t)(cid:17), (A16)
t GHZ 2
which yields the overlaps
OGHZ =(cid:104)GHZ|Eloc(cid:0)ρ (cid:1)|GHZ(cid:105)= 1(1+e−N(Γg+Γs)t) (A17)
sc t GHZ 2
OGHZ =(cid:104)GHZ|Ecol(cid:0)ρ (cid:1)|GHZ(cid:105)= 1(1+e−2N2γ(t)). (A18)
QG t GHZ 2
9
Appendix B: Atomic transport
Inthissectionweprovideasimpledescriptionoftheatomicmotioninatimedependentopticallattice. Considering
thetightbindingsituation,wheretheatomsarefirmlytrappedatanintensityextremaofthelattice,itisinprinciple
possible to displace the atoms by an arbitrary distance d. Working in the regime V0 (cid:29) V0, the s species can be
− π
displaced independently of the g species and thus, in principle, allows for creation of a superposition of type (for a
single atom)
|ψ(cid:105)=c |g(cid:105)ψ(x−x )+c |s(cid:105)ψ(x−x ), (B1)
g g s s
where we assume, that due to the tight binding the position of the atom is treated classically and can be described
by a classical field ψ(x), which has the property (cid:82) dxx|ψ(x−x )|2 =(cid:82) dxxδ(x−x )=x , where α=g,s. In other
α α α
words, with respect to the position in real space we treat the atom as a classical point like object, which is a justified
description for the system where separations of multiples of lattice spacings between |g(cid:105) and |s(cid:105) are achieved.
As only the s species is displaced, we thus seek solutions of the classical equation of motion associated with the
Hamiltonian
p2
H = s +V (x,t), (B2)
2m s
at
where V (x,t) ≈ V0cos2(kx+ϕ(t)) is given by (1b). The Hamilton equations of motion given by (B2), p˙ = ∂ H,
s − x
x˙ =−∂ H, can be combined to give
p
V0k
x¨ =− − sin(2kx +2kx (t)), (B3)
s m s latt
at
where x (t) is the lattice motion. As explained in the main text, the specific choice of the lattice motion is a result
latt
of an optimization with respect to a givenfigure of merit and subject tothe constraints such as a maximal achievable
separationorthetimescaleoftheexperiment. Forthesakeofconcretenessandmotivatedbytheexperimentalresults
of [22], we consider lattice motion with an initial sinusoidal acceleration ramp such that the maximal separation d
max
is reached after a time T. When d is reached, the atoms remain separated for a waiting time t before they are
max w
brought back together with the same acceleration ramp of duration T,
 (cid:104) (cid:105)
dmTax 2Tπ sin2π(TT−t) −(T −t) t<T
x (t)= d T ≤t≤t +T (B4)
latt max w
dmax− dmTax (cid:104)2Tπ sin2π(t−TT−tw) −(t−T −tw)(cid:105) tw+T <t≤tw+2T.
In order to control the atomic motion, one requires the atoms remain trapped in a given lattice site at all times,
i.e. that the trapping force is larger than the dynamical force m x¨ acting on the atoms. It is easy to check, that
at s
the relation (15) provides indeed the desired condition. By numerically solving (B3) we have verified, that when
the acceleration of the lattice a (cid:46) a , the atoms follow the lattice potential, while they are not able to follow for
max
a>a . In the following, we thus restrict only to the situation, where a(cid:46)a and the atomic motion d(t) can be
max max
identified with the lattice motion (B4).
1. Optimization of the atomic motion
Next, we would like to optimize the lattice motion of the form (B4), i.e. optimizing T and t with respect to the
w
ratior =Γ /Γ betweentheunconventionalandconventionaldecoherencerates,eq. (16). SinceΓ ∝(cid:82)τdtd(t)2,
QG sc QG 0
where τ is the total duration of the atomic round trip, one can define an effective distance d for any atomic motion
eff
d(t) and τ through
(cid:90) τ
d2 τ ≡ dtd(t)2. (B5)
eff
0
Taking d(t)=x (t), eq. (B4), one readily finds
latt
(cid:20) (cid:21)
2T
d2 =d2 1+ (α−1) , (B6)
eff max τ
10
where τ =2T +t and
w
1 (cid:90) T 1 (cid:18) 15(cid:19)
α= dtd(t)2 = 8+ ≈0.4. (B7)
d2 T 24 π2
max 0
Since Γ ∝d2 and Γ ∝Ω2/∆2,
QG eff sc
d2 ∆2
r ∝ eff . (B8)
Ω2
The maximal possible acceleration for a given Rabi frequency is given by (15) and at the same time is related to the
motion (B4) by
2πd
a = max.
max T2
Substituting (15) to (B8) yields
(cid:18) (cid:19)
T
r ∝d k∆ 1+2(α−1) T2, (B9)
max τ
where we have kept all the factors depending on the separation, detuning or time. The ratio r is thus maximized for
maximum possible detuning ∆, separation d and, subject to the constraint T ≤τ/2, for T =τ/2. Using T =τ/2
max
(i.e. t =0) in (B4) then yields the optimized atomic motion (14).
w