PHYSICAL REVIEW A 84, 052121 (2011)

Quantum superposition of massive objects and collapse models

Oriol Romero-IsartMax-Planck-Institut fur Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany

(Received 19 October 2011; published 28 November 2011)

We analyze the requirements to test some of the most paradigmatic collapse models with a protocol thatprepares quantum superpositions of massive objects. This consists of coherently expanding the wave function ofa ground-state-cooled mechanical resonator, performing a squared position measurement that acts as a doubleslit, and observing interference after further evolution. The analysis is performed in a general framework andtakes into account only unavoidable sources of decoherence: blackbody radiation and scattering of environmentalparticles. We also discuss the limitations imposed by the experimental implementation of this protocol usingcavity quantum optomechanics with levitating dielectric nanospheres.

DOI: 10.1103/PhysRevA.84.052121 PACS number(s): 03.65.Ta

I. INTRODUCTION

In the past few decades, seminal experiments have demon-strated that massive objects can be prepared in spatial su-perpositions of the order of its size. This has been realizedwith electrons [1], neutrons [2], atoms and dimers [3], smallvan der Waals clusters [4], fullerenes [5], and even withorganic molecules containing up to 400 atoms [6]. Theseexperiments are designed to observe the interference of matterwaves after passing, in essence, through a Young’s double slit.The possibility of observing these quantum pheonomena withyet-larger objects is extremely challenging. This is due to thegreat quantum control and isolation from the environment thatthese experiments require.

More recently, the field of cavity quantum electro- andoptomechanics [7–11] has opened the pathway to bringmuch more massive objects to the quantum regime, namelyobjects containing billions of atoms, thereby improving theprevious benchmark by many orders of magnitude. This allowsus to explore the physics of a completely new parameterregime. A first step toward this direction has been realizedin Refs. [12–14], where ground-state cooling of mechanicalresonators at the nano- and microscale has been achieved.Additionally, various researchers have proposed to exploit thecoherent coupling of the mechanical resonator with singlephotons or qubits to create quantum superpositions, see, forinstance, Refs. [15,16]. In these proposals, the superposition ofthe mechanical motion state is, typically, of the form |0〉 + |1〉,where |0〉 and |1〉 are, respectively, the ground state and thefirst excited state of the harmonic potential. In these states,the position is delocalized over distances of the order of thezero point motion, i.e., x0 = √

h/(2mω), where m is the massof the object and ω the frequency of the harmonic potential.Within the megahertz regime, objects containing nat atoms aredelocalized over distances of the order of 10−7n

−1/2at m, which

is subatomic for objects containing billions of atoms. This isin contrast with matter-wave experiments, where, despite thefact that objects have “only” hundreds of atoms, they can bedelocalized over distances larger than their size.

Remarkably, these experiments might be applied to theservice of a very fundamental goal, namely the exploration ofthe limits of quantum mechanics predicted by several collapsemodels [17–26]. The common idea of these models is theconjecture that the Schrodinger equation is an approximation

of a more fundamental equation, which breaks down whenobjects above a critical mass are delocalized over a criticaldistance. This prediction is very difficult to confront becauseof the following argument: standard decoherence [27,28],described within quantum mechanics, also predicts the im-possibility of delocalizing large objects due to the interactionwith the environment; thus, this masks the effects of collapsemodels. This poses a major challenge to corroborate collapsemodels, as the effects predicted by these must stand alonefrom decoherence processes and be exposed to potentialfalsification. This leads to the central questions of this work:How challenging is it to test collapse models while alsotaking into account unavoidable sources of decoherence? Isit preferable to have small objects delocalized over largedistances, as in matter-wave experiments, or rather largeobjects delocalized over small distances, as in experimentswith mechanical resonators?

The aim of this paper is to address the latter questions byanalyzing a prototypical experiment that bridges approachesfrom quantum-mechanical-resonators and matter-wave in-terferometry. This experiment relies, on the one hand, ontechniques of cavity electro-optomechanics to prepare a me-chanical resonator in the ground state of its harmonic potential.On the other hand, the experiment mimics matter-wave inter-ferometry, as the ground-state-cooled mechanical resonatoris released from the harmonic trap in such a way that itcoherently delocalizes over distances much larger than its zero-point motion x0. A subsequent measurement of the squaredposition, which is to be realized using techniques of quantum-mechanical resonators, collapses the state into a superpositionof different spatial locations, thereby acting as a Young’sdouble slit. Finally, the subsequent free evolution generatesan interference pattern. We remark that the implementationof this experiment using cavity quantum optomechanics withoptically levitating dielectric nanospheres [16,29–31] has beenrecently proposed in Ref. [32]. The present article analyzesthis proposal with a broader scope, namely it studies the effectof some of the most paradigmatic collapse models togetherwith unavoidable sources of decoherence. This allows us toobtain the environmental conditions, masses of the objects,and delocalization distances where collapse models can befalsified. These conditions are general and will be common toany physical implementation of the protocol.

052121-11050-2947/2011/84(5)/052121(17) ©2011 American Physical Society

ORIOL ROMERO-ISART PHYSICAL REVIEW A 84, 052121 (2011)

This article is organized as follows: in Sec. II we introduceand analyze the quantum-mechanical resonator double slitexperiment in a general fashion, neglecting decoherenceand without specifying the experimental implementation.The effect of unavoidable sources of decoherence, such asblackbody radiation and scattering of environmental particles,will be the subject of Sec. III. The effects of several collapsemodels in this experiment are discussed in Sec. IV, where wealso obtain the parameter regime needed to confront them.Finally, in Sec. V, we study the restrictions imposed by animplementation of this experiment using cavity optomechanicswith optically levitating dielectric nanospheres. We draw ourconclusions and provide further directions in Sec. VI.

II. MECHANICAL RESONATOR INTERFERENCE IN ADOUBLE SLIT

In this section we analyze a protocol that merges techniquesand insights from quantum-mechanical resonators and matter-wave interferometry; we call it mechanical resonator interfer-ence in a double slit (MERID). We analyze it without takinginto account standard decoherence (cf. Sec. III) and withoutspecifying its experimental implementation (cf. Sec. V).MERID is realized by applying the following steps (see Fig. 1):

(a) Prepare a mechanical resonator. For instance, trap amassive object of mass m, which is typically a sphere, into anharmonic potential with trap frequency ω.

(b) Cool the center-of-mass along one direction, say x, to,ideally, the ground state of the harmonic potential.

(c) Switch off the harmonic trap and let the wave functionexpand freely during some “time of flight” t1.

(d) Perform a measurement of x2, that is, of the squaredposition of the cooled degree of freedom. This measurementacts as a Young’s double slit since, given the outcome x2,the state collapses into a superposition of being at +x and at−x. The mechanical resonator is, thus, prepared in a spatialsuperposition separated by a distance d = 2|x|.

(e) Let the state evolve freely during a second time offlight t2.

(f) Perform a measurement of the center-of-massposition x.

(g) Repeat the experiment and collect the data for eachdouble slit distance d, corresponding to the result of thesquared position measurement. An interference pattern in thefinal position measurement is unveiled for each d.

A. Steps (a) and (b): Cooled initial state

These steps consist in preparing the object’s the center-of-mass motion along x in the ground state of an harmonicpotential with a trapping frequency ω [see Fig. 1(a)]. In theideal case, the wave function is given by

〈x|0〉 = 1[2πx2

0

]1/4 exp

[− x2

4x20

], (1)

where x0 = √h/(2mω) is the ground-state size and m is the

mass of the object. In realistic situations, the initial stateis given by a thermal state with mean occupation numbern = (exp[βhω] − 1)−1 (where β−1 = kBT , where kB is theBoltzmann constant and T is the effective one-dimensional

(a)

(b)

(c)

(d)t1 t2

ω

x0

σ

d

σd

xf

FIG. 1. (Color online) Schematic illustration of MERID. (a) Asphere of mass m is harmonically trapped, with frequency ω, andcooled into the ground state. The zero-point motion is given by x0. (b)The trap is switched off and the wave function expands freely duringsome time of flight t1. At this time, the width of the wave function isgiven by σ . (c) A squared position measurement is performed suchthat the wave function collapses into a superposition of two wavepackets of size σd separated by a distance d . Both σd and d depend onthe measurement outcome. (d) The superposition state evolves freelyduring a second time of flight t2. An interference pattern is formedwith peaks separated by xf .

center-of-mass temperature), which can be written in the Fockbasis as

ρ(0) =∞∑

n=0

nn

(1 + n)n+1|n〉〈n|. (2)

This state has the following moments: 〈x2(0)〉 = (2n + 1)x20 ,

〈p2(0)〉 = (2n + 1)h2/(4x20 ), and 〈[x(0),p(0)]+〉 = 0. We do

not discuss here how cooling is performed experimentally;see, however, Refs. [16,29–31,33] for optomechanical coolingtechniques [34–36] applied to optically levitating nanospheres.

B. Step (c): Expansion

This step [see Fig. 1(b)] consists in switching off the trapand letting the wave function evolve freely, that is, it evolveswith the unitary time evolution U0(t) = exp[−iH0t/h], whereH0 = p2/(2m). Considering the initial state to be the pureground state, the state after some time t1 is given by

〈x|U0(t1)|0〉 = 1

[2πσ 2]1/4exp

[− x2

4σ 2+ iφtof

x2

σ 2

], (3)

where σ 2 = x20 (1 + t2

1 ω2) is the size of the expanded wavefunction and φtof = ωt1/4 is the global phase accumulatedduring the free evolution.

C. Step (d): Double slit

In this step a squared position measurement of the stateat time t = t1 is performed, see Eq. (3), such that the statecollapses into

|ψ〉 ≡ MdU0(t1)|0〉||MdU0(t1)|0〉|| . (4)

052121-2

QUANTUM SUPERPOSITION OF MASSIVE OBJECTS AND . . . PHYSICAL REVIEW A 84, 052121 (2011)

TABLE I. Summary of the restrictions on the expansion time t1, the superposition size d , and the evolution time forming the interferencepattern t2 of the MERID experiment. Conditions (i)–(iv) are discussed in Sec. II and depend on the measurement strength. Conditions(v)–(viii) depend on the position-localization decoherence and are derived in Sec. III. Finally, condition (ix) is given by the optomechanicalimplementation of MERID, which is the subject of Sec. V. Recall the definitions of tmax in Eq. (19), of ξmax in Eq. (20), and of ξs(t1) in Eq. (18).

t1 d t2

(i) Highly probably outcome – <√

8σ –(ii) Superposition peaks resolved – >σ/

√χ –

(iii) Wave packets overlap �√

2t2χ/ω – –(iv) Fringes can be resolved – <2πht2/(mδx) –(v) Decoh. expansion if d � 2a Optimal: tmax <ξ (t1) � ξmax –(vi) Decoh. expansion if d � 2a �1/γ <ξs(t1) –(vii) Decoh. interference if d � 2a – <

√3/( t2) –

(viii) Decoh. interference if d � 2a – – �1/γ

(ix) Optomechanical implementation � min{√κ/g0,4g0/�0sc}/ω <ξs(t1) –

The measurement operator, Md , is assumed to have thefollowing form:

Md = exp[iφds(x/σ )2]

×{

exp

[−

(x − d

2

)2

4σ 2d

]+ exp

[−

(x + d

2

)2

4σ 2d

]}. (5)

This measurement has the potential to prepare a quantumsuperposition of Gaussian wave functions of width σd sep-arated by a distance d, with an added global phase thatwe discuss below. The state |ψ〉 presents a well-resolvedspatial superposition provided that d > 2σd . Also, one requires√

8σ > d in order to have a non-negligible probabilityto obtain the result d, that is, in order to ensure that|〈d/2|U0(t1)|d/2〉|2/|〈0|U0(t1)|0〉|2 > exp[−1]. We have sum-marized in Table I all the conditions required to successfullyrealize MERID that will be obtained throughout the article.These two obtained here are included as conditions (i) and (ii)by using the definition given below in Eq. (6). Motivated bythe optomechanical implementation of this measurement [seeSec. V where we derive Eq. (5)], let us define a dimensionlessparameter independent of the measurement result, whichcharacterizes the strength of the measurement and relates d

with σd as

χ ≡ σ 2

2σdd. (6)

For a given outcome d, the larger the value of χ , themore resolved the superposition. Figure 2 shows the positionprobability distribution of the state of Eq. (4) with d = σ/2for different measurement strengths χ .

Finally, note that a global phase φds is added during themeasurement. This phase, as well as the one accumulatedduring the time of flight, φtof in Eq. (3), plays an important role.The condition |φds + φtof|d2/(4σ 2) � 1 needs to be fulfilledin order to build the interference of the two wave packetscentered at x = d/2 and x = −d/2. This can be shown byanalyzing 〈p|ψ〉, that is, Eq. (4) in momentum space, fordifferent global phases; see Fig. 3. Recall that within freeevolution, the probability momentum distribution of a wavefunction at t1, has the same form as the position probabilitydistribution at much later times since x(t1 + t2) ≈ p(t1)t2/m.

More intuitively, the global phase adds some momentum tothe wave packets. Depending on the sign of this phase, thetwo wave packets either move apart or toward each other. Ifthe momentum given is too large, for the former case theywill separate with a velocity faster than their expansion rate,and, thus, they will never overlap. For the latter case, theyoverlap only during the “collision”; however, at this time, thewave packets have nearly not expanded and the fringes inthe interference pattern cannot be resolved, see the discussionbelow.

D. Steps (e) and (f): Interference

These steps consist in measuring the position distribution ofthe state obtained after letting the system evolve freely duringa second time of flight t2; this reads

|ψf 〉 ≡ U0(t2)MdU0(t1)|0〉||U0(t2)MdU0(t1)|0〉|| . (7)

The state |ψf 〉 presents interference peaks separated by adistance xf = 2πht2/(md) as long as |φds + φtof|d2/(4σ 2) �1. The peaks are clearly visible when the two wave packetsoverlap, that is, when d = t2h/(2σdm). Using Eq. (6) and

1.0 0.5 0.0 0.5 1.0

x σ

Ψx

2

FIG. 2. (Color online) |ψ(x)|2 = |〈x|ψ〉|2, see Eq. (4), is plottedfor d = σ/2 and measurement strength χ = 6 (dotted gray), χ = 10(dashed red), and χ = 20 (solid blue).

052121-3

ORIOL ROMERO-ISART PHYSICAL REVIEW A 84, 052121 (2011)

p

Ψp

2

FIG. 3. (Color online) |ψ(p)|2 = |〈p|ψ〉|2, see Eq. (4), is plottedin arbitrary units for d = σ/2, measurement strength χ = 50, andα ≡ |φds + φtof | equal to α = 0 (dotted gray), α = 50 (dashed red),α = 100 (dot-dashed blue), and α = 150 (solid purple).

σ 2 ≈ x20 t2

1 ω2 (valid at t1ω � 1), one obtains an upper boundfor t1 given by t1 �

√2t2χ/ω; this corresponds to condition

(iii) in Table I. Another condition is given by the requirementto resolve the interference fringes. Assuming a positionresolution of δx, one requires xf > δx, which provides anupper bound for the slit distance d given by d < 2πht2/(mδx);this sets condition (iv) in Table I. MERID is finished in step (g)where the protocol is repeated to obtain a different interferencepattern for each double slit length d.

III. DECOHERENCE

In the previous section we obtained conditions (i)–(iv) inTable I for a successful realization of MERID. Note that, sincet1 and t2 are unbounded, conditions (i)–(iv) do, in principle,allow for the preparation of arbitrarily large superpositions.This is the stage when one has to take into account the effectof decoherence, which is the subject of this section. We willconcentrate on unavoidable sources of decoherence, that is, onthe decoherence caused by the interaction with environmentalmassive particles (cf. Sec. III B 1), and the effects of blackbodyradiation (cf. Sec. III B 2). We start in Sec. III A by analyzinga general form of decoherence called position localization.We derive the limitations that this imposes on the expansiontime t1 (and, therefore, the superposition size d), as well as tothe visibility of the interference pattern. This form of decoher-ence includes the standard sources of decoherence mentionedabove as well as the effect of collapse models, which wediscuss in Sec. IV.

A. Position-localization decoherence

1. Master equation

The main feature of a position-localization decoher-ence is the exponential decay of position correlations, i.e.,〈x|ρ(t)|x ′〉 ∝ e−�t 〈x|ρ(0)|x ′〉, where � usually depends on|x − x ′|. This form is common both to the decoherencecaused by interaction with the environment [27,28] and to theexotic one caused by collapse models [17–26]. The qualitative

behavior of this source of decoherence is very well describedby the following master equation given in the position basis:

〈x|ρ(t)|x ′〉 = i

h〈x|[ρ,H ]|x ′〉 − �(x − x ′)〈x|ρ(t)|x ′〉, (8)

where we assume one dimension for simplicity. The decoher-ence rate function is defined by

�(x) = γ

(1 − exp

[− x2

4a2

]). (9)

This function depends on two parameters: the localizationstrength γ > 0, which has dimensions of frequency, and thelocalization distance a > 0, which has dimensions of length.The value of these parameters depends on the particular sourceof decoherence, such as gas scattering, blackbody radiation,or the one given by collapse models. This simple masterequation captures the important feature of position localizationdecoherence, namely after the saturation behavior (see Fig. 4)

�(x) ≈{

x2, x � 2a,

γ, x � 2a,(10)

where we have defined the localization parameter ≡γ /(4a2). That is, in the short-distance limit, |x − x ′| �2a, the position correlations decay (ignoring the coher-ent evolution given by the Hamiltonian) as 〈x|ρ(t)|x ′〉 ∝e− |x−x ′ |2t 〈x|ρ(0)|x ′〉, such that the decoherence rate dependsquadratically on |x − x ′|. In this limit, Eq. (8) reads

ρ(t) = i

h[ρ(t),H ] − [x[xρ(t)]]. (11)

The decoherence rate saturates in the long-distance limit|x − x ′| � 2a. In this regime, the rate is independent of|x − x ′| and the position correlations decay as 〈x|ρ(t)|x ′〉 ∝e−γ t 〈x|ρ(0)|x ′〉. For instance, in Sec. III B 1, we will see thatthis is the limit where the wavelength of the particles impingingthe object is smaller than the separation of a superposition state,such that a single scattering event resolves the position of theobject and provides which-path information.

In the rest of the article, our strategy will be to approximateeach source of position localization decoherence by the simple

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

x a

xγ

FIG. 4. (Color online) The correlation function �(x), see Eq. (9),is plotted (solid blue line). The short-limit approximation �(x) = x2

is also plotted for comparison (dashed gray line).

052121-4

QUANTUM SUPERPOSITION OF MASSIVE OBJECTS AND . . . PHYSICAL REVIEW A 84, 052121 (2011)

master equation of the form (9), obtaining the localizationstrength γ and distance a. Let us, therefore, analyze therestrictions that the master Eq. (9) impose on MERID.

2. Solution of the master equation

In MERID, different steps assume the free evolution givenby the Hamiltonian H0 = p2/(2m). Hence, we need to obtainthe solution of Eq. (8) for the free-dynamics case. This is givenby [17]

〈x|ρ(t)|x ′〉 =∫ ∞

−∞

dpdy

2πhe−ipy/h

×F(p,x − x ′,t)〈x + y|ρs(t)|x ′ + y〉, (12)

where ρs(t) denotes the evolution of the density matrix withthe Schrodinger equation only, that is, when �(x) = 0. Thefunction

F(p,x,t) = e−γ t exp

[γ

∫ t

0dτe−[(x−pτ/m)/(2a)]2

](13)

takes into account the effects of decoherence. We will usethis solution in the following to obtain different restrictions onMERID.

3. Free evolution: coherence length

We have seen that step (d) of MERID implements adouble slit. It is crucial to ensure that the squared positionmeasurement prepares a quantum superposition instead of astatistical mixture. For this to happen, the coherence lengthof the state before the measurement has to be larger thanthe slit separation d. The coherence length is obtained byanalyzing the decay of the position correlation functionC(x,t) ≡ 〈x/2|ρ(t)| − x/2〉 as a function of the distance.

Let us begin by using the solution Eq. (12) to obtainthe time evolution of the mean values and moments ofx and p. It is straightforward to observe that the meanvalues are not perturbed by the position localization decoher-ence, that is, 〈x(t)〉 = 〈x(t)〉s = 〈x(0)〉 + t〈p(0)〉/m, 〈p(t)〉 =〈p(t)〉s = 〈p(0)〉. However, decoherence does modify the timeevolution of the second-order moments

〈x2(t)〉 = 〈x2(t)〉s + 2 h2

3m2t3,

〈p2(t)〉 = 〈p2(t)〉s + 2 h2t, (14)

〈[x(t),p(t)]+〉 = 〈[x(t),p(t)]+〉s + 2 h2t2

m.

Here, 〈x2(t)〉s = 〈x2(0)〉 + 〈p2(0)〉t2/(2m), 〈p2(t)〉s =〈p2(0)〉, and 〈[x(t),p(t)]+〉 = 2〈p2(0)〉t/m. We remark thatthe extra diffusive term ∼t3 found in Eq. (14) for the positionfluctuations is a clear signature of a random force withoutdamping, which is the case of the position-localizationdecoherence. It is also interesting to note that Eqs. (14)depend only on and, therefore, could have been obtainedwith the simpler master equation Eq. (11) which is, however,only valid in the short-distance limit.

The position correlation function C(x,t) can be nowcomputed using Eq. (12) and Eq. (14). In particular, by taking

into account that ρs(t) is a Gaussian state, one can perform theintegration over y in Eq. (12) and obtain

C(x,t) =∫ ∞

−∞

dp

2πF(p,x,t)

× exp

[−〈p2〉sx2 + 〈x2〉sp2 − 〈[x,p]+〉sxp

2h2

]. (15)

Note that we do not explicitly write the time dependence ofthe moments in order to ease the notation. A simpler formulaof C(x,t) for the short-(long-) distance limit can be derivedby using the corresponding approximation in F(p,x,t). Thisleads to

C(x,t)

C(0,t)≈

{exp[−x2/ξ 2(t)], x � 2a,

exp[−x2/ξ 2s (t) − γ t], x � 2a.

(16)

We have defined the coherence lengths

ξ 2(t) = 8h2〈x2(t)〉4〈x2(t)〉〈p2(t)〉 − 〈[x(t),p(t)]+〉2

(17)

and

ξ 2s (t) = 8σ 2(t)

2n + 1. (18)

ξs is obtained by evaluating Eq. (17) with the unitary evolutiongiven by the Schrodinger equation alone. Recall the definitionof σ 2(t) = x2

0 (1 + t2ω2) and that for an initial thermal state,〈x2(0)〉 = (2n + 1)x2

0 . While ξs(t) increases monotonically intime, ξ (t) has a maximum at

tmax =[

3m(2n + 1)

2 hω

]1/3

, (19)

which yields

ξmax =√

2

[2hω

3m 2(2n + 1)

]1/6

. (20)

See Fig. 5 for a particular example. Notice that the maximumcoherence distance, according to Eq. (16), depends cruciallyon the saturation distance a of the position-localizationdecoherence.

As mentioned, the coherence distance imposes someconditions on MERID. In particular, the superposition sized has to be smaller than the coherence distance, namely onerequires C(d,t1)/C(0,t1) ∼ 1 in order to prepare a coherentsuperposition instead of a statistical mixture. This gives riseto conditions (v) and (vi) in Table I depending on the ratiod/(2a).

4. Visibility of the interference pattern

The position-localization decoherence can also compro-mise the visibility of the interference pattern in steps (e) and (f)of MERID. Using Eq. (12), one obtains that the time evolutionof the position distribution P (x,t) ≡ 〈x|ρ(t)|x〉 is given by

P (x,t) = 1

2πh

∫ ∞

−∞dpdx ′e

ipx

h F(p)e− ipx′h Ps(x

′,t)

= 1√2πh

∫ ∞

−∞dpe

ipx

h F(p)Ps(p,t), (21)

052121-5

ORIOL ROMERO-ISART PHYSICAL REVIEW A 84, 052121 (2011)

0 20 40 60 80 1000

10

20

30

40

50

60

70

t ms

ξnm

FIG. 5. (Color online) As an example, the coherence distanceξ (t), see Eq. (17), is plotted for a sphere of R = 50 nm at a bulktemperature of 200 K, taking into account the decoherence given byblackbody radiation; see Sec. III B 2. Other experimental parametersare taken from Table II.

where Ps(p,t) = ∫dxe−ipx ′/hPs(x,t)/

√2πh. The position

distribution without decoherence Ps(x,t) oscillates with awavelength given by xf = 2πht/(md), which corresponds tothe distances between the interference maxima. Thus, Ps(p,t)has peaks at pf = ±2πh/xf = md/t . Hence, the reductionof the interference peaks, which we use as a figure of meritfor the visibility of the interference pattern, is given byV(t) ≡ F(2πh/xf ) = exp [−t�], where

� = γ − γ

√πa

derf

[d

2a

]. (22)

Note that � ≈ d2/3 in the limit d � 2a and � ≈ γ in thelimit d � 2a. Therefore, the requirement �t2 � 1 establishesthe conditions (vii) and (viii) in Table I depending on the ratiod/(2a).

B. Unavoidable sources of standard decoherence

Let us now focus on the unavoidable decoherence givenby scattering of air molecules and blackbody radiation.Decoherence due to environmental scattering is a well-studiedtopic triggered by the work of Joos and Zeh [37]. Foran extensive review on these topics, we refer the readerto Refs. [27,28]. Here we review the results needed forour analysis. Localization due to environmental scattering isdescribed by a master equation of the type

〈x|ρ(t)|x′〉 = i

h〈x|[ρ,H ]|x′〉 − F (x − x′)ρ(x,x′), (23)

where the decoherence function F (x − x′) depends on thedistance |x − x′| and can be expressed as [28]

F (x) =∫ ∞

0dqρ(q)v(q)

∫dndn′

4π

×(1 − eiq(n−n′)·x/h)|f (qn,qn′)|2. (24)

The derivation assumes an infinitely massive object and thefact that the incoming particles are isotropically distributedin space. Here, ρ(q) denotes the number density of incomingparticles with magnitude of momentum equal to q, v(q) =

q/ma [v(q) = c] is the velocity of massive (massless) par-ticles, |n| = |n′| = 1, and f (qn,qn′) is the elastic-scatteringamplitude. For further details, see Chapter 3 of Ref. [28].The behavior of the function differs substantially dependingon the ratio between |x − x′| and the thermal wavelengthof the scattering particles λth. In the long-wavelength limit,λth � |x − x′|, F (x − x′) ∼ |x − x′|2, whereas in the short-wavelength limit, λth � |x − x′|, one obtains the saturationof F (x − x′) ∼ γ . That is, above some critical distance eachscattering event resolves the separation |x − x′| and therebyprovides which path information. This allows us to relatequalitatively and quantitatively the master equation (23) withthe simpler one (8) discussed in the previous subsections.This connection, which has been discussed previously inRefs. [38,39], is given by the following relations:

a = λth/2 and γ = λ2th . (25)

In the analysis of decoherence due to environmental scat-tering one typically employs the long-wavelength limit sinceit always provides upper bounds on decoherence rates, evenwhen one is in the short-wavelength limit. This was the case,for instance, in the analysis performed in the optomechanicaldouble slit proposal in Ref. [32]. As shown below, the upperbounds for the case of scattering of air molecules were tooloose, since one is in the saturation regime. This yielded therequirement of very low pressures. The analysis performedin the following takes into account the saturation effect andyields much more feasible vacuum conditions.

1. Air molecules

The thermal wavelength of a typical air molecule, which isassumed to be in thermal equilibrium with an environmentat temperature T , is given by λair

th = 2πh/√

2πmakbTe ≡2aair, where ma is its mass. Using ma ∼ 28.97 amu andTe ∼ 4.5 K, one obtains 2aair ∼ 0.15 nm. The localizationparameter associated with scattering of air molecules in thelong-wavelength limit is given by [28]

�air = 8√

2πmavPR2

3√

3h2, (26)

where v is the mean velocity of the air molecules, P thepressure of the gas, and R the radius of the sphere. Thus,using Eqs. (25) and (26) and the expression of λair

th , one obtains

γair = 16π√

2π√3

PR2

vma

. (27)

In the following, we will consider superpositions which are,at least, in the nanometer scale. Therefore, we will use theshort-wavelength limit d � 2a to account for the decoherenceeffect of air molecules. The effect of this decoherence isshown in Fig. 6, where the coherence time γ −1

air and thecorresponding coherence distance ξs(γ −1

air ) are plotted as afunction of the diameter of the sphere and for differentpressures. Note that these quantities define the conditions (vi)and (viii) in Table I. In particular, for sufficiently low pressures,large superpositions of the order of the size of the object arepermitted. We remark, again, that in Ref. [32] the saturationeffect was not taken into account, and this gave rise to pressuresof 10−16 Torr for spheres of 40 nm, which turns out to be a

052121-6

QUANTUM SUPERPOSITION OF MASSIVE OBJECTS AND . . . PHYSICAL REVIEW A 84, 052121 (2011)

10 11 Torr 10 12 Torr

10 13 Torr

10 14 Torr

10 15 Torr

0 500 1000 1500 20001

510

50100

5001000

D nm

γ air1

ms

10 11 10 12 Torr 10 13 Torr

10 14 Torr

10 15 Torr

ξs D

0 500 1000 1500 20001

10

100

1000

104

D nm

ξ sγ a

ir1nm

FIG. 6. (Color online) Coherence time 1/γair (upper panel) andthe corresponding coherence distance ξs(1/γair) (lower panel) takinginto account the decoherence of air molecules as a function of thesphere’s diameter and for environmental pressures of P = 10−11 Torr(solid blue line), 10−12 Torr (dashed red line), 10−13 Torr (dottedorange line), 10−14 Torr (dot-dashed green line), and 10−15 Torr (largedashed purple line). Other experimental parameters are taken fromTable II. In the lower panel, the thinner dashed gray line correspondsto the line ξs(1/γair) = D.

very loose upper bound when taking into account the saturationeffect. We will come back to this point in the optomechanicalimplementation of MERID in Sec. V.

2. Blackbody radiation

The thermal wavelength for massless particles is givenby λbb

th = π2/3hc/(kBTe) ≡ 2abb, which at temperatures T ∼4.5 K takes the value of λbb

th ∼ 1 mm. In this case, the long-wavelength limit can be employed since the superpositionsconsidered will be always smaller than λbb

th . Recall that inthis limit the relevant quantity is the localization parameter.This parameter has three contributions given by scattering,emission, and absorption of thermal photons, namely bb = bb,sc + bb,e + bb,a, which are given by

bb,sc = 8! × 8ζ (9)cR6

9π

[kBTe

hc

]9

Re

[εbb − 1

εbb + 2

]2

(28)

and

bb,e(a) = 16π5cR3

189

[kBTi(e)

hc

]6

Im

[εbb − 1

εbb + 2

]. (29)

100 200 300 400 5001

510

50100

5001000

Ti K

t max

ms

FIG. 7. (Color online) tmax taking into account blackbody radia-tion is plotted as a function of the internal temperature of the sphere.We used a sphere of radius of 50 nm although the dependence on theradius is negligible. Other experimental parameters are taken fromTable II.

We refer the reader to Refs. [28,29] for further details.Here, ζ (x) is the ζ Riemmann function, εbb is the averagedielectric constant, which is assumed to be time independentand relatively constant across the relevant blackbody spectrumand Ti is the bulk temperature of the object, which might differfrom Te.

From the three contributions, the emission localizationparameter is usually the dominant one since the internaltemperature is usually larger than the external one, for instance,due to laser absorption during the optical manipulation ofthe sphere. In Fig. 7 we plot the optimal time tmax, seeEq. (19), as a function of the internal temperature of theobject. The dependence of tmax on the size of the sphere isnegligible. Comparing Fig. 7 with the upper panel of Fig. 6,one concludes that, for low pressure, the decoherence due toblackbody radiation will be dominant, especially when theinternal temperature differs from the external one which issupposed to be cryogenic (a few Kelvins).

3. Limitations on the superposition size

Let us now summarize the operational parameter regime ofMERID, taking into account standard sources of decoherence.We compute the lower and upper bounds for d for allowedvalues of t1 and t2 according to Table I. We will use twosets of experimental parameters for the pressure, the internaltemperature, the measurement strength, and the common onesgiven in Table II. The first set, which is assumed to befeasible, assumes an environmental pressure of P = 10−12

Torr, internal temperature of the object Ti = 200 K, andmeasurement strength χ = 1000, whereas the challenging setassumes P = 10−16 Torr, Ti = 4.5 K, and χ = 106.

Figure 8 shows the upper bounds for t1 given by conditions(iii), (v), and (vi) in Table I, considering t2 = 0.1/γair. Forthe feasible set, t1 in the few milliseconds are possible forspheres up to a diameter of 250 nm. For the challenging set,much larger timescales of the order of seconds are possiblefor objects in the micrometer regime. In both cases, pressureis the limiting factor for the internal temperatures considered.Note that, in both cases, these values correspond to very long

052121-7

ORIOL ROMERO-ISART PHYSICAL REVIEW A 84, 052121 (2011)

TABLE II. Experimental parameters used in this article.

ρ εr ω n Te ma εbb δx F L λ Wc

2201Kg/m3 2.1 + i10−10 2π × 100 KHz 0.1 4.5K 28.97 amu 2.1 + i0.57 0.1 nm 1.3 × 105 2 μm 1064 nm 1.5 μm

coherence times comparing to typical quantum-mechanicalexperiments.

In Fig. 9 we plot the superposition size that fulfillsconditions (ii), (iv), (v), and (vii) in Table I for t2 = 0.1/γair

and t1 = min{√2t2χ/ω,tmax,0.05/γair}, which ensures thefulfillment of the restrictions imposed into t1 and t2. Hereafter,we will call this plot a “d vs. D” diagram. For the feasible set,superpositions larger than the size of the sphere are in principlepossible for objects of the order of 50 nm with much lowerpressures than those used in Ref. [32]. For the challengingset, larger objects and larger superpositions are obtained (notethe different scale in the plot). It is important to remark thatin both cases the limitation is given by condition (iv), whichreads d < 2πht2/(mδx) and is related to the resolution of theinterference fringes. This shows that while coherence times

P 10 12 TorrTi 200 K

103

50 100 150 200 2501.0

10.0

5.0

2.0

3.0

1.5

7.0

D nm

t 1m

s

P 10 16 TorrTi 4.5 K

105

0 200 400 600 800 1000 1200 14001

10

100

1000

104

105

D nm

t 1m

s

FIG. 8. (Color online) Upper bounds for t1 due to blackbodyradiation and scattering of air molecules as a function of the diameterof the sphere for P = 10−12 Torr, Ti = 200 K, and χ = 1000 (P =10−16 Torr, Ti = 4.5 K, and χ = 106) in the upper (lower line) panel.tmax (dashed red line), see Eq. (19), is condition (v) in Table I, 0.05/γair

(dotted brown line) is condition (vi), and√

2t2χ/ω (dot-dashed blueline) is condition (iii). We used t2 = 0.1/γair and the other parametersgiven in Table II. The shadowed region corresponds to t1, fulfillingall three conditions.

are very large, the dynamics resulting in the spreading of thewave function are very slow for large masses. This hints atpossible improvements of MERID using more efficiently thelong coherence times allowed by the unavoidable sources ofdecoherence considered here.

IV. COLLAPSE MODELS

In this section we discuss the possibility of using MERIDto test some of the most paradigmatic collapse models.Remark that any experimental evidence of these would imply abreakdown of the theory of quantum mechanics. In the

P 10 12 TorrTi 200 K

103

50 100 150 200 2501

2

5

10

20

50

100

200

D nm

dnm

P 10 16 TorrTi 4.5 K

105

0 200 400 600 800 1000 1200 14001

10

100

1000

104

D nm

dnm

FIG. 9. (Color online) Superposition distance d as a function ofthe diameter D = 2R of the sphere, taking into account blackbodyradiation and scattering of air molecules with P = 10−11 Torr, Ti =200 K, and χ = 1000 (P = 10−16 Torr, Ti = 4.5 K, and χ = 106) inthe upper (lower) panel and the others parameters given in Table II.Note the different scales for the upper and lower panels. Accordingto Table I, we plotted condition (ii) d > σ/

√χ (solid black line),

condition (iv) d < 2πht2/(mδx) (dot-dashed blue line), condition (v)d < ξ (t1) (dashed red line), and condition (vii) d <

√3/( t2) (dotted

orange line). The shadowed region corresponds to d , fulfilling all fourconditions and thin dashed gray line to d = D. We used t2 = 0.1/γair

and t1 = min{√2t2χ/ω,tmax,0.05/γair}.

052121-8

QUANTUM SUPERPOSITION OF MASSIVE OBJECTS AND . . . PHYSICAL REVIEW A 84, 052121 (2011)

following, we consider the unavoidable sources of decoher-ence discussed in the previous section and, on these grounds,we determine the experimental parameters required to falsifya given collapse model by the observation of the interferencepattern. We note that the corroboration of the collapse model ismore challenging than its falsification, since one must discardthat standard decoherence, in any of its forms, is responsiblefor the disappearance of the interference pattern.

We shall not review the extensive literature on collapsemodels; instead, we will focus on some of the most discussedones in the literature, and, for each of them, we will providea brief summary of their prediction. In particular, we shallexpress them in a common form, namely as a master equationdescribing position-localization decoherence; this will allowus to apply the results of Sec. III A and, thereby, to discuss thepossibility to test them using MERID.

A. Continuous spontaneous localization

We start with the continuous spontaneous localization(CSL) model [18,40], which is the best developed collapsemodel at present [19]. This model builds on the previous worksof Ghirardi, Rimini, and Weber (GRW) [17]; Pearle [41];and Gisin [42], and it bears some similarity to the works ofGisin [43] and Diosi [44,45]. The model is constructed byadding a nonlinear stochastic term to the Schrodinger equation.This term predicts a localization whose strength is directlyproportional to the mass of the object. At the same time, it isconstrained by the fact that the equation must reproduce allphenomenology of quantum mechanics for small objects. Thisintroduces two phenomenological constants that are boundedby experimental evidence.

More specifically, within the CSL model, the masterequation describing the wave function of N particles is givenin the position representation |x1, . . . ,xN 〉 ≡ |x〉 by [18,40,46]

〈x|ρ(t)|x′〉 = i

h〈x|[ρ,H ]|x′〉 − �CSL(x,x′)〈x|ρ(t)|x′〉, (30)

where

�CSL(x,x′) = −γ 0CSL

2

N∑i,j=1

mimj

m20

×[�(xi − xj ) + �(x′i − x′

j ) − 2�(xi − x′j )].

(31)

Here, m0 is the mass of a nucleon, γ 0CSL is the single nucleon

collapse rate, and

�(r) = exp

[− |r|2

4a2CSL

](32)

is the localization function with aCSL being the localizationdistance. Note that for the single-nucleon case, Eq. (31) reads

�CSL(x,x′) = γ 0CSL

(1 − exp

[−|x − x′|2

4a2CSL

]), (33)

which has the same form as Eq. (9). The parameters γ 0CSL

and aCSL are the two phenomenological constants of themodel. Their value is bounded by both experimental data and“philosophical” reasons; see Ref. [47] for a recent discussion.

The standard values originally proposed in the GRW model[17] are aCSL = 100 nm and γ 0

CSL = 10−16 Hz. However, thevalue of γ 0

CSL has been recently reconsidered by Adler and ispredicted to be 8 to 10 orders of magnitude larger [20,48] thanthe original one of 10−16 Hz, a prediction not yet confrontedby up-to-date experiments [20]. We will, however, considerhere the original values for the sake of comparison.

The decoherence factor of the CSL model, Eq. (31) can beobtained for the center of mass of a solid sphere of mass m,volume V , and homogeneous mass density. We use the resultsgiven in Ref. [46] where an analysis of the CSL model for thefree propagation of a solid mass is discussed. In this case, thedecoherence factor [cf. Eq. (31)] takes the form

�CSL(x,x′)

= −γ 0CSL

m2

m20

∫V

drdr′

V 2[�(r − r′) − �(r − r′ + x − x′)].

(34)

In order to approximate Eq. (34) to (9), we can extract thelocalization parameter in the expression given for the freeevolution of the position fluctuation, which is given by [46]

〈x2(t)〉 = 〈x2(t)〉s + γ 0CSLh

2f (R/aCSL)t3

6m20a

2CSL

. (35)

By comparing it with Eq. (14), one obtains

CSL = m2

m20

γ 0CSL

4a2CSL

f (R/aCSL), (36)

where the function f (x) is given by

f (x) = 6

x4

[1 − 2

x2+

(1 + 2

x2

)e−x2

], (37)

and has the following limits f (x → 0) = 1, f (1) = 0.62, andf (x → ∞) = 6/x4. Thus, recalling that = γ /(4a2), oneobtains the collapse rate

γCSL = m2

m20

γ 0CSLf (R/aCSL). (38)

Note that the rate γCSL grows quadratically with the numberof nucleons for spheres smaller than 2aCSL.

To grasp the strength of the exotic decoherence by thismodel, we plot in Fig. 10 the value of the coherence time1/( CSLd2) of a superposition of size d � 2aCSL as a functionof the sphere’s diameter D and the superposition distance d.Coherence times of the order of milliseconds are obtained forobjects of 300 nm and superpositions of tens of nanometers.Note that these coherence times would be strongly reduced byusing the enhancement of the localization rate γ 0

CSL predictedby Adler [20,48].

B. Quantum gravity

Ellis, Mohanty, Mavromatos, and Nanopoulos suggestedin Refs. [24,49] that quantum gravity (QG) can induce thecollapse of the wave function of sufficiently massive objects.They argue that the collapse is induced by the interaction ofthe massive object with topologically nontrivial space-timeconfigurations (wormholes) which are small compared tophysical scales but much larger than the Planck scale. From the

052121-9

ORIOL ROMERO-ISART PHYSICAL REVIEW A 84, 052121 (2011)

1 s

0.1 s

10 ms

1 ms

0.1 ms

CSL

0 100 200 300 400 500

20

40

60

80

100

120

D nm

dnm

FIG. 10. (Color online) Different values of the coherence time1/( CSLd2) as a function of the sphere’s diameter D and thesuperposition distance d . Other physical parameters, such as thedensity of the sphere, are taken from Table II.

point of view of quantum information theory, this decoherencemechanism can be understood as the result of the center ofmass of the object becoming entangled with some degrees offreedom belonging to these wormholes which are unaccessibleand, therefore, have to be traced out. Indeed, this providesa localization effect analogous to the one induced by theinteraction with the environment.

In this model, the quantum wormholes are assumed tobe in a Gaussian state in momentum space and have zeromean momentum with a spread given by � ∼ cm2

0/(hmP ) ∼10−3 m, where m0 is the mass of a nucleon and mP isthe Planck mass. For distances smaller than 1/�, the off-diagonal elements of the density matrix in position basis decayas 〈x|ρ(t)|x ′〉 ∝ exp[− 0

QG(x − x ′)2t]〈x|ρ(0)|x ′〉, where thelocalization parameter is given by

0QG = c4

h3

m60

m3P

. (39)

Note that the localization distance is very large since it isgiven by aQG = 1/(2�) ∼ 103 m, hence, the decoherence doesnot saturate within the typical wave-function spreadings. Thecollapse rate for the single nucleon is thus given by γ 0

QG =4a2

QG 0QG. It is remarkable that this model [24,49], which is

based on quantum gravity, converts the CSL model into aparameter-free model. The extension of the single nucleoncase to the solid sphere can be obtained by retrieving the resultgiven in Eq. (38). Based on this, the localization parameter ofa solid sphere predicted by this model is given by

QG = c4

h3

m2m40

m3P

, (40)

where we have used f (R/aQG) = 1 since R � aQG for thespheres considered here.

0.1 s

10 ms

1 ms

0.1 ms

0.01 ms

QG

0 100 200 300 400 500

20

40

60

80

100

D nm

dnm

FIG. 11. (Color online) Different values of the coherence time1/( QGd2) as a function of the sphere’s diameter D and thesuperposition distance d . Other physical parameters are taken fromTable II.

As previously, we plot in Fig. 11 different values of thecoherence time 1/( QGd2) as a function of D and d. Bycomparing these results with the ones given by the CSL modelin Fig. 10, we notice that the decoherence effect is slightlystronger but, nonetheless, very similar.

C. von Neumann-Newton equation

In the past 30 years, many authors have investigated thepossible role of the Newtonian gravity in the collapse of thewave function. From these, the independent but similar worksof Diosi and Penrose (DP) are the most famous ones [22,23,25,44,50–52]. The prediction of the model can be casted intothe so-called von Neumann-Newton equation, which can beexpressed as [23,44,50]

〈x|ρ(t)|x′〉 = i

h〈x|[ρ,H ]|x′〉 − �DP(x,x′)〈x|ρ(t)|x′〉 (41)

where

�DP(x,x′) = Ug(x,x) − Ug(x′,x′) + 2Ug(x,x′)2h

. (42)

Here Ug(x,x′) is the Newtonian interaction between two massdensities corresponding to two spheres centered at position xand x′, respectively. This reads

Ug(x,x′) = −G

∫f (r|x)f (r′|x′)

|r − r′| drdr′, (43)

where f (r|x) is the mass density at location r for the spherecentered at x. For a rigid homogeneous ball, the mass densityis uniform and equals f = 3M/(4πR3). In this case, the

052121-10

QUANTUM SUPERPOSITION OF MASSIVE OBJECTS AND . . . PHYSICAL REVIEW A 84, 052121 (2011)

decoherence function of Eq. (42) depends on the relativedistance |x − x′| and presents the following limits [50]:

�DP(x) ={

Gm2/(2R3h)x2, x � R,

6Gm2/(5Rh), x � R.(44)

The quadratic dependence at short distances allows us toidentify the localization parameter of the model

DP = Gm2

2R3h, (45)

as well as the saturation distance at 2aDP = R.The strength of this model is much weaker than the CSL and

the QG; see Fig. 12, where 1/( DPd2) is plotted as a function of

D and d. Coherence times of the order of milliseconds are onlyobtained for objects of few microns prepared in superpositionswithin the micrometer scale.

Let us point out that the decoherence rate given by thiscollapse model can be strongly enhanced by considering themass density at the microscopic level [23]. For instance, bymodeling the fine structure beyond the constant average massas a conglomerate of identical small balls of mass m0 andradius r0, the parameters of the model are given by [23] 2aDP =r0 and

DP =(

R

r0

)3

DP. (46)

Thus, since r0 is typically chosen much smaller than R,the localization parameter is greatly enhanced. This is used,for instance, in the Marshall proposal to test the Penrosemodel using the small delocalization of a micromirror [15,53].However, this choice is controversial since it not only convertsthe parameter-free model into a one-parameter (r0) model, but

10 s 1 s 0.1 s

10 ms

1 ms

d 2aDP

DP

0 1000 2000 3000 4000 50000

1000

2000

3000

4000

5000

D nm

dnm

FIG. 12. (Color online) Different values of the coherence time1/( DPd

2) as a function of the sphere’s diameter D and thesuperposition distance d . The dashed gray line marks the limit wherethe coherence time is not longer valid since, for d > 2aDP, it wouldthen be given by 1/γDP. Other physical parameters are taken fromTable II.

the von-Neumann-Newton equation also becomes divergentfor pointlike particles [23,52,54]. This yields unphysicalresults such as the nonconservation of energy, especially whenr0 � 100 nm (see Refs. [23,54]). For this reason, while it isnot clear how to choose the mass distribution, in this article weassume the well-behaved case of a solid homogeneous density;this comes at the price of making formidable the possibility offalsifying the model, as shown below.

D. Imprecise space-time

Finally, we also consider the K model, named afterKarolyhazy, who introduced one of the first collapse models asearly as the 1960s [55,56]. The model builds on the insight thatthe sharply determined structure of space-time is incompatiblewith quantum mechanics and general relativity: According toquantum mechanics, the position and the velocity of an objectcannot have deterministic values simultaneously, while generalrelativity states that the space-time structure is determined bythe positions and velocities of the masses.

We base our approach on the article of Frenkel [21], whoprovides a very clear review of the K model and its relation tothe CSL model. The prediction of the K model in the so-called“no-breathing limit” [21] is given by the following masterequation:

ρ(t) = i

h[ρ(t),H ] − K [x[xρ(t)]]. (47)

This corresponds to a position-localization decoherence witha localization distance 2aK → ∞, a localization rate γK → 0,and a localization parameter γK/(4a2

K ) → K . The localiza-tion parameter for a solid sphere of mass m is given by [21]

K = h

8ma4c

, (48)

where

ac ={(

RlP

)3/2lC, R > aK,(

lClP

)2lC, R < aK.

(49)

Here lP =√

Gh/c3 is the Planck length and lC = h/(mc) theCompton wavelength. Note that this model is also parameterfree.

As shown in Fig. 13, the strength of this model is alsoweaker than the CSL and the QG and only slightly strongerthan the DP. Coherence times of the order of millisecondsare obtained for spheres with diameter between one andtwo micrometers prepared in superpositions smaller than onemicrometer.

E. Experimental test

Let us now address the possibility of testing these col-lapse models using MERID. The localization parameter andthe localization distance of each model are summarizedin Table III.

For the sake of comparison, we plot the ratio betweentheir localization parameter with the localization parameterprovided by blackbody radiation in Fig. 14, where we haveassumed a bulk temperature of Ti = 4.5 K (recall Sec. III B 2).While all collapse models provide a stronger localization rate

052121-11

ORIOL ROMERO-ISART PHYSICAL REVIEW A 84, 052121 (2011)

10 s1 s

0.1 s

10 ms

1 ms

K

0 500 1000 1500 20000

500

1000

1500

2000

D nm

dnm

FIG. 13. (Color online) Different values of the coherence time1/( Kd2) as a function of the sphere’s diameter D and thesuperposition distance d . Other physical parameters are taken fromTable II.

than the blackbody radiation, the CSL and QG model aremany orders of magnitude stronger than the DP and the Kmodel. Actually, the standard decoherence given by blackbodyradiation is comparable to the one predicted by the DP andK model for a bulk temperature of 20 K. Since standarddecoherence will limit the superpositions to be smaller thanthe localization distance, the localization parameter is theonly relevant parameter.

We start with the stronger collapse models: the CSL andthe QG. We will use the d vs. D diagram (Fig. 9) to determinehow they can be falsified using MERID. From the boundslisted in Table I, the decoherence given by the collapsemodel contributes only to conditions (v) d < ξ (t1) and (vii)d <

√3/( t2). Hence, we recalculate these two bounds using

= bb + CM, where CM is the contribution given by thecollapse model; this is summarized in Table III. The timest1 and t2 depend only on the standard decoherence and are,thus, chosen as described in the caption of Fig. 9. Figure 15shows the d vs. D diagram, including either the CSL or the QGcollapse model. As explained in the caption of the figure, thegreen region is the parameter regime where the collapse modelcan be falsified. This shows that both the CSL and the QGcollapse model can be tested, for instance, at P = 10−14 Torr,

TABLE III. Summary of the decoherence parameters predictedby different collapse models. Recall that for the CSL model we takethe original value of γ 0

CSL ≈ 10−16 Hz.

2a

CSL ∼200 nm γ 0CSLm2f (R/aCSL)/(4a2

CSLm20)

QG 1/� ∼ 103 m c4m2m40/(h3m3

P )DP R Gm2/(2R3h)K ∞ h/(8ma4

c )

QG

CSL

K

DP

0 200 400 600 800 1000 1200 14001

100

104

106

108

1010

1012

D nm

BB

FIG. 14. (Color online) Localization parameter for the differentcollapse models (CSL, QG, K, and DP) in units of the localizationparameter provided by blackbody radiation, assuming a bulk temper-ature of Ti = 4.5 K, as a function of the diameter of the sphere. Otherparameters are taken from Table II.

Ti = 100 K, and χ = 1000 for spheres with a diameter rangingfrom 100 to 500 nm approximately. As expected, this regionis larger for the QG model than for the CSL model withγ 0

CSL = 10−16 Hz. The appearance of the green region ismainly due to the tighter bound given by condition (vii)d <

√3/( t2), which is imposed in order to preserve the

visibility of the interference pattern. A simulation of theinterference pattern is also plotted in Fig. 15 for a particularpoint in the diagram; see the figure’s caption. The simula-tion is done by numerically solving the master equation atthe different steps of MERID. Therefore, both the CSL (withthe conservative value of γ 0

CSL = 10−16 Hz) and the QGcollapse model can be falsified with the successful observationof the interference pattern if MERID were implemented at thethe green region of the parameter regime.

These results are in strong contrast with the ones obtainedfor the much weaker exotic decoherence given by the DPand K collapse models. Indeed, the green region is zero forboth cases even at much higher vacuum conditions and lowbulk temperatures. However, we have shown in Fig. 14 thatthe localization parameter of these collapse models is largerthan the one given by blackbody radiation at Ti = 4.5 K, and,thus, their additional exotic decoherence should be observable.However, the problem in this case is that the time required toobserve this decoherence using MERID is longer than thecoherence time allowed by the scattering of air molecules1/γair (see Fig. 6). More specifically, the reduction in thevisibility for d � 2aCM and t2 � 1/γair is given by

V(t2) = exp[−( bb + CM)d2t2/3]. (50)

Assuming bb � CM, the visibility can be approximated toV(t2) ≈ exp[− CMd2t2/3], and, thus, it will be reduced attimes

3

CMd2� t2 � 1

γair, (51)

where we added the second inequality to emphasize that t2 hasto be smaller than the coherence time allowed by scattering ofair molecules. By inspection of Figs. 6, 12, and 13, one realizesthat these two conditions are extremely challenging to be

052121-12

QUANTUM SUPERPOSITION OF MASSIVE OBJECTS AND . . . PHYSICAL REVIEW A 84, 052121 (2011)

CSL

P 10 14 TorrTi 100 K

104

100 200 300 400 5001

2

5

10

20

50

100

200

D nm

dnm

QG

P 10 14 TorrTi 100 K

104

100 200 300 400 5001

2

5

10

20

50

100

200

D nm

dnm

2 1 0 1 20.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x D

xρ

xD

FIG. 15. (Color online) Operational parameter regime of MERIDwhere the CSL and the QG collapse model can be tested. The uppertwo panels show the d vs. D diagram as in Fig. 9 with P = 10−14 Torr,Ti = 100 K, and χ = 1000. The green region is the nonoverlappingallowed region, taking into account only standard decoherence andthe CSL (first panel) or QG (second panel) collapse model. The dot isat d = 30 nm and D = 100 nm. For this value, the third lower panelplots the simulated interference pattern, taking into account standarddecoherence only (dotted gray line), the CSL model (dashed blueline), and the QG model (solid red line).

fulfilled for the DP and K model. Therefore, we conclude thatthe DP and the K model cannot be tested using the present formof MERID. However, the fact that the localization parameter ofthese collapse models is larger than the one given by blackbodyradiation at cryogenic bulk temperatures hints at the fact thatan improved version of MERID, in which, for instance, the

free dynamics are accelerated by using repulsive potentials,could meet this challenge.

V. OPTOMECHANICAL DOUBLE SLIT

The last section of this article is devoted to the analysisof the restrictions that an optomechanical implementation ofthe squared position measurement imposes on MERID. Theimplementation of MERID using cavity optomechanics withlevitating dielectric spheres has been recently proposed inRef. [32]. Here, we provide a thorough derivation and we betterremark the conditions that need to be fulfilled. For furtherliterature on cavity optomechanics with levitating dielectrics,we refer the reader to Refs. [16,29,31,32].

In the following we focus on step (d) of the protocol, wherethe sphere is assumed to enter a small cavity, ideally alignedsuch that the mean position along the cavity axis 〈x〉 of thesphere is at the node of a cavity mode. In this configuration,the optomechanical coupling is quadratic with x. This impliesthat the output light of the cavity contains information aboutx2, and therefore, this can be measured by homodyning thelight. The optomechanical Hamiltonian reads [31]

H (t) = p2

2m+ hga†ax2 + ihE(t)(a − a†). (52)

The first term describes the kinetic energy of the sphere alongthe cavity axis (note that that there is no harmonic potentialsince the particle does not need to be trapped during the shortinteraction required to measure x2). The third term describesa time-dependent driving at frequency ωL which equals to thecavity resonant frequency ωc, which is used to parametrizethe short light pulse. Finally, the second one is the importantterm describing the optomechanical coupling when the sphereis placed at the node of the cavity mode. We have definedthe creation (annihilation) operators of the cavity modes a†

(a), the dimensionless position operator x = x/σ , and theoptomechanical coupling rate given by g = g0σ

2/x20 , where

g0 = εcx20k3

c cV

4Vc

(53)

in the case of a nanosphere [16,29,31,32]. Here, εc ≡3Re [(εr − 1)/(εr + 2)] depends on the relative dielectricconstant εr , kc = ωc/c, and Vc = πW 2

c L/4 is the cavityvolume, where W is the waist of the cavity mode and L

the length of the cavity. As discussed in Ref. [32], notethat g enhances g0 by a potentially very large factor σ 2/x2

0 ,depending on the size of the wave packet. The interactiontime is assumed to be very small so the interaction is inthe regime of pulsed optomechanics [57]. We do not takeinto account the optimization of the pulse shape [57] andsimply consider a time-dependent driving frequency given byE(t) = √

2κnphξ (t), where ξ (t) is a flat-top function of length

T and amplitude ∼ 1/√

T such that∫ T

0 ξ 2(t)dt = 1, κ is thedecay rate of the cavity, and nph is the total number of photonsthat the light pulse carries. The decay rate of the cavity has acontribution given by the finesse F of the empty cavity and bylight scattering, and it reads [29,32]

κ = 2π

2FL+ cε2

c V2k4

c

16πVc

. (54)

052121-13

ORIOL ROMERO-ISART PHYSICAL REVIEW A 84, 052121 (2011)

A. Measurement operator and strength

Let us show here that after a short interaction with thelight pulse, the measurement of the phase quadrature of theoutput light realizes a measurement of x2 with some givenmeasurement strength χ . Pulsed optomechanics [57] consistsin implementing a very short interaction time T ∼ κ−1 suchthat

〈p2〉2m

T

h= (2n + 1)ωT

4� 1. (55)

This allows us to neglect the kinetic term in Eq. (52), whichyields

H (t) ≈ hga†ax2 − ihE(t)(a† − a). (56)

In order to obtain the output light quadrature we make useof the input-output formalism [58]. The Langevin equationassociated to a is given by

a(t) = −(igx2 + κ)a(t) + E(t) +√

2κain(t), (57)

where ain is the input cavity noise operator. We further assumethat κ � g, such that one can adiabatically eliminate the cavitymode by setting a(t) = 0. This leads to

a(t) ≈ [E(t) +√

2κain(t)]

(1

κ− igx2

κ2

). (58)

By using the input output relation aout(t) = √2κa(t) −

ain(t) and defining the phase quadrature P Lout(t) ≡ i[a†

out(t) −aout(t)]/

√2, one obtains the relation

P Lout(t) ≈ P L

in(t) + χ (t)x2, (59)

where χ (t) ≡ 2gE(t)/(κ√

κ), and we have neglected the smallterm ∼2gx2XL

in/κ . A balanced homodyne measurement of theoutput field performs a quantum measurement of the time-integrated output quadrature given by [32]

P Lout ≡ 1√

T

∫ T

0P L

out(t)dt = P Lin + χx2. (60)

An important result is the value of the measurement strength,which is given by

χ ≈ 2√

2g√

nph

κ. (61)

An optimization of the pulse shape provides a different pref-actor which slightly increases the measurement strength (seeRefs. [57,59]). If the measurement of the optical phase yieldsthe measurement outcome pL, the measurement operatordescribing the collapse of the center-of-mass state of the sphereis given by [32,57,59]

M = exp[iφdsx

2 − (pL − χx2

)2 ]. (62)

As a result of the measurement operator of Eq. (62), asuperposition of two wave packets, separated by a distanced = 2σ

√pL/χ and a width given by approximately σd ∼

σ/(4√

pLχ) = σ 2/(2dχ ), is prepared. This is, thus, in fullagreement with the treatment of Sec. II C. Furthermore, the

global phase accumulated during the interaction with theclassical part of the field is given by

φds = −∫ T

0g〈a†(t)a(t)〉dt

= −∫ T

0E2(t)

(1

κ2+ g2〈x4〉

κ4

)dt ≈ −2gnph

κ, (63)

where the second term can be neglected in the regime κ � g.

B. Restrictions to MERID

The double slit implementation imposes the followingrestriction on MERID. First, recall that the phase [Eq. (63)]needs to be compensated by the phase accumulated during thetime of flight φtof = t1ω/4. Thus, φtof + φds ∼ 0 is fulfilledwhen the total number of photons in the light pulse is

nph = ωt1κ

8g0

x20

σ 2≈ κ

8g0t1ω. (64)

Here we have used again that σ 2 ≈ x20 t

21 ω2 at times

t1ω � 1. Inserting equality Eq. (64) into the definition of themeasurement strength Eq. (69) leads to

χ ≈ (t1ω)3/2

√g0

κ. (65)

Additionally, standard decoherence due to light scatteringduring the light-mechanics interaction [29,32] is preventedif the following conditions are met. This decoherence is alsoof the localization type, with a decoherence rate given by�sc(t) = 0

scE2(t)σ 2/κ for distances smaller than the optical

wavelength, where the localization parameter is [29,32]

0sc = ε2

c

6π

c

Vc

V 2k6c . (66)

This form of decoherence is prevented as long as∫ T

0 �sc(t)dt � 1, which gives rise to the following conditionon t1

t1 � 4g0

ω�0sc

, (67)

where we have used Eq. (64) and we have defined �0sc ≡ 0

scx20 .

Bear in mind that the adiabatic elimination used in thederivation is valid as long as κ � g = g0t

21 ω2, which leads

to a further constraint, namely t1 � ω−1√κ/g0. Thus, theoptomechanical implementation of step (d) of MERID yieldsan additional upper bound on t1 given by

t1 � 1

ωmin

{√κ

g0,4g0

�0sc

}≡ tOM

1 . (68)

This is incorporated in Table I as condition (ix). Also, byinserting Eq. (68) into Eq. (65) we obtain an upper bound forthe measurement strength given by

χ � min

{(κ

g0

)1/4

,8g2

0√κ(�0

sc)3

}≡ χmax. (69)

These conditions give a strong limitation for the overallperformance of MERID which crucially depend on the qualityof the optical cavity employed. In Ref. [32], it was suggested

052121-14

QUANTUM SUPERPOSITION OF MASSIVE OBJECTS AND . . . PHYSICAL REVIEW A 84, 052121 (2011)

10 20 30 40 50 60 70 800

2

4

6

8

D nm

t 1m

s

10 20 30 40 50 60 70 80

10

20

30

40

50

60

D nm

χm

ax

FIG. 16. (Color online) In the upper panel, t1 [see Eq. (68)] isplotted as a function of the diameter of the sphere. The solid blueline is the upper bound 4g0/(ω�0

sc) and the dashed red line theω−1(κ/g0)1/4, which is the bound given by the adiabatic condition.The gray region shows allowed t1 values. In the lower panel, χmax

[see Eq. (69)] is plotted. We used a fiber-based Fabry-Perot opticalcavity; see the main text and Table II.

to use the recently developed fiber-based Fabry-Perot cavitiesof length of 2 μm and finesse F ≈ 1.3 × 105 [60]. As shownin Fig. 16, for this cavity, upper bounds for t1 of the orderof milliseconds and a corresponding χmax of several tensare obtained for spheres smaller than 100 nm. To see theimplications for the realization of MERID, we plot in Fig. 17the d vs. D diagram for a pressure of P = 10−13 Torrand a bulk temperature of Ti = 200 K (which is reasonableconsidering the heating produced by laser absorption [29,32]).For these parameters, t1 = tOM

1 /4 guarantees the fulfillmentof the bounds on t1 given in Table II. It is remarkable thateven taking into account the restrictions imposed by theoptomechanical implementation, spheres with a diameter oftens of nanometers that contain of the order of 107 atomscan be prepared in superpositions of the order of their size.Moreover, even the CSL model with a localization ratefrequency given by 104γ 0

CSL, which is orders of magnitudeslower than the enhancement predicted by Adler [20,48], canbe falsified. The result shown in Fig. 17 is very similar tothe one given in Ref. [32]. Note, however, that here wehave used a pressure three orders of magnitude larger dueto the saturation effect omitted in Ref. [31]. This renders

104γ CSL0

CSLP 10 13 TorrTi 200 K

t1 t1OM 4

t2 10 2 γ air

20 30 40 50 600

20

40

60

80

100

D nm

dnm

FIG. 17. (Color online) The d vs. D diagram (see Fig. 9) isplotted, taking into account the additional limitations imposed byan optomechanical implementation of the squared position measure-ment. We used the experimental parameters of Table II, a pressureof P = 10−13 Torr, an internal bulk temperature of Ti = 200 K,t1 = tOM

1 /4, and t2 = 10−2/γair.

the implementation of MERID using cavity optomechanicswith levitating spheres less challenging. Finally, we remarkthe recent proposal given in Ref. [61] to test the CSL modelusing an all-optical time-domain Talbot-Lau interferometer forclusters with masses exceeding 106 amu.

VI. CONCLUSIONS

In summary, we have shown that by combining techniquesand insights from quantum-mechanical resonators and matter-wave interferometry, one can prepare large spatial quantumsuperpositions of massive objects comparable to their size.The protocol consists of cooling a mechanical resonator toits ground state, switching off the harmonic potential to letthe wave function coherently expand, preparing a spatialquantum superposition by performing a measurement of thesquared position observable, and observing interference bymeasuring the position after further free evolution. We havefocused on solid spheres with diameters ranging from tens ofnanometers to few micrometers. We have taken into accountunavoidable sources of decoherence such as the scattering ofenvironmental massive particles and the emission, absorption,and scattering of blackbody radiation. Both sources providecoherence times of the order of milliseconds within reasonablevalues for pressures and temperatures. At low pressures,decoherence due to blackbody radiation is dominant when thebulk temperature is larger than the cryogenic environmentaltemperature. Additional limitations are given by the slowfree dynamics involved for these massive objects. For largermasses, the wave function takes longer to coherently expandand to build a visible interference pattern after a superpositionhas been prepared.

We have also argued that this protocol can be applied to testsome of the most paradigmatic collapse models. In particular,we have analyzed the continuous spontaneous localizationmodel (CSL), a model based on quantum gravity (QG), theDiosi-Penrose model (DP), and the Karolyhazy model (K). TheCSL and the DP are much stronger than the DP and K model

052121-15

ORIOL ROMERO-ISART PHYSICAL REVIEW A 84, 052121 (2011)

and can be falsified using reasonable experimental parameters.In particular, the famous continuous spontaneous localizationmodel can be tested using the original and conservative choiceof parameters. However, the DP and K models are muchmore challenging to falsify despite the fact that they predicta decoherence which is stronger than the one provided bystandard decoherence at low bulk temperatures. Nevertheless,these models are strongly limited by the fact that the freedynamics of the large masses required is too slow. We remarkthat for the Diosi-Penrose model we did not consider the strongenhancement provided by taking into account the mass densityat the microscopic level. The latter, in addition to being acontroversial choice, turns the model into a one-parametermodel given by the mass resolution parameter r0. Note thatif this parameter is taken into account, the protocol proposedhere provides unprecedented lower bounds to its value. SeeRef. [62] as a possible future development to test the DP andthe K models in a medium-sized space mission.

We have also addressed the optomechanical implementa-tion of the protocol presented, namely MERID. We focusedon the squared position measurement required to performthe double slit experiment, and we have considered cavityoptomechanics with optically levitating nanospheres. We haveshown that the overall performance of the protocol is limited bythis implementation, since both the global phase added duringthe interaction and light scattering set upper bounds on theexpansion time and the measurement strength. Nevertheless,with recently developed fiber-based cavities and for spheresof the order of tens of nanometers, superpositions of the order

of their size could be prepared. This provides unprecedentedbounds to the continuous spontaneous localization model thatcan be used to falsify the enhancement of the localization ratepredicted by Adler.

There are various directions to further pursue the workpresented here. First, the study of the implementation ofMERID using cavity optomechanics with suspended disks[63]. In this setup, the mechanical frequency can also bevaried since the tight harmonic potential is achieved by opticaltrapping, but the scattering of light is strongly reduced whenthe laser waist is smaller than the disk. This comes at the priceof inducing decoherence due to the coupling with internalelastic modes. Second, the possibility of using repulsivepotentials to exponentially increase the time scales of the freedynamics and, thus, to efficiently use the long coherence timesgiven by scattering of air molecules and blackbody radiation.Naturally, this has to be done without incorporating additionalsources of decoherence. In any case, we believe that thesynergy between the fields of quantum-mechanical resonatorsand matter-wave interferometry will allow to explore in thenear future the limits of quantum mechanics at unprecedentedscales, an exciting possibility indeed.

ACKNOWLEDGMENTS

I am grateful to J. I. Cirac, A. C. Pflanzer, F. Blaser,R. Kaltenbaek, N. Kiesel, and M. Aspelmeyer for usefuldiscussions. I acknowledge support from the Alexander vonHumboldt Stiftung.

[1] C. Davisson and L. H. Germer, Nature 119, 558 (1927).[2] H. v. J. Halban and P. Preiswerk, C. R. Acad. Sci. 203, 73 (1936).[3] I. Estermann and O. Stern, Z. Phys. 61, 95 (1930).[4] W. Schollkopf and J. P. Toennies, Science 266, 1345 (1994).[5] M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van Der Zouw,

and A. Zeilinger, Nature 401, 680 (1999).[6] S. Gerlich, S. Eibenberger, M. Tomandl, S. Nimmrichter,

K. Hornberger, P. J. Fagan, J. Tuxen, M. Mayor, and M. Arndt,Nat. Commun. 2, 263 (2011).

[7] T. J. Kippenberg and K. J. Vahala, Science 321, 1172 (2008).[8] M. Aspelmeyer and K. Shwab, New J. Phys. 10, 095001 (2008).[9] F. Marquardt and S. M. Girvin, Physics 2, 40 (2009).

[10] I. Favero and K. Karrai, Nat. Photon. 3, 201 (2009).[11] M. Aspelmeyer, S. Groblacher, K. Hammerer, and N. Kiesel,

J. Opt. Soc. Am. B 27, A189 (2010).[12] A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak,

M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang,M. Weides et al., Nature 464, 697 (2010).

[13] J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman,K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, andR. W. Simmonds, Nature 475, 359 (2011).

[14] J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill,A. Krause, S. Groeblacher, M. Aspelmeyer, and O. Painter,e-print arXiv:1106.3614.

[15] W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, Phys.Rev. Lett. 91, 130401 (2003).

[16] O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, New J.Phys. 12, 033015 (2010).

[17] G. C. Ghirardi, A. Rimini, and T. Weber, Phys. Rev. D 34, 470(1986).

[18] G. C. Ghirardi, P. Pearle, and A. Rimini, Phys. Rev. A 42, 78(1990).

[19] A. Bassi and G. C. Ghirardi, Phys. Rep. 379, 257(2003).

[20] S. L. Adler and A. Bassi, Science 325, 275 (2009).[21] A. Frenkel, Found. Phys. 20, 159 (1990).[22] R. Penrose, Gen. Relativ. Gravit. 28, 581 (1996).[23] L. Diosi, J. Phys. A: Math. Theor. 40, 2989 (2007).[24] J. Ellis, N. E. Mavromatos, and D. V. Nanopoulos, Phys. Lett. B

293, 37 (1992).[25] L. Diosi, Phys. Lett. A 105, 199 (1984).[26] A. J. Legger, J. Phys. Condens. Matter 14, R415 (2002).[27] E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, and

I.-O. Smatescu, Decoherence and the Appearance of a ClassicalWorld in Quantum Theory (Springer, New York, 2003).

[28] M. A. Schlosshauer, Decoherence and the Quantum-to-Classical Transition (Springer, Berlin, 2007).

[29] D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye,O. Painter, and H. J. Kimble, Proc. Natl. Acad. Sci. USA 107,1005 (2010).

[30] P. F. Barker and M. N. Shneider, Phys. Rev. A 81, 023826(2010).

052121-16

QUANTUM SUPERPOSITION OF MASSIVE OBJECTS AND . . . PHYSICAL REVIEW A 84, 052121 (2011)

[31] O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant,N. Kiesel, M. Aspelmeyer, and J. I. Cirac, Phys. Rev. A 83,013803 (2011).

[32] O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N.Kiesel, M. Aspelmeyer, and J. I. Cirac, Phys. Rev. Lett. 107,020405 (2011).

[33] P. F. Barker, Phys. Rev. Lett. 105, 073002 (2010).[34] F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, Phys.

Rev. Lett. 99, 093902 (2007).[35] I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. K. Kippenberg,

Phys. Rev. Lett. 99, 093901 (2007).[36] C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer,

Phys. Rev. A 77, 033804 (2008).[37] E. Joos and H. D. Zeh, Z. Phys. B 59, 223 (1985).[38] M. R. Gallis and G. N. Fleming, Phys. Rev. A 42, 38

(1990).[39] B. Vacchini, J. Phys. A: Math. Theor. 40, 2463 (2007).[40] P. Pearle, Phys. Rev. A 39, 2277 (1989).[41] P. Pearle, Phys. Rev. D 13, 857 (1976).[42] N. Gisin, Phys. Rev. Lett. 52, 1657 (1984).[43] N. Gisin, Helv. Phys. Acta 62, 363 (1989).[44] L. Diosi, Phys. Rev. A 40, 1165 (1989).[45] L. Diosi, Phys. Lett. A 129, 419 (1988).[46] B. Collett and P. Pearle, Found. Phys. 33, 1495 (2003).[47] W. Feldmann and R. Tumulka, e-print arXiv:1109.6579.[48] S. L. Adler, J. Phys A: Math. Theor. 40, 2935 (2007).

[49] J. Ellis, S. Mohanty, and D. V. Nanopoulos, Phys. Lett. B 221,113 (1989).

[50] L. Diosi, Phys. Lett. A 120, 377 (1987).[51] I. M. Moroz, R. Penrose, and P. Tod, Classical Quantum Gravity

15, 2733 (1998).[52] L. Diosi, Braz. J. Phys. 35, 260 (2005).[53] D. Kleckner, I. Pikovski, E. Jeffrey, L. Ament, E. Eliel, J. van den

Brink, and D. Bouwmeester, New J. Phys. 10, 095020 (2008).[54] G. C. Ghirardi, R. Grassi, and A. Rimini, Phys. Rev. A 42, 1057

(1990).[55] F. Karolyhazy, Nuovo Cimento 42, 390 (1966).[56] F. Karolyhazy, Magy. Fiz. Foly. 22, 23 (1974).[57] M. R. Vanner, I. Pikovski, M. S. Kim, C. Brukner, K. Hammerer,

G. J. Milburn, and M. Aspelmeyer, Proc. Natl. Acad. Sci. U.S.A.108, 16182 (2011).

[58] C. W. Gardiner and P. Zoller, Quantum Noise (Springer, Berlin,2004).

[59] M. R. Vanner, e-print arXiv:1106.0763.[60] D. Hunber, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hansch,

and J. Reichel, New J. Phys. 12, 065038 (2010).[61] S. Nimmrichter, K. Hornberger, P. Haslinger, and M. Arndt,

Phys. Rev. A 83, 043621 (2011).[62] R. Kaltenbaek, G. Hechenblaikner, N. Kiesel, O. Romero-Isart,

U. Johann, and M. Aspelmeyer (in press).[63] D. E. Chang, K. K. Ni, O. Painter, and H. J. Kimble, e-print

arXiv:1101.0146.

052121-17