Nano-object interferometry

Internal decoherence in nano-object interferometry due to phonons

C. Henkel1 and R. Folman2

1)Universität Potsdam, Institut für Physik und Astronomie, 14476 Potsdam, Germany2)Ben-Gurion University of the Negev, Department of Physics, Beer Sheva 84105, Israela)

(*Electronic mail: henkel@uni-potsdam.de)

(Dated: 27 April 2022)

We discuss the coherent splitting and recombining of a nanoparticle in a mesoscopic “closed-loop” Stern-Gerlachinterferometer in which the observable is the spin of a single impurity embedded in the particle. This spin, wheninteracting with a pulsed magnetic gradient, generates the force on the particle. We calculate the internal decoherencewhich arises as the displaced impurity excites internal degrees of freedom (phonons) that may provide Welcher Weginformation and preclude interference. We estimate the constraints this decoherence channel puts on future interferenceexperiments with massive objects. We find that for a wide range of masses, forces and temperatures, phonons do notinhibit Stern-Gerlach interferometry with micro-scale objects. However, phonons do constitute a fundamental limit onthe splitting of larger macroscopic objects if the applied force induces phonons.

I. INTRODUCTION

Quantum Mechanics (QM) and General Relativity (GR),the latter being the current theory of gravity, are the two pil-lars of modern physics. The quantum nature of gravity, orthe unification of these two pillars, has been an open ques-tion of utmost importance for decades now. While theory hasnot been able to find a satisfactory solution to this question, itis of paramount importance for experiments to deliver hints.We discuss here a nano-object interferometer aimed at deliver-ing such hints as they emerge from the interface between QMand GR. Such an interferometer may probe gravity-relatedideas, from the mainstream quantum of gravitation1, namedthe graviton, to speculative ideas, such as those of Penroseconcerning gravitationally induced collapse (see, e.g., Ref. 2)or the short-range correction to gravity, the so-called fifthforce. For an in-depth overview, see for example Refs. 3–9.

Another specific motivation for such an interferometer re-lates to the foundations of QM. It will push the limits to whichthe accuracy of QM is tested, by several orders of magnitude:both in the amount of mass that is being put in a spatial su-perposition, and in the size of the spatial splitting. Anothercontribution to the foundations of QM would be the ability totest continuous spontaneous localization models. These are ofcrucial importance in the search for extensions to QM. For amore complete review of underlying concepts, see for exam-ple Refs. 10 and 11.

Finally, a third motivation is quantum technology. Inaddition to the quantum computer, there is metrology. Anano-object interference experiment will bring about cutting-edge metrological capabilities, e.g., in the measurement ofgravity12, including geodesic studies and mineral searches, orin acceleration sensing.

Let us briefly note that the aims of such an endeavor aredifficult to be obtained by laser-based matter interferometry.First, laser pulses can be absorbed and scattered by the nano-

a)Also at Ilse Katz Institute for Nanoscale Science and Technology, BeerSheva, Israel

object (nOb) and, as already shown in experiments, they heatup the object and can lead to its opaqueness or destruction.13

Light scattering also increases the spatial decoherence rate, ifit can resolve the two paths. Second, laser-based matter-waveinterferometry requires an appropriate optical transition, andthis severely restricts its applicability to massive solid objects.Such optical transitions have been suggested14, but they sufferfrom two major drawbacks: (a) they typically require cryo-genic temperatures and as noted the light impinging on theobject heats it and cooling a free-space object is extremelyhard; (b) they give very low recoil velocities and thousandsof these transitions would be required for significant splittingin a short time. For example, for a 10−20 kg nOb, the opti-cal recoil velocity is on the order of 10−8 m/s whereas a con-tinued acceleration of say 100m/s2 for a short period of say100 µs gives a velocity six orders of magnitude higher. Herewe consider a Stern-Gerlach interferometer (SGI) which uti-lizes magnetic gradients rather than light pulses to generatethe spatial superposition.

The state of the art for nOb interferometry utilises a beamof heavy molecules impinging on a grating. The heaviestmolecules put in a superposition to date consist of up to 2000atoms.15,16 The SGI may go far beyond this limit. In the spiritof molecular-beam experiments, several proposals exist forslit-type experiments with higher masses, specifically includ-ing solid nObs.10,17,18 In these proposed experiments, none ofwhich has been realized yet, the signal is a spatial interfer-ence pattern, reminiscent of the double-slit interference pat-tern. These proposals face several challenges. For example,for spatial interference patterns to form, a long time-of-flight(TOF) is required, and as the decoherence rate of delocalizedmassive-object states is expected to be high, a prolonged TOFseems impractical. Furthermore, for many of these configu-rations, the periodicity of the interference pattern is expectedto be extremely small, so that high spatial resolution is re-quired for detection, a resolution which may be beyond avail-able technology, especially for high-efficiency detection.

The SGI considered here is a completely different route forrealizing such an interferometer, as shown in Fig. 1. A singlespin embedded in the nano-object is first put in a superposi-

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Nano-object interferometry 2

tion of opposite spins. When moving through a magnetic-fieldgradient, the two spin orientations experience opposite forces,and this splits the wavefunction of the entire nOb into twowave-packets (WPs), effectively generating spin-momentumentanglement. Reversing the opposite forces is then used tobring the two WPs back together again. Such an interferom-eter has three crucial advantages: (a) the splitting is active,namely, a real force is utilized, and it does not depend on ex-panding the WP; consequently, large splitting distances maybe achieved in a short time; (b) the signal forms independentlyof any TOF and, as the recombination is also active, it may bedone in a short time; (c) the observable is spin population, asin modern atom interferometry; in contrast to spatial fringes,its detection does not require high spatial resolution.

0 T1/2 2T1/2 𝗍𝗂𝗆𝖾

𝖢𝗈𝖬

𝖼𝗈𝗈𝗋𝖽𝗂

𝗇𝖺𝗍𝖾

Ψ(z, {uq}, 0) |1⟩ + |2⟩2

Ψ 1(z,{u q},

t) |1⟩

Ψ2 (z, {uq }, t) |2⟩

ΔZmax

Ψ1(z, {uq}, 2T1/2) |1⟩+ Ψ2(z, {uq}, 2T1/2) |2⟩

FIG. 1. Sketch of the Stern-Gerlach interferometer (SGI), adaptedfrom Fig. 1 in Ref. 19. Along the two paths, the wavepackets be-have distinctly different with respect to the center-of-mass coordinate(CoM) z and the amplitudes {uq} of the internal phonon modes. Themaximum splitting ∆Zmax is reached at T1/2 which is half the timerequired to close the loop.Adapted with permission from Y. Margalit, O. Dobkowski, Z. Zhou,O. Amit, Y. Japha, S. Moukouri, D. Rohrlich, A. Mazumdar, S. Bose,C. Henkel, and R. Folman, “Realization of a complete Stern-Gerlachinterferometer: Towards a test of quantum gravity,” Science Adv. 7,eabg2879 (2021). Copyright (2021) The Authors under licence CCBY 4.0, DOI:10.1126/sciadv.abg2879.

We recently demonstrated the coherence of a Stern-Gerlach“closed-loop” interferometer with ultracold atoms and dis-cussed the possibility of realizing such an interferometer witha nano-particle.19 Decoherence, or lack of coherence, comesfrom two very different physical processes, the first originat-ing in the coupling of the nOb to the environment, and thesecond originating in the fact that the loop is not completelyclosed, coined the “Humpty-Dumpty” effect (see Ref. 19 andreferences therein). The latter type of decoherence, indepen-dent of the environment, depends on the final overlap of sev-eral wave functions, that of the external degrees of freedom(position, momentum and rotation, for the latter, see for ex-ample Ref. 20 and references therein), as well as internal de-grees of freedom, including spin and phonons, the latter beingthe topic of this work. If the wave functions of the WPs alongthe two paths become orthogonal, no interference is possible.Orthogonality may arise from the non-identical phonon exci-tations due to the applied Stern-Gerlach force along the twopaths the WPs take. As we show in the following, this is de-pendent on the nOb size and temperature and on the applied

force (i.e., the magnetic gradient).As we would like to focus solely on decoherence due to

phonons, in the following we assume that all other types ofdecoherence are negligible. However, let us briefly note thedifferent types of decoherence which should be considered,and the works which have already addressed them. To beginwith, spin coherence is crucial if spin is the observable of theinterferometer. The T1 and T2 times of the spin must be longerthan the interferometer time. For the nitrogen-vacancy cen-ter embedded in a nano-diamond as discussed in this work,a room-temperature coherence time of a few hundred µs wasobserved.21 Blackbody radiation (BBR) is a source of spatialdecoherence.10,22–25 Collisions with background gas are alsoan important source of decoherence.14,24 Any charging of thenOb would again cause decoherence through the electrostaticcoupling to the environment.26 See also Ref. 19 and referencestherein.

Many works have already considered the importance of in-ternal temperatures for lower decoherence in such nOb inter-ferometers. Specifically, concerning BBR, it has been sug-gested that internal state cooling of the neutral test masses inaddition to external cooling in opto-mechanical cavities wouldprove to be greatly beneficial.27 Furthermore, quite a fewworks dealt with the quantum dynamics of nOb phonons.28–30

However, as far as we know, this is the first treatment ofphonons as a source of decoherence.

II. MODEL

A. Hamiltonian

We envision a nano-object embedded with a single spin,such as a nano-diamond with a single nitrogen-vacancy cen-ter. The model for the nano-object is based on microscopicmodels of magnetic materials, see, e.g., Ref. 31. The atomsof the object are characterized by their positions ri and mo-menta pi. For our purposes, only a single atom, say at sitenumber s, carries a spin S. For definiteness we assume S = 1

2 .The Hamiltonian is a sum of mechanical and magnetic termsH = H1 +H2(t) with

H1 = ∑i

p2i

2m+ ∑〈i, j〉

V (ri− r j) (1)

H2(t) =−µS ·B(rs, t) (2)

where m is the single-atom mass (assumed identical for sim-plicity) and µ the magnetic moment (the spin S is taken di-mensionless). We do not take into account collective magneticinteractions, such as the diamagnetic interaction32,33, as theyare not expected to excite phonons.

The second sum in Eq. (1) can be restricted to nearest neigh-bor sites i, j and involves the bond potential

V (ri− r j) =K2(ri− r j)

2 + anharmonic terms (3)

with a common spring constant K. From this model, we get

Nano-object interferometry 3

TABLE I. Typical orders of magnitude for nano-particles made fromdiamond.

magnetic gradienta 106 T/mparticle mass 106−1010 amusize (diameter)b L = 10−200nmmagnetic moment µ 1 µB = h×14GHz/Tacceleration 6000−0.6m/s2

lowest phonon modec

ω1/2π = c/2L 900−45GHzde Broglie wavelengthd λcmphonon coherence lengthe λph

Tcm = 293K : 10−14−10−16 mTph = 4K : 10−13−10−15 m

a Achievable with atom chips 1 µm away from a 1×1µm2 wire with109 A/cm2 current density.

b Unit cell with a = 3.6Å and 8 C atoms.c Speed of sound c = 17.5km/s.d Estimated as λcm = h(MkBTcm)

−1/2 with the center-of-mass temperatureTcm.

e Estimated as λph = h(MkBTph)−1/2 with the internal (phonon)

temperature Tph.

for the center-of-mass coordinate R = (1/N)∑i ri and its cor-responding momentum P = ∑i pi the equation of motion

P =−∑i

∂H∂ri

= µ ∑α

Sα

∂Bα

∂rs(4)

where the interaction potential Eq. (3) drops out by Newton’sactio = reactio. Typical accelerations are given in Table I.

We start for the phonons with a simple one-dimensionalmodel with N atoms in a linear chain having a total massof mN = M. If we interpret the coordinates zi as the devi-ation from equilibrium positions spaced by the equilibriumbond length (lattice constant) a, we get a chain model whosephonon spectrum in the harmonic approximation is given by(force-free or Neumann boundary conditions)

ω(q) =

√4Km

sinqa2, q =

π

L{0,1,2, . . . ,Nx−1} (5)

where L = aN is the total length of the chain. The mode am-plitudes uq (illustrated in Fig. 2) can be computed from theprojection

uq =2N ∑

nzn cos[(n+ 1

2 )qa] , q 6= 0 (6)

and for them, we get the equations of motion

d2uq

dt2 =−ω2(q)uq +

2µ

Mcos[(s+ 1

2 )qa]∑α

Sα

∂Bα

∂ zs(7)

where the integer s labels the equilibrium position of the im-purity spin (the left end of the chain is at s = 0).34 Finally, thespin itself precesses according to the Larmor equation

dSdt

=µ

hS×B(rs, t) (8)

We consider here the simplest setting where the magneticfield is aligned to the z-axis and is a linear function Bz(z, t) =

0 5 10 15

lattice site

0

2

4

6

8

10

frequencyω(q)/ω

1

FIG. 2. Sketch of phonon modes (standing waves) for a linear chain.The total length is covered by an integer multiple of half the wave-length. The vertical red line marks the position of the impurityspin. Frequencies scaled to the fundamental tone ω1 = πc/L withthe speed of sound c and length L = 17 unit cells. The phonon am-plitude is represented by the displacements of the circles, but notethat in this linear model, this is actually a longitudinal phonon. Thezero-frequency mode corresponds to a displacement of the chain asa whole (center of mass mode).

B0+b(t)z. Indeed, this is the experimental configuration usedby the atomic SGIs19. The linear field significantly simplifiesthe dynamics, as there is no coupling of the CoM, spin andphonon degrees of freedom. This is valid provided the ratio|b(t)R/B0| remains small enough to neglect the contributionsfrom transverse components of the magnetic field (required bydivB = 0).

In the “spin up” or “down” configurations, the spin vectorS(t) = ± 1

2 ez will thus be stationary (no precession, no spinflips). The Stern-Gerlach force becomes independent of thenano-object position, and we may solve Eqs. (4, 7) easily:

P(t) = P(0)± µ

2

∫ t

0dt ′ b(t ′) (9)

Z(t) = Z(0)+1M

∫ t

0dt ′P(t ′) (10)

uq(t) = uq(0)cosωqt +uq(0)

ωqsinωqt (11)

± µ

Mωqcos[(s+ 1

2 )qa]∫ t

0dt ′ b(t ′)sin[ωq(t− t ′)]

The± sign of the spin projection thus determines the directionof the momentum, as well as the sign of the phonon amplitude.The equations of motion being linear, these expressions areequally valid, whether the variables are treated classically oras quantum operators (Heisenberg picture).

Nano-object interferometry 4

III. OVERLAP AND CONTRAST

The solution presented in Eqs. (9–11) determines the con-trast of the spin interference signal in the following way. Thekey element is the overlap between the wavepackets for thecenter-of-mass degree of freedom (DoF) and the phonon DoFsthat have evolved with either sign ± of the spin projection.

A. Center of mass

To illustrate our method of calculation, we first calculatethe overlap for the center of mass (CoM). As noted in the in-troduction, we will eventually assume that it is perfect, so thatloss of contrast is only due to orthogonality between phononexcitations.

Consider for the CoM state an initial pure state |ψ〉, and re-write the solution for momentum P(t) and position Z(t) as theaction of some displacement operator D± where the spin sign± corresponds to opposite displacement directions:

spin up: P(t) = D†+P(0)D+

Z(t) = D†+[Z(0)+P(0)t/M]D+

spin down: P(t) = D†−P(0)D−

Z(t) = D†−[Z(0)+P(0)t/M]D− (12)

The overlap in question can then be written as

Tr(D+|ψ〉〈ψ|D†

−)= 〈ψ|D†

−D+|ψ〉 (13)

This goes down to zero contrast in a Gaussian fashion whenthe splitting exceeds the width of the initial state |ψ〉, as ex-pected for orthogonal states.

Since the inverse displacement D†− is the same as D+, the

displacement operators can be combined into one operator Dthat depends on the relative splittings ∆Z(t), ∆P(t) betweenthe spin-down and spin-up trajectories. From this viewpoint,the overlap may be understood as the amplitude to stay in theinitial state |ψ〉 after applying the operator D. Such an ampli-tude is known in scattering as the Debye-Waller factor (see,e.g., Ref. 35 and 36), and may also be related to the zero-phonon line in molecular spectroscopy37,38.

It is now a well-known identity (sometimes called the Blochformula) that the expectation value of D is related to theWigner representation W (z, p) of the wave function |ψ〉, no-tably its double Fourier transform39–42

〈ψ|D|ψ〉= χ(k,s) =∫

dxdpW (z, p)ei(kz−sp). (14)

The latter is also known as characteristic function for sym-metrically ordered products. Its arguments correspond to thedisplacements in phase space: k = ∆P(t)/h, s = ∆Z(t)/h thatcan be read off from Eq. (9).

The value χ(0,0) = 1 corresponds to full contrast when thedisplacements in position and momentum are exactly zero.This defines the target conditions for a “closed loop” in phasespace. Any deviations from them therefore characterize the

accuracy that is needed to close the interferometer. In interfer-ometry, it is well known that a non-closed loop generates a so-called separation phase proportional to ∆P(t) and ∆Z(t).43,44

The expression (14) corresponds to the average of the corre-sponding phase factor over the initial position and momentumdistribution.

The initial distribution is indeed likely to be nonpure, andthe advantage of using the Wigner characteristic function isthat it can be carried simply through, if in Eqs. (13, 14) weare dealing with a density operator ρ rather than the pure state|ψ〉. If we assume for simplicity that the initial Wigner func-tion is a Gaussian with variances σ2

p = MkBTcm (kinetic tem-perature Tcm for the center-of-mass DoF) and σ2

z , we get thecontrast reduction factor

Ccm = |tr(Dρ)|= |χ(∆P(t)/h,∆Z(t)/h)|= exp

[− 1

2 ∆P(t)2σ

2z /h2− 1

2 ∆Z(t)2σ

2p/h2] (15)

The characteristic (rms) width of this Gaussian in ∆Z(t) isgiven by the spatial coherence length h/σp = λcm that coin-cides simply with the thermal de Broglie wavelength λcm ofthe nOb (see Table I). This suggests formidable precision re-quirements for closing the phase-space loop. It turns out, how-ever, that one may confine the nOb in some potential and coolits CoM motion down to the ground state, thereby increasingits coherence length.45–48

B. Internal degrees of freedom (phonons)

The decoherence due to phonons can be estimated in a sim-ilar way. Consider first a fixed phonon mode with frequencyωq 6= 0. Along the two interferometer arms, the amplitude uqof this mode suffers a differential displacement in phase spacegiven by, from Eq. (11):

∆uq(t) =2µ

Mωqcos[(s+ 1

2 )qa]∫ t

0dt ′ b(t ′)sin[ωq(t− t ′)]

∆uq(t) =2µ

Mcos[(s+ 1

2 )qa]∫ t

0dt ′ b(t ′)cos[ωq(t− t ′)] (16)

Now consider this mode to be initially in thermal equilibriumat the (internal) temperature Tph. Its contribution to the en-ergy in Eq. (1) is Eq =

14 M(u2

q +ω2q u2

q), the magnetic interac-tion energy being an irrelevant constant in a slowly varyingfield. The initial phonon mode amplitude thus has a GaussianWigner function with variances σ2

u,q = 2kBTph/(Mω2q ) in dis-

placement and σ2u,q = 2kBT/M in the corresponding velocity.

These variances actually provide the classical limit only: atlow temperatures, the replacement

kBTph 7→ 12 hωq coth

βωq

2, β =

hkBTph

(17)

applies to get the correct Wigner function.49 The width in am-plitude σu,q at zero temperature for the lowest phonon modeis, at realistic temperatures, comparable to the thermal phononcoherence length λph given in Table I.

Nano-object interferometry 5

This Wigner function is invariant under the rotation inphase space (first line of Eq. (11)), this is why we may fo-cus on the displacements (16). For the contrast reduction dueto phonon mode q, we need the variables kq,sq in the dis-placement operator Dq = exp[i(kquq− squq)]. The amplitudeoperators uq and uq satisfy the commutation relations

[uq, uq′ ] =2ihM

δqq′ (18)

that follow from Eq. (6), and we find kq = M∆uq(t)/2h, sq =M∆uq(t)/2h. The overlap for the mode q thus generates acontrast

Cq = exp[− 1

2 k2qσ

2u,q +

12 s2

qσ2u,q

]= exp

[−

Mωq

8h

(∆uq(t)2 +

∆uq(t)2

ω2q

)coth

βωq

2

](19)

We can here read off the characteristic phonon coherence“length” relative to which the splitting ∆uq(t) of the phononamplitude between the two interferometer arms must be nulli-fied. For low-frequency modes (i.e., hωq� kBTph), this scaleis simply given by the thermal de Broglie wavelength withλph = h/(MkBTph)

1/2 (in the pm range or below, see Table I).The splitting in the phase space of phonon amplitude and mo-mentum can be visualized as in Fig. 3. (A similar picturewould represent the results obtained in Sec. III A for the CoMmode.) We also recall the alternative interpretation in termsof a Debye-Waller factor: the contrast depends on the prob-ability that the excitation of the mode uq in one path of theWP relative to the other does not differ by a single phononquantum.

phonon mode amplitude

phon

on m

ode 

mom

entu

m

FIG. 3. Splitting in the phase space of a fixed phonon mode. Thecontrast in the interferometer is determined by the overlap betweenthe distributions illustrated by black contours (with color shading)and white contours.

The two terms in the parenthesis of Eq. (19) can be com-bined into a complex, finite-window Fourier transform. Intro-ducing the time-dependent acceleration a(t) = µ

M b(t) and theintegral (with dimension velocity)

a(ωq, t) =∫ t

0dt ′ a(t ′)eiωq(t ′−t) (20)

we can write

∆uq(t)2 +∆uq(t)2

ω2q

=4

ω2q

cos2[(s+ 12 )qa]|a(ωq, t)|2 (21)

The extension to the full phonon spectrum is immediate ifwe assume that the variables {uq, uq} for q in the Brillouinzone describe the normal modes of the chain of atoms. Inthe harmonic approximation, this is obviously true, and theinitial state thus factorizes into a product of thermal states pernormal mode. The overlaps per mode multiply, and we get thephonon-based contrast reduction in the form

Cph = exp[−∑

q

M2hωq

coth( 1

2 βωq)

cos2[(s+ 12 )qa]|a(ωq, t)|2

](22)

A similar technique has been used for the dephasing of a qubitwhose energy splitting couples to a phonon bath.50,51

In the following section, we discuss the sum over all modesin Eq. (22). To get a qualitative picture, consider first the be-haviour of the terms in the sum as a function of phonon fre-quency ωq. A sketch is provided in Fig. 4. The main fea-ture is the Fourier spectrum a(ωq, t = 2T1/2) of the accel-eration that is nonzero in a range of frequencies 0 < ω .2π/T1/2. Here, the half duration T1/2 of the interferome-ter loop corresponds to the moment of maximum wavepacketsplitting. The vertical lines with symbols illustrate the posi-tions of the phonon eigenfrequencies, they are approximatelyharmonics of the ‘fundamental tone’ ω1/2π = c/2L wherec is the speed of sound. The curves provide an upper limitto the summands in Eq. (22), the symbols give smaller val-ues because they take into account the phonon mode am-plitude cos[(s+ 1

2 )qa]. Impurity spins located near the nObcenter present two advantages: their overlap with the funda-mental mode is small there (see Fig. 2), and their spin co-herence time is maximal because they avoid enhanced sur-face noise.52 Spins of nitrogen-vacancy (NV) centers in nano-diamonds have already exhibited a room-temperature coher-ence time of 200 µs.21 While it may be assumed that signif-icant material engineering will improve these numbers53 (thestate of the art for room-temperature bulk is 3ms54), even200 µs is enough for a chip-based Stern-Gerlach interferome-ter to achieve significant splitting. Utilising well-known NVtechniques (e.g., recent work of the Ben-Gurion University ofthe Negev group55–57), we do not see any fundamental spin-related obstacles.

C. Discrete phonon spectrum (small objects)

The object is small if its fundamental tone is much higherthan the inverse duration of the splitting pulse, i.e., ω1T1/2�2π . With T1/2 = 30 µs and c ≈ 20km/s (diamond), thisapplies for objects with L < 30cm, i.e. for any realisticsmall particle. The amount of orthogonality ‘hidden’ in thephonon amplitudes along the spin up and down paths of thewavepacket is then determined by the tails of the Fourier spec-trum a(ω,2T1/2) of the nOb acceleration (see Fig. 4). Weconsider for definiteness three simple protocols for the pulsedmagnetic gradient. They have in common a zero net velocityshift (in order to close the loop for the center-of-mass DoF)and their duration 2T1/2. Moving for simplicity the moment

Nano-object interferometry 6

0 2 4 6 8 10frequency  q/2  [c/2L]

weig

ht

1/T1/2 3/2T1/2

0.0 0.5 1.0 1.5 2.0time t / T1/2

acce

lera

tion

N = 17 chainprofile 0

profile 1profile 2

FIG. 4. Sketch of the contribution of individual phonon modes tothe interference contrast Cph. We plot the terms under the sum inEq. (22) as a function of phonon frequency, scaled to the fundamen-tal tone ω1/2π = c/(2L). The solid lines give an upper limit, dis-crete phonons are marked by vertical lines, the symbols below theupper limit (�, ×, ◦) include the squared amplitude of the phononstanding wave at the spin site s = 7 ' N/2. In the indigo shading,hωq ≥ kBTph with the phonon temperature Tph. Profiles 0, 1, 2: ac-celeration protocols a0(t), a1(t), a2(t) of Eqs. (23–25), the last one(green) having the smoothest switching-on and being the most adia-batic, see inset. The width of the spectrum is inversely proportionalto the duration T1/2 of the closed loop (arrows). Parameters: chainwith N = 17 atoms, duration of applied forces compared to soundroundtrip 2T1/2 = 1.4L/c, (internal) temperature kBTph = 8.1 hω1.

of maximum splitting to t = 0, we take for −T1/2 ≤ t ≤ T1/2:

a0(t) =±amax (square profile, see inset Fig. 4) (23)

a1(t) = amax[−1+6(t/T1/2)2−5(t/T1/2)

4] (24)

a2(t) =−amax

2[cos(πt/T1/2)+ cos(2πt/T1/2)] (25)

Their Fourier transforms have envelopes that scale with1/ωn+1 (n = 0,1,2):

a0(ω, t) =amax

ω

[2sin(ωT1/2)−4sin(ωT1/2/2)

](26)

a1(ω, t) =−16amaxT1/2

[cos(ωT1/2)

(ωT1/2)2

(1− 15

(ωT1/2)2

)−

sin(ωT1/2)

(ωT1/2)3

(6− 15

(ωT1/2)2

)](27)

a2(ω, t) =−3π2amaxT1/2

sin(ωT1/2)

(ωT1/2)3

×(

1− π2

(ωT1/2)2

)−1(1− 4π2

(ωT1/2)2

)−1

(28)

For small nObs, one expects that high frequency modes shouldfollow adiabatically the slowly varying acceleration and re-turn to their initial state. The values of the Fourier trans-form ai(ωq, t) then estimate how large is the deviation fromadiabaticity. The protocol 2 shows the smallest results witha2(ωq, t) ∼ 1/ω3

q because its acceleration has the smoothest(most adiabatic) onset. The difference between the temporal

profiles can be traced back to the experience of a traveller, aseither an old-fashioned or a modern train takes off. As theparticle increases in size, the phonon modes shift to lower fre-quencies and adiabatic following becomes less easier. By thismechanism, the interference contrast gets worse.

The overall contrast reduction is very tiny, however, ascan be seen in Fig. 5. The symbols give the sum in Eq. (22)for one-dimensional nObs of increasing length. The overallscaling can be understood by focusing on the fundamentalphonon mode, i.e., the orthogonality for ω1 = πc/L. We write− logCph = f S with the prefactor

f =(amaxT1/2)

2kBTphM

h2ω2

1=

(amaxT1/2

ω1λph

)2

∝ L3 (1D chain)

(29)where the phonon coherence ‘length’ λph (measuring actuallya displacement) turns out to be equal to the thermal de Brogliewavelength for the chain’s total mass M ∼ L, but taken at theinternal temperature Tph. Adopting the approximation that thefundamental tone is in the classical regime, hω1 � kBTph (atroom temperature, frequencies < 6THz or size L > 3nm), thefactor f ∼ Tph makes the scaling with the phonon temperatureexplicit. The other factor S is a dimensionless sum over thephonon spectrum and takes the form

S = ∑q

βω21

2ωqcoth

βωq

2cos2[(s+ 1

2 )qa]∣∣∣∣ a(ωq, t)amaxT1/2

∣∣∣∣2' An

(ω1T1/2)2n+2 ∑q(ω1/ωq)

2n+4∝ L2n+2 (30)

with n = 0,1,2. In the second line, we took cos2[(s +12 )πa/L] ≤ 1 as upper limit and applied an upper bound forthe Fourier spectra (28), leading to the numbers A0 = 36,A1 = (16/π)2, A2 = 9. For an equidistant phonon spec-trum, the sum in the second line evaluates to ζ (2n + 4) ≈1.08232,1.01734,1.00408, being dominated by its first term.

The resulting lower limit for Cph = exp(− f S) is shown inFig. 5 as dashed gray lines, while the symbols give the fullsum computed numerically, showing good agreement. Byplacing the impurity spin near the center of the nOb, the ex-citation of the fundamental phonon is reduced, improving thecontrast compared to Eq. (30). For the nano-diamond con-sidered here, the temperature is such that the upper part of thephonon spectrum is in the quantum regime, hωq > kBT , wherecoth(βωq/2)→ 1 stays above its classical limit. Due to thesteep power laws, these modes make a negligible contribution,however, and do not exceed the estimate (30).

The main message of this plot is that the actual values forthe contrast Cph are extremely close to unity: the interfer-ence contrast is not reduced at all by the excitation of phononmodes in a linear chain. In practice, the limits will be ratherset by the magnetic gradient (values given at the top of Fig. 5)and the spin coherence time (typically ∼ 200 µs).21 The caseof a fixed force (rather than acceleration) is considered inFig. 6 below.

We have checked that a mismatch of the final velocity doesnot qualitatively change these results. For that, we consid-ered models with adjusted coefficients in such a way that

Nano-object interferometry 7

10 100 103 104

particle size L [nm]

10 56

10 47

10 38

10 29

10 20

10 11

10 2

107lo

g C p

h

linear chain,  L5

profile 1,  L7

profile 2,  L9

3D sphere, L7

profile 1, L9

profile 2, L11

100 103 104 105 106 107 magnetic gradient [T/m]

FIG. 5. Exponent in interference contrast Cph for a one-dimensionalchain (symbols and gray dashed) and a three-dimensional, round ob-ject (dash-dotted). log denotes the natural logarithm; in the gray-shaded area, the contrast drops below 10%. Lattice parameters fordiamond at room temperature, maximum acceleration kept fixed atamax = 100m/s2, duration 2T1/2 = 60 µs, spin position near the cen-ter, (internal) temperature Tph = 293K. The data sets marked ‘lin-ear chain’ and ‘3D sphere’ correspond to the closed-loop accelera-tion protocol a0(t) [Eq. (23)], while ‘profile 1 and 2’ correspond toEqs. (24–25), respectively. The top scale gives the magnetic gradientrequired to achieve the acceleration amax for a 3D object, it scaleswith its volume L3.

the initial acceleration shows the same behaviour, but its in-tegral ∆v is nonzero. It turns out that for any small parti-cle and reasonably slow gradients (in the sense given above,c/2L > 1/T1/2), the high-frequency asymptote of the Fourierspectrum a1(ω,2T1/2) shows the same scaling, only its ampli-tude is changed. For our purposes, the previously discussedclosed-loop case ∆v = 0 is thus sufficient.

D. Three-dimensional particle

The exact calculation of phonon modes for a three-dimensional assembly of a few thousand atoms or more be-comes challenging. We sketch here the modifications thatare needed with respect to the linear chain. The wave vec-tor q becomes three-dimensional, and a mode function likeuq cos[(n+ 1

2 )qa] becomes a vector-valued function uqfq(rn)with mode amplitude uq. In the projection formula Eq. (6), Nis now the total number of atoms in the nOb, and in Eq. (7),M becomes the total mass. The Stern-Gerlach force on theimpurity spin (at rs) defines a direction that selects an angu-lar pattern of emitted phonons via a scalar product with thelocal “polarisation vector” fq(rs) of the phonon mode. Theshape of the nOb and its boundary conditions determine theallowed values of q within the Brillouin zone of the crystalstructure. While an exact calculation is possible (numerically)for small clusters, we focus in the following on the region nearthe Γ point with an approximately linear dispersion. The valueq = 0 corresponds again to the center-of-mass mode, and weare interested in the smallest non-zero sound frequency ω1(think of the fundamental pitch of a musical instrument). Onehas to allow for different values of the speed of sound c (longi-

tudinal and transverse). In a nOb with cubic shape, the modesare separable, and we have ω1 = πc/L with the linear sizeL. In a spherical object of diameter D, adopting a continuummodel and solving the Helmholtz equation with a Neumannboundary condition, we find a dipole mode proportional tothe gradient of cos(θ)

[sin(qr)/r2−qcos(qr)/r

](a spherical

Bessel function) at ω1 = cq with q≈ 4.1632/D. We note thatexperiments on resonances observed with icosahedral clustersin quasi-crystalline materials show good agreement with sucha continuum analysis, even though the cluster diameters areas small as 1nm.58 A more complete analysis of the acousticmodes of a nano-sphere can be found in Ref. 29.

For a small particle (recall the typical limit L < 30cm),all phonon frequencies are way beyond the cutoff frequency1/T1/2 of the acceleration spectrum a(ωq,2T1/2), and the low-est phonon mode gives the dominant contribution. Its sec-ond harmonic already contributes only a few percent, depend-ing on the protocol. This being said, we get the followingrough estimate for the interference contrast for a small three-dimensional nOb:

− logCph 'M

2hω1coth

βω1

2|ez · f1(rs)|2|a(ω1, t)|2

= (· · ·)MkBTa2max

h2ω6

1 T 21/2

= (· · ·) (∆Zmax)2

λ 2ph

(L

cT1/2

)6

(31)

where ez gives the direction of the Stern-Gerlach force, andf1(rs) is the fundamental phonon mode (normalized to unitmaximum amplitude), evaluated at the position of the spin rs.In the second line, (· · ·) is a numerical factor we expect to beof order unity, we adopted protocol a1(t) and assumed hω1�kBT . The scaling with the linear dimension L of the particlenow gives the exponent L9 (L7 for a0(t) and L11 for a2(t)), seedash-dotted lines in Fig. 5.

The second form of Eq. (31) makes contact to the semi-nal estimation of decoherence due to Zurek59: the maximumspatial splitting (at half-loop) ∆Zmax ' amaxT 2

1/2 of the nObwavepacket is compared to the phonon coherence length λph.This huge ratio would preclude any realistic contrast, were itnot compensated by the high power of the small ratio L/cT1/2,as long as the object size L is smaller than the travelling dis-tance of sound during the closed loop.

In Fig. 6, the contrast is shown for a different setting ofparameters: here, the magnetic gradient is such that thewavepacket splits to a fixed fraction (∼ 10%) of the nOb diam-eter. This would be a typical requirement for a Stern-Gerlachinterferometer with two particles that interact via their mu-tual gravitational attraction, as suggested for probing quantumgravity.5,6 We adjusted the protocol time T1/2 to get a fixed ve-locity splitting ∆vmax. In the non-shaded area of the plot, therequired values are below the limits 106 T/m and 100 µs set bythe current experimental device in the Ben-Gurion Universityof the Negev group. (In the shaded area, the achieved splittingis less than the targeted value.) The contrast is again excellent,and comparing to Fig. 5, these settings permit to split some-what larger particles. The protocols a0, a1, and a2 now differby factors independent of the particle size and scale all with

Nano-object interferometry 8

∼ L5

∼ L3

∼ L

FIG. 6. Interference contrast for a three-dimensional particle. Mag-netic gradient (upper scale) and duration of the loop are adjusted suchthat the maximum spatial splitting is a fixed fraction (10%) of thenOb size, and the maximum velocity splitting is 1mm/s. In the grayshaded area, the required magnetic gradient and half-loop time areset to realistic maximum values (106 T/m and 100 µs21) so that thetargeted splitting noted above is not achieved. Other parameters as inFig. 5. The three data sets correspond to the closed-loop protocols ofEqs. (23–25), respectively, they show the same scaling ∼ L5 with theparticle diameter L in the non-shaded area. The scaling laws differ inthe gray area, as indicated.

the power L5. In fact, in this setting, the ratio L/(cT1/2) turnsout to be constant (∼ ∆vmax divided by the speed of sound), aslong as one stays below the upper limit to T1/2.

E. Further insight: macroscopic particle

We finally consider the limit that the acoustic modes be-come dense on the scale 2π/T1/2 of the acceleration spec-trum. Although experiments will be extremely challenging(size L > 30cm), we include this case as a reference.

The sum over phonon wave vectors may be replaced by anintegral (recall the spacing ∆q = π/L). Going directly to thethree-dimensional case, the exponent in the contrast Eq. (22)becomes (average |ez · fq(rs)|2 ≈ 1

2 , 3 acoustic branches)

− logCph '3ML3

2h

∫ d3q(2π)3ωq

cothβωq

2|a(ωq, t)|2

' 3ML3kBTπ h2c3

∫ dω

2π|a(ω, t)|2 (32)

In the second line, we focused on the acoustic part of the dis-persion relation and took the high-temperature limit. This is agood approximation, since the integration range is effectivelylimited by the bandwidth 2π/T1/2 of the acceleration spec-trum. The integral gives, according to the Parseval-Plancherelformula:

∫ dω

2π|a(ω,2T1/2)|2 =

T1/2∫−T1/2

dt a(t)2 =Cna2maxT1/2 (33)

with C0 = 1 for a0(t) and C1 = 256/315, C2 = 1/2. Note thatin this regime, the three protocols are essentially equivalent.

In terms of the phonon coherence length λph, the contrast be-comes

− logCph ' (· · ·)(

∆Zmax

λph

)2( LcT1/2

)3

(34)

where again ∆Zmax ' amaxT 21/2 and (· · ·) is a numerical coef-

ficient of order one. Since now the last factor is larger thanunity (large particle), coherent splitting can only occur overdistances smaller than λph � 10−15 m. For the mass regionstated in the beginning of this section, non-negligible splittingwill thus be impossible.

It is interesting to note that when the acceleration is ex-pressed by the maximum force, Fmax = Mamax, the objectmass M drops out of this estimate. In terms of magnetic gra-dient and protocol time, the requirement Cph ≥ 10% gives theinequality

T1/2

µs

(bmax

T/m

)2

≤ (· · ·)1015 [ρ/(g/cm3)][c/(103 m/s)]3

(µ/µB)2(Tph/300K)(35)

This upper limit is larger than the parameters available in cur-rent experiments, but only by a few orders of magnitude. Themain challenge seems to be that the corresponding forces onlarge objects are too small to split a wavepacket by a signifi-cant fraction of the object size (see Fig. 6).

IV. DISCUSSION AND CONCLUSION

In this work we examined the coherent splitting of a nano-object with an embedded single spin, in a full-loop Stern-Gerlach interferometer. We addressed the question of whetherthe internal degrees of freedom (phonons) pose a problem, asthey are excited by kicks in opposite directions for the twowavepackets. This may cause distinguishability to the pointof orthogonality.

We looked at 1D and 3D models, assuming phonon nor-mal modes in the harmonic approximation, and examined thescaling of the contrast reduction with the particle size at fixedacceleration or fixed maximal separation. We looked at thecoherence drop solely due to phonons, assuming that the clos-ing (overlap) of other degrees of freedom (position, momen-tum, rotation) is perfect. We took care to make use of realisticexperimental numbers for the magnetic gradients and spin co-herence time inside a nano-object.

As expected, we find that achieving complete overlap ofthe phonon state is impossible. However, the suppression ofcoherence for practical numbers is found to be minimal. Weexamined several temporal profiles of the magnetic field andfound that the smoother the profile (namely, slower onset ofthe magnetic field), the higher the eventual coherence. Thisis caused by an increasingly adiabatic behavior of the phononmodes. We also find that a non-zero temperature does notsignificantly alter the contrast. However, we find that once wego for higher masses, phonons indeed suppress the possibilityof coherent interferometry, and this may indeed turn out to bea fundamental limit for creating massive superpositions in thefuture.

Nano-object interferometry 9

The final conclusion is therefore that phonon dynamicsare not an inhibiting factor for a large range of parame-ters in nano-object Stern-Gerlach interferometry, making thismethod highly relevant for the fundamental experiments notedin the Introduction.

Data Availability. The data that supports the findings ofthis study are available within the article.

Conflict of Interest. The authors have no conflicts to dis-close.

Acknowledgments. This work was funded in part by theIsrael Science Foundation (grant nos. 856/18, 1314/19,3151/20, and 3470/21), by the Israel Innovation Authoritywithin the QuantERA network (project LEMAQUME, grantno. 76974), and by the Deutsche Forschungsgemeinschaftthrough the DIP program (grant no. Fo 703/2-1).

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