Tutorials � Macroscopic superpositions to test quantum gravity

International school on Quantum Sensors for Fundamental Physics QSFP 2021

Welcome to the tutorials!

What is the gravitational �eld generated by a quantum system?

The material is split into 4 exercises:

1. Basic interferometry and gravitational phases

2. Gravitational �eld of a Superposition

3. Decoherence in matter-wave interferometers with nanoparticles

4. State-of-the-art-experiments and future prospects

Parts (a),(b) of each exercise contains the core material, while part (c) is given as a bonus exercise.If you have any questions feel free to contact us at marko.toros(at)glasgow.ac.uk, julen.pedernales(at)gmail.com,

a.pontin(at)ucl.ac.uk, giulio.gasbarri(at)uab.ac.

2

1) BASIC INTERFEROMETRY AND GRAVITATIONAL PHASES

We consider a non-relativistic mass m with initial kinetic energy E0 = p20/2m in three di�erent situations depictedin Fig. 1. The key message of this exercise is that the interferometric phases are due to potential di�erences.

Figure 1: (a) Potential di�erence V induces a momentum change p0 → p0 + ∆p. (b) Interferometer in the Earth'sgravitational �eld (g is the gravitational acceleration). (c) Interferometer on a rotating platform rotating with angularfrequency ω.

(a) The particle traverses a potential di�erence V . Starting from energy conservation

p202m

=p212m

+ V, (1)

and assuming V is such that the corresponding momentum change is small (i.e. p1 − p0 = ∆p� p0), computethe relative change in momentum

∆p

p0= −1

2

V

E0. (2)

(b) The phase di�erence ∆φ between the paths indicated by purple/green arrows can be computed as a

∆φ =1

~

∮p · dr =

∆pl

~, (3)

where l is the length of the horizontal segment (the phase di�erence arises from the gravitational potential

di�erence V = mgh). Using Eq. (2) compute the Collella-Overhauser-Werner phase di�erence:

∆φCOW = −m2gλ

2π~2×A, (4)

where the area of the interferometer is A = lh, and the matterwave wavelength is λ = 2π~/(p0).

(c) The phase di�erence ∆φ between the paths indicated by purple/green arrows can be again computed usingEq. (3). The potential di�erence is given by V = ω · L, where L = r × p0 is the angular momentum. UsingEqs. (2) and (3) compute the Sagnac phase di�erence:

∆φSagnac =2mω

~×A, (5)

where the area of the interferometer is A = πr2. Find the condition for ∆φCOW/∆φSagnac ∼ 1.

References:[a] H. Rauch et al, Neutron Interferometry, Oxford University Press, USA (2015)[b] R. Colella et al, Phys. Rev. Lett. 34, 1472 (1975)[c] S. A. Werner et al, Phys. Rev. Lett. 42, 1103 (1979)

3

2) GRAVITATIONAL FIELD OF A SUPERPOSITION

We consider a non-relativistic particle s1 with mass m1 in a spatial superposition state:

|φ1〉 =|α〉+ |β〉√

2, (6)

where 〈x1|α〉 = δ(x1− d) and 〈x1|β〉 = δ(x1 + d). We suppose for simplicity that |φ1〉 will not evolve thus providing aconstant source for the gravitational �eld. We further consider a test particle s2 with mass m2 in an initial gaussianstate centered in the origin, i.e.

〈x2|φ2〉 =1

(2πσ2)1/4e−

x224σ2 . (7)

Figure 2: The particle s1, which generates the gravitational �eld is in a superposition of the states α and β. Particles2, which is used as test mass, is in a localized state located in the center.(a) Quantum Case: The total gravitational �eld is a linear combination of gravitational �elds produced by the statesα and β. (b) Semi-classical treatment of gravity: the total gravitational �eld acting on m2 is produced by the massdensity m1(|α(x)|2 + |β(x)|2)/2.

(a) In a quantum scenario we would expect the gravitational �eld to be in a superposition of the gravitational �eldsgenerated independently by |α〉 and |β〉. Suppose the initial state is |φ1〉|φ2〉.

1) Convince yourself that the state after the evolution will take the following form:

|φqg, t〉 =1√2

(|φα, t〉|α〉+ |φβ , t〉|β〉) , (8)

where the states of the test particle satisfy the following dynamics:

i~∂t|φα, t〉 =

{p22

2m2+Gm1m2

|x2 − d|

}|φα, t〉

i~∂t|φβ , t〉 =

{p22

2m2+Gm1m2

|x2 + d|

}|φβ , t〉

(9)

.

4

Hint: the state |φα〉 feels the gravitational potential generated by |α〉 while the state |φβ〉 feels the gravitationalpotential generated by |β〉.

2) Assuming that the gravitational potential can be approximated as

Gm1m2

|x2 ± d|' Gm1m2

|d| ∓ Gm1m2

|d|2 x2, (10)

it can be shown that

〈x2|φα, t〉 =1

{2πσ2[1 + (~t/m2σ)2)]}1/4 exp

(−

(x2 + Gm1

2|d|2 t2)2

4σ2(1 + ( ~tm2σ

)2)

)(11)

〈x2|φβ , t〉 =1

{2πσ2[1 + (~t/m2σ)2)]}1/4 exp

(−

(x2 − Gm1

2|d|2 t2)2

4σ2(1 + ( ~tm2σ

)2)

). (12)

Verify that the states |φα, t〉 and |φβ , t〉 are localized around the following positions

〈φα, t|x2|φα, t〉 = −Gm1

2|d|2 t2, 〈φβ , t|x2|φβ , t〉 =

Gm1

2|d|2 t2 (13)

while the mean value of the total state is still zero, i.e.

〈φqg, t|x2|φqg, t〉 =1

2(〈φα, t|x2|φα, t〉+ 〈φβ , t|x2|φβ , t〉) = 0. (14)

(b) In a semi-classical scenario the gravitational �eld is not allowed to be in a superposition of two di�erent con�g-urations. Let us suppose the gravitational �eld is sourced by the mass density m1|φ1(x)|2.Assume the initial state of the test and source particle to be again |φ1〉|φ2〉.

1) Convince yourself that, after the evolution, the state will take the following form:

|φcl, t〉 = |φ2,cl, t〉|α〉+ |β〉√

2(15)

and |φ2,cl, t〉 is governed by the following dynamical equation:

i~∂t|φ2,cl, t〉 =

{p22

2m2+ Vcl(x2)

}|φ2,cl, t〉 (16)

where Vcl(x) = 12 (Gm1m2

|x2−d| + Gm1m2

|x2+d| ).

2) Under the assumption in Eq. (10) show that:

Vcl(x) =Gm1m2

|d| . (17)

Under this approximation can be shown that the state at time t reads

〈x2|φ2,cl, t〉 =1

{2πσ2[1 + (~t/m2σ)2)]}1/4 exp

(− x22

4σ2(1 + ( ~tm2σ

)2)

). (18)

Show that:

〈φcl, t|x2|φcl, t〉 = 0. (19)

5

(c) In the scenario described in Fig. 2, measuring position of the test particle is not enough to discriminate betweena quantum and a semi-classical scenario but measuring variance of the test particle maybe help on testing thetwo hypothesis. Show that:

〈φqg, t|x22|φqg, t〉 = σ2

(1 +

(~tm2σ

)2)

+

(Gm1

2|d|2 t2

)2

(20)

〈φcl, t|x22|φcl, t〉 = σ2

(1 +

(~tm2σ

)2). (21)

References:[d] R. Feynman, et al. Feynman lectures on gravitation. CRC Press, 2018[e] L. Diosi, Phys. Lett. 105A (1984) 199-202[f] R. Penrose Found. Phys. 44, 557-575 (2014)[g] D.Carney et al Class. Quantum Grav. 36 034001(2019)

6

3) DECOHERENCE IN MATTER-WAVE INTERFEROMETERS WITH NANOPARTICLES

Consider the one dimensional interferometric loop in Fig. 3a. The center of mass of a particle of mass m is preparedin a pure quantum state described by a Gaussian wave packet of width x0 and zero momentum. The wave packet is split

in two components, which travel in opposite directions with constant momenta −~k and ~k until they reach a separationdistance d at time τ/2. At this moment, the wave packets are re�ected and their momenta are inverted such that theynow travel towards each other. At time τ , when the centers of the wave packets overlap, we measure the position ofthe particle. If the process is coherent, an interference pattern will be observed in the probability distribution of the

center of mass position, which is of the form P (x) = 1N e− x2

2σ2τ cos2(kx~ + φ), where k =∣∣∣~k∣∣∣, σ2

τ = x20[1 + ( ~τx202m

)2], N

is a normalization constant and φ is a relative phase picked up between the components of the superposition as theytravel di�erent arms of the interferometric loop.

-10 -8 -6 -4 -2 0 2 4 6 8 10

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Figure 3: Matter-wave interferometer

a) To reconstruct the probability distribution, the experiment needs to be repeated a statistically signi�cant numberof times. Consider that the experiment is not initialized in a pure state of 0 momentum, but that it contains a �nitetemperature. One way to treat the problem is to consider that in each run of the experiment the initial wave packetwill have a velocity v picked at random from a Boltzmann distribution according to the particles temperature. In one

dimension, for a free particle of mass m such a distribution is given by µB(v) =√

m2πKBT

e− mv2

2KBT , where KB is Boltz-

man's constant. Compute the e�ect of the initial temperature T of the center-of-mass motion on the visibility of the re-

constructed interference pattern. [Hint:∫dae− a2

2σ2a e−(x−a)2

σ2 cos2[k(x−a)] = Ae− x2

2(σ2a+σ2) [1 + e− 2k2σ2aσ

2

σ2a+σ2 cos(

2kσ2

σ2a+σ

2x)

],

with A a constant]

b) Another source of noise that can erase the interference pattern is given by the interferometric phase φ. If thetwo arms of the interferometric loop observe di�erent �elds due to the presence of some uncontrolled environment,a relative phase will build up and this will shift the interference pattern. Shot-to-shot �uctuations of this phase willagain reduce the visibility of the reconstructed interference pattern. Consider that the particle has an electric dipolemoment given by the presence of a negative charge at the center of the particle and its compensating positive chargeat the surface of the particle, as shown in Fig. 3b. Estimate the phase generated by a single electron sitting in the xaxis at some distance D from the interferometer. For simplicity, you can assume that D � d,R such that the �eldcan be linearized across the size of the interferometer and the dipole can be assumed to observe a uniform electric�eld. You can also assume that the dipole is aligned with the electric �eld. [Hint: Remember the energy of an

electric dipole moment ~p in an electric �eld ~E is given by U = −~p ~E, and the phase picked by each component of thesuperposition can be computed as the time integral of its energy ϕ =

∫ τ0dtU(t)/~. ]

c) Consider that the particle has a radius of R = 250 nm and that we are able to prepare it in a superposition ofdistance d = R, in a time τ = 0.5 s. What is the minimal volume around the interferometer that we need to ensure

is free of any electrical charges? [Hint: In S.I. units e− ≈ −1.6 · 10−19 C, ~ ≈ 1.05 · 10−34 J s, 14πε0

≈ 9 · 109 kgm3

s2C2 ]

7

4) STATE-OF-THE-ART-EXPERIMENTS

This section will focus on two experimental results achieved in the past year. The �rst is the cooling to the motionalground state of a mesoscopic silica nanoparticle in an optical tweezer [h]. The second is a direct measurement of thegravitational attraction between two mg scale gold spheres [i]. The objective here is to perform order of magnitudeestimations to get a feeling of the typical di�culties in these experiments.

Figure 4: (a) Schematic overview of the ground state cooling experiment (taken from Ref. [h]). An optical tweezerprovides a 3D trapping potential for a spherical silica nanoparticle. The motion along the x-axis is cooled by theinteraction with an optical cavity. (b) The Cavendish-like torsional balance use for the direct measurement of thegravitational coupling between mt and ms. The Newtonian force is modulated by changing the separation betweenthe two masses.

(a) The levitated oscillator of Ref. [h] is a silica nanosphere of radius RS = 70 nm and density ρ = 1850Kg. Let'sconsider a one dimensional harmonic oscillator with a trap frequency of ωm/2π = 160 kHz. At the beginning ofthe experiment, in absence of cooling, the oscillator is in a classical thermal state.

1) From the equipartition theorem calculate the position variance σ2x and the associated displacement rms.

2) How does it compare to zero point �uctuations xxpf =√

~2mωm

?

3) Calculate the mean thermal occupation nth in the high temperature limit.

4) The main sources of decoherence are due to thermal noise and photons recoil. The former is due to collisionswith the background gas which generates a damping:

γgas ' 0.8Pressure [mbar]

ρRS(22)

The �uctuation dissipation theorem connects this damping to a force noise. Expressed in terms of a heatingrate this is given by Γth = γgasnth.

Photon recoil is due to scattering of the trapping �eld which generates a force F ' ~knsc = Psc/c. Here, nsc isthe number of scattered photons and Psc is the scattered power. As for the thermal noise this force is stochasticand its heating rate can be calculated to be:

Γrec =2

5

ωlωm

Pscmc2

(23)

where ωl is the trapping �eld angular frequency.

Assuming a pressure of 10−7mbar and a scattered power Psc = 10µW (λ = 1064nm) calculate Γth and Γrec.

8

5) In absence of cooling the evolution of the mean energy can be predicted by the following equation (Fokker-Planck):

n = −γgasn+∑i

Γi. (24)

In the short time scale limit calculate the average number of oscillations in the time it take for n to increase byone assuming n(0) = no �

∑i Γi/γgas.

(b) The torsion balance in Fig. 4 can be described as an harmonic oscillator with a resonance frequency ωm/2π =3.6mHz and mechanical quality factor Q = ωm/γ = 5. The Newtonian force between the test mass mt ' 90mgand the source mass ms ' 90mg is modulated by changing the separation of the mass centers from 2.5mm to5.8mm.

1) Calculate the expected force at the smallest separation;

2) To get an idea of the e�ect of the surrounding gravitational disturbances calculate how far has to be anexperimenter (say 80Kg) to generate a similar force. What about a nearby train (say a carriage weighting15000Kg)?

3) Assuming the oscillator is thermal the minimum detectable rms force is given by Fmin =√

4kBTmeffγBW ,where T is the temperature of the environment, kB is the Boltzman constant, meff ' 2mt is the e�ectivemass of the torsion balance, γ the damping and BW is the measurement bandwith. Assuming a measurementbandwidth of 0.1mHz (roughly 3 hours) what is the signal to noise ratio?

4) Among the disturbances that have to be considered the Casimir force is one of the most problematic for smallseparations. For a metallic sphere near a conductive plane the Casimir force is given by:

Fcas =RSkBTζ(3)

8d2s(25)

where ζ(3) ' 1.2 is the Riemann Zeta function Rs is the sphere radius and ds = 250µm is the sphere-Faradayshield separation. One can verify that for the current con�guration Fcas is negligible when compared to thegravitational attraction.

However, it can become dominant for future experiments aiming to replicate the measurement with test massesapproaching the Planck mass (mp ' 2.2 × 10−8Kg). Assuming a symmetric con�guration, masses separationr = 4RS and ds = 2RS calculate the radius at which the Casimir force equals the Newtonian force (gold densityρ = 19000Kg/m3). How does the resulting mass compare to the Planck mass?

Constants:kB = 1.38 · 10−23 J/K~ = 1.05 · 10−34 J sG = 6.67 · 10−11Nm2/kg2

References:[h] U. Delic et al, Science 367, 892 (2020)[i] T. Westphal et al, Nature 591, 225 (2021)