Mass-independent scheme for enhancing spatial quantum superpositions Run Zhou1Ryan J. Marshman2Sougato Bose3and Anupam Mazumdar1 1Van Swinderen Institute University of Groningen 9747 AG Groningen The Netherlands._2

2025-04-27 0 0 826.52KB 12 页 10玖币

侵权投诉

Mass-independent scheme for enhancing spatial quantum superpositions

Run Zhou,1Ryan J. Marshman,2Sougato Bose,3and Anupam Mazumdar1

1Van Swinderen Institute, University of Groningen, 9747 AG Groningen, The Netherlands.

2Centre for Quantum Computation and Communication Technology, School of Mathematics

and Physics, University of Queensland, Brisbane, Queensland 4072, Australia

3Department of Physics and Astronomy, University College

London, Gower Street, WC1E 6BT London, United Kingdom.

(Dated: February 27, 2023)

Placing a large mass in a large spatial superposition, such as a Schr¨odinger Cat state is a sig-

niﬁcant and important challenge. In particular, the large spatial superposition (O(10 −100) µm)

of mesoscopic masses (m∼ O(10−14 −10−15) kg) makes it possible to test the quantum nature of

gravity via entanglement in the laboratory. To date, the proposed methods of achieving this spatial

delocalization are to use wavepacket expansions or quantum ancilla (for example spin) dependent

forces, all of whose eﬃcacy reduces with mass. Thus increasing the spatial splitting independent

of the mass is an important open challenge. In this paper, we present a method of achieving a

mass-independent enhancement of superposition via diamagnetic repulsion from current-carrying

wires. We analyse an example system which uses the Stern-Gerlach eﬀect to creating a small ini-

tial splitting, and then apply our diamagnetic repulsion method to enhance the superposition size

O(400 −600) µm from an initial modest split of the wavefunction. We provide an analytic and

numeric analysis of our scheme.

I. INTRODUCTION

Gravity is special as it is not yet evident whether

gravity is a classical or a quantum entity [1]. It is

often thought that any quantum gravitational eﬀects

will become important only when we approach the

Planck length or the time scale; making it impossible

to probe directly. Furthermore, cosmological data,

such as perturbations in the cosmic microwave back-

ground radiation [2] and the potential B-modes for

future detection may not help to settle this outstand-

ing issue [3]. Both astrophysical and cosmological

sources contain many uncertainties [4].

Despite all these challenges, a tabletop experiment

has recently been proposed to explore the quantum

origin of gravity with the help of quantum superposi-

tion and quantum entanglement in the infrared [5–7],

see also [8,9]. The protocol is known as the quantum

gravity induced entanglement of masses (QGEM),

which evidences both quantum superposition of ge-

ometries [10,11], as well as the exchange of virtual

gravitons [6], that is, spin-2 graviton exchange [7]),

see also [12]. Recently, a protocol has been created to

entangle the matter with that of the Standard Model

photon in a gravitational optomechanical setup [13],

see also [14]. This will probe not only the light bend-

ing due to the gravitational interaction but will also

probe the spin-2 nature of the graviton mediated en-

tanglement [13]. Note that the entanglement is purely

a quantum observable, which measure the quantum

correlation in complementary bases, and has no clas-

sical analogue whatsoever.

One of the key challenges towards realising the

QGEM protocol experimentally is to create a large

spatial superposition δz ∼ O(10−100) µm for a large

mass object (m∼ O(10−14 −10−15 kg), see for details

in [5,15] and in a free falling setup [5,16]. It is also

well known that creating a large superposition has

many further fundamental applications; one can test

the foundations of quantum mechanics in presence of

gravity [17–21], a purely quantum gravitational ver-

sion of the equivalence principle [22], falsifying spon-

taneous collapse mechanisms [19,20], placing a bound

on decoherence mechanisms [23–29], quantum sen-

sors [16,30], probing physics of a ﬁfth fundumental

force and the axion [31], and probing gravitational

waves [30].

Atom interferometers are well-known to create a

large baseline superposition [32–34], but at masses

well below what is required to test the quantum

nature of gravity. To date, macromolecules repre-

sent the heaviest masses placed in a superposition

of spatially distinct states [35,36]. There are phys-

ical schemes to obtain tiny superpositions of large

masses [37] or moderate sized (∼10 nm - 1µm) su-

perpositions of ∼10−19 −10−17 kg masses [24,25,38–

49]. However, we require a large spatial superposition

of heavy (m∼ O(10−17 −10−14) kg) masses, with the

current likely scheme utilising the Stern-Gerlach ef-

fect [50–55]. In fact, a proof of principle experiment

has already been conduced using atoms, showing that

such a Stern-Gerlach Interferometer (SGI) for mas-

sive objects can indeed be realised [53].

The crucial problem now is how to achieve a large

spatial separation for these larger mass particles. To

the best of our knowledge, there are so far only

two realistic types of schemes to separate wave pack-

ets, one using wavepacket expansions in conjunction

with slits/measurements [24,35,36,45–49] and the

other using spin-dependent forces [53–55]. However,

both schemes inevitably become progressively worse

as the mass increases. This paper presents a mass-

arXiv:2210.05689v4 [quant-ph] 24 Feb 2023

independent experimental scheme to create a large

spatial superposition.

There are already some very attractive experi-

mental ideas which have been proposed to create

large superposition for masses in the range of m∼

O(10−17 −10−14) kg, see Refs. [5,30], and Refs. [54–

56]. The latter references considered the eﬀect of in-

duced magnetic potential in the diamond like crystal

along with the nitrogen valence (NV)-centred poten-

tial. However, it was also noticed that the diamag-

netic term in the Hamiltonian inhibits the spatial su-

perposition despite taking into account that the co-

herence of the NV centre can be maintained for a long

enough time [54,56]. This was mainly due to the

fact that the diamagnetic term creates an harmonic

trap and the particle tends to move towards the cen-

tre of the potential, therefore inhibiting the growth

in the superposition size. This issue was tackled by

employing new dynamical techniques such as cata-

pulting the trajectory by assuming non-linear pro-

ﬁle for the magnetic ﬁeld [55]. In the current paper,

we will explore a yet new possibility of utilising this

unavoidable induced diamagnetic eﬀect to enhancing

the superposition size. In our analysis, we will use the

simple, example model of a magnetic ﬁeld sourced by

current-carrying wires.

Our aim will be to enhance the superposition size

using only modest currents, but will require one sim-

ple assumption. We will assume that we have al-

ready created an initial spatial superposition of the

nanocrystal, as could be made using a Stern-Gerlach

setup, see [54–56]. In particular, we will take for

demonstration the initial superposition size ∆z0∼

1µm and present how this can be increased to a size of

∆z∼ O(100 −500)µm. We will also assume that the

interferometer setup is one dimensional. And, once

we create the superposition, we will assume that the

superposition can be closed using the same splitting

mechanism, i.e. by the Stern-Gerlach mechanism,

see [55]. It is for this reason we present this as an

enhancement of a spatial superposition. We will pro-

vide both analytic and numerical analyses. We will

take the modest currents ∼ O(1) A, and will restrict

the example operating time to be within ∼0.1 s while

still aiming to maximise the superposition size.

The maximum size of the superposition will not de-

pend on the mass, thus making this as an attractive

scheme for creating macroscopic quantum superposi-

tion for any range of masses, provided the initial small

superposition is ﬁrst created. We will show that there

are two possible conﬁgurations; one where the trajec-

tory is triangular and the other where the trajectory

traces its path after interacting with the wire with a

closest impact parameter, discussed below.

In our current analysis we will assume that the

nanocrystal is not rotating. Recent paper has anal-

ysed the rotation of the crystal in Refs. [57–59]. We

will need revisit this issue separately once we fully

analyse the largest superposition size we can obtain.

This is similar to the spin coherence problem and

will require a dedicated study. We will also not dis-

cuss how to close the superposition as it is possible to

employ the Stern-Gerlach technique to close the tra-

jectory, see [55]. Another generic issue is the phonon

vibration causing the decoherence, see [60]. We will

assume that the desired level of internal cooling can

be obtained to mitigate the phonon induced decoher-

ence. Given all these challenges can be tackled, we

ask how large spatial superposition can we achieve

via the diamagnetic contribution.

The paper is organised as follows. In section II,

we will discuss the setup with the Hamiltonian. In

section III, we will discuss diamagnetic repulsion. In

section IV, we will provide the analytical understand-

ing of the interaction of the nanocrystal with the ﬁxed

wires. The analysis is very similar to central poten-

tial problem in classical mechanics. In section V we

will present and discuss our numerical results, and in

section VI, we will conclude our paper.

II. SETUP

The Hamiltonian of a diamond embedded in a spin

in a magnetic ﬁeld is [56,61]

H=ˆ

2m+~Dˆ

S2−χρm

2µ0

B2−ˆ

µ·B,(1)

where ˆ

µ=−gµBˆ

Sis the spin magnetic moment,

g≈2 is the Land`e g-factor, µB=e~

2meis the Bohr

magneton, ˆ

Sis spin operator, ˆ

pis momentum op-

erator, Bis the magnetic ﬁeld. m,~,D,χρand

µ0are scalars representing the mass of the diamond,

the reduced Planck constant, NV zero-ﬁeld splitting,

magnetic susceptibility, and vacuum permeability, re-

spectively. By applying the microwave pulse with an

appropriate resonance frequency, the electron spin in

the diamond can be coupled with the nuclear spin

and the spin magnetic ﬁeld interaction can be ignored

[62]. We will assume that this has happened prior to

the ampliﬁcation of the superposition size.

Let us ﬁrst consider the magnetic ﬁeld generated

by a current-carrying wire

B=µ0I×er

2πr ,(2)

where Iis the current carried by a straight wire. ris

the vertical distance from the wire to a point in space.

eris the unit vector corresponding to the distance r.

At this point, the potential energy of the diamond in

the magnetic ﬁeld is given by

U=−χρm

2µ0

B2=1

2mαI2

r2,(3)

where α=−χρµ0

4π2. Combining Eq.(3) and Eq.(2),

the acceleration of the diamond can be obtained as

follows

adia =−1

m∇U=αI2

r3er,(4)

It can be seen from Eq.(4) that the acceleration

caused by the diamagnetic eﬀect (the third term of

Eq.(1)) is mass independent. This means we can get

a desired acceleration of a diamagnetic object by set-

ting an appropriate distance between the object and

the wire and an appropriate current ﬂowing through

the wire regardless of the mass of the object. This

property provides the possibility of obtaining a large

superposition size for massive object.

We will use the trajectory of a classical wave packet

to represent the expectation value of the position of

the wave packet.

III. DIAMAGNETIC REPULSION

The scheme of increasing the superposition size by

diamagnetic repulsion is mainly divided into three

steps.

•First, we will create a small, initial spatial split-

ting between the two wave packets, potentially

using a Stern-Gerlach apparatus.

•Then, a micrwave pulse is used to map the elec-

tron spin state to the nuclear spin state, thus

allowing us to ignore the spin magnetic ﬁeld in-

teraction.

•Finally, the wave packets enter the magnetic

ﬁeld generated by the current-carrying wires

and are further separated under the diamag-

netic repulsion, thus achieving a large superpo-

sition size.

The ﬁrst and second steps are the initial state of

the system that we assume. The following analysis is

focused on the third step, that is, how to use diamag-

netic repulsion to enhance the superposition state.

The motion of the wave packet in the magnetic ﬁeld

can be divided into two stages (see Fig.1).

•Stage-I. The wave packet is incident parallel to

the x-axis with an initial velocity v0and its tra-

jectory is deﬂected by the action of the diamag-

netic repulsion. Then the wave packet moves

towards the top (bottom) wire until the dis-

tance from the x axis reaches a maximum. The

spatial distance between the two wave packets

also reaches a maximum.

•Stage-II. The wave packet trajectory is de-

ﬂected again near the top (bottom) wire and

then returns to its initial position. There are

two kinds of trajectories that can go back to

their initial positions, the triangle trajectory

and the inverse trajectory. For triangular tra-

jectory, the wave packet returns to its initial po-

sition in the second stage by the shortest path

(see Fig.1(a)). For inverse trajectory, the wave

packet returns to its initial position in the sec-

ond stage along the path it took in the ﬁrst

stage (see Fig.1(c)).

IV. ANALYTIC RESULTS

In the ﬁrst stage, the motion of a wave packet in

the magnetic ﬁeld can be viewed as the scattering

process of a diamagnetic particle in the magnetic ﬁeld

and then solved analytically using polar coordinates.

Since the acceleration of the particle is along the ra-

dial direction, the equation of motion can be written

d2r

dt2−rdθ

dt2

=αI2

r3,(5)

where the pole is located at the center of the splitting

wire and the polar axis is along the x-axis direction.

In this case, θis the polar angle. The angular mo-

mentum of the particle

L=|L|=mr2dθ

dt.(6)

By using d/dt = (L/mr2)(d/dθ) and noting that the

angular moment is conserved and the initial moment

is given by:

L=mv0b, (7)

where v0is the initial velocity (parallel to the po-

lar axis direction), bis the vertical distance of the

initial position with respect to the polar axis and is

called the impact parameter. Substituting Eq.(7) into

Eq.(5), we obtain

d2u

dθ2=−ku, (8)

where u= 1/r, and

k= 1 + αI2

0b2.(9)

The solution to Eq.(8) is given by

u= C cos(√kθ −θ0),(10)

where C and θ0are constants determined by the ini-

tial conditions. We will solve for C in the Appendix.

Assuming that a particle is incident parallel to the po-

lar axis from inﬁnity and scattered to inﬁnity. When

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Mass-independentschemeforenhancingspatialquantumsuperpositionsRunZhou,1RyanJ.Marshman,2SougatoBose,3andAnupamMazumdar11VanSwinderenInstitute,UniversityofGroningen,9747AGGroningen,TheNetherlands.2CentreforQuantumComputationandCommunicationTechnology,SchoolofMathematicsandPhysics,UniversityofQueenslan...

展开>> 收起<<

Mass-independent scheme for enhancing spatial quantum superpositions Run Zhou1Ryan J. Marshman2Sougato Bose3and Anupam Mazumdar1 1Van Swinderen Institute University of Groningen 9747 AG Groningen The Netherlands._2.pdf

共12页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Mass-independent scheme for enhancing spatial quantum superpositions Run Zhou1Ryan J. Marshman2Sougato Bose3and Anupam Mazumdar1 1Van Swinderen Institute University of Groningen 9747 AG Groningen The Netherlands._2

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: