Quantum mechanical modeling of the multi-stage SternGerlach experiment conducted by Frisch and Segr e S. S uleyman Kahraman1Kelvin Titimbo1Zhe He1Jung-Tsung Shen2and Lihong V. Wang1

2025-04-29 0 0 1003.21KB 11 页 10玖币

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Quantum mechanical modeling of the multi-stage Stern–Gerlach experiment

conducted by Frisch and Segr`e

S. S¨uleyman Kahraman,1, †Kelvin Titimbo,1, †Zhe He,1Jung-Tsung Shen,2and Lihong V. Wang1, ∗

1Caltech Optical Imaging Laboratory, Andrew and Peggy Cherng Department of Medical Engineering,

Department of Electrical Engineering, California Institute of Technology,

1200 E. California Blvd., MC 138-78, Pasadena, CA 91125, USA

2Department of Electrical and Systems Engineering,

Washington University in St. Louis, St. Louis, MO 63130, USA

(Dated: August 28, 2024)

The multi-stage Stern–Gerlach experiment conducted by Frisch and Segr`e includes two cascaded

quantum measurements with a nonadiabatic ﬂipper in between. The Frisch and Segr`e experiment

has been modeled analytically by Majorana without the nuclear eﬀect and subsequently revised

by Rabi with the hyperﬁne interaction. However, the theoretical predictions do not match the

experimental observation accurately. Here, we numerically solve the standard quantum mechanical

model, via the von Neumann equation, including the hyperﬁne interaction for the time evolution

of the spin. Thus far, the coeﬃcients of determination from the standard quantum mechanical

model without using free parameters are still low, indicating a mismatch between the theory and

the experiment. Non-standard variants that improve the match are explored for discussion.

Keywords: spin-ﬂip transitions, electron spin, quantum dynamics, nonadiabatic transitions, hyperﬁne inter-

action.

I. INTRODUCTION

The quantum measurement problem tackles the conun-

drum of wave function collapse and the Stern–Gerlach

(SG) experiment is considered as the ﬁrst observation

of a quantum measurement [1–5]. While the SG obser-

vation was interpreted as proof of quantization of spin

[6–8], cascaded quantum measurements provide a more

stringent test of theories [3, 9]. Frisch and Segr`e (FS)

conducted the ﬁrst successful multi-stage SG experiment

[1, 10–12] after improving the apparatus from Phipps and

Stern [13]. Even though more recent multi-stage SG ex-

periments have been conducted, they diﬀer in the mecha-

nisms of polarizing, ﬂipping, and analyzing the spin [14–

21]. Most experiments designed for precise atomic mea-

surements use a narrow-band resonant (adiabatic) ﬂipper

[14] while the FS experiment uses a wide-band nonadia-

batic ﬂipper.

The FS experiment was suggested by Einstein [7, 12,

22], studied analytically by Majorana [23, 24] and later

by Rabi [25]. Majorana investigated the nonadiabatic

transition of the electron spin through a closed-form an-

alytical solution, which is now widely used to analyze any

two-level systems [26]. Rabi revised Majorana’s deriva-

tion by adding the hyperﬁne interaction but still could

not predict the experimental observation accurately. De-

spite additional theoretical studies into similar problems

involving multilevel nonadiabatic transitions [26–32], an

exact solution with the hyperﬁne interaction included has

not been obtained.

Among the more recent multi-stage SG experiments

†These authors contributed equally.

∗Corresponding email:lvw@caltech.edu

[15–20], the study most similar to the FS experiment uses

a sequence of coils to obtain the desired magnetic ﬁeld

[15, 16]. The models in these works not only simpliﬁed

the mathematical description of the magnetic ﬁelds gen-

erated by the coils but also ﬁt free parameters to predict

the experimental observations. We choose to model the

FS experiment over other similar experiments because

of the simplicity of the nonadiabatic spin ﬂipper and its

historical signiﬁcance.

Here, we numerically simulate the FS experiment us-

ing a standard quantum mechanical model via the von

Neumann equation without tuning any parameters and

compare the outcome with the predictions by both Majo-

rana and Rabi. Even though our approach is a standard

method of studying such spin systems, our results do

not match the experimental observations. This discrep-

ancy indicates that either our understanding of the FS

experiment is lacking or the standard theoretical model

is insuﬃcient. Recent studies have modeled the FS ex-

periment using an alternative model called co-quantum

dynamics [33–35] without resorting to free parameters.

We believe it is essential to bring the FS experiment to

the attention of the research community.

This paper is organized as follows. In Sec. II, we

present the experimental conﬁguration used by Frisch

and Segr`e to measure the fraction of electron spin ﬂip. In

Sec. III, we introduce the von Neumann equation and the

Hamiltonian for the nuclear-electron spin system. Nu-

merical results for the time evolution of the spins and

the ﬁnal electron spin-ﬂip probability are shown here. In

Sec. V, we compare the numerical results with previous

solutions. Finally, Sec. VI is left for conclusions. Non-

standard variants of the quantum mechanical model are

explored in the appendices to stimulate discussion.

arXiv:2210.11553v5 [quant-ph] 26 Aug 2024

II. DESCRIPTION OF THE FRISCH–SEGR`

EXPERIMENT

The schematic of the setup used in the Frisch–Segr`e

experiment [10, 11] is redrawn in Figure 1. There, mag-

netic regions 1 and 2 act as Stern–Gerlach apparatuses,

SG1 and SG2, respectively, with strong magnetic ﬁelds

along the +zdirection. In SG1, stable neutral potassium

atoms (39K) eﬀused from the oven are spatially separated

by the magnetic ﬁeld gradient according to the orienta-

tion of their electron magnetic moment µe. The magnet-

ically shielded space containing a current-carrying wire

forms the inner rotation (IR) chamber. The shielding re-

duces the fringe ﬁelds from the SG magnets down to the

remnant ﬁeld Br= 42 µT aligned with +ˆz. Inside the IR

chamber, the current-carrying wire placed at a vertical

distance za= 105 µm below the atomic beam path cre-

ates a cylindrically symmetric magnetic ﬁeld. The total

magnetic ﬁeld in the IR chamber equals the superposition

of the remnant ﬁeld and the magnetic ﬁeld created by the

electric current Iwﬂowing through the wire. The precise

magnetic ﬁeld outside the IR chamber was not reported

[10, 11]. After SG1, the atoms enter the IR chamber;

we approximate the motion to be rectilinear and con-

stant along the yaxis. The rectilinear approximation of

atomic motion within the IR chamber is acceptable since

the total displacement due to the ﬁeld gradients is neg-

ligible, approximately 1 µm. Along the beam path, the

magnetic ﬁeld is given by

Bexact =µ0Iwza

2π(y2+z2

a)ey+Br−µ0Iwy

2π(y2+z2

a)ez,(1)

where µ0is the vacuum permeability; the trajectory of

the atom is expressed as y=vt, where vis the speed of

the atom and the time is set to t= 0 at the point on

the beam path closest to the wire. The right-handed and

unitary vectors {ex,ey,ez}describe the directions of the

Cartesian system.

The magnetic ﬁeld inside the IR chamber has a

current-dependent null point below the beam path at

coordinates (0, yNP ,−za), with yNP =µ0Iw/2πBr. In the

vicinity of the null point, the magnetic ﬁeld components

are approximately linear functions of the Cartesian co-

ordinates. Hence, the magnetic ﬁeld can be approxi-

mated as a quadrupole magnetic ﬁeld around the null

point [10, 23]. Along the atomic beam path, the approx-

imate quadrupole magnetic ﬁeld is [33, 35]

Bq=2πB2

µ0Iw

zaey+2πB2

µ0Iw

(y−yNP )ez.(2)

For the study of the time evolution of the atom inside

the IR chamber both of the ﬁelds, Bexact and Bq, are

considered below.

After the IR chamber, a slit transmits one branch of

electron spins initially polarized by SG1 and blocks the

other branch. The slit was positioned after the intermedi-

ate stage to obtain a sharper cut-oﬀ [10]. In the forthcom-

ing theoretical model, we track only the top transmitted

Oven

Magnetic

shielding

Microscope

Hot wire

Slit

4 cm

Beam path

Magnetic

region 1

Magnetic

region 2

Trench Trench

Edge Edge

FIG. 1. Redrawn schematic of the original setup [10, 11].

Heated atoms in the oven eﬀuse from a slit. First, the atoms

enter magnetic region 1, which acts as SG1. Then, the atoms

enter the region with magnetic shielding (i.e., the inner ro-

tation chamber) containing a current-carrying wire W. Next,

a slit selects one branch. Magnetic region 2 acts as SG2.

The hot wire is scanned vertically to map the strength of the

atomic beam along the zaxis. The microscope reads the po-

sition of the hot wire.

branch with spin down, mS=−1/2, at the entrance of

the IR chamber and ignore the blocked branch. How-

ever, the opposite choice of mS= +1/2yields exactly the

same results in this model. The atoms that reach SG2 are

further spatially split into two branches corresponding to

the electron spin state with respect to the magnetic ﬁeld

direction. The ﬁnal distribution of atoms is measured by

scanning a hot wire along the zaxis while monitored by

the microscope. The probability of ﬂip is then measured

at diﬀerent values of the electric current Iw.

III. THEORETICAL DESCRIPTION

The time evolution of the noninteracting atoms in the

beam traveling through the IR chamber of the Frisch–

Segr`e experiment is studied using standard quantum me-

chanics. The whole setup is modeled in multiple stages

as illustrated in Figure 2. First, the output of SG1 and

the slit is modeled as a pure eigenstate of the electron

spin measurement in the zdirection. Since the gradient

of the strong ﬁeld in SG1 is not high enough, nuclear spin

eigenstates do not separate during the ﬂight. Hence, the

nuclear state is assumed to be unaﬀected by SG1 and the

slit. During the ﬂight from SG1 to the entrance of the

IR chamber, the state is assumed to vary adiabatically

as in Figure 2. The ﬁelds in the transition regions were

not reported; but when the IR chamber was turned oﬀ,

Iw= 0 A, no ﬂipping was observed after SG2 [11]. There-

fore, it can be assumed that outside the IR chamber, the

state evolves adiabatically. Later, the atom enters the

IR chamber designed to induce nonadiabatic transitions.

The evolution of the state in the IR chamber is modeled

using the von Neumann equation, which is solved using

numerical methods. During the ﬂight from the exit of

the IR chamber to SG2, the state is assumed to vary

adiabatically as in Figure 2. Finally, SG2 measures the

probabilities in diﬀerent electron spin eigenstates in the

zdirection according to the Born principle.

The density operator formalism is used for its capabil-

ity to represent mixed states in quantum systems, oﬀer-

ing a more complete description than pure states alone

[36, 37]. The time evolution of the density operator ˆρis

governed by the von Neumann equation [36–39]:

∂ˆρ(t)

∂t =1

iℏ[ˆ

H(t),ˆρ(t)] ,(3)

where ˆ

H(t) is the Hamiltonian of the system and ℏis the

reduced Planck constant. For the time-dependent Hamil-

tonian ˆ

H(t), we introduce the instantaneous eigenstates

|ψj(t)⟩and eigenenergies Ej(t) such that

H(t)|ψj(t)⟩=Ej(t)|ψj(t)⟩,(4)

where jcan take a ﬁnite number of values for the

spin system considered here. In the basis of the in-

stantaneous eigenstates of the Hamiltonian, from (3)

the matrix elements of the density operator, ρj,k(t) =

⟨ψj(t)|ˆρ(t)|ψk(t)⟩, evolve according to

∂ρj,k(t)

∂t ="1

iℏ(Ej(t)− Ek(t)) − ⟨ψj(t)|∂|ψj(t)⟩

∂t +⟨ψk(t)|∂|ψk(t)⟩

∂t #ρj,k(t)

r̸=q

⟨ψl(t)|∂ˆ

H(t)

∂t |ψk(t)⟩

Ek(t)− El(t)ρj,l(t)−X

r̸=p

⟨ψj(t)|∂ˆ

H(t)

∂t |ψl(t)⟩

El(t)− Ek(t)ρl,k(t).(5)

In particular, the elements in the diagonal ρj,j (t), cor-

responding to the probabilities of ﬁnding the quantum

system in the eigenstate of the Hamiltonian, follow

∂ρj,j (t)

∂t = 2 Re 

X

r̸=p

⟨ψl(t)|∂ˆ

H(t)

∂t |ψj(t)⟩

Ej(t)− El(t)ρj,l(t)

.(6)

In the adiabatic approximation, since the time deriva-

tive of the Hamiltonian is small compared to the energy

diﬀerence, we set [40, 41]

⟨ψl(t)|∂ˆ

H(t)

∂t |ψj(t)⟩

Ej(t)− El(t)→0.(7)

Therefore, for the adiabatic evolution, the populations in

the instantaneous eigenstates do not change over time

∂ρj,j (t)

∂t = 0 .(8)

If the system’s Hamiltonian changes quickly relative to

the energy gap, the above approximation fails, leading to

nonadiabatic transitions.

Let us consider the quantum system for a neutral al-

kali atom, composed of the spin S=1/2of the valence

electron and the spin Iof the nucleus. In an external

magnetic ﬁeld B, the electron Zeeman term ˆ

Hedescribes

the interaction between the electron magnetic moment

and the ﬁeld via [41]

He=−ˆ

µe·B,(9)

where ˆ

µeis the quantum operator for the electron mag-

netic moment. Furthermore, ˆ

µe=γeˆ

S, where γedenotes

the gyromagnetic ratio of the electron; the electron spin

operator ˆ

S=ℏ

2ˆ

σ, with the Pauli vector ˆ

σconsisting of

the Pauli matrices {σx, σy, σz}. Substitutions yield

He=−γe

ℏ

2ˆ

σ·B.(10)

The (2S+ 1)-dimensional Hilbert space He=

span (|S, mS⟩), with mS=−S, . . . , S, and |S, mS⟩being

the eigenvectors of ˆ

Sz.

The nuclear Zeeman Hamiltonian ˆ

Hndescribes the in-

teraction of the nuclear magnetic moment with the ex-

ternal magnetic ﬁeld:

Hn=−ˆ

µn·B,(11)

where ˆ

µn=γnˆ

Idenotes the quantum operator for the

nuclear magnetic moment, γnthe nuclear gyromagnetic

ratio for the atomic specie, and ˆ

Ithe nuclear spin quan-

tum operator for spin I. Therefore, we can write ˆ

I=ℏ

2ˆ

τ,

with ˆ

τbeing the generalized Pauli vector constructed

with the generalized Pauli matrices of dimension 2I+ 1,

namely {τx, τy, τz}, satisfying [τj, τk] = 2iϵjklτl. Substi-

tutions produce

Hn=−γn

ℏ

2ˆ

τ·B.(12)

A basis for the (2I+ 1)-dimensional Hilbert space Hn

can be obtained from the eigenvectors of ˆ

Iz, such that

Hn= span (|I, mI⟩) with mI=−I,...,I.

The interaction between the magnetic dipole moments

of the nucleus and the electron gives the hyperﬁne struc-

ture (HFS) term ˆ

HHFS . In terms of the electron and nu-

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Quantummechanicalmodelingofthemulti-stageStern–GerlachexperimentconductedbyFrischandSegr`eS.S¨uleymanKahraman,1,†KelvinTitimbo,1,†ZheHe,1Jung-TsungShen,2andLihongV.Wang1,∗1CaltechOpticalImagingLaboratory,AndrewandPeggyCherngDepartmentofMedicalEngineering,DepartmentofElectricalEngineering,CaliforniaI...

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Quantum mechanical modeling of the multi-stage SternGerlach experiment conducted by Frisch and Segr e S. S uleyman Kahraman1Kelvin Titimbo1Zhe He1Jung-Tsung Shen2and Lihong V. Wang1

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