Nonstandard Deformed Oscillators from Deformed Heisenberg Algebras

Nonstandard Deformed Oscillators
from $\boldsymbol{q}$\- and $\boldsymbol{p,q}$-Deformations of Heisenberg
Algebra

Alexandre M. GAVRILIK † and Ivan I. KACHURIK †‡

A.M. Gavrilik and I.I. Kachurik

† Bogolyubov Institute for Theoretical Physics, 14-b Metrolohichna Str., Kyiv,
03680 Ukraine<EMAIL_ADDRESS><EMAIL_ADDRESS>
‡ Khmelnytskyi National University, 11 Instytutska Str., Khmelnytskyi, 29016
Ukraine

Received February 12, 2016, in final form May 06, 2016; Published online May
12, 2016

For the two-parameter $p,q$-deformed Heisenberg algebra introduced recently
and in which, instead of usual commutator of $X$ and $P$ in the l.h.s. of
basic relation $[X,P]={\rm i}\hbar$, one uses the $p,q$-commutator, we
established interesting properties. Most important is the realizability of the
$p,q$-deformed Heisenberg algebra by means of the appropriate deformed
oscillator algebra. Another uncovered property is special extension of the
usual mutual Hermitian conjugation of the creation and annihilation operators,
namely the so-called $\eta(N)$-pseudo-Hermitian conjugation rule, along with
the related $\eta(N)$-pseudo-Hermiticity property of the position or momentum
operators. In this work, we present some new solutions of the realization
problem yielding new (nonstandard) deformed oscillators, and show their
inequivalence to the earlier known solution and the respective deformed
oscillator algebra, in particular what concerns ground state energy.

deformed Heisenberg algebra; position and momentum operators; deformed
oscillators; structure function of deformation; deformation parameters; ground
state energy

81R50; 81S05; 81Vxx; 17B37

##  1 Introduction

The obtaining and analysis of modified/generalized versions of the famous
Heisenberg uncertainty relation or principle corresponds to diverse
generalizations of the standard Heisenberg algebra (HA) with basic relation
$[X,P]={\rm i}\hbar$ for the position and momentum operators. This direction
of research is under development for more than three decades (with some early
works quoted as [7, 12, 28, 29, 34, 37, 38]), and still remains to be a hot
topic, see, e.g., [5, 8, 13, 16, 17, 32] for more recent papers. Physically,
deformations of the Heisenberg algebra and the related generalized uncertainty
relation find their motivation in quantum gravity, in string theory,
noncommutative geometry etc. An overview of different approaches can be found,
e.g., in [14, 27]. What concerns the particular existing variants of deformed
Heisenberg algebra (DHA), we have to note that predominantly there were DHAs
with deformed r.h.s. of the basic relation for $X$ and $P$ that involved a
function of momentum, or some function of the Hamiltonian. However, within
more exotic approach followed in [12, 38], deformation of HA emerged in the
l.h.s. of basic relation, by using “$q$-commutator” instead of the usual
commutator.

Recently, a hybrid so-called “two-sided” deformation of HA has been introduced
[16]. In that DHA, first, the commutator in the l.h.s. of its defining
relation is replaced with $q$-commutator like in [12, 38] or with
$q,p$-commutator as in [16]. In addition, the r.h.s. of basic relation is
modified by an extra term involving Hamiltonian times a pre-factor denoted as
$\mu$. For this two-sided or $q,p,\mu$-deformed DHA it was shown that,
similarly to the “left-handed” $q$\- and $q,p$-deformed ones, this more
general deformed algebra of the operators $X$ and $P$ can as well be realized
by (i.e., mapped onto) definite nonstandard deformed oscillator algebra (DOA).

Unusual thing concerning just the $q,p,\mu$-DHA is that the deformation factor
(parameter) $\mu$ in front of the Hamiltonian $H$ in the r.h.s. of basic
relation becomes inevitably depending not only on the deformation parameter(s)
$q$ or $q,p$ of the $q$\- or $q,p$-commutator in the l.h.s., but also on the
particle number operator $N$, see [16].

The explicit connection of the DHA with the respective DOA discloses an
unexpected property [17] of the operators involved. We mean the option that
the creation and annihilation operators $a^{+}$ and $a^{-}$ may be not the
(mutual) Hermitian conjugates, but the so-called $\eta$-pseudo-Hermitian
conjugates of one another, with $\eta\equiv\eta(N)$ – certain operator
function of the number operator $N$. Accordingly, the position operator $X$ or
the momentum operator $P$, or both, were shown to possess not the customary
Hermiticity, but the property of being $\eta(N)$-pseudo-Hermitian [17].

The latter fact was uncovered in the framework of the particular DOA presented
explicitly through its structure function of deformation (DSF) $\Phi(N)$. That
was found in closed form in [16]. However, further analysis shows that this
solution given by the DSF $\Phi(N)$ is not unique, and other possibilities may
exist. The goal of this paper is to present other admissible solutions which
as well provide realization of DHA by some (different) DOA. After presenting
the new solutions, we demonstrate that these are inequivalent to the first
one, found in [16] and given by $\Phi(N)$. As a major feature, they show
differing values of the ground state energy. The latter can be either lower or
higher than the familiar ground state energy $E_{0}=\frac{1}{2}\hbar\omega$ of
the usual quantum harmonic oscillator111Note that this same ground state
energy $E_{0}=\frac{1}{2}\hbar\omega$ is shared by such well-known deformed
oscillators as Arik–Coon (AC) $q$-oscillator [3], Biedenharn–Macfarlane (BM)
$q$-oscillator [6, 31], Tamm–Dancoff (TD) type $q$-oscillator [10], and the
$p,q$-deformed oscillator [4, 9]..

##  2 $\boldsymbol{q}$-deformed Heisenberg algebra

An alternative approach to deform the Heisenberg algebra was considered in the
literature. Namely, in [12, 38] different approach of deforming HA was studied
such that a deformation is introduced in the commutator in the l.h.s. of basic
relation (this is the $q$-deformed HA)

$\displaystyle XP-qPX={\rm i}\hbar.$ (2.1)

For convenience, in all our treatment we put $\hbar=1$. Below, in the first
subsection we mainly follow [12]. We require that the equality (2.1) is
connected with special deformed oscillator algebra whose generating elements
$a^{+}$, $a^{-}$ and $N$ (the creation/annihilation operators, which are not
necessarily strict conjugates of each other, and the excitation number
operator) satisfy

$\displaystyle[N,a^{+}]=a^{+},\qquad[N,a^{-}]=-a^{-},$ (2.2) $\displaystyle
H(N)a^{-}a^{+}-G(N)a^{+}a^{-}=1.$ (2.3)

It is meant that the operator functions $H(N)$ and $G(N)$ admit formal power
series expansion.

###  2.1 From DHA to DOA

Like in [12] we express the position and momentum operators in terms of
$a^{-}$, $a^{+}$ as

$\displaystyle X\equiv f(N)a^{-}+g(N)a^{+},\qquad P\equiv{\rm
i}\big{(}k(N)a^{+}-h(N)a^{-}\big{)}$ (2.4)

where $f(N)$, $g(N)$, $h(N)$, $k(N)$ are some functions of the operator $N$.

On the base of (2.2), for any function ${\cal F}(N)$ possessing formal power
series expansion we have

$\displaystyle{\cal F}(N)a^{\pm}=a^{\pm}{\cal F}(N\pm 1),\qquad[{\cal
F}(N),a^{\pm}a^{\mp}]=0.$ (2.5)

Proceeding like in [12] and using (2.3)–(2.5), we deduce

$\displaystyle\frac{h(N+1)}{h(N)}=q\frac{f(N+1)}{f(N)},\qquad\frac{k(N-1)}{k(N)}={q}\frac{g(N-1)}{g(N)}$
(2.6)

as well as the expressions

$\displaystyle H(N)=f(N)k(N+1)+qh(N)g(N+1),$ (2.7) $\displaystyle
G(N)=g(N)h(N-1)+qk(N)f(N-1).$ (2.8)

The DSF $\Phi(N)$, see, e.g., [33], determines both the bilinears (cf. (2.5))

$\displaystyle a^{+}a^{-}=\Phi(N),\qquad a^{-}a^{+}=\Phi(N+1),$

and the commutation relation

$\displaystyle[a^{-},a^{+}]=\Phi(N+1)-\Phi(N).$ (2.9)

It also gives the action formulas in the $\Phi(n)$-deformed analog of Fock
space

$\displaystyle
N|n\rangle=n|n\rangle,\qquad|n\rangle=\frac{(a^{+})^{n}}{\sqrt{\Phi(n)!}}|0\rangle,\qquad
a^{-}|0\rangle=0,$

where $\Phi(n)!\equiv\Phi(n)\Phi(n-1)\cdots\Phi(2)\Phi(1)$ and, in addition,
$\Phi(0)!=1$. In that space we have

$\displaystyle a^{+}|n\rangle=\sqrt{\Phi(n+1)}|n+1\rangle,\qquad
a^{-}|n\rangle=\sqrt{\Phi(n)}|n-1\rangle.$ (2.10)

The DSF is obtainable from the above functions $H(n)$ and $G(n)$ using the
recipe [33]

$\displaystyle{\Phi(n)}=\frac{G(n-1)!}{H(n-1)!}\left(\frac{1}{H(0)}+\sum^{n-1}_{j=1}\frac{H(j-1)!}{G(j)!}\right).$
(2.11)

Here the factorials are defined similarly to $\Phi(n)!$.

###  2.2 Solutions of the relations (2.6)

We need the solutions of (2.6) which then, using (2.7) and (2.8), yield the
corresponding operator functions $G(N)$ and $H(N)$. So let us list some of
them.

_Solution_ A. This is

$\displaystyle f(N)=k(N)=\frac{1}{\sqrt{2}}{q}^{N},\qquad
h(N)=g(N)=\frac{1}{\sqrt{2}}{q}^{2N},$ (2.12)

from which

$\displaystyle H(N)=\frac{1}{2}{q}^{2N+1}\big{(}1+q^{2N+2}\big{)},\qquad
G(N)=\frac{1}{2}{q}^{2N}\big{(}1+{q}^{2N-2}\big{)}=q^{3}H(N-2).$ (2.13)

From (2.11) and (2.13) we obtain the DSF

$\displaystyle{\Phi_{q}^{(1)}(n)}=\frac{2q^{-n}}{(1+q^{2n-2})(1+q^{2n})}\left(1+\frac{q^{n}-q^{-n+1}}{q-1}\right)=\frac{2q^{-n}[n]_{q}(1+q^{-n+1})}{(1+q^{2n-2})(1+q^{2n})},$
(2.14)

where $[n]_{q}\equiv(1-q^{n})/(1-q)$. The obtained DSF (2.14) implies that now
we have, besides the relation (2.3), also the alternative form (2.9) of the
commutation relation.

In addition, putting (2.12) into (2.4), for the operators $X$ and $P$ we
obtain

$\displaystyle
X=\frac{1}{\sqrt{2}}\bigl{(}q^{2N}a^{+}+q^{N}a^{-}\bigr{)},\qquad P=\frac{\rm
i}{\sqrt{2}}\bigl{(}q^{N}a^{+}-q^{2N}a^{-}\bigr{)}.$ (2.15)

Note that this DSF (2.14) coincides with the deformation structure function
formerly found in [16] (see also [12]). However, new solutions are possible,
see the cases ${\bf B}$–${\bf D}$.

_Solution_ B. This has the form

$\displaystyle f(N)=k(N)=\frac{1}{\sqrt{2}}{q}^{-2N},\qquad
h(N)=g(N)=\frac{1}{\sqrt{2}}{q}^{-N},$ (2.16)

which then yields

$\displaystyle H(N)=\frac{1}{2}{q}^{-2N}\big{(}1+q^{-2N-2}\big{)},\qquad
G(N)=\frac{1}{2}{q}^{-2N+1}\big{(}1+{q}^{-2N+2}\big{)}=q^{-3}H(N-2).\\!\\!\\!$
(2.17)

Then from (2.11) and (2.17) we obtain the DSF (which casts (2.3) into the form
(2.9))

$\displaystyle{\Phi_{q}^{(2)}(n)}=\frac{2q^{5n-3}[n]_{q}(1+q^{-n+1})}{(1+q^{2n-2})(1+q^{2n})}=q^{3(2n-1)}{\Phi_{q}^{(1)}(n)}.$
(2.18)

Moreover, putting (2.16) into (2.4), for the operators $X$ and $P$ we obtain

$\displaystyle
X=\frac{1}{\sqrt{2}}\big{(}q^{-N}a^{+}+q^{-2N}a^{-}\big{)},\qquad P=\frac{\rm
i}{\sqrt{2}}\big{(}q^{-2N}a^{+}-q^{-N}a^{-}\big{)}.$ (2.19)

_Solution_ C. In this case we find

$\displaystyle f(N)=\frac{1}{\sqrt{2}}{q}^{-2N},\qquad
k(N)=\frac{1}{\sqrt{2}}{q}^{N},\qquad g(N)=\frac{1}{\sqrt{2}}{q}^{2N},\qquad
h(N)=\frac{1}{\sqrt{2}}{q}^{-N},$ (2.20)

from which we infer

$\displaystyle H(N)=\frac{1}{2}{q}^{-N+1}\big{(}1+q^{2N+2}\big{)},\qquad
G(N)=\frac{1}{2}{q}^{N+1}\big{(}1+{q}^{-2N+2}\big{)}=H(N-2).$ (2.21)

From (2.11) and (2.21) we obtain the DSF (which casts (2.3) into the form
(2.9))

$\displaystyle{\Phi_{q}^{(3)}(n)}=\frac{2q^{2n-3}[n]_{q}(1+q^{-n+1})}{(1+q^{2n-2})(1+q^{2n})}=q^{3(n-1)}{\Phi_{q}^{(1)}(n)}.$
(2.22)

In addition, putting (2.20) into (2.4), for the operators $X$ and $P$ we find

$\displaystyle
X=\frac{1}{\sqrt{2}}\big{(}q^{2N}a^{+}+q^{-2N}a^{-}\big{)},\qquad P=\frac{\rm
i}{\sqrt{2}}\big{(}q^{N}a^{+}-q^{-N}a^{-}\big{)}.$ (2.23)

_Solution_ D. That reads

$\displaystyle f(N)=\frac{1}{\sqrt{2}}{q}^{N},\qquad
k(N)=\frac{1}{\sqrt{2}}{q}^{-2N},\qquad g(N)=\frac{1}{\sqrt{2}}{q}^{-N},\qquad
h(N)=\frac{1}{\sqrt{2}}{q}^{2N},$ (2.24)

so that

$\displaystyle H(N)=\frac{1}{2}{q}^{N}\big{(}1+q^{-2N-2}\big{)},\qquad
G(N)=\frac{1}{2}{q}^{-N}\big{(}1+{q}^{2N-2}\big{)}=H(N-2).$ (2.25)

From (2.11) and (2.25) we obtain the DSF (which casts (2.3) into the form
(2.9))

$\displaystyle{\Phi_{q}^{(4)}(n)}=\frac{2q^{2n}[n]_{q}(1+q^{-n+1})}{(1+q^{2n-2})(1+q^{2n})}=q^{3n}{\Phi_{q}^{(1)}(n)}=q^{3}{\Phi_{q}^{(3)}(n)}.$
(2.26)

At last, putting (2.24) into (2.4), for the operators $X$ and $P$ we obtain

$\displaystyle X=\frac{1}{\sqrt{2}}\big{(}q^{-N}a^{+}+q^{N}a^{-}\big{)},\qquad
P=\frac{\rm i}{\sqrt{2}}\big{(}q^{-2N}a^{+}-q^{2N}a^{-}\big{)}.$ (2.27)

######  Remark 2.1.

Each of the solutions A–D in the limit $q\to 1$ yields $f=g=h=k=1/{\sqrt{2}}$,
$G=H=1$. Then we recover the structure function $\Phi(n)=n$ of the usual
oscillator, along with known relations $X=(a^{+}+a^{-})/{\sqrt{2}}$, $P={\rm
i}(a^{+}-a^{-})/{\sqrt{2}}$.

######  Remark 2.2 (concerning pseudo-Hermiticity).

From any of the relations (2.15), (2.19), (2.23), (2.27) in can be deduced
that the position operator and the momentum operator cannot be the both
Hermitian simultaneously. Say, using the relation identical to (2.15) it has
been shown in [17] that (i) if $P$ is chosen to be Hermitian, then $X$ turns
out to be $\eta_{X}(N)$-pseudo-Hermitian; (ii) if $X$ is fixed to be
Hermitian, then $P$ is $\eta_{P}(N)$-pseudo-Hermitian; (iii) in general, the
both of $X$, $P$ are non-Hermitian and, moreover, $X$ is $\eta_{X}(N)$-pseudo-
Hermitian and $P$ is $\eta_{P}(N)$-pseudo-Hermitian. For more details see
[17].

Now let us examine the behavior of all the structure functions as reflected in
the energy eigenspectrum. With the Hamiltonian and the energy eigenspectrum
given as

$\displaystyle{\cal
H}=\frac{1}{2}(aa^{+}+a^{+}a)=\frac{1}{2}\bigl{(}\Phi_{q}(N+1)+\Phi_{q}(N)\bigr{)},\qquad
E_{q}(n)=\frac{1}{2}\bigl{(}\Phi_{q}(n+1)+\Phi_{q}(n)\bigr{)},$

we obtain the energies $E(n)$ for the respective DSFs. In Fig. 1, the
structure functions $\Phi_{q}^{(1)}(n)$, $\Phi_{q}^{(2)}(n)$ and
$\Phi_{q}^{(3)}(n)$ from (2.15), (2.19) and (2.23) are shown for the
particular value $q=1.015$ of the deformation parameter. Note that the 4th DSF
$\Phi_{q}^{(4)}(n)$, due to the simple relation with $\Phi_{q}^{(3)}(n)$, see
(2.26), and the chosen $q=1.015$, looks almost the same as $\Phi_{q}^{(3)}(n)$
and so is not shown in Fig. 1.

Figure 1: Deformation structure functions versus $n$ at fixed value $q=1.015$:
here SF1, SF2 and SF3 denote respectively $\Phi_{q}^{(1)}(n)$,
$\Phi_{q}^{(2)}(n)$ and $\Phi_{q}^{(3)}(n)$ from (2.15), (2.19) and (2.23).

######  Remark 2.3 (ground state energies).

It is of interest to examine the ground state (at $n=0$) energy eigenvalues
for each of the cases A–D. As seen from (2.14), (2.18), (2.22), (2.26), each
of the DSFs $\Phi_{q}^{(1)}(n)$, $\ldots$ , $\Phi_{q}^{(4)}(n)$ give zero at
$n=0$, due to the common factor $[n]_{q}$ such that $[0]_{q}=0$. Therefore the
ground state energy is given by (one half of) the corresponding DSF value at
$n=1$. This way we obtain the following data for $E_{q}^{(j)}(0)$

$\displaystyle
E_{q}^{(1)}(0)=\frac{1}{2}\Phi_{q}^{(1)}(1)=\frac{q^{-1}}{1+q^{2}},\qquad$
$\displaystyle
E_{q}^{(2)}(0)=\frac{1}{2}\Phi_{q}^{(2)}(1)=\frac{q^{2}}{1+q^{2}},$
$\displaystyle
E_{q}^{(3)}(0)=\frac{1}{2}\Phi_{q}^{(3)}(1)=\frac{q^{-1}}{1+q^{2}},\qquad$
$\displaystyle
E_{q}^{(4)}(0)=\frac{1}{2}\Phi_{q}^{(4)}(1)=\frac{q^{2}}{1+q^{2}}.$

Assuming $q>1$ (like in Fig. 1) we find that, in comparison with the zero-
level energy $E_{q}(0)|_{q=1}=\frac{1}{2}$ of the usual quantum oscillator,
here we have both the increased and the lowered ground state energies of
deformed oscillators

$\displaystyle E_{q}^{(1)}(0)=E_{q}^{(3)}(0)<\frac{1}{2},\qquad{\rm
whereas}\quad E_{q}^{(2)}(0)=E_{q}^{(4)}(0)>\frac{1}{2}.$

On the other hand, if $q<1$ we find

$\displaystyle E_{q}^{(1)}(0)=E_{q}^{(3)}(0)>\frac{1}{2},\qquad{\rm
while}\quad E_{q}^{(2)}(0)=E_{q}^{(4)}(0)<\frac{1}{2}.$

######  Remark 2.4 (concerning accidental degeneracy).

As seen from the behavior of the structure functions pictured in Fig. 1, the
considered deformed oscillators may possess, at certain corresponding values
of the deformation parameter $q$, diverse cases of accidental pairwise energy
level degeneracy (note that such kind of degeneracy in one dimension is
peculiar for certain class of deformed oscillators that was formerly studied
in [21, 22, 24], along with the modified versions in [11, 18, 23]). For
instance, for the deformed oscillator given by the DSF $\Phi_{q}^{(1)}(n)$
from (2.14) there exists certain value $q=q(n)|_{n=2}$ such that the
degeneracy $E_{q}(0)=E_{q}(2)$ is realized. In the case of general $n$ such
value is to be found by solving the equation

$\displaystyle
E_{q}^{(1)}(n)-E_{q}^{(1)}(0)=\frac{1}{2}\big{[}\Phi_{q}^{(1)}(n+1)+\Phi_{q}^{(1)}(n)\big{]}-\frac{q^{-1}}{1+q^{2}}=0$

or, in a more expanded form, the equation

$\displaystyle
q^{4n}\big{(}q^{2}+q^{-2}\big{)}+q^{3n}(q-1)\big{(}q^{2}+q^{-3}\big{)}-q^{2n}\big{(}q^{3}+q^{-3}-2\big{)}$
$\displaystyle\qquad{}+q^{n}\big{(}q-q^{-1}\big{)}-q-q^{-1}-\frac{1-q^{-1}}{1+q^{2}}q^{2n}\big{(}1+q^{2n-2}\big{)}\big{(}1+q^{2n}\big{)}\big{(}1+q^{2n+2}\big{)}=0.$

For instance, if one fixes $n=10$, the solution for
$E_{q}^{(1)}(10)-E_{q}^{(1)}(0)=0$ is $q_{10}=1.0913$. Likewise, if $n=90$ the
degeneracy $E_{q}^{(1)}(90)=E_{q}^{(1)}(0)$ occurs at $q_{90}=1.015148$. Note
that $q_{90}<q_{10}$, i.e., the larger is $n$ the lesser is respective value
$q_{n}$.

In a similar fashion, for each of the DSFs $\Phi^{(j)}(n)$, $j=1,2,3,4$, one
can consider these and many other cases of degeneracies (and find the relevant
values of $q$), e.g., such as $E_{q}(1)=E_{q}(7)$, $E_{q}(2)=E_{q}(5)$, the
neighboring $E_{q}(3)=E_{q}(4)$ and so on.

###  2.3 General approach to the relations (2.6)

Let us take (2.6) in the equivalent form

$\displaystyle\frac{h(n+1)}{f(n+1)}=q\frac{h(n)}{f(n)}.$

The latter, by denoting the ratio as $\frac{h(n)}{f(n)}\equiv d(n)$, implies
$d(n+1)=qd(n)$ from which we infer $d(n)=q^{n}d(0)$. That yields the relation

$\displaystyle h(n)=q^{n}f(n)d(0),$

and similarly (putting $c(0)\equiv{g(0)}/{k(0)}$) we find the relation

$\displaystyle g(n)=q^{n}k(n)c(0).$

With account of the latter two relations, from (2.7) and (2.8) we infer the
desired functions $G(N)$ and $H(N)$:

$\displaystyle G(N)=qf(N-1)k(N)\bigl{(}1+c(0)d(0)q^{2N-2}\bigr{)},$ (2.28)
$\displaystyle H(N)=f(N)k(N+1)\bigl{(}1+c(0)d(0)q^{2N+2}\bigr{)},$ (2.29)

along with

$\displaystyle H(0)=f(0)k(1)\bigl{(}1+c(0)d(0)q^{2}\bigr{)}$

(of course, there is natural simplifying choice $c(0)=d(0)=1$). The
expressions (2.28) and (2.29) can be used for obtaining the corresponding DSF.

By introducing ${\cal R}(N)\equiv f(N-1)k(N)$ we can present (2.28) and (2.29)
in the form

$\displaystyle G(N)=q{\cal R}(N)\bigl{(}1+c(0)d(0)q^{2N-2}\bigr{)},\qquad
H(N)={\cal R}(N+1)\bigl{(}1+c(0)d(0)q^{2N+2}\bigr{)},$

with the condition $G(N)|_{q\to 1}=H(N)|_{q\to 1}=1$, see (2.3) and Remark
2.1. But that means ${\cal R}(N)={\cal R}(N+1)$ if $q=1$, or ${\cal
R}(N)|_{q\to 1}={\rm const}$.

##  3 A $\boldsymbol{q,p}$-deformed HA and $\boldsymbol{q,p}$-oscillators

An extended two-parameter deformation of HA, see [16], obeys the basic
relation

$\displaystyle pXP-qPX={\rm i}\hbar,\qquad p\neq q,\qquad p\neq 1,\qquad q\neq
1.$ (3.1)

Denote $Q\equiv\frac{q}{p}$. In analogy to (2.6), (2.7) and (2.8) we obtain
the formulas

$\displaystyle\frac{\tilde{f}(N+1)}{\tilde{f}(N)}=Q^{-1}\frac{\tilde{h}(N+1)}{\tilde{h}(N)},\qquad\frac{\tilde{g}(N-1)}{\tilde{g}(N)}=Q^{-1}\frac{\tilde{k}(N-1)}{\tilde{k}(N)},$
(3.2)

and the relations

$\displaystyle\tilde{H}(N)=p\tilde{f}(N)\tilde{k}(N+1)+q\tilde{h}(N)\tilde{g}(N+1),$
$\displaystyle\tilde{G}(N)=p\tilde{g}(N)\tilde{h}(N-1)+q\tilde{k}(N)\tilde{f}(N-1).$

Now we seek solutions of (3.2). For each of them, we also give the
corresponding $\tilde{H}(N)$, $\tilde{G}(N)$, the operators $X$, $P$, and the
DSF.

_Solution_ ${\bf\tilde{A}}$. The first solution is

$\displaystyle\tilde{f}(N)=\tilde{k}(N)=\frac{1}{\sqrt{2}}Q^{N},\qquad\tilde{h}(N)=\tilde{g}(N)=\frac{1}{\sqrt{2}}Q^{2N}$

that leads to the result

$\displaystyle\tilde{H}(N)=\frac{1}{2}pQ^{2N+1}\big{(}1+Q^{2N+2}\big{)},\qquad\tilde{G}(N)=\frac{1}{2}pQ^{2N}\big{(}1+Q^{2N-2}\big{)}=Q^{3}\tilde{H}(N-2),$
(3.3)

along with

$\displaystyle X=\frac{1}{\sqrt{2}}\big{[}Q^{2N}a^{+}+Q^{N}a^{-}\big{]},\qquad
P=\frac{\rm i}{\sqrt{2}}\big{[}Q^{N}a^{+}-Q^{2N}a^{-}\big{]}.$

The DSF $\Phi_{q,p}(N)$ (it determines the relation of $a^{\pm}a^{\mp}$ with
$N$ like in (2.9) as well as the action formulas for $a^{+}$, $a^{-}$, see
(2.10)) is inferred from equation (2.11) using the functions $\tilde{H}(n)$
and $\tilde{G}(n)$ in (3.3). The result is

$\displaystyle{\Phi_{q,p}^{(1)}(n)}=\frac{2p^{-1}Q^{-n}}{(1+Q^{2n-2})(1+Q^{2n})}\left(1+\frac{Q^{n}-Q^{-n+1}}{Q-1}\right)$
$\displaystyle\hphantom{{\Phi_{q,p}^{(1)}(n)}}{}=\frac{2q^{-n}p^{5n-3}}{(q^{2n-2}+p^{2n-2})(q^{2n}+p^{2n})}\left(1+\frac{[2n-1]_{q,p}}{(qp)^{n-1}}\right)=\frac{2p^{-1}Q^{-n}[n]_{Q}(1+Q^{1-n})}{(1+Q^{2n-2})(1+Q^{2n})},\\!\\!\\!$
(3.4)

where

$\displaystyle[x]_{q,p}\equiv\frac{p^{x}-q^{x}}{p-q},\qquad[x]_{Q}\equiv\frac{1-Q^{x}}{1-Q}.$

This DSF determines the nonstandard two-parameter deformed oscillator
coinciding with that found in [16]. It is ($q\leftrightarrow p$)-nonsymmetric
and thus obviously differs from the well known $q,p$-oscillator [9, 4] whose
structure function $\Phi_{q,p}(n)=[n]_{q,p}$ is ($q\leftrightarrow
p$)-symmetric.

_Solution_ ${\bf\tilde{B}}$. The next solution is

$\displaystyle\tilde{f}(N)=\tilde{k}(N)=\frac{1}{\sqrt{2}}Q^{-2N},\qquad\tilde{h}(N)=\tilde{g}(N)=\frac{1}{\sqrt{2}}Q^{-N},$

that leads to the result

$\displaystyle\tilde{H}(N)=\frac{1}{2}pQ^{-2N}\bigl{(}1+Q^{-2N-2}\bigr{)},$
$\displaystyle\tilde{G}(N)=\frac{1}{2}pQ^{-2N+1}\bigl{(}1+Q^{-2N+2}\bigr{)}=Q^{-3}\tilde{H}(N-2),$
(3.5)

along with

$\displaystyle
X=\frac{1}{\sqrt{2}}\big{[}Q^{-N}a^{+}+Q^{-2N}a^{-}\big{]},\qquad P=\frac{\rm
i}{\sqrt{2}}\big{[}Q^{-2N}a^{+}-Q^{-N}a^{-}\big{]}.$

The DSF $\Phi_{q,p}(N)$ (which determines the relation of $a^{\pm}a^{\mp}$
with $N$ like in (2.9) as well as the action formulas for $a^{+}$, $a^{-}$,
see (2.10)) is inferred from equation (2.11) using the functions
$\tilde{H}(n)$ and $\tilde{G}(n)$ in (3.5). The result is

$\displaystyle\Phi_{q,p}^{(2)}(n)=\left(\frac{q}{p}\right)^{3(2n-1)}\Phi_{q,p}^{(1)}(n).$
(3.6)

This DSF determines the second, nonstandard ($q\leftrightarrow
p$)-nonsymmetric two-parameter deformed oscillator.

_Solution_ ${\bf\tilde{C}}$. The next solution is

$\displaystyle\tilde{f}(N)=\frac{1}{\sqrt{2}}Q^{-2N},\qquad\tilde{k}(N)=\frac{1}{\sqrt{2}}Q^{N},\qquad\tilde{g}(N)=\frac{1}{\sqrt{2}}Q^{2N},\qquad\tilde{h}(N)=\frac{1}{\sqrt{2}}Q^{-N},$
(3.7)

that leads to the result

$\displaystyle\tilde{H}(N)=\frac{1}{2}pQ^{1-N}\bigl{(}1+Q^{2N+2}\bigr{)},\qquad\tilde{G}(N)=\frac{1}{2}pQ^{N+1}\bigl{(}1+Q^{-2N+2}\bigr{)}=\tilde{H}(N-2),$
(3.8)

along with

$\displaystyle
X=\frac{1}{\sqrt{2}}\bigl{[}Q^{2N}a^{+}+Q^{-2N}a^{-}\bigr{]},\qquad
P=\frac{\rm i}{\sqrt{2}}\bigl{[}Q^{N}a^{+}-Q^{-N}a^{-}\bigr{]}.$

The DSF $\Phi_{p,q}(N)$ (it provides the relation of $a^{\pm}a^{\mp}$ with $N$
like in (2.9) and the action formulas for $a^{+}$, $a^{-}$, see (2.10)) is
inferred from equation (2.11) using the functions $\tilde{H}(n)$ and
$\tilde{G}(n)$ from (3.8). The result is

$\displaystyle\Phi_{q,p}^{(3)}(n)=\left(\frac{q}{p}\right)^{3(n-1)}\Phi_{q,p}^{(1)}(n).$
(3.9)

This DSF determines yet another nonstandard, ($q\leftrightarrow
p$)-nonsymmetric deformed oscillator.

_Solution_ ${\bf\tilde{D}}$. This solution is of the form

$\displaystyle\tilde{f}(N)=\frac{1}{\sqrt{2}}Q^{N},\qquad\tilde{k}(N)=\frac{1}{\sqrt{2}}Q^{-2N},\qquad\tilde{g}(N)=\frac{1}{\sqrt{2}}Q^{-N},\qquad\tilde{h}(N)=\frac{1}{\sqrt{2}}Q^{2N},$

that leads to the result

$\displaystyle\tilde{H}(N)=\frac{1}{2}pQ^{N}\bigl{(}1+Q^{-2N-2}\bigr{)},\qquad\tilde{G}(N)=\frac{1}{2}pQ^{-N}\bigl{(}1+Q^{2N-2}\bigr{)}=\tilde{H}(N-2),$
(3.10) $\displaystyle
X=\frac{1}{\sqrt{2}}\bigl{[}Q^{-N}a^{+}+Q^{N}a^{-}\bigr{]},\qquad P=\frac{\rm
i}{\sqrt{2}}\bigl{[}Q^{-2N}a^{+}-Q^{2N}a^{-}\bigr{]}.$

The DSF $\Phi_{q,p}(N)$ (it determines the relation of $a^{\pm}a^{\mp}$ with
$N$ like in (2.9), and the action formulas for $a^{+}$, $a^{-}$, see (2.10))
is inferred using equation (2.11) with the functions $\tilde{H}(n)$ and
$\tilde{G}(n)$ from (3.10). The result is

$\displaystyle\Phi_{q,p}^{(4)}(n)=\left(\frac{q}{p}\right)^{3n}\Phi_{q,p}^{(1)}(n).$
(3.11)

The latter DSF determines the forth, nonstandard, two-parameter deformed
oscillator obviously nonsymmetric under $q\leftrightarrow p$.

Note that at $p\to 1$ the results obtained here for the $p,q$-deformed HA
(3.1) reduce to those of the preceding section (say, (3.4) reduces to (2.14),
etc.), whereas for the case $p=q\neq 1$ we come to the structure function
$\phi(n)=\frac{n}{q}$, the familiar operators
$X=\frac{1}{\sqrt{2}}(a^{+}+a^{-})$ and $P=\frac{\rm
i}{\sqrt{2}}(a^{+}-a^{-})$ along with $[a^{-},a^{+}]=1/q$ and
$\tilde{H}(N)=H=q$, $\tilde{G}(N)=G=q$. Obviously, we deal again with the
usual harmonic oscillator, but the spacing in its (linear) energy spectrum is
$\frac{1}{q}$-scaled.

######  Remark 3.1.

Using the two-parameter family of ($p\leftrightarrow q$)-symmetric
$p,q$-oscillators from [4, 9] one can infer, see [24], a whole “plethora” of
one-parameter $q$-deformed oscillators. This variety includes such well-known
or “standard” ones as Biedenharn–Macfarlane [6, 31] (if $p=q^{-1}$), Arik–Coon
[3] (if $p=1$), and Tamm–Dancoff [10] (if $p=q$) $q$-oscillators. Now, quite
analogously, by imposing diverse functional dependences $p=\xi(q)$ it is
possible to deduce from each of the new ($p\leftrightarrow q$)-nonsymmetric
$\tilde{\bf A}$–$\tilde{\bf D}$ families found in this section, see (3.4),
(3.6), (3.9) and (3.11), the corresponding alternative “plethoras” of non-
standard $q$-deformed oscillators, of which only relatively simple examples
(got by setting $p=1$) are given as the solutions ${\bf A}$–${\bf D}$ above
(in Section 2.2).

######  Remark 3.2.

The parameters $p$ and $q$ in the defining relation (3.1) of the $q,p$-DHA may
be either real or complex. The issue of which particular (complex) values of
$p$, $q$ are admissible was discussed in [17, Section 4] – clearly that
depends on the adopted rules of (pseudo)Hermiticity of $X$ and $P$. On the
other hand, the deformed oscillators obtained in Section 2.2 and given by the
DSFs (2.14), (2.18), (2.22) and (2.26) admit only real values of the
deformation parameter $q$. However, from each of these DSFs, say
${\Phi_{q}^{(1)}(n)}$ from (2.14), we can construct the related
($q\leftrightarrow q^{-1}$)-symmetric deformed oscillator by combining this
$q$-DSF with its “cousin” ${\Phi_{q^{-1}}^{(1)}(n)}$. That yields the
symmetrized (and factorized) DSF

$\displaystyle{\Phi_{\rm
symm.}^{(1)}(n)}\equiv\frac{1}{2}\bigl{(}{\Phi_{q}^{(1)}(n)}+{\Phi_{q^{-1}}^{(1)}(n)}\bigr{)}=\frac{q^{-n}[n]_{q}(1+q^{-n+1})}{(1+q^{2n-2})(1+q^{2n})}+\frac{q^{n}[n]_{q^{-1}}(1+q^{n-1})}{(1+q^{-2n+2})(1+q^{-2n})}$
$\displaystyle\hphantom{{\Phi_{\rm
symm.}^{(1)}(n)}}{}=\frac{(q^{3n-1}+q^{-3n+1})(q^{\frac{n-1}{2}}+q^{\frac{-n+1}{2}})}{(q^{n}+q^{-n})(q^{n-1}+q^{-n+1})}[[n]]_{q^{1/2}},\qquad[[X]]_{q}\equiv\frac{q^{X}-q^{-X}}{q-q^{-1}},$

which admits (real and) the complex form222For some profits of dealing with
complex deformation parameter(s) see the last paragraph in [11, Section 7]
(with references therein), and also the works [26, 35, 36]. of $q$, namely
$q={\rm e}^{{\rm i}\theta}$, $0\leq\theta<\pi$. The same recipe applies to
$\Phi_{q}^{(j)}(n)$, $j=2,3,4$. It is also clear how to proceed in the case of
two-parameter deformed oscillators of Section 3. Namely, each of those DSFs
yields the corresponding ($q\leftrightarrow p$)-symmetric DSF and its deformed
oscillator, by adding ${\Phi_{q,p}^{(j)}(n)}$ to its “cousin”
${\Phi_{p,q}^{(j)}(n)}$, $j=1,\dots,4$. Then the parameters may be either real
or complex such that $p=\bar{q}=r{\rm e}^{-{\rm i}\theta}$.

At last let us note that the DSF ${\Phi_{q}^{(j)}(n)}$ and its “cousin” DSF
${\Phi_{q^{-1}}^{(j)}(n)}$ are inferred from different copies of DHA.
Likewise, ${\Phi_{q,p}^{(j)}(n)}$ and its “cousin” ${\Phi_{p,q}^{(j)}(n)}$ are
linked with the (differing) $q,p$-DHA and $p,q$-DHA respectively.

##  4 Discussion and outlook

Our main results are contained in Sections 2, 3 and give solutions of the
mapping DHA to DOA problem which allow to present the $q,p$-deformed (“left-
sided”) HA in terms of respective non-standard deformed oscillators determined
through their respective structure functions $\Phi^{j}_{q}(n)$ and
$\Phi^{j}_{q,p}(n)$ where $j=1,2,3,4$. Note that the aspects concerning
(pseudo-Hermitian) mutual conjugation of $a^{-}$ and $a^{+}$, as well as
special non-Hermiticity (i.e., $\eta(N)$-pseudo-Hermiticity) of $X$, $P$ can
be examined by a detailed analysis, in analogy to what was done in [17].

Remembering that in [17] we used as starting point not only $q$\- and
$p,q$-deformed Heisenberg algebras but also the two-sided $p,q,\mu$-deformed
DHA, it would be useful to undertake more detailed and complete (than in [17])
study of the problem of finding solutions which map the $p,q,\mu$-deformed DHA
onto certain $p,q,\mu$-deformed oscillators.

Both the BM-type, AC-type $q$-oscillators and the $p,q$-oscillators [3, 6, 9,
10, 31], along with more exotic deformed oscillators [18, 24, 28], are used to
construct the respective one- and two-parameter deformed analogs of Bose gas
model (see [1, 2, 19, 25, 26] and references therein) which find interesting
applications including phenomenological ones [15, 20, 30]. So it is of
interest to develop, starting from the $q$\- or $p,q$-deformed oscillators
explored in this paper, the corresponding new deformed (certainly non-
standard) models with either thermodynamics or statistical distributions and
correlations in the focus. Also, there is an interesting issue of modified
versions of the Heisenberg uncertainty relation (with its expected physical
implications) for the above studied deformations of HA. All these topics are
worth of detailed future studies.

### Acknowledgements

This work was partially supported by the Special Programme of Division of
Physics and Astronomy of NAS of Ukraine. The authors also thank the anonymous
referees for valuable remarks that have led to the improved version of the
manuscript.

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††thanks: These authors contributed equally.††thanks: These authors
contributed equally.††thanks: These authors contributed equally.††thanks:
These authors contributed equally.

# Numerical modeling of the multi-stage Stern–Gerlach experiment by Frisch and
Segrè using co-quantum dynamics via the Bloch equation

Kelvin Titimbo  Caltech 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  David C. Garrett  Caltech 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  S. Süleyman Kahraman  Caltech
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  Zhe
He  Caltech 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  Lihong V. Wang<EMAIL_ADDRESS>Caltech 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

###### Abstract

We numerically study the spin flip in the Frisch–Segrè experiment, the first
multi-stage Stern–Gerlach experiment, within the context of the novel co-
quantum dynamics theory. We model the middle stage responsible for spin
rotation by sampling the atoms with the Monte Carlo method and solving the
dynamics of the electron and nuclear magnetic moments numerically according to
the Bloch equation. Our results show that, without using any fitting
parameters, the co-quantum dynamics closely reproduces the experimental
observation reported by Frisch and Segrè in 1933, which has so far lacked
theoretical predictions.

spin-flip transition, electron spin, Bloch equation, co-quantum dynamics.

##  I Introduction

The Stern–Gerlach experiment was a crucial benchmark for the early development
of quantum mechanics [1, 2] and is still presented in introductory books as
evidence of quantization and the existence of the electron spin angular
momentum [3, 4, 5, 6, 7]. The so-called Stern–Gerlach apparatus (SG) has been
used to illustrate the projection of the quantum wave function onto its
eigenstates along the quantization axis, which is given by the direction of a
strong inhomogeneous magnetic field. Despite quantum mechanics being
fundamental for explaining physical phenomena in almost any branch of physics,
there is no generally accepted theory for how the wave function collapses [8,
9, 10, 11].

Recently, a novel theory denoted co-quantum dynamics (CQD) has been proposed
to describe the evolution and collapse of electron spins in alkali atoms
interacting with an external magnetic field, $\mathbf{B}$ [12, 13]. In CQD,
the electron magnetic moment $\mathbf{\bm{\mu}}_{e}$ is termed the principal
quantum, whereas the nuclear magnetic moment $\mathbf{\bm{\mu}}_{n}$ is termed
the co-quantum. The evolution of both quanta is modeled by the Bloch equation,
and the collapse of the principal quantum is treated by adding an induction
term. In addition to $\mathbf{B}$, the magnetic field $\mathbf{B}_{n}$
generated by the nuclear magnetic moment also acts upon
$\mathbf{\bm{\mu}}_{e}$; similarly, the magnetic field $\mathbf{B}_{e}$
generated by the electron magnetic moment acts upon $\mathbf{\bm{\mu}}_{n}$.

We apply CQD to model the multi-stage Stern–Gerlach experiment conducted by R.
Frisch and E. Segrè [14, 15]. Their apparatus is composed of two Stern–Gerlach
stages and an intervening stage referred to as the inner rotation chamber. The
middle stage is characterized by the presence of a weak but rapidly varying
magnetic field that rotates the electron magnetic moment. Despite attempts by
E. Majorana [16] and I.I. Rabi [17, 18, 19], theoretical descriptions of this
experiment deviate from the Frisch–Segrè experimental observation [12, 20]. To
date, this divergence remains unresolved by existing theories [20, 21, 22].
However, a closed-form approximation of CQD matches the experimental data with
a high coefficient of determination without the use of fitting parameters
[12]. The approximation models the dynamics by means of the Schrödinger
equation coupled with the CQD concept; this approach has been corroborated
numerically [23]. Here, we validate CQD by numerically solving the Bloch
equation instead to estimate the fraction of spin flip in the Frisch–Segrè
experiment and achieve a high coefficient of determination.

The manuscript is organized as follows. The experimental setup used by Frisch
and Segrè is described in Section II. The CQD theory for the experiment is
described in Section III. First, we introduce the equations of motion for the
electron and nuclear magnetic moments. Second, we describe all stages in the
Frisch–Segrè experiment with emphasis on the intermediate stage. We compare
our numerical results with both the experimental observation and the
analytical solution reported previously [12]. In Section IV, we make final
remarks about our findings. In the Appendices, we mathematically derive the
approximated external magnetic field (Appendix A) and summarize the closed-
form analytical formula obtained using CQD (Appendix B).

##  II Experiment

A schematic of the multi-stage Stern–Gerlach apparatus, first constructed by
T.E. Phipps and O. Stern [24] and later improved by Frisch and Segrè [14, 15],
is shown in Fig. 1. The experimental setup combines two magnet pairs, SG1 and
SG2, which each act as a standard Stern–Gerlach apparatus [2, 25, 7]. Between
them, a magnetically shielded inner rotation chamber contains a homogeneous
remnant magnetic field and a cylindrically symmetric magnetic field generated
by a current-carrying wire. The aim of the experiment is to observe spin flips
in ground-state alkali atoms due to non-adiabatic rotations in the middle
stage.

Figure 1: Schematic of the Frisch–Segrè apparatus as in [14, 15]. The atomic
beam from the oven is sent to SG1, whose inhomogeneous magnetic field points
along the $z$-axis. The blue color indicates the magnetic south pole, whereas
the red color indicates the magnetic north pole. Then, the atomic beam enters
the magnetically shielded inner rotation chamber containing a remnant
homogeneous magnetic field plus a magnetic field generated by the electric
current $I_{w}$ flowing through a wire $W$ along the $-x$-direction. A slit
post-selects one of the initially aligned branches of the electron magnetic
moment. Finally, the atoms travel through SG2, and the fraction of spin flip
is measured using a hot wire.

Alkali atoms are appealing for this experiment since their ground state has a
closed shell with only one valence electron having an orbital angular momentum
$\mathbf{L}=0$. As a consequence, the total angular momentum for the electron
of the atom equals the spin $\mathbf{S}$, with $S=\nicefrac{{1}}{{2}}$, and
the spin–orbit coupling vanishes. However, the electron–nuclear spin
interaction, also called the hyperfine interaction, arises from the coupling
between the nuclear magnetic moment $\mathbf{\bm{\mu}}_{n}$ and the magnetic
field generated by the electron magnetic moment $\mathbf{\bm{\mu}}_{e}$, or
equivalently vice versa. The coupling strength depends on the relative
orientations of the magnetic moments, which are related to the spin angular
momenta as

$\displaystyle\mathbf{\bm{\mu}}_{e}$ $\displaystyle=\gamma_{e}\mathbf{S},$
$\displaystyle\mathbf{\bm{\mu}}_{n}$ $\displaystyle=\gamma_{n}\mathbf{I},$ (1)

where $\gamma_{e}$ is the electron gyromagnetic ratio, $\gamma_{n}$ is the
nuclear gyromagnetic ratio, and $\mathbf{I}$ is the nuclear spin.

Figure 2: Illustration of three representative electron magnetic moments of
atoms traversing the Frisch–Segrè apparatus. At the oven, the electron
magnetic moments are randomly oriented, SG1 aligns them to $\uparrow$ and
$\downarrow$ with respect to the orientation of the magnetic field. Both
branches enter the inner rotation chamber, where the varying magnetic field
rotates the electron magnetic moments. The upper branch is selected using a
slit, and the atomic beam is sent to SG2 where the electron magnetic moments
are again aligned. Finally, both branches are measured to quantify the
fraction of spin flip.

A beam of alkali atoms is easily generated since alkali metals have sufficient
vapor pressure at temperatures of only a few hundred degrees Celsius [26, 27].
Let us consider the beam of atoms emerging from an oven and propagating along
the $y$ direction as shown in Fig. 1. The atomic beam first enters SG1, the
strong magnetic field ($\mathbf{B}_{0}\sim$0.5\text{\,}\mathrm{T}\text{/}$$)
along the $z$-axis defines the quantization axis. The initially isotropically
oriented electron magnetic moments are quickly aligned parallel $\uparrow$ or
antiparallel $\downarrow$ to the magnetic field direction. The gradient of the
magnetic field deflects the atoms into two branches. Despite no branch being
physically selected before the inner rotation chamber, in the following
description, we track only the branch with magnetic moments parallel to the
field [15]. Figure 2 depicts the general behavior of electron magnetic moments
through the experimental setup.

Next, the split beam is sent into an inner rotation chamber, which has an
innermost diameter $d$. Despite the shielding from the fringe fields of SG1
and SG2, in the innermost chamber a weak remnant magnetic field
$\mathbf{B}_{r}$ persists, predominantly aligned along the $z$-axis. A wire
with electric current $I_{w}$, flowing along the $-x$ direction, is placed at
a vertical distance $z_{a}$ below the selected atomic beam path. Figure 3a
shows the main elements and the coordinate system inside the chamber. For an
infinitely long straight wire, the magnetic field generated by the current,
$\forall\,y,z$ in the chamber, is given by

$\mathbf{B}_{w}=\frac{\mu_{0}\,I_{w}}{2\pi\left[(z+z_{a})^{2}+y^{2}\right]}\left[(z+z_{a})\,\hat{\mathbf{y}}-y\,\hat{\mathbf{z}}\right]\
,$ (2)

where $\mu_{0}$ denotes the vacuum permeability. Therefore, the total field in
the innermost shielded region is $\mathbf{B}=\mathbf{B}_{w}+\mathbf{B}_{r}$.
As illustrated in Fig. 3b, the magnetic field cancels at the null point (NP),
at position
$\mathbf{r}_{\text{\tiny{NP}}}=y_{\text{\tiny{NP}}}\hat{\mathbf{y}}-z_{a}\hat{\mathbf{z}}$,
with

$y_{\text{\tiny{NP}}}=\frac{\mu_{0}\,I_{w}}{2\pi B_{r}}\ .$ (3)

(a)

(b)

Figure 3: Magnetic field inside the inner rotation chamber. The beam
propagates along the $y$-axis at constant speed $v$. The top panel shows the
Cartesian reference frame, the position of the current-carrying wire (not
drawn to scale), and the contributions from the remnant field $\mathbf{B}_{r}$
and the wire field $\mathbf{B}_{w}$ to the total magnetic field at position
$\mathbf{r}$. The bottom panel shows the field lines of the total magnetic
field with a null point (NP) below the beam path. In this example from the
Frisch–Segrè experiment [14], ${B}_{r}=$42\text{\,}\mathrm{\SIUnitSymbolMicro
T}\text{/}$$, $I_{w}=$0.02\text{\,}\mathrm{A}\text{/}$$, and
$z_{a}=$105\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$$.

In the region near the NP, a first-order Taylor expansion around the point
$(y_{\text{\tiny{NP}}},-z_{a})$ approximates the magnetic field as a
quadrupole (see Appendix A for details):

$\mathbf{B}\xrightarrow[\text{vicinity}]{\text{\tiny{NP}}}\mathbf{B}_{q}=\dfrac{2\pi
B_{r}^{2}}{\mu_{0}\,I_{w}}\biggl{[}(z+z_{a})\hat{\mathbf{y}}+(y-y_{\text{\tiny{NP}}})\hat{\mathbf{z}}\biggr{]}.$
(4)

After the atom interacts with the field inside the inner rotation chamber, the
upper branch is selected using a slit while the lower branch is blocked, as
illustrated in Fig. 2. Frisch and Segrè claimed that placing the slit
immediately after instead of before the inner rotation chamber improved the
beam quality [14].

Finally, the selected atomic branch enters SG2. This second Stern–Gerlach
stage again aligns the electron magnetic moment of each atom in parallel
$\uparrow$ and antiparallel $\downarrow$ directions with respect to the strong
magnetic field $\mathbf{B}_{0}$ along the $+z$-axis. A hot wire records the
final distribution of the atoms, which is used to quantify the fraction of
spin flip of the electron magnetic moment resulting from the inner rotation
chamber. The experiment is repeated with varying wire currents $I_{w}$,
producing a fraction of spin flip as a function of the current.

##  III Theory

Here, we use CQD [13, 12] to describe the multi-stage Stern–Gerlach experiment
by Frisch and Segrè. The formalism provides a mechanism of collapse of the
electron spin in atoms interacting with an external magnetic field based on
the combined dynamics of the electron and nuclear magnetic moments (1). The
Bloch equation governs the evolution of the magnetic moments in the inner
rotation chamber.

###  III.1 Magnetic moment evolution

The evolution of the magnetic moments in an external magnetic field are
described by the Bloch equation [28]:

$\derivative{\hat{\mathbf{\bm{\mu}}}}{t}=\gamma\,\hat{\mathbf{\bm{\mu}}}\crossproduct\mathbf{B}\,,$
(5)

where $\hat{\mathbf{\bm{\mu}}}$ denotes the unit vector of the magnetic
moment, $\gamma$ is the gyromagnetic ratio, and $\mathbf{B}$ is the magnetic
flux density. This equation governs an undamped precession of the magnetic
moment about the magnetic field $\mathbf{B}$. While usually considered a
classical formalism, the Bloch equation has been recently shown to yield the
space-independent von Neumann equation [13].

For the principal quantum $\mathbf{\bm{\mu}}_{e}$ and the co-quantum
$\mathbf{\bm{\mu}}_{n}$, the Bloch equation (5) becomes

$\displaystyle\derivative{\hat{\mathbf{\bm{\mu}}}_{e}}{t}$
$\displaystyle=\gamma_{e}\,\hat{\mathbf{\bm{\mu}}}_{e}\crossproduct(\mathbf{B}+\mathbf{B}_{n})\,,$
(6) $\displaystyle\derivative{\hat{\mathbf{\bm{\mu}}}_{n}}{t}$
$\displaystyle=\gamma_{n}\,\hat{{\bf\it\bm{\mu}}}_{n}\crossproduct(\mathbf{B}+\mathbf{B}_{e})\,,$
(7)

where $\gamma_{e}$ and $\gamma_{n}$ denote the gyromagnetic ratios of the
electron and the nucleus, respectively, and $\mathbf{B}$ the external magnetic
field. CQD introduces the
$\mathbf{\bm{\mu}}_{e}\text{\textendash}\mathbf{\bm{\mu}}_{n}$ interaction via
each other’s torque-averaged magnetic field, given by [12]

$\displaystyle\mathbf{B}_{e}$ $\displaystyle=\dfrac{5\mu_{0}\,\mu_{e}}{16\pi
R^{3}}\hat{\mathbf{\bm{\mu}}}_{e}=B_{e}\,\hat{\mathbf{\bm{\mu}}}_{e}\,,$ (8)
$\displaystyle\mathbf{B}_{n}$ $\displaystyle=\dfrac{5\mu_{0}\,\mu_{n}}{16\pi
R^{3}}\hat{\mathbf{\bm{\mu}}}_{n}=B_{n}\,\hat{\mathbf{\bm{\mu}}}_{n}\,,$ (9)

where $R$ is the van der Waals radius of the atom, and $\mu_{e}$ and $\mu_{n}$
are the magnitudes of the magnetic moments for the electron and the nucleus,
respectively.

The orientations of the magnetic moments in $\mathbb{R}^{3}$ are conveniently
described using spherical coordinates. With the polar and azimuthal angles,
$\theta_{e}$ and $\phi_{e}$ for $\mathbf{\bm{\mu}}_{e}$ and $\theta_{n}$ and
$\phi_{n}$ for $\mathbf{\bm{\mu}}_{n}$, we write the unit vectors as

$\hat{\mathbf{\bm{\mu}}}_{e}=\pmqty{undef},$ (10)
$\hat{\mathbf{\bm{\mu}}}_{n}=\pmqty{undef}.$ (11)

Meanwhile, the external field is written as

$\mathbf{B}=\pmqty{undef}.$ (12)

Finally, substituting (8)–(12) into (6) and (7), we obtain a set of
differential equations determining the evolution of the unit vectors of
$\mathbf{\bm{\mu}}_{e}$ and $\mathbf{\bm{\mu}}_{n}$:

$\displaystyle\dot{\theta}_{e}$
$\displaystyle=-\gamma_{e}\left[B_{y}\cos(\phi_{e})-B_{x}\sin(\phi_{e})+B_{n}\sin(\theta_{n})\sin(\phi_{n}-\phi_{e})\right]\
,$ (13a) $\displaystyle\dot{\phi}_{e}$
$\displaystyle=-\gamma_{e}\left[B_{z}+B_{n}\cos(\theta_{n})-\cot(\theta_{e})\left[B_{x}\cos(\phi_{e})+B_{y}\sin(\phi_{e})+B_{n}\sin(\theta_{n})\cos(\phi_{e}-\phi_{n})\right]\right]\
,$ (13b) $\displaystyle\dot{\theta}_{n}$
$\displaystyle=-\gamma_{n}\left[B_{y}\cos(\phi_{n})-B_{x}\sin(\phi_{n})+B_{e}\sin(\theta_{e})\sin(\phi_{e}-\phi_{n})\right]\
,$ (13c) $\displaystyle\dot{\phi}_{n}$
$\displaystyle=-\gamma_{n}\left[B_{z}+B_{e}\cos(\theta_{e})-\cot(\theta_{n})\left[B_{x}\cos(\phi_{n})+B_{y}\sin(\phi_{n})+B_{e}\sin(\theta_{e})\cos(\phi_{e}-\phi_{n})\right]\right]\
.$ (13d)

Given the dynamical equations for the electron and nuclear magnetic moments of
a single atom, we model their evolution in the inner rotation chamber of the
Frisch–Segrè experiment.

###  III.2 Multi-stage Stern–Gerlach experiment by Frisch and Segrè

In the Frisch–Segrè experiment, neutral potassium-39 atoms
(${}^{39}\mathrm{K}$) with an electron configuration of
$[\mathrm{Ar}]\>4\mathrm{s}^{1}$ were used. The valence electron in its ground
state is fully specified in the Russell-Saunders notation as
$4\,^{2}\mathrm{S}_{\nicefrac{{1}}{{2}}}$. Table 1 lists the values of the van
der Waals radius, spins, magnetic moments, and gyromagnetic ratios for the
${}^{39}\mathrm{K}$ atom. Accordingly, we compute the magnitudes of the
magnetic fields generated by the electron and nuclear magnetic moments from
(8) and (9):

$\displaystyle B_{e}$
$\displaystyle=$55.806\,267\,22(2)\text{\,}\mathrm{mT}\text{/}$,$ (14a)
$\displaystyle B_{n}$
$\displaystyle=$11.8842(3)\text{\,}\mathrm{\SIUnitSymbolMicro T}\text{/}$\ .$
(14b)

Table 1:  Values for potassium-39 [29, 30, 31, 32, 33]. ${}^{39}\mathrm{K}$
---
|  | Property | Value
Atom | $R$ |  $275\text{\,}\mathrm{pm}\text{/}$
Valence |  electron | $S$ | $\phantom{\Bigl{[}}\frac{1}{2}\phantom{\Bigl{]}}$
$\mu_{e}$ |  $9.284\,767\,704\,3(28)\text{\times}{10}^{-24}\text{\,}\mathrm{J}\text{/}\mathrm{T}$
$\gamma_{e}$ |  $-1.760\,859\,630\,23(53)\text{\times}{10}^{11}\text{\,}\mathrm{rad}\text{/}\text{(}\mathrm{s}\text{\,}\mathrm{T}\text{)}$
Nucleus | $I$ | $\phantom{\Bigl{[}}\frac{3}{2}\phantom{\Bigl{]}}$
$\mu_{n}$ |  $1.977\,23(4)\text{\times}{10}^{-27}\text{\,}\mathrm{J}\text{/}\mathrm{T}$
$\gamma_{n}$ |  $1.250\,061\,2(3)\text{\times}{10}^{7}\text{\,}\mathrm{rad}\text{/}\text{(}\mathrm{s}\text{\,}\mathrm{T}\text{)}$

####  III.2.1 Oven

First, heating ${}^{39}\mathrm{K}$ atoms in an oven with a small opening
produces an effusive beam of non-interacting thermal atoms. We define the
$y$-axis along the beam as shown in Fig. 1.

The initial orientations of both the electron and nuclear magnetic moments of
a single traveling atom are isotropically distributed. Therefore, their
angular probability density functions are respectively

$\displaystyle
p_{e,\mathrm{oven}}(\theta_{e,\mathrm{oven}},\phi_{e,\mathrm{oven}})$
$\displaystyle=\frac{1}{4\pi},$ $\displaystyle
p_{n,\mathrm{oven}}(\theta_{n,\mathrm{oven}},\phi_{n,\mathrm{oven}})$
$\displaystyle=\frac{1}{4\pi}\ .$

To implement a Monte Carlo simulation, the polar and azimuthal angles are
numerically generated from independent realizations of

$\displaystyle\theta_{e,\mathrm{oven}}$
$\displaystyle=2\arcsin(\sqrt{\zeta_{1}}),$
$\displaystyle\phi_{e,\mathrm{oven}}$ $\displaystyle=2\pi\zeta_{2},$ (15)
$\displaystyle\theta_{n,\mathrm{oven}}$
$\displaystyle=2\arcsin(\sqrt{\zeta_{3}}),$
$\displaystyle\phi_{n,\mathrm{oven}}$ $\displaystyle=2\pi\zeta_{4},$ (16)

with $\zeta_{j}$, $j=1,\dots,4$, being random numbers uniformly sampled from
zero to one.

####  III.2.2 Stern–Gerlach apparatus 1

Emerging from the oven, the atoms enter a Stern–Gerlach apparatus consisting
of an inhomogeneous magnetic field with a strong gradient along the $z$-axis,
as illustrated in Fig. 1. Usually, the amplitude of the magnetic field is
$B_{0}\sim$0.5\text{\,}\mathrm{T}\text{/}$$; thus, $B_{0}>B_{e}\gg B_{n}$. For
such a strong field, $\mathbf{\bm{\mu}}_{e}$ and $\mathbf{\bm{\mu}}_{n}$
couple more strongly to the external magnetic field than to each other, and
can be considered as independently precessing with Larmor frequencies
$\omega_{e}$ and $\omega_{n}$, for electron and nuclear magnetic moments,
respectively, about the external field direction. From the values in Table 1,
the ratio between the nuclear and electron Larmor frequencies
$\nicefrac{{\omega_{n}}}{{\omega_{e}}}\approx$7\text{\times}{10}^{-5}$$. In
consequence, while the flight time guarantees the collapse of the electron
magnetic moment, it is too short for the nuclear magnetic moment to collapse
[12].

Therefore, the electron magnetic moments of the single atoms, initially
isotropically distributed, split into two branches with well-defined opposite
alignments along the $z$-axis. We characterize the orientations of the aligned
$\mathbf{\bm{\mu}}_{e}$ by setting $\theta_{e,0}=0$ and $\theta_{e,0}=\pi$ in
(10), respectively, as illustrated in Fig. 2. Meanwhile, since
$\mathbf{\bm{\mu}}_{n}$ does not collapse, CQD assumes that $\theta_{n}$ does
not vary significantly during the flight. CQD shows that at the end of the SG1
stage, the $\mathbf{\bm{\mu}}_{n}$ orientation in each of the branches is
redistributed into an anisotropic probability density function [12]:

$p_{n}(\theta_{n,0},\phi_{n,0})=\begin{cases}\frac{1}{4\pi}\left(1-\cos(\theta_{n,0})\right)&\text{if
}\theta_{e,0}=0,\\\ \frac{1}{4\pi}\left(1+\cos(\theta_{n,0})\right)&\text{if
}\theta_{e,0}=\pi.\end{cases}$ (17)

####  III.2.3 Inner rotation chamber

The inner rotation chamber consists of three hollow iron shielding spheres.
Without shielding, the fringe field was
$\sim$0.4\text{\,}\mathrm{T}\text{/}$$; with shielding, the measured remnant
field along the $+z$-axis was reported to be
$B_{r}=$42\text{\,}\mathrm{\SIUnitSymbolMicro T}\text{/}$$ [14]. Inside the
chamber, an electric current $I_{w}$ flowing through a wire produces a
spatially varying magnetic field (2), resulting in the total field shown in
Fig. 3b.

We select the branch of the atomic beam with $\mathbf{\bm{\mu}}_{e}$ aligned
to the $+z$-axis according to the original experiment [15], i.e.,
$\theta_{e,0}=0$, as shown in Fig. 2. Therefore, the probability density
function (17) becomes

$p_{n}(\theta_{n,0},\phi_{n,0})=\frac{1}{4\pi}\left(1-\cos(\theta_{n,0})\right).$
(18)

To continue the Monte Carlo simulation, the polar and azimuthal angles
$(\theta_{n,0},\phi_{n,0})$ are sampled from the above anisotropic probability
density function:

$\displaystyle\theta_{n,0}$ $\displaystyle=2\arcsin(\zeta_{1}^{1/4}),$
$\displaystyle\phi_{n,0}$ $\displaystyle=2\pi\,\zeta_{2},$ (19)

where $\zeta_{1}$ and $\zeta_{2}$ are two independent random numbers sampled
uniformly from zero to one.

Each atom in the beam propagates with velocity
$\mathbf{v}=v\,\hat{\mathbf{y}}$, and we approximate the motion to be
rectilinear and uniform such that its position on the propagation axis is
$y=vt$. The atom is simulated from $-\nicefrac{{d}}{{2}}$ to
$\nicefrac{{d}}{{2}}$ as shown in Fig. 3a. Here,
$v=$800\text{\,}\mathrm{m}\text{/}\mathrm{s}$$ and
$d=$16.3\text{\,}\mathrm{mm}\text{/}$$ [14]. Therefore, the time in the inner
rotation chamber $t\in\left[$-10.2\text{\,}\mathrm{\SIUnitSymbolMicro
s}\text{/}$,$10.2\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$\right]$. The
total magnetic field along the $y$-axis varies according to

$\mathbf{B}=\frac{\mu_{0}I_{w}}{2\pi\left[z_{a}^{2}+(vt)^{2}\right]}\left[z_{a}\,\hat{\mathbf{y}}-vt\,\hat{\mathbf{z}}\right]+B_{r}\,\hat{\mathbf{z}}\
.$ (20)

The wire is positioned along the $x$-axis at a distance
$z_{a}=$105\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$$ below the beam
path. The electric current $I_{w}$ varies from
$0.01\text{\,}\mathrm{A}\text{/}$ to $0.5\text{\,}\mathrm{A}\text{/}$ [14,
15].

It is known that a time-dependent magnetic field can rotate the magnetic
moment of the electron [16, 19, 34, 35]. To determine whether the process is
adiabatic or non-adiabatic, we compute the so-called adiabaticity parameter,
defined as the ratio between the absolute values of the Larmor frequency
$\omega_{e}$ and the rotational speed of the field in the $yz$ plane
$\Omega_{B}$ [16]:

$k=\absolutevalue{\frac{\omega_{e}}{\Omega_{B}}}=\absolutevalue{\dfrac{\gamma_{e}B}{\derivative{t}\\!\left[\arctan\left({B_{z}}/{B_{y}}\right)\right]}}\
.$ (21)

If $\omega_{e}\gg\Omega_{B}$ ($k\gg 1$), $\mathbf{\bm{\mu}}_{e}$ is able to
follow the changing orientation of the magnetic field, and the rotation is
said to be adiabatic. Otherwise, $\mathbf{\bm{\mu}}_{e}$ may be unable to
follow the changing orientation of the magnetic field; if so, the rotation is
said to be non-adiabatic.

From (20), the adiabaticity parameter becomes

$k=\Biggl{|}\dfrac{\gamma_{e}\mu_{0}I_{w}}{2\pi
z_{a}v}\dfrac{\sqrt{(vt)^{2}+z_{a}^{2}}}{\left(1-\dfrac{4\pi
B_{r}vt}{\mu_{0}I_{w}}\right)}\\\ \times\left[1-\dfrac{4\pi
B_{r}vt}{\mu_{0}I_{w}}+\left(\dfrac{2\pi
B_{r}}{\mu_{0}I_{w}}\right)^{2}\left((vt)^{2}+z_{a}^{2}\right)\right]^{3/2}\Biggr{|}.$
(22)

Figure 4 shows the variation of $k$ around the center of the cavity computed
from (22) for a set of representative electric currents $I_{w}$.

We note that in the vicinity of the wire position ($t=0$) and before the atoms
reach the null point at time

$t_{\text{\tiny{NP}}}=\frac{y_{\text{\tiny{NP}}}}{v}=\frac{\mu_{0}\,I_{w}}{2\pi
vB_{r}},$ (23)

the adiabaticity parameter is much greater than unity at the peaks.
Accordingly, as the $z$ component of the magnetic field changes its
orientation (see Fig. 3), the electron magnetic moment follows the varying
$\mathbf{B}$-field adiabatically. Therefore, the electron magnetic moment
flips adiabatically, $\theta_{e,0}:0\mapsto\pi$, in this region. In
comparison, Majorana stated that the flip is due to the orientation reversal
of the quadrupole field along the flight path [16]. For simplicity, we ignore
the contribution from the nucleus (14b) to $k$ because its much weaker field
does not significantly affect the magnitudes of the adiabatic peaks in Fig. 4

Figure 4: Adiabaticity parameter $k$ along the beam path for various electric
currents. Adiabatic rotation of the electron magnetic moment occurs around the
peak ($k\gg 1$), whereas non-adiabatic rotation happens around the trough
(near the null point $t_{\text{\tiny{NP}}}$ of the magnetic field). The
vertical black dashed line indicates the time when the atom is above the wire
($t=0$).

Then, the electron magnetic moment is rotated non-adiabatically around the
corresponding trough in Fig. 4, which is near the null point. We make use of
the quadrupole approximation for the magnetic field to describe this effect
near the null point, $\mathbf{B}_{q}$ in (4), rewritten for the beam path as

$\displaystyle B_{x}$ $\displaystyle=0\,,$ (24a) $\displaystyle B_{y}$
$\displaystyle=\frac{2\pi B_{r}^{2}}{\mu_{0}I_{w}}z_{a}\,,$ (24b)
$\displaystyle B_{z}$ $\displaystyle=\frac{2\pi
B_{r}^{2}}{\mu_{0}I_{w}}v\left(t-t_{\text{\tiny{NP}}}\right)\,.$ (24c)

The resulting system of coupled differential equations is reduced to

$\displaystyle\dot{\theta}_{e}$
$\displaystyle=-\gamma_{e}\bigl{[}B_{y}\cos(\phi_{e})$
$\displaystyle\qquad\qquad+B_{n}\sin(\theta_{n})\sin(\phi_{n}-\phi_{e})\bigr{]}\
,$ (25a) $\displaystyle\dot{\phi}_{e}$
$\displaystyle=-\gamma_{e}\Bigl{[}B_{z}+B_{n}\cos(\theta_{n})-\cot(\theta_{e})\bigl{[}B_{y}\sin(\phi_{e})$
$\displaystyle\qquad\qquad+B_{n}\sin(\theta_{n})\cos(\phi_{e}-\phi_{n})\bigr{]}\Bigr{]}\
,$ (25b) $\displaystyle\dot{\theta}_{n}$ $\displaystyle=0\ ,$ (25c)
$\displaystyle\dot{\phi}_{n}$
$\displaystyle=-\gamma_{n}\Bigl{[}B_{z}+B_{e}\cos(\theta_{e})-\cot(\theta_{n})\bigl{[}B_{y}\sin(\phi_{n})$
$\displaystyle\qquad\qquad+B_{e}\sin(\theta_{e})\cos(\phi_{e}-\phi_{n})\bigr{]}\Bigr{]}\
.$ (25d)

Note that we have imposed the condition $\dot{\theta}_{n}=0$ since the
postulates of the CQD theory require small variations of the polar angle of
the nuclear magnetic moment. This condition corresponds to the selection rule
for nuclear spin transitions $\Delta m_{I}=0$ [36, 12].

To solve the above differential equations, we sample the initial orientation
of $\mathbf{\bm{\mu}}_{n}$ from (19) and sample $\phi_{e,0}$ uniformly, in a
way similar to (15), while setting $\theta_{e,0}=\pi$. The concurrence of two
vastly different time scales,
$\nicefrac{{\omega_{n}}}{{\omega_{e}}}\approx$7\text{\times}{10}^{-5}$$, for
the dynamics of $\hat{\mathbf{\bm{\mu}}}_{e}$ and
$\hat{\mathbf{\bm{\mu}}}_{n}$, leads to stiffness prone to numerical
instability [37, 38]. We chose the Radau methods [39, 40] implemented in Julia
[41, 42] to solve the differential equations (25) numerically. Our source code
is posted online [43]. An alternative approach, to avoid the stiffness,
approximates the nuclear Larmor frequency as a constant and accelerates the
solution using a variable transformation [23]

Figure 5 shows the characteristic dynamics of $\theta_{e}$ inside the inner
rotation chamber for a representative set of currents. To facilitate
comparison, we introduce $\Delta t=t-t_{\text{\tiny{NP}}}$, such that the time
$\Delta t=0$ at the current-dependent null point (23). The dynamics are
characterized by an oscillatory behavior; the polar angle $\theta_{e}$ varies
rapidly in the vicinity of the null point ($\Delta t=0$) due to strong non-
adiabatic rotation. Then, the oscillations are damped due to the increasing
adiabaticity parameter for $\Delta t>0$. However, since $\theta_{e}$ has not
asymptotically converged to a definite value, we numerically approximate its
final value $\theta_{e,f}$ by averaging the oscillations over the last
$2\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$.

Figure 5: Dynamics of $\theta_{e}$ for different electric currents $I_{w}$.
The initial conditions are $\theta_{e,0}=\pi$ and $\phi_{e,0}=0$ for the
electron magnetic moment, whilst $\theta_{n,0}=\frac{5}{8}\pi$ and
$\phi_{n,0}=\frac{11}{10}\pi$ for the nuclear magnetic moment.

####  III.2.4 Stern–Gerlach apparatus 2

After traveling through the inner rotation chamber, the atom enters the second
Stern–Gerlach apparatus as shown in Fig. 1. The strong magnetic field along
the $z$-axis realigns the electron magnetic moment. The final orientation
$\theta_{e,f}\longmapsto\theta_{e,D}$ follows the branching condition
postulated by the CQD theory [12]:

$\theta_{e,D}=\begin{cases}0&\text{if}\quad\theta_{e,f}<\theta_{n,0},\\\
\pi&\text{if}\quad\theta_{e,f}>\theta_{n,0}.\end{cases}$ (26)

Therefore, the measured polar angle $\theta_{e,D}$ takes on either $0$ or
$\pi$.

####  III.2.5 Fraction of spin flip

The spin flip corresponds to those atoms with $\theta_{e,D}=\pi$, as depicted
in Fig. 2. We numerically solved the Bloch equations for $N=$15\,000$$ atoms
for each current $I_{w}$. Thus, the fraction of spin flip, for a given
$I_{w}$, is computed using

$\mathcal{W}_{\mathrm{num}}=\frac{1}{N}\sum_{i=1}^{N}\left[\theta^{(i)}_{e,D}=\pi\right],$
(27)

where $i$ denotes the $i$th-atom sampled and $\left[P\right]$ is the Iverson
bracket, which is defined to take on the value 1 when the statement $P$ is
true and 0 otherwise.

Figure 6 shows the resulting fraction of spin flip obtained from the numerical
simulation in comparison with the experimental results reported by Frisch and
Segrè [14]. In addition, the closed-form analytical prediction from the CQD
theory is included (see Appendix B) [12].

Figure 6: Fraction of spin flip for the multi-stage Stern–Gerlach experiment
by Frisch and Segrè. The red circles are the experimental data [14], the blue
dashed line is the closed-form analytical solution from CQD [12], and the gray
crosses show the numerical solution from CQD.

The coefficient of determination for our numerical simulation is
$R_{\mathrm{num}}^{2}=0.945$, compared with $R^{2}_{\mathrm{ana}}=0.962$ from
the analytical solution [12]. The statistical errors associated with the
numerical results are smaller than the symbol size in Fig. 6. These results
show a close agreement of CQD with the experimental data [14], which has not
yet been achieved using standard quantum mechanical treatments [22].

##  IV Conclusions

We used CQD to numerically model the Frisch–Segrè experiment without using any
fitting parameters. The numerical problem is stiff because of the two vastly
different characteristic time scales [12]. The solver of the differential
equations must be carefully chosen and validated. We described all the stages
of the experiment, with emphasis on the inner rotation chamber. Using the
Monte Carlo method, we sampled the spatial orientations of the electron and
nuclear magnetic moments of the atoms in the atomic beam. Our findings show
that the fraction of spin flip obtained by means of CQD closely reproduces the
reported experimental results, with a coefficient of determination of
$R^{2}_{\mathrm{num}}=0.945$. The obtained results have not been explained by
standard quantum mechanical approaches [22], despite attempts by Majorana [16,
34, 35] and Rabi [17, 18, 19], whose theoretical formulae deviate from the
experimental results.

Hence, this work supports CQD as a model for both the evolution and the
collapse of the electron spin in atoms in the presence of external magnetic
fields. Some other reported experiments involving multi-stage Stern–Gerlach
experiments [44, 45] could be explored using the present formalism.
Notwithstanding, we hope that this work will encourage further experiments to
verify CQD.

###### Acknowledgements.

This project has been made possible in part by grant number 2020-225832 from
the Chan Zuckerberg Initiative DAF, an advised fund of the Silicon Valley
Community Foundation.

## Supplemental Material

Our source code written in Julia is available online [43].

##  Appendix A Approximation to quadrupole field

The total magnetic field inside the inner rotation chamber,
$\mathbf{B}=\mathbf{B}_{w}+\mathbf{B}_{r}$, can be approximated as an ideal
quadrupole field. The Taylor expansion of $\mathbf{B}$ in the $yz$ plane
around the null point $(y_{\text{\tiny{NP}}},-z_{a})$ reads

$\mathbf{B}=\mathbf{B}\Big{|}_{\text{\tiny{NP}}}+\partialderivative{\mathbf{B}}{y}\Big{|}_{\text{\tiny{NP}}}(y-y_{\text{\tiny{NP}}})+\partialderivative{\mathbf{B}}{z}\Big{|}_{\text{\tiny{NP}}}(z+z_{a})\\\
+\dfrac{1}{2}\partialderivative[2]{\mathbf{B}}{y}\Big{|}_{\text{\tiny{NP}}}(y-y_{\text{\tiny{NP}}})+\dfrac{1}{2}\partialderivative[2]{\mathbf{B}}{z}\Big{|}_{\text{\tiny{NP}}}(z+z_{a})\\\
+\partialderivative{\mathbf{B}}{y}{z}\Big{|}_{\text{\tiny{NP}}}(y-y_{\text{\tiny{NP}}})(z+z_{a})+\cdots\,.$
(28)

Up to the first order, for the hyperbolic field approximation, the
coefficients are

$\displaystyle\mathbf{B}\Big{|}_{\text{\tiny{NP}}}$ $\displaystyle=0\,,$ (29a)
$\displaystyle\partialderivative{\mathbf{B}}{y}\Big{|}_{\text{\tiny{NP}}}$
$\displaystyle=\frac{2\pi B_{r}^{2}}{\mu_{0}I_{w}}\,\hat{\mathbf{z}}\,,$ (29b)
$\displaystyle\partialderivative{\mathbf{B}}{z}\Big{|}_{\text{\tiny{NP}}}$
$\displaystyle=\frac{2\pi B_{r}^{2}}{\mu_{0}I_{w}}\,\hat{\mathbf{y}}\,.$ (29c)

Substituting (29) into (28), the quadrupole magnetic field can be written as

$\mathbf{B}_{q}=\frac{2\pi
B_{r}^{2}}{\mu_{0}I_{w}}(z+z_{a})\>\hat{\mathbf{y}}+\frac{2\pi
B_{r}^{2}}{\mu_{0}I_{w}}(y-y_{\text{\tiny{NP}}})\>\hat{\mathbf{z}}.$ (30)

This corresponds to the magnetic field in (4).

##  Appendix B Closed-form solution of the co-quantum dynamics theory

Using the CQD theory, it is possible to derive a closed-form formula for the
probability of spin flip in the inner rotation chamber in the presence of the
quadrupole magnetic field and the nuclear magnetic moment as we described for
the experiment conducted by Frisch and Segrè [12].

In terms of the parameters of the experiment, the fraction of spin flip, shown
as the blue dashed line in Fig. 6, is given by

$\mathcal{W}_{\mathrm{ana}}(I_{w})=\exp\left[-\sqrt{\left(\dfrac{c_{r0}}{I_{w}}\right)^{2}+c_{rs}^{2}}-c_{rr}I_{w}^{3}\right],$
(31)

where

$\displaystyle c_{r0}$
$\displaystyle=\absolutevalue{\gamma_{e}}\dfrac{2\pi^{2}z_{a}^{2}}{\mu_{0}v}\left(B_{r}+B_{n}\cos(\expectationvalue{\theta_{n}})\right)^{2},$
(32a) $\displaystyle c_{rs}$
$\displaystyle=\absolutevalue{\gamma_{e}}\dfrac{\pi
z_{a}}{v}B_{n}\sin(\expectationvalue{\theta_{n}}),$ (32b) $\displaystyle
c_{rr}$ $\displaystyle=\dfrac{\mu_{0}^{3}\gamma_{e}^{2}\gamma_{n}}{32\pi
v^{3}}\dfrac{B_{e}\left(B_{n}\sin(\expectationvalue{\theta_{n}})\right)^{5}}{\left(B_{r}+B_{n}\cos(\expectationvalue{\theta_{n}})\right)^{6}}.$
(32c)

The coefficients represent three physical effects identified in the solution,
i.e., null-point rotation, rotation saturation, and resonant rotation,
respectively. The mean polar angle of the nuclear magnetic moment is computed
from the probability density function (18) as follows:

$\expectationvalue{\theta_{n}}=\int\differential{\theta_{n}}\differential{\phi_{n}}\sin(\theta_{n})p_{n}(\theta_{n},\phi_{n})=\frac{5}{8}\pi.$
(33)

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