Quantum mechanical modeling of the multi-stage Stern–Gerlach experiment conducted by Frisch and Segrè

The multi-stage Stern–Gerlach experiment conducted by Frisch and Segrè includes two cascaded quantum measurements with a nonadiabatic flipper in between. The Frisch and Segrè 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.

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 analytical solution, which is now widely used to analyze any two-level systems [26].Rabi revised Majorana's derivation by adding the hyperfine interaction but still could not predict the experimental observation accurately.Despite additional theoretical studies into similar problems involving multilevel nonadiabatic transitions [26][27][28][29][30][31][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][16][17][18][19][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 generated 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 using a standard quantum mechanical model via the von Neumann equation without tuning any parameters and compare the outcome with the predictions by both Majorana and Rabi.Even though our approach is a standard method of studying such spin systems, our results do not match the experimental observations.This discrepancy indicates that either our understanding of the FS experiment is lacking or the standard theoretical model is insufficient.Recent studies have modeled the FS experiment using an alternative model called co-quantum dynamics [33][34][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 Segrè 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.Numerical 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.Nonstandard variants of the quantum mechanical model are explored in the appendices to stimulate discussion.

II. DESCRIPTION OF THE FRISCH-SEGR È EXPERIMENT
The schematic of the setup used in the Frisch-Segrè experiment [10,11] is redrawn in Figure 1.There, magnetic regions 1 and 2 act as Stern-Gerlach apparatuses, SG1 and SG2, respectively, with strong magnetic fields along the +z direction.In SG1, stable neutral potassium atoms ( 39 K) effused from the oven are spatially separated by the magnetic field gradient according to the orientation of their electron magnetic moment µ e .The magnetically shielded space containing a current-carrying wire forms the inner rotation (IR) chamber.The shielding reduces the fringe fields from the SG magnets down to the remnant field B r = 42 µT aligned with +ẑ.Inside the IR chamber, the current-carrying wire placed at a vertical distance z a = 105 µm below the atomic beam path creates 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 I w flowing 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 constant along the y axis.The rectilinear approximation of atomic motion within the IR chamber is acceptable since the total displacement due to the field gradients is negligible, approximately 1 µm.Along the beam path, the magnetic field is given by where µ 0 is the vacuum permeability; the trajectory of the atom is expressed as y = vt, where v is 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 {e x , e y , e z } 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, y NP , −z a ), with y NP = µ0Iw /2πBr.In the vicinity of the null point, the magnetic field components are approximately linear functions of the Cartesian coordinates.Hence, the magnetic field can be approximated as a quadrupole magnetic field around the null point [10,23].Along the atomic beam path, the approximate quadrupole magnetic field is [33,35] For the study of the time evolution of the atom inside the IR chamber both of the fields, B exact and B q , 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 intermediate stage to obtain a sharper cut-off [10].In the forthcoming theoretical model, we track only the top transmitted 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 rotation 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 z axis.The microscope reads the position of the hot wire.
branch with spin down, m S = − 1 /2, at the entrance of the IR chamber and ignore the blocked branch.However, the opposite choice of m S = + 1 /2 yields 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 z axis while monitored by the microscope.The probability of flip is then measured at different values of the electric current I w .

III. THEORETICAL DESCRIPTION
The time evolution of the noninteracting atoms in the beam traveling through the IR chamber of the Frisch-Segrè experiment is studied using standard quantum mechanics.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 z direction.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, I w = 0 A, no flipping was observed after SG2 [11].Therefore, 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 z direction according to the Born principle.
The density operator formalism is used for its capability to represent mixed states in quantum systems, offering a more complete description than pure states alone [36,37].The time evolution of the density operator ρ is governed by the von Neumann equation [36][37][38][39]: where Ĥ(t) is the Hamiltonian of the system and ℏ is the reduced Planck constant.For the time-dependent Hamiltonian Ĥ(t), we introduce the instantaneous eigenstates |ψ j (t)⟩ and eigenenergies E j (t) such that where j can take a finite number of values for the spin system considered here.In the basis of the instantaneous eigenstates of the Hamiltonian, from (3) the matrix elements of the density operator, ρ j,k (t) = ⟨ψ j (t)|ρ(t)|ψ k (t)⟩, evolve according to In particular, the elements in the diagonal ρ j,j (t), corresponding to the probabilities of finding the quantum system in the eigenstate of the Hamiltonian, follow In the adiabatic approximation, since the time derivative of the Hamiltonian is small compared to the energy difference, we set [40,41] Therefore, for the adiabatic evolution, the populations in the instantaneous eigenstates do not change over time 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 alkali atom, composed of the spin S = 1 /2 of the valence electron and the spin I of the nucleus.In an external magnetic field B, the electron Zeeman term Ĥe describes the interaction between the electron magnetic moment and the field via [41] where μe is the quantum operator for the electron magnetic moment.Furthermore, μe = γ e Ŝ, where γ e denotes the gyromagnetic ratio of the electron; the electron spin operator Ŝ = ℏ 2 σ, with the Pauli vector σ consisting of the Pauli matrices {σ x , σ y , σ z }.Substitutions yield The (2S + 1)-dimensional Hilbert space H e = span (|S, m S ⟩), with m S = −S, . . ., S, and |S, m S ⟩ being the eigenvectors of Ŝz .The nuclear Zeeman Hamiltonian Ĥn describes the interaction of the nuclear magnetic moment with the external magnetic field: where μn = γ n Î denotes the quantum operator for the nuclear magnetic moment, γ n the nuclear gyromagnetic ratio for the atomic specie, and Î the nuclear spin quantum operator for spin I. Therefore, we can write Î = ℏ 2 τ , with τ being the generalized Pauli vector constructed with the generalized Pauli matrices of dimension 2I + 1, A basis for the (2I + 1)-dimensional Hilbert space H n can be obtained from the eigenvectors of Îz , such that 2. Schematic of the model considered in this study.The system consists of two measurements with SG1 and SG2.The inner rotation chamber that induces nonadiabatic transitions is modeled with the time-dependent von Neumann equation.The evolution from the end of SG1 and the filter to the entrance of the rotation chamber is modeled as an adiabatic evolution.
Similarly, the evolution from the exit of the inner rotation chamber to the entrance of SG2 is modeled as adiabatic evolution.
clear spin operators, the Hamiltonian is written as where a HFS reflects the coupling strength.Then, the total Hamiltonian of the combined system consisting of the electron and nuclear spins under an external magnetic field is The (2S + 1)(2I + 1)-dimensional Hilbert space for the combined nuclear-electron spin system is H = H n ⊗H e = span (|m I , m S ⟩); where for simplicity of notation we have dropped the S and I labels.The Frisch-Segrè experiment used 39 K; for this isotope, the nuclear spin is I = 3 /2, the nuclear gyromagnetic ratio is γ n = 1.250 061 2(3) × 10 7 rad/(s T), and the experimentally measured hyperfine constant is a exp = 230.859860 1(3) MHz [42].The terms of the nuclear-electron spin Hamiltonian Ĥtotal in ( 14) are explicitly expressed as [43,44] in the {|m I , m S ⟩} basis, where σ0 and τ0 are the 2dimensional and 4-dimensional identity matrices, respectively.This Hamiltonian has been validated numerically by comparing the eigenvalues with the solutions from the exact Breit-Rabi formula [45] with respect to the external field.

A. Adiabatic evolution
As depicted in Figure 2, the system undergoes an adiabatic evolution from SG1 (polarizing magnet) to the entrance of the rotation chamber, t ∈ [t SG1 , t i ], as well as from the exit of the inner rotation chamber to SG2 (the analyzing magnet), t ∈ [t f , t SG2 ].From (7), we have The quantum state of the atoms after SG1 and the filter is a pure state for the electron but maximally mixed for the nuclear spin [25].Hence, the density matrix is diagonal following The measurement probabilities at SG2 can be directly obtained from the state at t f from

B. Nonadiabatic evolution
In the IR chamber, the external magnetic field is not homogeneous; instead, along the beam path the magnetic field rapidly changes its direction and magnitude.The IR chamber of the FS experiment was specially designed to induce nonadiabatic variations of the magnetic field [10,13].For such behavior the field has to be sufficiently weak and the variation of its direction sufficiently fast, such that the frequency of rotation of the magnetic field is large compared to the Larmor frequency [23,24].The conditions for nonadiabatic rotations are satisfied near y = y NP along the beam path [35].
An exact closed-form analytical time-dependent solution for the density operator in the IR chamber has not been obtained.To calculate a numerical solution, we discretize the von Neumann equation (3).We used several different differential equation solvers to validate the solutions [46,47], one of which is the second-order Runge-Kutta method [48]: where ∆t is the temporal step size.

IV. RESULTS
Historically, the first attempt to describe nonadiabatic rotations was made by Majorana with a model involving only the electron Zeeman term [23].In Section IV A, the same model is solved numerically while ignoring any nuclear effect.Improving on Majorana's solution, Rabi considered the effect of the nuclear spin [25].Here, we explore the same model numerically in Section IV B. Our results indicate that in the IR chamber, the field strength is weak enough that the effect of the nuclear spin cannot be neglected.As the atom nears the null point, the field strength reduces.When eigenenergies converge, the rapid field rotation triggers a nonadiabatic transition.In the quadrupole field approximation, the nonadiabatic transition can be described through the Landau-Zener-Stückelberg-Majorana (LZSM) model [23,26].

A. Excluding hyperfine interaction
We first consider the case when the Hamiltonian Ĥ is Ĥe in (10), excluding the nuclear Zeeman and hyperfine effects.The analytical asymptotic solution for this model was found using the quadrupole field approximation by Majorana [23] and applied to the Frisch-Segrè experiment [10].The flip probability is given by the well-known Landau-Zener-Stuckelberg-Majorana (LZSM) model: where the adiabaticity parameter is defined as (see Appendix A for details) [23,26] Here, we numerically solve this model for both the exact and quadrupole fields.In modeling adiabatic evolution as discussed in Section III, we introduce an instantaneous eigenstate |ψ mS (t)⟩ with the associated instantaneous eigenenergy E mS (t) = m S γ e ℏ| ⃗ B(t)|.As the atom nears the null point, the instantaneous eigenenergies become asymptotically degenerate, and the state transitions nonadiabatically between the instantaneous eigenstates as visualized in Figure 3.
Figure 4a shows the evolution of ψ1 /2 (t) ρ(t) ψ1 /2 (t) over the flight of the atom in the IR chamber, where I w = 0.1 A. The evolution based on the quadrupole field approximation closely follows that based on the exact field, indicating the accuracy of the field approximation.(b) Flip probability of the electron spin versus the wire current.The numerical simulations match Majorana's prediction [23] but not the experimental observation [10].
Figure 4b shows the flip probability of the electron spin observed in SG2 as ψ1 /2 for the exact and quadrupole fields at different wire currents.The numerical prediction using the quadrupole approximation agrees exactly with Majorana's analytical prediction [23] and closely with the numerical prediction using the exact field.The coefficients of determination R 2 between the numerical predictions and the experimental data are, however, −18.9 and −19.9 for the exact and quadrupole fields, respectively.Therefore, this model does not predict the experimental observations.

B. Including hyperfine interaction
We now generalize the Hamiltonian Ĥ to Ĥtotal in ( 14) by including both the nuclear Zeeman and hyperfine effects.The total spin for the system assumes the values of F = I ± 1 2 .Let the instantaneous eigenstate |ψ j (t)⟩ be |ψ m I ,mS (t)⟩.
Building on Majorana's work, Rabi developed an asymptotic solution that incorporates the influence of the nuclear spin [25].Rabi's approach has been visualized in Figure 5. Pairs of eigenstates with different F values are too far apart in eigenenergies to nonadiabatically transition to each other.In contrast, the eigenstates with the same F values become asymptotically degenerate as the atom approaches the null point.Therefore, eigenstates within each F manifold nonadiabatically couple to each other, and the approximation in (7) no longer holds.Rabi solved the nonadiabatic transition in each F manifold as a single rotation in a (2F + 1)-dimensional Hilbert space.The flip probability between individual eigenstates (m F → m ′ F ) after a rotation can be calculated using the Wigner-d matrix [25,49]: where the angle of rotation can be found through The adiabaticity parameter k ′ is approximated as which accounts for the modified Landé g-factor.
The total transition probability between the m F = −F state and the m F ̸ = −F states is found with a summation: where the prefactor denotes the initial populations, which are assumed to be equal.The maximally mixed initial state is given by Figure 6a illustrates the evolution of the populations, ⟨ψ m I ,mS (t)|ρ(t)|ψ m I ,mS (t)⟩, around the null point.The state follows the instantaneous eigenstates adiabatically during the majority of the flight.Above the wire, there is a small nonadiabatic evolution that recovers.Hence, the majority of the nonadiabatic transition occurs around the null point.Apart from the behavior immediately above the wire, the assumption that the state follows the instantaneous eigenstates outside the null point is accurate.For a lower current, where the null point is closer 5. Eigenenergy-based visualization of the nonadiabatic transition of the nucleus and electron spins.The initial state is in a mixture of mS = −1/2 states.As the atom approaches the null point, the state mostly adiabatically transitions into low-field eigenstates.Near the null point, when the F = 2 states have similar eigenenergies, nonadiabatic transitions occur.As the atom leaves the null point region, the state again follows adiabatically to the high-field eigenstates.However, a portion of the population reaches the mS = +1/2 states due to the nonadiabatic transition.When the quadrupole field is used, the nonadiabatic transition can be modeled by Rabi's derivation [25].
to the wire, the behavior right above the wire affects the transition near the null point.Therefore, the quadrupole approximation for the low currents is inaccurate (to be shown below).
Figure 6b shows the flip probabilities predicted by the numerical solution in comparison to Rabi's analytical solution [25] and the experimental observation [10].The inaccuracy of the quadrupole approximation explains the discrepancy between the numerical solutions in Figure 6b.Furthermore, the numerical solution with the quadrupole approximation matches Rabi's solution as expected.The coefficients of determination R 2 of our model for the exact and quadrupole fields in relation to the experimental observation are 0.19 and −0.67, respectively.Our standard quantum mechanical model or Rabi's solution, even if hyperfine interaction is considered, does not predict the experimental observation well in the high current regime.

V. DISCUSSION
As a natural extension of the first SG experiment, FS aimed to implement cascaded quantum measurements using two SG apparatuses with a nonadiabatic spin flipper in between.Since the SG apparatuses here cannot distinguish nuclear eigenstates [10], it was conceivable for Majorana to ignore the nuclear effect [23].However, as shown in Section IV A, such an electron-only model can- not predict the FS observations.Along the beam path, the atom enters a magnetically weak region where the effect of the nuclear spin is critical as shown in Section IV B. While factoring in nuclear effects greatly enhances the theory-experiment match, it still fails to model the highly nonadiabatic regime.Regarding the decrease of the flip probability as a function of increasing current in the high-current regime, FS mentioned that it could be observed if there was a remnant field along the propagation axis [11].In Appendix A, we tried to fit the observations with the remnant field as a free parameter disregarding the reported value.Furthermore, the peak flip probability in the experimental data exceeds 1 /4 while the theoretical models here cannot.There are at least two possibilities for the theoretical model to exceed 1 /4.Either the initial nuclear state is not maximally mixed as in (28) or the hyperfine interaction strength, a HFS , is orders of magnitude smaller.Surprisingly, recent "semi-classical" studies [33][34][35] have been able to predict the Frisch-Segrè observations without fitting.
In Appendix B, we have considered various other initial nuclear states.Since SG1 cannot distinguish nuclear eigenstates, one might question how the nuclear state must be modeled after SG1 and the slit.In Appendix C, we consider different hyperfine interaction strengths.The hyperfine interaction term, first written by Fermi, uses an interaction strength, a HFS , theoretically calculated via the Fermi contact interaction [50,51].However, for any atom except the hydrogen isotopes, the theoretical values are orders of magnitude different than the spectroscopically measured values [14,42].While this discrepancy might be due to the ill-defined nature of the Fermi contact interaction [52], using such theoretical values to model the FS experiment can result in flip probabilities higher than 1 /4.Ultimately, none of the variants of standard quantum mechanical approaches in the appendices follow the reported experiment but have been explored to stimulate discussion.

VI. CONCLUSIONS
Simulating the FS experiment [10,11] using a standard quantum mechanical model has yielded the following conclusions: • The FS observations cannot be replicated by modeling only the electron spin without hyperfine coupling (Section IV A).Considering hyperfine coupling significantly improves the predictions (Section IV B).
• The FS observations cannot be closely replicated by modeling the atom as a pair of electron and nuclear spins without modifying the reported experimental parameters (Appendix A), the initial nuclear state (Appendix B), or the hyperfine interaction coefficient (Appendix C).
Based on some of the non-standard variants that can improve the model prediction of the FS observation (see Appendices), one might question the following: • What is the nuclear spin state before and after each SG apparatus?How does the nuclear spin state affect the electron spin-flip?
• Do we need a more sophisticated model of the atom (especially, the nucleus) to understand and predict the SG and nonadiabatic FS experiments?
• Do the SG apparatuses in the FS experiment truly follow the Born principle?Is there a (hidden) vari-able, such as the nuclear spin, affecting the FS measurement?
Despite the prevalent understanding of multi-stage Stern-Gerlach experiments, our models here fall short of accurately explaining the initial experiments [10,11].Later multi-stage SG experiments include different designs of the SG apparatuses and spin flippers, and the associated models used free parameters for fitting [15][16][17][18][19][20].Given the foundational importance of the multi-stage SG experiments as cascaded quantum measurements, we believe that the mismatch between the theory and the experiment merits further investigation.
One has to solve the von Neumann equation for each initial state considered.Various pure states have been tried here; however, the coefficients of determination are not better than that with the mixed state.
Notwithstanding, modifying initial states for the Frisch-Segrè experiment is unconventional in the literature [25].

Appendix C: Modified HFS coefficients
Up to now, we have used the experimentally measured HFS coefficient value, a exp = 230.859860 1(3) MHz [42], which does not accurately predict the experimental observation by Frisch and Segrè.Here, we attempt to improve the match by modifying the hyperfine coefficient.
One way to calculate the HFS coefficient is to use the Fermi contact interaction as follows [41,43,50,57]: where ψ(r) denotes the spatial wave function of the electron.The wave function for the 4s 1 electron in 39 K does not have an exact solution.However, various approximations are available [33,58,59], yielding the following HFS coefficients: where R = 275 pm is the van der Waals radius for 39 K.
Another set of values for a HFS are obtained on the basis of an alternative averaging method [33]: All of these values along with the experimental value, a exp , have been considered.Using the maximally mixed initial state, the best match is obtained for a HFS = a 3 with R 2 = 0.20 for the exact field.Fitting for the initial state using the exact field as in Appendix B yields the best match for a HFS = a 1 with R 2 = 0.77.
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 rotation 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 z axis.The microscope reads the position of the hot wire.
H n = span (|I, m I ⟩) with m I = −I, . . ., I.The interaction between the magnetic dipole moments of the nucleus and the electron gives the hyperfine structure (HFS) term ĤHFS .In terms of the electron and nu-

FIG. 3 .
FIG.3.Eigenenergy-based visualization of the nonadiabatic transition of the electron spin.The initial electron spin is in the mS = −1/2 state.As the atom nears the null point, the field strength reduces.When eigenenergies converge, the rapid field rotation triggers a nonadiabatic transition.In the quadrupole field approximation, the nonadiabatic transition can be described through the Landau-Zener-Stückelberg-Majorana (LZSM) model[23,26].

FIG. 4 .
FIG.4.Electron spin only model, where hyperfine interaction is ignored.(a) Time evolution of ψ1 /2 (t) ρ(t) ψ1 /2 (t) for the exact and quadrupole fields at the wire current Iw = 0.1 A. (b) Flip probability of the electron spin versus the wire current.The numerical simulations match Majorana's prediction[23] but not the experimental observation[10].

FIG. 6 .
FIG.6.Nucleus-electron spin model, where the nuclear effects are included.(a) Evolution of the populations for the exact field within the IR chamber at Iw = 0.3 A. Near the null point, the populations exchange between ψ −3/2,−1/2 (t) and states with mS = +1/2.Far from the null point, the evolution of the state is mostly adiabatic.(b) Flip probability of the electron spin for the exact and quadrupole fields.The numerical predictions do not match the experimental observation well.