The Stern-Gerlach effect, discovered a century ago, has become a paradigm of quantum mechanics. Surprisingly there has been little evidence that the original scheme with freely propagating atoms exposed to gradients from macroscopic magnets is a fully coherent quantum process. Specifically, no full-loop Stern-Gerlach interferometer has been realized with the scheme as envisioned decades ago. Furthermore, several theoretical studies have explained why such an interferometer is a formidable challenge. Here we provide a detailed account of the first full-loop Stern-Gerlach interferometer realization, based on highly accurate magnetic fields, originating from an atom chip, that ensure coherent operation within strict constraints described by previous theoretical analyses. Achieving this high level of control over magnetic gradients is expected to facilitate technological as well as fundamental applications, such as probing the interface of quantum mechanics and gravity. While the experimental realization described here is for a single atom, future challenges would benefit from utilizing macroscopic objects doped with a single spin. Specifically, we show that such an experiment is in principle feasible, opening the door to a new era of fundamental probes.
We analyze idealized sequential Stern-Gerlach experiments with higher spin particles. This analysis serves at least two purposes: The widely discussed spin-1/2 case leads to some misunderstandings which hopefully is removed by the higher spin discussion. Secondly, Wigner rotation matrices for generic spins become conceptually more transparent with this physical example. We also give compact formulas for the probabilities in terms of the angle between the sequential SG apparatuses for generic spins. We work out the spin-$1/2$, spin-$1$ and spin-$2$ cases explicitly. Since there are some confusi…
It is often stated that complex numbers are essential in quantum theory. In this article, the need for complex numbers in quantum theory is motivated using the results of tandem Stern-Gerlach experiments
Stern-Gerlach experiment is a paradigm of measurement theory in quantum mechanics. Notwithstanding several analysis given in literature, no clear understanding of the apparent collapse has been given so far. Indeed, one can imagine a Stern-Gerlach device where all environmental effects are removed and ask the question on how the measurement goes on. In this letter we will prove that a the Stern-Gerlach device behaves as a true measurement apparatus, as expected by the Copenaghen interpretation, by the Ehrenfest theorem. In this way we recover, by other means, a limit on the Stern-Gerlach devic…
We propose a new setup to demonstrate quantum eraser, using spin-1/2 particles in a modified Stern-Gerlach setup, with a double slit. The "which-way" information can be erased simply by applying a transverse magnetic field with an additional magnet, resulting in the emergence of two complementary staggered interference patterns. Use of the classic Stern-Gerlach setup, and the unweaving of the washed out interference without any coincident counting, is what makes this proposal novel.
We study the birefringence of the quantized polarized light in a magneto-optically manipulated atomic ensemble as a generalized Stern-Gerlach Effect of light. To explain this engineered birefringence microscopically, we derive an effective Shrödinger equation for the spatial motion of two orthogonally polarized components, which behave as a spin with an effective magnetic moment leading to a Stern-Gerlach split in an nonuniform magnetic field. We show that electromagnetic induced transparency (EIT) mechanism can enhance the magneto-optical Stern-Gerlach effect of light in the presence of a con…
Spin is an important quantum degree of freedom in relativistic quantum information theory. This paper provides a first-principles derivation of the observable corresponding to a Stern-Gerlach measurement with relativistic particle velocity. The specific mathematical form of the Stern-Gerlach operator is established using the transformation properties of the electromagnetic field. To confirm that this is indeed the correct operator we provide a detailed analysis of the Stern-Gerlach measurement process. We do this by applying a WKB approximation to the minimally coupled Dirac equation describin…
Emergence of classicality from quantum mechanics, a hotly debated topic, has had no satisfactory resolution so far. Various approaches including decoherence and gravitational interactions have been suggested. In the present work, the Schrödinger-Newton model is used to study the role of semi-classical self-gravity in the evolution of massive spin-1/2 particles in a Stern-Gerlach experiment. For small mass, evolution of the initial wavepacket in a spin superposition shows a splitting in the magnetic field gradient into two trajectories as in the standard Stern-Gerlach experiment. For larger mas…
We present a unique matter-wave interferometer whose phase scales with the cube of the time the atom spends in the interferometer. Our scheme is based on a full-loop Stern-Gerlach interferometer incorporating four magnetic field gradient pulses to create a state-dependent force. In contrast to typical atom interferometers which make use of laser light for the splitting and recombination of the wave packets, this realization uses no light and can therefore serve as a high-precision surface probe at very close distances.
We propose a state estimation scheme for spins, using a modified setup of the Stern-Gerlach experiment, in which a beam of neutral spin-1/2 point particles interacts with a quadrupolar magnetic field. The proposed linear inversion estimation procedure, based on a quadrant intensity detector, requires a suitable initial spatial state of the beam. The statistical characterization of the estimator of the initial spin state allows us not only to associate an error to the estimated parameters, but also to define a measure for comparing estimation procedures corresponding to different Stern-Gerlach …
The widely accepted interpretation of the Stern-Gerlach experiment assumes the objective atomic trajectories (the "classical trajectories") in front of the screen. Following this interpretation, we perform an {\it ab initio} analysis of the experiment and conclude that the objective trajectories do not physically exist. The alternative to our conclusion is substantially to change the model of the experiment.
We present a feasibility study with several magnetic field configurations for creating spin-dependent forces that can split a low-energy ion beam by the Stern-Gerlach effect. To the best of our knowledge, coherent spin-splittings of charged particles have yet to be realised. Our proposal is based on ion source parameters taken from a recent experiment that demonstrated single-ion implantation from a high-brightness ion source combined with a radio-frequency Paul trap. The inhomogeneous magnetic fields can be created by permanently magnetised microstructures or from current-carrying wires with …
We numerically study the spin flip in the Frisch$\unicode{x2013}$Segrè experiment, the first multi-stage Stern$\unicode{x2013}$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, w…
We argue that the non-adiabatic evolution of spin states in Stern-Gerlach apparatus can blur the manifestation of path spin entanglement. This fact introduces a contradiction to the usual perception of spin measurement even in a formally ideal situation. Through quantitative calculation, we have pointed out the specific reason behind the breakdown of adiabatic evolution to be the spatial inhomogeneity of applied magnetic field. The angle $θ$ between the $z$ component of magnetic moment of the particle and applied magnetic field is also a determining factor for the category of evolution of spin…
The Stern-Gerlach experiment is the fundamental experiment in order to exhibit the quantization of spin and understand the measurement problem in quantum mechanics. However, although the Stern-Gerlach experiment plays an essential role in the teaching of quantum mechanics, no complete analysis of this experiment using Pauli spinors is presented in the pedagogical literature. This paper presents such an analysis and develops implications for the theory of quantum measurement. We first propose an analytic expression of both the wave function and the probability density in the Stern-Gerlach exper…
Atomic interferometers have been studied as a promising device for precise sensing of external fields. Among various configurations, a particular configuration with a butterfly-shaped geometry has been designed to sensitively probe field gradients. We introduce a Stern-Gerlach (SG) butterfly interferometer by incorporating magnetic field in the conventional butterfly-shaped configuration. Atomic trajectories of the interferometer can be flexibly adjusted by controlling magnetic fields to increase the sensitivity of the interferometer, while the conventional butterfly interferometer using Raman…
On the basis of a variant of the EPR-Bohm example, we show that the no-signaling condition can be employed as a useful tool for deriving a constraint on a suitably defined measure of the `nonidealness' of a Stern-Gerlach(SG) setup. In this demonstration, a key ingredient is provided by the characteristics of the exact solution of the time-dependent Schroedinger equation as applied to a most general SG setup.
The motion of neutral particles with magnetic moments in an inhomogeneous magnetic field is described in a semi-classical framework. The concept of Coherent Internal States is used in the formulation of the semiclassical approximation from the full quantum mechanical expression. The classical trajectories are defined only for certain spin state, that satusfy the conditions for being Coherent Internal State. The raalibility of Stern-Gerlach experiments to measure spin projectons is assessed in this framework.
The motion of neutral particles with magnetic moments in an inhomogeneous magnetic field is described in a quantum mechanical framework. The validity of the semi-classical approximations which are generally used to describe these phenomena is discussed. Approximate expressions for the evolution operator are derived and compared to the exact calculations. Focusing and spin-flip phenomena are predicted. The reliability of Stern-Gerlach experiments to measure spin projections is assessed in this framework.
We give a simple example of the tight connection between entanglement and coherence for pure bipartite systems showing the double role played by entanglement; it allows for the creation of superpositions of macroscopic objects but at the same time makes subsystems lose their quantum mechanical coherence. For this we study the time evolution of the spin coherence in the Stern-Gerlach (SG) experiment. We also show that, contrary to the naive intuition, the spin coherence is lost before the two beams become separated in the spatial coordinates.
There is no consensus among today's physicists about how to describe the gravitational interaction properly in a quantum framework. We propose in this paper an experimental test aimed at revealing the existence of a non-linear self-interaction à la Schrodinger-Newton (S-N). In this test, a mesoscopic spin 1/2 microsphere is freely falling in a Humpty-Dumpty Stern-Gerlach interferometer. After clarifying the role of the scaling of the interaction in function of the amplitudes of the up and down spin components of the microsphere, it is shown that self-gravity induces a measurable phase shift be…
We show that the performances of a Stern-Gerlach apparatus can be improved by using a magnetic field profile for the atomic spin evolution designed through shortcut to adiabaticity technique. Interestingly, it can be made more compact - for atomic beams propagating at a given velocity - and more resilient to a dispersion in velocity, in comparison with the results obtained with a standard uniform rotation of the magnetic field. Our results are obtained using a reverse engineering approach based on Lewis-Riesenfeld invariants. We discuss quantitatively the advantages offered by our configuratio…
Consider a spin s prepared in a pure state. It is shown that, generically, the moduli of the (2s+1) spin components along three directions in space determine the state unambiguously. These probabilities are accessible experimentally by means of a standard Stern-Gerlach apparatus. To reconstruct a pure spin state is therefore possible on the basis of 3(2s+1) measured intensities.
The motion of a particle with a spin in spherical harmonic oscillator potential with spin-orbit interaction is studied. We have focus our attention on spatial motion of wave packets, giving a description complementary to motion of spin discussed already in [1]. The particular initial conditions studied here lead to the most transparent formulas and can be treated analytically. A strong analogy with the Stern-Gerlach experiment is suggested. [1] R.Arvieu and P.Rozmej, Phys.Rev.A50 (1994) 4376.
The following is a translation of the paper by Walther Gerlach and Otto Stern that reported the first evidence for the quantisation of atoms in a magnetic field. The atoms have quantum states corresponding to a limited number of possible angles between the directions of the angular momenta of the atoms and the magnetic field, also called space quantisation. Wording and layout have been chosen to be as close to the original as possible. For context we recommend the recent review.
Originally conceived as a gedankenexperiment, an apparatus consisting of two Stern--Gerlach apparatuses joined in an inverted manner touched on the fundamental question of the reversibility of evolution in quantum mechanics. Theoretical analysis showed that uniting the two partial beams requires an extreme level of experimental control, making the proposal in its original form unrealizable in practice. In this work we revisit the above question in a numerical study concerning the possibility of partial-beam recombination in a spin-coherent manner. Using the Suzuki--Trotter numerical method of …
Probing quantum mechanics, quantum aspects of general relativity along with the sensing and the constraining of classical gravity can all be enabled by unprecedented spatial sizes of superpositions of massive objects. In this paper, we show that there is a feasible setup sourced by realizable magnetic field gradients ${\cal O}(10-100)$ Tm$^{-1}$ to construct a large spatial superposition of ${\cal O}(10^{-4}-10^{-8})$ m for masses ${\cal O}(10^{-16}-10^{-14})$ kg over a time period of up to $0.1-10$ seconds.
The density matrix of a spin s is fixed uniquely if the probabilities to obtain the value s upon measuring n.S are known for 4s(s+1) appropriately chosen directions n in space. These numbers are just the expectation values of the density operator in coherent spin states, and they can be determined in an experiment carried out with a Stern-Gerlach apparatus. Furthermore, the experimental data can be inverted providing thus a parametrization of the statistical operator by 4s(s+1) positive parameters.
We describe an interactive computer program that simulates Stern-Gerlach measurements on spin-1/2 and spin-1 particles. The user can design and run experiments involving successive spin measurements, illustrating incompatible observables, interference, and time evolution. The program can be used by students at a variety of levels, from non-science majors in a general interest course to physics majors in an upper-level quantum mechanics course. We give suggested homework exercises using the program at various levels.
We use a theory termed co-quantum dynamics (CQD) to numerically model spin flip in the multi-stage Stern$\unicode{x2013}$Gerlach (SG) experiment conducted by R. Frisch and E. Segrè. This experiment consists of two Stern$\unicode{x2013}$Gerlach apparatuses separated by an inner rotation chamber that varies the fraction of spin flip. To this day, quantum mechanical treatments inadequately predict the Frisch$\unicode{x2013}$Segrè experiment. Here, we account for electron-nuclear interactions according to CQD and solve the associated Schrödinger equation. Our simulation outcome agrees with the Fri…
The Stern-Gerlach effect is well-known as spin-dependent splitting of a beam of atoms with magnetic moments by a magnetic-field gradient. Here, we show that an induced gauge potential may lead to a similar effect for chiral molecules. In the presence of three inhomogeneous light fields, the center-of-mass of a three-level chiral molecule is subject to an optically induced gauge potential, and the internal dynamics of the molecules can be described as an adiabatic evolution in the reduced pseudo-spin subspace of the two lowest energy levels. We demonstrate numerically that such an induced gauge…
We employ the Stern-Gerlach experiment to highlight the basics of a minimalist, non-interpretational top-down approach to quantum foundations. Certain benefits of the here highlighted "quantum structural studies" are detected and discussed. While the top-down approach can be described without making any reference to the fundamental structure of a closed system, the hidden variables theory ?a la Bohm proves to be more subtle than it is typically regarded.
We present a fully quantum scheme to perform 2D atomic lithography based on a cross-cavity optical Stern-Gerlach setup: an array of two mutually orthogonal cavities crossed by an atomic beam perpendicular to their optical axes, which is made to interact with two identical modes. After deriving an analytical solution for the atomic momentum distribution, we introduce a protocol allowing us to control the atomic deflection by manipulating the amplitudes and phases of the cavity field states.
We combine the Ramsey interferometry protocol, the Stern-Gerlach detection scheme, and the use of elongated geometry of a cloud of fully polarized cold cesium atoms to measure the selected component of the magnetic field gradient along the atomic cloud in a single shot. In contrast to the standard method where the precession of two spatially separated atomic clouds is simultaneously measured to extract their phase difference, which is proportional to the magnetic field gradient, we here demonstrate a gradiometer using a single image of an expanding atomic cloud with the phase difference imprin…
Quantum nonlocality and quantum steering are fundamental correlations of quantum systems which can not be created using classical resources only. Nonlocality describes the ability to influence the possible results of measurements carried out in distant systems, in quantum steering Alice remotely steers Bob's state. Research in nonlocality and steering possess a fundamental interest for the development of quantum information and in many applications requiring nonlocal resources like quantum key distribution. On the other hand, the Stern-Gerlach experiment holds an important place in the history…
Although Heisenberg's uncertainty principle is represented by a rigorously proven relation about intrinsic indeterminacy in quantum states, Heisenberg's error-disturbance relation (EDR) has been commonly believed as another aspect of the principle. However, recent developments of quantum measurement theory made Heisenberg's EDR testable to observe its violations. Here, we study the EDR for Stern-Gerlach measurements. In a previous report [arXiv:1910.07929], it has been pointed out that their EDR is close to the theoretical optimal. The present note reports that even the original Stern-Gerlach …
In the optical Stern-Gerlach effect the two branches in which the incoming atomic packet splits up can display interference pattern outside the cavity when a field measurement is made which erases the which-way information on the quantum paths the system can follow. On the contrary, the mere possibility to acquire this information causes a decoherence effect which cancels out the interference pattern. A phase space analysis is also carried out to investigate on the negativity of the Wigner function and on the connection between its covariance matrix and the distinguishability of the quantum pa…
The Stern-Gerlach experiment has played an important role in our understanding of quantum behavior. We propose and analyze a modified version of this experiment where the magnetic field of the detector is in a quantum superposition, which may be experimentally realized using a superconducting flux qubit. We show that if incident spin-$1/2$ particles couple with the two-state magnetic field, a discrete target distribution results that resembles the distribution in the classical Stern-Gerlach experiment. As an application of the general result, we compute the distribution for a square waveform o…
In the context of the quantum mechanical modelling of a measurement process using the Stern-Gerlach setup, we critically examine the relationship between the notion of `distinguishability' of apparatus states defined in terms of the inner product and spatial separation among the emerging wave packets. We show that, in general, the mutual orthogonality of these wave packets does not necessarily imply their unambiguous spatial separation even in the asymptotic limit. A testable scheme is formulated to quantify such departures from `idealness' for a range of relevant parameters.
We show that the edge of a two-dimensional topological insulator can be used to construct a solid state Stern-Gerlach spin-splitter. By threading such a Stern-Gerlach apparatus with a magnetic flux, Ahranov-Bohm like interference effects are introduced. Using ferromagnetic leads, the setup can be used to both measure magnetic flux and as a spintronics switch. With normal metallic leads a switchable spintronics NOT-gate can be implemented. Furthermore, we show that a sequence of such devices can be used to construct a single-qubit $SU(2)$-gate, one of the two gates required for a universal quan…
We design a Stern-Gerlach apparatus that separates quasispin components on the lattice, without the use of external fields. The effect is engineered using intrinsic parameters, such as hopping amplitudes and on-site potentials. A theoretical description of the apparatus relying on a generalized Foldy-Wouthuysen transformation beyond Dirac points is given. Our results are verified numerically by means of wavepacket evolution, including an analysis of Zitterbewegung on the lattice. The necessary tools for microwave realizations, such as complex hopping amplitudes and chiral effects, are simulate…
Using the optical Stern-Gerlach model, we have recently shown that the non-local correlations between the internal variables of two atoms that successively interact with the field of an ideal cavity in proximity of a nodal region are affected by the atomic translational dynamics. As a consequence, there can be some difficulties in observing violation of the Bell's inequality for the atomic internal variables. These difficulties persist even if the atoms travel an antinodal region, except when the spatial wave packets are exactly centered in an antinodal point.
Although Heisenberg's uncertainty principle is represented by a rigorously proven relation about intrinsic uncertainties in quantum states, Heisenberg's error-disturbance relation (EDR) has been commonly believed to be another aspect of the principle. Based on the recent development of universally valid reformulations of Heisenberg's EDR, we study the error and disturbance of Stern-Gerlach measurements of a spin-1/2 particle. We determine the range of the possible values of the error and disturbance for arbitrary Stern-Gerlach apparatuses with the orbital degree prepared in an arbitrary Gaussi…
In this paper, a quantum Stern-Gerlach thought experiment is introduced where, in addition to the intrinsic angular momentum of an atom, the magnetic field is also treated quantum mechanically. A freely falling spin polarised Bose-Einstein condensate passes close to a flux-qubit and interacts with the quantum superimposed magnetic field of the flux-qubit. Such an interaction results a macroscopic quantum entanglement of the path of a Bose-Einstein condensate with the magnetic flux quantum state of the flux-qubit. In this paper, three regimes of coupling between the flux-qubit and a freely fall…
We study the Quantum Measurement Process in a Stern-Gerlach setup with the spin of a silver atom as the quantum system and the position as the apparatus. The system and the apparatus are treated quantum-mechanically using unitary evolution. The new ingredient in our analysis is the idea that the probes determining the position of the silver atom are limited in resolution. We show using a Wigner matrix that due to the coarseness of the detection process, the pure density matrix appears to evolve to an impure one. We quantify the information gained about the spin in a coarse position measurement…
We propose and develop the thesis that the quantum theoretical description of experiments emerges from the desire to organize experimental data such that the description of the system under scrutiny and the one used to acquire the data are separated as much as possible. Application to the Stern-Gerlach and Einstein-Podolsky-Rosen-Bohm experiments are shown to support this thesis. General principles of logical inference which have been shown to lead to the Schrödinger and Pauli equation and the probabilistic descriptions of the Stern-Gerlach and Einstein-Podolsky-Rosen-Bohm experiments, are use…
Electromagnetically induced transparency allows for light transmission through dense atomic media by means of quantum interference. Media exhibiting electromagnetically induced transparency have very interesting properties, such as extremely slow group velocities. Associated with the slow light propagation are quasiparticles, the so-called dark polaritons, which are mixtures of a photonic and an atomic contribution. We here demonstrate that these excitations behave as particles with a nonzero magnetic moment, which is in clear contrast to the properties of a free photon. It is found that light…
Spin preparation and spin detection are fundamental problems in spintronics and in several solid state proposals for quantum information processing. Here we propose the mesoscopic equivalent of an optical polarizing beam splitter (PBS). This interferometric device uses non-dispersive phases (Aharonov-Bohm and Rashba) in order to separate spin up and spin down carriers into distinct outputs and thus it is analogous to a Stern-Gerlach apparatus. It can be used both as a spin preparation device and as a spin measuring device by converting spin into charge (orbital) degrees of freedom. An importan…
We propose a method to separate enantiomers using parity violation in the Casimir-Polder potential between chiral mirrors and chiral molecules. The proposed setup involves a molecular beam composed of chiral molecules passing through a planar cavity consisting of two chiral mirrors. Enantiomers of opposite handedness are deflected differently due to a chiral dependence of the Casimir-Polder potential resulting in the separation of the enantiomers. Our setup provides an alternative experimental tool for enantiomer separation, as well as shedding light on the fundamental properties of the Casimi…
In this paper, we propose a simplified model of optical Stern-Gerlach effect based on coherent coupling between clock transition of alkaline-earth single atoms and a traveling-wave light. It is demonstrated that spin-orbit coupling induced chiral motion in atom deflection appears under the strong atom-light interaction. The strong optical driving removes perturbation from the Doppler effect and back action effect to access the coherent system. In this process, superposition of distant matter waves connected to the arbitrary distribution of atom internal state could be predicted, which is impor…
The possibility of obtaining the initial pure state in a usual Stern-Gerlach experiment through the recombination of the two emerging beams is investigated. We have extended the previous work of Englert, Schwinger and Scully \cite{ISG1} including the fluctuations of the magnetic field generated by a properly chosen magnet. As a result we obtained an attenuation factor to the possible revival of coherence when the beams are perfectly recombined . When the source of the magnetic field is a SQUID(superconducting quantum interference device) the attenuation factor can be controlled by external cir…
