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arXiv:1608.04073 · Physical Review A

Quantum Stern-Gerlach experiment and path entanglement of a Bose-Einstein condensate

Mandip Singh

Published Apr 18, 2017• 6 citations
arXivHTMLPDF

Abstract

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 considered to be a quantum mechanical field. A free falling spin polarized 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 free falling Bose-Einstein condensate are discussed. This paper also explains how to produce a path entangled Bose-Einstein condensate where the condensate can be located at physically distinct locations simultaneously. This paper highlights new insights about the foundations of the quantum Stern-Gerlach experiment.

DOI
10.1103/physreva.95.043620

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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 considered to be a quantum mechanical field. A free falling spin polarized 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 free falling Bose-Einstein condensate are discussed. This paper also explains how to produce a path entangled Bose-Einstein condensate where the condensate can be located at physically distinct locations simultaneously. This paper highlights new insights about the foundations of the quantum Stern-Gerlach experiment.
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 Gaussian waveform of the incident fermion. This analysis allows us to demonstrate theoretically: (1) the quantization of the intrinsic angular momentum of a spin-1/2 particle, and (2) a correlation between EPR pairs leading to nonlocality, without necessarily collapsing the particle’s spin wavefunction.
The Stern-Gerlach experiment (SGE) has been known for decades, but a complete theory that explains all facets of it remains enigmatic. In its classic realization, each spin state of an atom follows a unique path or trajectory in an SGE. The enigmatic part is the final output of SGE. Feynman envisaged the outcome to be an entangled state, a superposition of $|\text{spin}\phantom{\rule{4.pt}{0ex}}\text{state}\ensuremath{\rangle}\ensuremath{\bigotimes}|\text{corresponding}\phantom{\rule{4.pt}{0ex}}\text{path}\ensuremath{\rangle}$. Later works described the final state of SGE as a mixed state, a weighted average of states belonging to separate beams or trajectories. Notably, Schwinger, Englert, and Scully compared the difficulty of obtaining a coherent output (a pure state) to ``putting together Humpty Dumpty'' again. Yet recent experiments using Bose-Einstein condensates indicate the emerging beams are partly coherent, favoring Feynman's description. In this work, we revisit SGE using a recent version of the quantum master equation. We show that both outcomes are probable depending on the time of flight and the timescale of the environmental fluctuations. We predict the output is initially coherent and grows gradually into a mixed state. Moreover, we also estimate how the magnetic-field gradient helps broaden the beam waist. Moreover, our treatment implies that the wave function of an atom passing through a Stern-Gerlach device does not undergo an instantaneous collapse; rather, a gradual decay of coherences results in a mixed state, resulting in an irreversible process.
Wave mechanics and interference have always been at the heart of quantum mechanics. Many ofthe most striking quantum effects, at their heart, arise from the wave nature that quantum particles exhibit. In this thesis, I describe two experiments which showcase two different interference effects using a rubidium Bose-Einstein condensate. The first experiment examines the ways geometric phases enter into two-dimensional spin systems by studying their effect on an atomic Ramsey interferometer. The addition of an off-resonantly coupled level causes the location of interference minima to shift on frequency scales comparable to relevant Rabi frequencies of the system, as well as potentially creating locations with increased or decreased sensitivity to magnetic noise. The second experiment is a weak Stern-Gerlach experiment, showing how the components of a superposition state, with different magnetic moments, can interfere to show a greatly increased shift due to a magnetic gradient. The effect is observed using the second-order magnetic sensitivity of the clock states of 87Rb, with a greater than fivefold amplification of the momentum shift created by the magnetic gradient.
Abstract We consider a macroscopic spin qubit based on spin–orbit coupled Bose–Einstein condensates, where, in addition to the spin–orbit coupling (SOC), spin dynamics strongly depends on the interaction between particles. The evolution of the spin for freely expanding, trapped, and externally driven condensates is investigated. For condensates oscillating at the frequency corresponding to the Zeeman splitting in the synthetic magnetic field, the spin Rabi frequency does not depend on the interaction between the atoms since it produces only internal forces and does not change the total momentum. However, interactions and SOC bring the system into a mixed spin state, where the total spin is inside rather than on the Bloch sphere. This greatly extends the available spin space making it three-dimensional, but imposes limitations on the reliable spin manipulation of such a macroscopic qubit. The spin dynamics can be modified by introducing suitable spin-dependent initial phases, determined by the SOC, in the spinor wave function.
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 θθθ between the zzz component of magnetic moment of the particle and applied magnetic field is also a determining factor for the category of evolution of spin states. Adiabaticity always can be restored by choosing a sufficiently small value of θθθ. Considering azimuthal inhomogeneity, we have examined the nature of path spin entanglement in the context of Stern-Gerlach experiment.