Englert, Schwinger, and Scully (Foundations of Physics 18, 1045 (1988)) asked whether the two partial beams of a Stern–Gerlach apparatus (SGA) can be reunited so that the original spin superposition is recovered — the 'Humpty-Dumpty' problem of spin coherence. Their simplified one-dimensional model showed that uncontrolled run-to-run variations of the magnetic-field gradient must be held to roughly one part in 10510^5 (about one part in 10810^8 for realistic beam phase-space factors), and the completion of this account was left open: their treatment takes the SGA magnetic field as an external classical field and does not account for the quantum properties of the electromagnetic fields generating it, and the quantitative effect of the neglected three-dimensional dynamics and a full nonideal calculation were likewise outstanding. Later work settled much of the classical part: the authors' general theory (Z. Phys. D 10, 135 (1988)) and a full three-dimensional Maxwell-compliant wave-propagation simulation by Paraniak and Englert (Symmetry 13, 1660 (2021)) show that perfect restoration is impossible even under perfect classical control once the beams are well separated, and coherent Stern–Gerlach interferometers have been realized on atom chips with time-dependent gradients. One partial quantum treatment exists: de Oliveira and Caldeira (Phys. Rev. A 73, 042502 (2006)) quantized the field source in the one-dimensional geometry and found the resulting coherence attenuation negligible for realistic parameters. What remains open is the general quantum-field completion: treating the SGA fields (and their sources) as quantum systems beyond that special-case dissipative model, deriving the fundamental quantum limits on preparing and reproducing the field gradient, quantifying the coherence loss caused by the quantum fluctuations of these fields, and combining these limits with the known classical tolerances into a complete quantitative account of how much spin coherence can actually be restored.

Background

A Stern–Gerlach apparatus (SGA) splits a beam of spin-1/2 atoms into two partial beams according to the projection of the atomic magnetic moment along the gradient of an applied magnetic field. When the incoming atoms are in a coherent spin superposition (say, an eigenstate of σx\sigma_x, spin 'sideways' to the apparatus axis), the splitting entangles the spin with the atomic position: the two spin components travel on macroscopically different paths. The Humpty-Dumpty question — named after the nursery-rhyme egg whom 'all the King's horses and all the King's men couldn't put together again' — is whether the two partial beams can be reunited by further Stern–Gerlach magnets so that a single coherent packet emerges and the original spin superposition is recovered. A scheme of two inverted SGAs appears in Bohm's 1951 textbook, which concluded that recombination is 'in principle attainable' though requiring 'fantastic' precision, and Wigner posed the corresponding operational question in 1963.

Englert, Schwinger, and Scully gave the first quantitative treatment. In their simplified model the Hamiltonian is H=p2/(2m)−E(t)σz−F(t)z σzH=p^2/(2m)-\mathscr{E}(t)\sigma_z-F(t)z\,\sigma_z, where F(t)F(t) is the spin-dependent force from the field gradient and E(t)=μB(t)\mathscr{E}(t)=\mu B(t) sets the Larmor precession (the spin's rotation about the local field direction, with μ\mu the magnetic moment). Solving the Heisenberg equations exactly, they found that the restored spin coherence, measured by ⟨σx⟩\langle\sigma_x\rangle, is a Larmor-phase factor times a spatial-overlap factor exp⁡[−12(Δz/δz)2]exp⁡[−12(Δp/δp)2]\exp[-\frac{1}{2}(\Delta z/\delta z)^2]\exp[-\frac{1}{2}(\Delta p/\delta p)^2], where Δz,Δp\Delta z,\Delta p are residual position and momentum offsets between the recombined partial beams and δz,δp\delta z,\delta p are the incoming wave-packet spreads. For a minimum-uncertainty Gaussian packet, keeping the coherence deficit Q≲0.01Q\lesssim 0.01 requires 12(εs2+εo2)=Q\frac{1}{2}(\varepsilon_s^2+\varepsilon_o^2)=Q, with εs\varepsilon_s the Larmor-phase error and εo\varepsilon_o the orbital-error parameter. Uncontrolled run-to-run variations δF\delta F of the gradient must then obey δF/F0≅10−5\delta F/F_0\cong10^{-5} for representative parameters; for a realistic slit-defined beam with phase-space factor D=(δz δp)/(ℏ/2)∼103D=(\delta z\,\delta p)/(\hbar/2)\sim10^3 the requirement tightens to ∼10−8\sim10^{-8}.

The authors presented this as a best-case, one-dimensional model. A complete account still lacked the following: (i) the quantum properties of the electromagnetic fields that generate the SGA — the fundamental quantum limits on how precisely the field gradient can be prepared and reproduced, a calculation taking field quantization into account that the trilogy did not perform; (ii) the neglected dynamical effects — the spin-dependent change of the longitudinal velocity and the resulting unequal transit times, small spin-dependent reflection probabilities at SGA boundaries, the transverse field components BxB_x and ByB_y with the resulting time dependence of σz(t)\sigma_z(t), and the zz-dependence of the force (their field was linearized in zz), which causes unequal path lengths; (iii) the practical feasibility of the prescribed gradient precision; and (iv) a full three-dimensional, nonideal calculation quantifying how much additional coherence loss these effects introduce.

The question matters beyond the foundations of measurement and reversibility. A Stern–Gerlach interferometer (SGI) — splitting and recombining matter waves with magnetic gradients only — is a leading architecture for proposed fundamental tests, including interferometry with massive objects and tests of the quantum nature of gravity. The original transversal SGI (atoms crossing static SG magnets) and the modern full-loop variant (time-dependent gradient pulses, e.g. from currents in an atom chip, a planar wire device producing controllable magnetic fields near a surface) share the same coherence budget: the achievable spin coherence, which sets the interferometer visibility — the contrast of the interference fringes — is the central resource, and its fundamental limits — technical, dynamical, and quantum — determine feasibility.

Problem Statement

Complete the quantitative account of spin-coherence restoration in Stern–Gerlach recombination at the level of the quantum theory of the apparatus itself, as follows.

(1) Quantize the electromagnetic fields, or their macroscopic sources, that generate the Stern–Gerlach apparatus, and derive the fundamental quantum limits on how precisely the field gradient can be prepared and reproduced from run to run — the quantum-field completion that the 1988–1989 Englert–Schwinger–Scully trilogy left outstanding (those papers treat the SGA magnetic field as an external classical field and do not account for the quantum origin and fluctuations of the field), going beyond the special-case dissipative SQUID-source model of de Oliveira and Caldeira (Phys. Rev. A 73, 042502 (2006)). This requires identifying the operational quantum noise (for example fluctuations of the field quadratures — the phase-space amplitudes of the field modes — and of the photon number of the source, or quantum shot noise of the gradient-generating currents) that limits run-to-run reproducibility beyond the classical tolerance δF/F0≅10−5\delta F/F_0\cong10^{-5} on the uncontrolled run-to-run variation of the gradient-induced spin-dependent force (tightening to about 10−810^{-8} for realistic beam phase-space factors D=(δz δp)/(ℏ/2)∼103D=(\delta z\,\delta p)/(\hbar/2)\sim10^3, the ratio of the incoming beam's phase-space area to its minimum-uncertainty value).

(2) Quantify the spin-coherence loss induced by the quantum fluctuations of these fields during splitting and recombination, and combine it with the known classical limitations — the one-dimensional tolerance analysis of the 1988 paper and the three-dimensional irreversibility established for Maxwell-compliant classical fields (Paraniak and Englert 2021) — into a bound on the actually achievable restored spin coherence in a realistic nonideal apparatus, including uncontrolled run-to-run gradient variations and small spin-dependent reflection at apparatus boundaries.

Determine, with supporting derivations and quantitative estimates, how much initial spin coherence can be restored after Stern–Gerlach splitting-and-recombination when both the classical three-dimensional dynamics and the quantum nature of the apparatus fields are taken into account, and which class of limitation dominates in which parameter regime.

References

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