Magnetically Levitated Schrödinger Cat States

• Magnetically levitated Schrödinger cat states are non-classical superpositions where mesoscopic magnetic particles exist simultaneously in distinct spatial locations, providing a platform for quantum tests.
• They are generated by coherently coupling spin states with center-of-mass motion using high magnetic gradients and precise timing, achieving superposition separations from sub-micron to tens of microns.
• These states serve as experimental probes for quantum gravity, collapse models, and force sensing, leveraging ultra-high vacuum, cryogenic cooling, and advanced magnetic control to mitigate decoherence.

Magnetically levitated Schrödinger cat states are non-classical superpositions in which a mesoscopic or macroscopic magnetic particle exists simultaneously in spatially distinct locations, realized via magnetic trapping, control of spin-state superpositions, and measurement protocols that entangle or disentangle motional and spin degrees of freedom. These states provide a unique experimental platform for testing quantum mechanics at new mass and length scales, for probing proposed modifications to quantum theory, and for investigating quantum aspects of gravity and inter-particle forces. The following sections synthesize the key methodologies, theoretical constructs, experimental challenges, and future applications of magnetically levitated Schrödinger cat states, as elucidated in recent foundational works.

1. Experimental Realization of Magnetically Levitated Schrödinger Cats

Magnetically levitated cat states are typically formed using ferromagnetic, ferrimagnetic, or diamagnetic nanocrystals levitated in ultra-high vacuum traps and cooled to deep cryogenic temperatures. The primary implementations fall into ion/Paul traps for charged, ferromagnetic nanoparticles (e.g., FePt, YIG) (Rahman, 2018, Rahman, 2019), and static field-based diamagnetic traps for neutral nanodiamonds hosting color centers (Marshman et al., 2023). The key trap characteristics are:

• Harmonic confinement: The effective center-of-mass (COM) Hamiltonian is harmonic to lowest order, $$H_{\rm CM} = \frac{\hat{p}^2}{2m} + \frac{1}{2} m \vec{\omega}^2 \vec{x}^2$$.
• High magnetic gradients: Pulsed gradients $dB/dx \sim 10^3$ – $10^6$ T/m are used to couple internal spin to the COM motion, imparting spin-dependent forces.
• Ground-state cooling: Feedback cooling, buffer-gas cooling, or dilution refrigeration reduces the COM temperature to $T_{\rm cm} \sim 1$ mK or below, enabling spatial spread well below the eventual superposition scale.

A representative example in FePt is a $20$ nm-radius nanosphere ($m\sim2\times10^{-20}$ kg) in a 150 kHz Paul trap at 300 mK, achieving initial position uncertainty $\sqrt{\langle x^2 \rangle} \approx 1$ nm (Rahman, 2018). For NV-hosting diamonds ($200$ nm diameter), diamagnetic traps with oscillator frequency $2\pi\times100$ Hz to $2\pi\times1$ kHz are employed (Marshman et al., 2023).

2. Spin–Motion Coupling and Cat-State Generation

Central to cat-state creation is the coherent coupling of a spin or qubit degree of freedom to the particle's COM motion in a spatially inhomogeneous magnetic field. The effective Hamiltonian is generically

$$\hat{H} = \frac{\hat{p}^2}{2m} + V_{\rm trap}(\hat{x}) - g_L\mu_B \hat{S}_z B(\hat{x}),$$

with $B(\hat{x}) = B_0 + \hat{x} (dB/dx)$, and $\hat{S}_z$ the spin operator (macrospin for magnetic materials or electronic spin for NV centers). The spin-state preparation differs by platform:

• Ferromagnetic nanoparticles: Exploit all-optical helicity-dependent switching (AOHDS) to prepare macrospin superpositions $$|\psi_{\rm spin}\rangle = (|\uparrow_M\rangle + |\downarrow_M\rangle)/\sqrt{2}$$ via ultrashort linearly polarized laser pulses (Rahman, 2018).
• Diamagnetic nanodiamonds: Employ microwave $\pi/2$ pulses to create $(|+1\rangle + |-1\rangle)/\sqrt{2}$ superpositions on the NV center (Marshman et al., 2023).
• YIG nano-crystals: Use tunnel-split ground-state doublets arising from uniaxial anisotropy to realize even-parity cat states $\ket{\phi_0} = (\ket{+S_z} + \ket{-S_z})/\sqrt{2}$ (Rahman, 2019).

After spin superposition, a magnetic gradient pulse is applied, resulting in each spin projection experiencing a force $F_\pm = \mp g_L\mu_B S_z (dB/dx)$, which accelerates the spatial wavepackets apart. The separation after time $t$ is

$$\Delta x(t) = \frac{g_L\mu_B S_z}{4m} t^2 \frac{dB}{dx}$$

(for FePt and YIG), or tailored via quarter-period pulses in harmonic potential for diamonds. With engineered pulse sequences (single step, echo, or interferometric closure), coherent two-arm mesoscopic superpositions with separations $\Delta x \sim 0.1$ – $25\,\mu$m are attainable, exceeding the particle diameter by several orders of magnitude (Rahman, 2018, Rahman, 2019, Marshman et al., 2023).

3. Protocols for Interferometry, Expansion, and Entanglement

Cat-state interferometry proceeds via carefully timed sequences of trap release, gradient application, and spin readout:

• Sequential protocol: (1) Trap off, (2) spin superposition, (3) magnetic gradient ON (arms split), (4) gradient OFF, (5) spin disentangling, (6) reversed gradient (arms recombine), (7) free flight and interference/measurement (Rahman, 2018, Rahman, 2019).
• Diamagnetic protocols: Symmetric Stern-Gerlach field pulses (e.g., $t_p/4 -$ gradient, free oscillation, $t_p/4$ - gradient closure) are used to bring arms together in both position and momentum (Marshman et al., 2023).
• Exponential expansion: Exponentially growing superpositions are created by switching from harmonic to inverted harmonic potentials, resulting in coherent branch separations $\Delta x \sim \sqrt{2}\,\delta x \exp(t_{-})$, with precise unitary control enabling recombination and phase readout sensitive to forces or gravity-mediated entanglement (Braccini et al., 21 Aug 2024).

In two-mass entanglement protocols, parallel cat-state interferometers are coupled via magnetic dipole–dipole (induced by background fields), Casimir–Polder, or hypothetical gravitational interactions. The interaction Hamiltonians produce entangling phases that are detected via population measurements on the joint spin state after interferometric closure. A positive-partial-transpose (PPT) entanglement test can be directly computed from the reduced density matrix (Marshman et al., 2023, Braccini et al., 21 Aug 2024).

4. Decoherence Mechanisms and Macroscopicity

Cat-state longevity is limited by environmental and technical decoherence:

• Spin decoherence: $T_2 \sim 10\,\mu$s (FePt/YIG), $T_2 \sim 1-30$ s (NV centers), setting an upper bound on interaction and superposition times (Rahman, 2018, Rahman, 2019, Marshman et al., 2023).
• COM decoherence: Ultra-high vacuum ($P < 10^{-12}$ Pa) at cryogenic temperature ($T < 1$ K) suppresses gas collision rates ($\Gamma_{\rm gas} \ll 1$ per protocol duration); blackbody and photon recoil are negligible under protocol timescales (Rahman, 2018, Marshman et al., 2023, Rahman, 2019).
• Magnetic noise: Demands $\delta B/B < 10^{-6}$ over protocol windows ($\sim 0.1$ s) for negligible spin dephasing (Marshman et al., 2023).
• Collapse models: Continuous Spontaneous Localization (CSL) predicts spatial decoherence rate $$\Gamma_{\rm CSL} = \lambda_{\rm CSL}(m/m_0)^2 [1 - e^{-\Delta z^2/(4 r_c^2)}]$$, tightly constrained by successful generation of micron-scale cats (Rahman, 2019).

Macroscopicity measures, e.g., $\mu = \log_{10}[|(m/m_e)^2 t_f \ln f|]$, reach $\mu \approx 16$–29, surpassing molecular interference records, especially for core–shell mass architectures (Rahman, 2018, Rahman, 2019).

5. Readout and Verification Techniques

Cat-state verification relies on interference, spin correlation, and population measurements:

• Fringe visibility: Overlap of recombined arms produces high-contrast interference fringes, with fringe period $P_0 = 2\pi\hbar t_f/(m x_0)$ (e.g., $P_0 \approx 60$ nm for $t_f = 100$ ms, $x_0 = 20$ nm) (Rahman, 2018).
• Spin measurement: Projective spin measurements (e.g., via polarized optical pulses or SQUID pickup of ferromagnetic resonance) yield probabilities $P(\pm S_x) = \frac{1}{2}[1 \pm \cos \beta_g]$, with phase $\beta_g$ encoding gravitational or interaction-induced effects (Rahman, 2019).
• Entanglement witnesses: For dual interferometers, the entanglement witness $\mathcal{W}$ is defined by measured spin populations and predicted joint phase evolution, witnessing entanglement if $\mathcal{W} < 0$ (Marshman et al., 2023).

In the GIE context (gravitationally induced entanglement), the two-mode Gaussian evolution of the coupled masses is solved exactly; a negative eigenvalue in the partially transposed density matrix indicates non-classical correlations (Braccini et al., 21 Aug 2024).

6. Applications: Fundamental Tests, Sensing, and Quantum Gravity

Magnetically levitated Schrödinger cat states serve as testbeds for:

• Quantum collapse models: Protocols with $\Delta z \gg r_c$, $t_{\rm coh} \gtrsim 1/\Gamma_{\rm CSL}$, and observed coherence directly constrain CSL parameters by excluding fast localizations (Rahman, 2019).
• Force sensing: Exponentially expanding superpositions enhance force sensitivity, as the interferometric phase $\phi_F \sim e^{t_-}$ allows rapid, high-resolution detection of weak forces, outperforming Heisenberg-limited protocols (Braccini et al., 21 Aug 2024).
• Modifications of gravity: Dual-setup interferometry can be used to detect quantum gravity-induced entanglement (GIE), Casimir–Polder potentials, or gravity as a classical mediator. The phase accrual $\phi_g = G m^2 t/(\hbar d)$ encodes gravitational interaction strength (Rahman, 2019, Marshman et al., 2023, Braccini et al., 21 Aug 2024).
• Macroscopic quantum phenomena: Increasing the mass (via core-shell structures) and size of superpositions extends the test regime, with quantum-to-classical transition mechanisms and self-gravity decoherence (Diósi–Penrose) accessible for $m\sim10^{-13}$ kg and $\tau_{\rm Penrose}\sim 0.1$ ms (Rahman, 2019).

7. Experimental Parameters and Feasibility

Current proposals and state-of-the-art experiments converge on the following feasible parameter range, collated from the primary studies:

Property	Values (FePt/YIG)	Values (Diam. NV)	Typical Reference
Particle radius	10–20 nm	100 nm	(Rahman, 2018, Rahman, 2019, Marshman et al., 2023)
Mass	$2\times10^{-20}$–$3\times10^{-19}$ kg	$3\times10^{-19}$ kg	–
Trap frequency	$2\pi \times 10^2$–$10^6$ Hz	$2\pi \times 10^2$–$10^3$ Hz	–
Magnetic gradient	$10^3$–$10^6$ T/m	$10^6$–$10^7$ T/m	–
Superposition size $\Delta x$	$5$–$25\,\mu$m	$0.1$–$1\,\mu$m	–
Spin coherence $T_2$	$10\,\mu$s–$30$ s	$1$–$30$ s	–
Cat mass for gravity tests	up to $10^{-13}$ kg	up to $10^{-13}$ kg (core-shell)	(Rahman, 2019)
Ultra-high vacuum	$<10^{-12}$ Pa	$<10^{-14}$ Pa	–
Base temperature	$<1$ K	$<1$ K	–

Practical implementation demands precision in timing ($\delta t < 1\,\mu$s), stability in magnetic fields ($\delta B/B < 10^{-6}$), and low spin-flip error rates ($\ll 1\%$), all compatible with modern cryogenic, vacuum, and quantum control technology (Marshman et al., 2023, Braccini et al., 21 Aug 2024).

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Magnetically levitated Schrödinger cat states have matured into a viable platform for macroscopic quantum experiments, offering high mass and spatial scale, robust spin control, and sensitivity to both canonical and speculative extensions of quantum theory. Continued progress in materials purity, measurement sensitivity, and noise suppression is expected to unlock further regimes of non-classicality and enable quantum tests of gravity and novel force laws.