Test of Causal Nonlinear Quantum Mechanics by Ramsey Interferometry with a Trapped Ion

Quantum mechanics requires the time evolution of the wave function to be linear. While this feature has been associated with the preservation of causality, a consistent causal nonlinear theory was recently developed. Interestingly, this theory is unavoidably sensitive to the full physical spread of the wave function, rendering existing experimental tests for nonlinearities inapplicable. Here, using well-controlled motional superpositions of a trapped ion, we set a stringent limit of $5.4\times {10}^{-12}$ on the magnitude of the unitless scaling factor ${\tilde{\epsilon }}_{\gamma }$ for the predicted causal nonlinear perturbation.

Figure 1

Experimental implementation. (a) A ${}^{40}{\mathrm{Ca}}^{+}$ ion is trapped using a combination of rf and dc electric fields. In a time-averaged sense, the confinement is well modeled by a three-dimensional harmonic potential. (b) Motion along the $x$ direction is excited by resonantly coupling to the internal electronic Zeeman sublevels of the $4^2{S}_{1/2}\leftrightarrow 3^2{D}_{5/2}$ transition using narrow band light near 729 nm. The degeneracy of the Zeeman states is broken through application of a strong magnetic field of $\approx 4\mathrm{G}$. Measurement is performed by scattering photons of the short-lived $4^2{S}_{1/2}\leftrightarrow 4^2{P}_{1/2}$, which are then focused onto an EMCCD camera. (c) The experimental pulse sequence. Pulses that address the qubit are colored gray and those that address the blue sideband, blue. After preparing the state $|S,0⟩$, the first pair of pulses is used to generate the state $|\psi (t=0)⟩=|D⟩({\alpha }_0|0⟩+{\alpha }_1|1⟩)$. This is then allowed to freely evolve for a time $\tau$, accumulating a relative phase of $\mathrm{\Phi }(\tau )$, which is sensitive to the proposed causal nonlinear perturbation. Afterward, the information is mapped onto the qubit with a blue sideband pulse and then the expectation value of the Pauli spin operator $\cos ({\xi }_L){\sigma }_x+\sin ({\xi }_L){\sigma }_y$ is measured. (d) An illustration of the two-step process for generating the state $|\psi (t=0)⟩$, as described in more detail in the main text. The key feature is the fact that the state $|D,0⟩$ is transparent to the resonant blue sideband drive as illustrated in (e). This allows us to map an arbitrary qubit state onto the ground and first excited state of the vibrational mode.

Figure 2

Experimental performance.—(a) Measured $P(\tau )$, as described by Eq. (4) (red). The black dashed line is the predicted decay envelope taking into account only heating of the vibrational mode at a rate of $10\mathrm{quanta}/\mathrm{s}$. The reasonable agreement between the predicted and measured decay suggests that the Ramsey signal contrast is dominated by this heating process. (b) The black circles represent the sample standard deviation from repeated measurements of $\mathrm{\Delta }{\varphi }_{\mathrm{NL}}(\tau )$ taken at various interrogation times and normalized to an integration time of 1 s. The blue shaded region bounds the simulated predictions assuming only QPN and a heating rate between 7 and $13\mathrm{quanta}/\mathrm{s}$ (lower and upper edge of the region, respectively). The dark blue line corresponds to $10\mathrm{quanta}/\mathrm{s}$.

Figure 3

Results. (a) (top to bottom) The measured contrast, frequency and offset over a full day of data collection. The blue circles represent data taken at an interrogation time of 15 ms and the red circles were taken at a time of 15 ms divided by the golden ratio ($\approx 9.3\mathrm{ms}$). (b) The distribution of ${\tilde{\epsilon }}_{\gamma }$ estimated from the data. The mean value is $5\pm 5.4\times {10}^{-12}$. The black curve is a Gaussian fit to the distribution.