Experimental Speedup of Quantum Dynamics through Squeezing

We show experimentally that a broad class of interactions involving quantum harmonic oscillators can be made stronger (amplified) using a unitary squeezing protocol. While our demonstration uses the motional and internal states of a single trapped ${}^{25}{\mathrm{Mg}}^{+}$ ion, the scheme applies generally to Hamiltonians involving just a single harmonic oscillator as well as Hamiltonians coupling the oscillator to another quantum degree of freedom such as a qubit, covering a large range of systems of interest in quantum information and metrology applications. Importantly, the protocol does not require knowledge of the parameters of the Hamiltonian to be amplified, nor does it require a well-defined phase relationship between the squeezing interaction and the rest of the system dynamics, making it potentially useful in instances where certain aspects of a signal or interaction may be unknown or uncontrolled, such as searches for novel forms of dark matter.

Popular Summary

Quantum dynamics—the way quantum systems interact with each other and their environment—is the basis of all quantum information and sensing. However, the strength of these interactions is often fixed by experimental or environmental constraints. Methods that amplify or speedup quantum dynamics could enable faster entangling gates for quantum computing, for example, or increase the sensitivity of a quantum sensor. Quantum squeezing, a technique that reduces uncertainty in one observable by transferring the uncertainty into a different observable, has been used to amplify quantum dynamics, but successful amplification required detailed foreknowledge of the dynamics to be amplified so that the correct observable could be squeezed. Here, we demonstrate a squeezing-based amplification method that does not require advance knowledge of the dynamics to be amplified.

The protocol uses repeated squeezing and unsqueezing of two conjugate observables of a quantum harmonic oscillator (such as position and momentum) to amplify quantum dynamics that involve the harmonic oscillator, resulting in uniform amplification regardless of where the dynamics are relative to the squeezing. Our experimental demonstration uses the motion of a single trapped magnesium ion as a quantum harmonic oscillator, and we amplify two different types of interactions: small motional displacements caused by applied electric fields and a laser-beam-induced coupling between the ion’s motion and internal levels.

Our new amplification procedure represents an advance in the speedup of quantum dynamics, expanding the kind of systems that can benefit from quantum amplification.

Figure 1

Pulse sequences for different quantum amplification schemes using unitary squeezing. (a) Pulse sequence for the phase-sensitive amplification scheme used in Ref. [11]. (b) Pulse sequence for Hamiltonian amplification of a coherent displacement. (c) Pulse sequence for Hamiltonian amplification of a general interaction as in Eq. (5), which must be appropriately Trotterized [36, 37].

Figure 2

Schematic of Hamiltonian amplification of a coherent displacement. We plot a series of schematic representations of the motional phase space distribution, before (dotted gray outline) and after (orange) each numbered pulse. The arrows indicate squeezing (gray) and displacement (blue) operations, respectively. In the final panel, the red arrow shows the total unamplified displacement, while the black arrow shows the amplified displacement, which points in the same direction, but is larger by a factor $G(r)$.

Figure 3

HA gain versus displacement phase. Experimental data are shown as red circles, with error bars denoting 68% confidence intervals. The red line (light red band) shows the theoretical prediction (68% confidence interval) for the HA gain $G=\cosh (r)$ given the experimentally determined value of $r=1.38(8)$. The theoretical phase-dependent gain of the scheme in Fig. 1 given the same $r$ is shown for comparison as the black line. The gray line denotes $G=1$. The data point plotted at $\phi /2\pi =1$ is a copy of the data point at $\phi /2\pi =0$.

Figure 4

Hamiltonian amplification of the Jaynes-Cummings (JC) interaction. Population in $|\tilde{\uparrow }⟩$ as a function of the JC interaction duration for a Hamiltonian amplification sequence with $N=6$. For the amplified case (red data points and fit line), the duration of the squeezing pulses is not included (see the main text). The JC interaction induces Rabi oscillations between the $|\tilde{\uparrow }⟩|n=0⟩$ and $|\tilde{\downarrow }⟩|n=1⟩$ states with a rate $\mathrm{\Omega }$. Increasing the squeezing parameter from $r=0$ to $r=1.1$ increases $\mathrm{\Omega }$ by a factor of 1.56(2). The solid lines are fits to exponentially decaying sinusoids. Error bars indicate 68% confidence intervals.

Figure 5

Wall clock speedup. We plot the ratio of total durations with and without HA, $T_{\mathrm{HA}}/T_0$, for a range of values of $r$. The two panels show the cases without (a) and with (b) Trotterization during HA. The black dotted lines indicate the wall clock speedup threshold $T_{\mathrm{HA}}/T_0=1$.

Figure 6

Pulse sequences and calibration data for amplification of coherent displacements. GSC, ground-state cooling; BSB, blue sideband analysis pulse. Displacement and squeezing operators are the same as in Fig. 1. The lower plot in each panel shows the measured population in $|\downarrow ⟩$ (purple circles) as a function of BSB analysis pulse duration for various initial motional states. The dark blue lines show the result of model-free fits to the data from Eq. (D1), as described in the text, and the light blue lines show the model-based fits used to extract the various parameters ($r$, $\overline{n}$, $\alpha$). Both the Fock state plots and BSB flops demonstrate good agreement between the model-based and model-free fits, validating our choice of fit models. The upper plot in each panel shows the corresponding Fock state populations extracted by fitting to the BSB oscillations. The red lines show the best-fit Fock state populations from a model-based fit to the BSB oscillations, while the yellow bars show the Fock state populations extracted from a model-free fit. The vertical black lines show 68% confidence intervals on the model-free fitted populations. Diagonal black lines connect the tops of the yellow bars to guide the eye and highlight deviations between the model-based and model-free fitted populations. (a) Ground-state cooling calibration. We find a mean occupation of $\overline{n}=0.06(1)$. (b) Displacement calibration. Following ground-state cooling, the motional state is displaced by an applied resonant excitation pulse split into two $3.5$-$\text{μ}\mathrm{s}$ segments ($7\text{μ}\mathrm{s}$ total), yielding ${\alpha }_i=0.55(2)$. (c) Squeezing calibration. Following ground-state cooling, the parametric drive is applied for $2.75\text{μ}\mathrm{s}$ to generate a squeezed state of motion. (d) Antisqueezing calibration and squeezing coherence verification. The ion is cooled to its motional ground state, the parametric drive is applied for $2.75\text{μ}\mathrm{s}$, and then the parametric drive is applied for an additional $2.75\text{μ}\mathrm{s}$ with a ${180}^{\circ }$ phase shift. Fitting the resulting state to a model for a thermal state, we find that $\overline{n}=0.09(1)$. (e) Hamiltonian amplification experiment. Final motional state after HA of a coherent displacement with $N=1$, using two displacement pulses of $3.5$-$\text{μ}\mathrm{s}$ duration and four squeezing or antisqueezing pulses of $2.75$-$\text{μ}\mathrm{s}$ duration. The data in Fig. 3 are produced by comparing the calibrated unamplified displacement as seen in (b) to the displacement extracted in (e) for different relative phases between the displacement and squeezing pulses.

Figure 7

HA gain versus displacement phase for $N=3$. This plot shows the same experiment as in Fig. 3, but with $N=3$ rather than $N=1$. This means that the displacement is split into six equal pieces rather than two, which, due to experimental imperfections, results in decreased amplification but improved phase insensitivity. Experimental data are shown as red circles, with error bars denoting 68% confidence intervals. The red line (light red band) shows the theoretical prediction (68% confidence interval) for the HA gain $G=\cosh (r)$ given the experimentally determined value of $r=1.38(8)$. The theoretical phase-dependent gain of the scheme in Fig. 1 given the same $r$ is shown for comparison as the black line. The gray line denotes $G=1$. The data point plotted at $\phi /2\pi =1$ is a copy of the data point at $\phi /2\pi =0$.