Phonon Pair Creation by Inflating Quantum Fluctuations in an Ion Trap

Quantum theory predicts intriguing dynamics during drastic changes of external conditions. We switch the trapping field of two ions sufficiently fast to tear apart quantum fluctuations, i.e., create pairs of phonons and, thereby, squeeze the ions’ motional state. This process can be interpreted as an experimental analog to cosmological particle creation and is accompanied by the formation of spatial entanglement. Hence, our platform allows one to study the causal connections of squeezing, pair creation, and entanglement and might permit one to cross-fertilize between concepts in cosmology and applications of quantum information processing.

Figure 1

Particle creation out of quantum fluctuations in harmonic oscillators and the early Universe. (a) The ground state wave function ${\psi }_k(x)$ (black solid line) of a harmonic oscillator has finite energy and spatial spread (variance) ${(\mathrm{\Delta }x)}^2\propto 1/{\mathrm{\Omega }}_k$. When ${\mathrm{\Omega }}_k$ is changed to a different value (dashed parabola), ${\psi }_k(x)$ evolves accordingly. (b) If the change of ${\mathrm{\Omega }}_k$ is fast, ${\psi }_k(x)$ cannot follow adiabatically and represents an excited, squeezed state (solid line) within the new potential, that is characterized by a decreased spread ${(\mathrm{\Delta }x)}^2$ compared to the new ground state (dashed line). (c) Schematic of curved space-time during cosmic inflation. Quantum fluctuations are depicted as a pair of virtual particles at some time $t_0$. Because of inflation the two virtual particles are torn apart until their distance (related to the physical wavelength of the corresponding mode, shaded area) becomes too large for them to recombine and annihilate (at $t_1$). Thereafter the two virtual particles have become real and move into opposite directions with momenta $\pm \hslash k$. (d) In the harmonic oscillator analog, the creation of multiple particle pairs with momenta $\pm \hslash k$ is evidenced by a squeezed state that contains excitations of even numbers of particles only.

Figure 2

Accessible experimental parameter regime in our setup. (a) Frequency of the harmonic oscillator mode ${\omega }_2$ as a function of $U$; data points depict experimental results (error bars smaller than symbol size), the solid line shows a model fit to the data. The selected frequency range (shaded area) is compared to the analog scale parameter $a_2$ (right-hand axis), which corresponds to 1.6 $e$-foldings of space. (b) Real-time switching of ${\omega }_2$ (solid line) within $t_{\mathrm{ramp}}=1\mu \mathrm{s}$ and corresponding time evolution of $a_2$ (dashed line, right-hand axis). (c) The fast variation ${\dot{\omega }}_2/{\omega }_2^2$ (solid line) is accompanied by significant time derivatives of the scale parameter ${\dot{a}}_2$ (dashed line, right-hand axis). (d) Numerical simulation of the quantum dynamics. Quantum fluctuations are inflated and squeezing $r$ (solid line) emerges. This corresponds to the pairwise creation of ${\overline{n}}_{\mathrm{sq}}$ phonons (dashed line, right-hand axis).

Figure 3

Experimental results for particle creation and purification of squeezing with an echo sequence consisting of two pulses separated by a free evolution duration $t_{\mathrm{free}}$. (a) Phase space illustration of the anticipated echo effect: (i) The initial vacuum state, (ii) squeezed and displaced by the first pulse, (iii) rotates on a circular path in phase space. (iv) After an optimal duration $t_{\pi }=\pi /{\omega }_2$ (or odd multiples) the residual coherent displacement is reversed by the second pulse. (v) The final state shows enhanced squeezing excitation. (b) Experimental results for squeezed (circles) and coherent (squares) excitation as a function of $t_{\mathrm{free}}$, error bars indicate the standard error of the mean. Numerical simulations (lines) indicate the different oscillation rates of $r(t_{\mathrm{free}})$ and $\alpha (t_{\mathrm{free}})$; see Ref. [10]. (c) Phonon number distribution (data points, error bars indicate three standard error of the mean) for $t_{\mathrm{free}}=30.2\mu \mathrm{s}$. From a model fit (bars) we extract $r=0.83(8)$ and $|\alpha |=0.29(15)$ and the corresponding Wigner function (inset, phases omitted for clarity). The squeezed state evidences entanglement in the spatial degrees of freedom (d.o.f.) of the two ions, represented by the superposition $|{}^{•}{}_{•}⟩+|{}_{•}{}^{•}⟩$; see main text.