Quantum Harmonic Oscillator State Control in a Squeezed Fock Basis

We demonstrate control of a trapped-ion quantum harmonic oscillator in a squeezed Fock state basis, using engineered Hamiltonians analogous to the Jaynes-Cummings and anti-Jaynes-Cummings forms. We demonstrate that for squeezed Fock states with low $n$ the engineered Hamiltonians reproduce the $\sqrt{n}$ scaling of the matrix elements which is typical of Jaynes-Cummings physics, and also examine deviations due to the finite wavelength of our control fields. Starting from a squeezed vacuum state, we apply sequences of alternating transfer pulses which allow us to climb the squeezed Fock state ladder, creating states up to excitations of $n=6$ with up to 8.7 dB of squeezing, as well as demonstrating superpositions of these states. These techniques offer access to new sets of states of the harmonic oscillator which may be applicable for precision metrology or quantum information science.

Figure 1

Energy-eigenstate populations of (a) $|\zeta ,0⟩$, (b) $|\zeta ,1⟩$, (c) $|\zeta ,2⟩$, and (d) $|\zeta ,3⟩$. The height of the yellow bars indicates the probability of finding the ion in the $n$th energy eigenstate, with error bars included as standard error on the mean extracted from fits. The blue points indicate the expected populations for the squeezed Fock states with $r=1$. As can be seen from the increased error bars, for Fock states above $k=16$, it is difficult to separate contributions of neighboring states. This is due to the limited number of Rabi oscillations that we can perform in our experiment and the reduced separation between the Rabi oscillations on neighboring energy eigenstates.

Figure 2

Rabi frequency scaling in the squeezed Fock basis. (a) Example data with Rabi oscillation fits using a Gaussian decay to the spin populations versus time for (i) the $|\uparrow ⟩|\zeta ,0⟩\leftrightarrow |\downarrow ⟩|\zeta ,1⟩$ and (ii) the $|\downarrow ⟩|\zeta ,5⟩\leftrightarrow |\uparrow ⟩|\zeta ,6⟩$ transitions. (b) The measured ratio ${\mathrm{\Omega }}_{n,n-1}/{\mathrm{\Omega }}_0$ obtained from similar data for squeezed Fock states up to $n=7$ with $r=1$. Squares are measurements made using ${\widehat{H}}_{+}$ and circles for ${\widehat{H}}_{-}$. Theoretical curves are also plotted for $\sqrt{n}$ (dashed line) and for the modified theory taking into account the finite Lamb-Dicke parameter (solid line). (c) A plot of measured ${\mathrm{\Omega }}_{n,n-1}/{\mathrm{\Omega }}_0-\sqrt{n}$, showing the deviations from the $\sqrt{n}$ behavior. The statistical error bars do not account for the deviations of the measurement from the theory (solid line), indicating that the error sources are primarily systematic. In simulations, we see that a finite detuning between half the difference frequency of the lasers and the trap frequency increases the Rabi frequency (see Supplemental Material for more details [16]). For comparison, the dotted curve indicates the matrix elements for the $n$th state of the energy eigenbasis for the same Lamb-Dicke parameter.

Figure 3

Measurements performed on a superposition of squeezed Fock states. (a) Population analysis in the squeezed Fock basis, with $p(n)$ close to 0.5 for both $n=0$ and $n=2$. The bar heights and error bars are extracted from fits to Rabi oscillation data with the ${\widehat{H}}_{+}$ Hamiltonian. Blue points are the expected values for the desired superposition. (b) Energy-eigenstate population analysis, showing that the primary occupied levels are those with even parity. (c) Population of the $\downarrow$ internal state as a function of the phase ${\varphi }_a$ of a red-sideband analysis pulse, indicating the phase coherence between the superposed states. The contrast gives a lower bound on the fidelity of the superposition state.