Optomechanical many-body cooling to the ground state using frustration

We show that the vibrations of an ion Coulomb crystal can be cooled to the zero-point motion through the optomechanical coupling with a high-finesse cavity. Cooling results from the interplay between coherent scattering of cavity photons by the ions, which dynamically modifies the vibrational spectrum, and cavity losses, that dissipate motional energy. The cooling mechanism we propose requires that the length scales of the crystal and the cavity are mismatched so that the system is intrinsically frustrated, leading to the formation of defects (kinks). When the pump is strong enough, the anti-Stokes sidebands of all vibrational modes can be simultaneously driven. These dynamics can be used to prepare ground-state chains of dozens of ions within tens of milliseconds in state-of-the-art experimental setups. In addition, we identify parameter regimes of the optomechanical interactions where individual localized modes can be selectively manipulated, and monitored through the light at the cavity output. These dynamics exemplify robust quantum reservoir engineering of strongly correlated mesoscopic systems and could find applications in optical cooling of solids.

Figure 1

The axial vibrations of an ion chain are cooled to the ground state through the optomechanical coupling with a cavity. The cavity has a standing-wave mode of frequency ${\omega }_c$, which is pumped by a laser at frequency ${\omega }_p<{\omega }_c$ and dissipates photons at the rate $\kappa$. The figure shows a schematic representation of particle positions in (a) the sliding phase and (b) the pinned phase in the optical potential; these phases are found for incommensurate lengths of the ion chain and the cavity wavelength. Due to this competition, in the pinned phase defects called kinks are formed, which are essential for coupling all modes simultaneously to the cavity. (c) For low pump strengths (here the pump strength is $\eta =1.5\kappa$) the particles are in the sliding phase and the vibrational frequencies can be individually resolved, as visible in the spectrum at the cavity output, $S\left(\nu \right)$ (in units of $1/\mathrm{Hz}$), as a function of the frequency shift $\nu$ from the pump frequency (see the Appendix for details and Sec. 3c for the parameters). Each mode can thus be separately cooled by resonant coherent scattering processes when a pump photon at frequency ${\omega }_p$ is absorbed and a cavity photon at frequency ${\omega }_c$ is emitted, carrying away the energy of a vibrational excitation (determined by ${\mathrm{\Delta }}_{\mathrm{eff}}$, which is the effective detuning of the laser pump from the cavity field including the frequency shift due to the presence of the ions, see text for details), as sketched in (e). For a strong pump (here $\eta =300\kappa$) the particles are in the pinned phase, the bandwidth of the modes (d) much smaller than the cavity linewidth $\kappa$ and (f) coherent scattering can simultaneously drive all anti-Stokes side bands, thus cooling the chain.

Figure 2

Axial displacements of the ions for each normal mode $\alpha$, for parameters such that the chain is (a) in the sliding phase, (b) in the pinned phase, close to the sliding-to-pinned transition, and (c) deep in the pinned phase (modes are ordered from the lowest frequency at the bottom to the highest at the top). The displacements are given by the values of the matrix $M_j^{\alpha }$ in Eq. (19) as a function of the ion $j$ for 11 ions, where $j$ labels the ions from left to right. The red horizontal lines indicate the value of zero axial displacement (with respect to the equilibrium position of each ion), about which the corresponding value of $M_j^{\alpha }$ is reported. (a) and (c) have been calculated for the same parameters of the cavity output spectra in Figs. 1 and 1, with $\eta =1.5\kappa$ and $\eta =300\kappa$ respectively, while subplot (b) is with $\eta =55\kappa$. The presence of the kink in (b)–(c) significantly modifies the normal modes.

Figure 3

Mean asymptotic excitation number ${\overline{n}}_{\alpha }$, averaged over all modes with nonzero coupling to the cavity fluctuations, for $N=11$ ions (see text for the relevant parameters). The dashed line indicates the sliding-pinned transition, to the left of which the symmetric modes of the chain are decoupled from the cavity. Phonon numbers above 1 are indicated by the brightest color in the figure.

Figure 4

Analysis of cavity cooling for ${\mathrm{\Delta }}_c=-8.5\kappa$ (the remaining parameters are given in the text). (a) Coupling coefficients $|{\chi }_{\alpha }|$ between each phonon mode $\alpha$ and cavity fluctuations as a function of $\eta$, ordered from lowest frequency mode (black line), to highest frequency mode (yellow/light gray line). (b) Frequencies of the generalized phonon-photon modes as a function of the pump strength. For this detuning there is a resonance between the photon fluctuation frequency $-{\mathrm{\Delta }}_{\mathrm{eff}}$ (blue dashed line) and the vibrational modes at $\eta =250\kappa$ (green/dark gray vertical line); we investigate the cooling at this point and at $\eta =300\kappa$ (orange/light gray vertical line). (c) Mean steady-state occupation of each vibrational chain mode as a function of $\eta$ and (d) at $\eta =250\kappa$ (green/dark gray bars) with black lines representing the analytic results from Eq. (27). The orange (light gray) bars are at a $20\%$ fluctuation of the pump strength to $\eta =300\kappa$. (e) Cooling rates ${\mathrm{\Gamma }}_{\tilde{\alpha }}$ (in log scale) for each of the generalized modes, for the same values of $\eta$ as in (d). For these parameters, the last eigenmode has a dominantly photonic character, while each of the first 11 generalized modes is composed mainly by a particular vibrational mode that is mixed with cavity fluctuations. The black lines indicate the analytic estimates of the cooling rates from Eq. (26) for the corresponding vibrational modes. (d) and (e) demonstrate that the chain can be robustly cooled close to its ground state for all modes simultaneously.

Figure 5

(a) Mean steady-state occupation averaged over all vibrational modes as a function of the chain size, for $N=11,51,81$. For each $N$ the distance between the ions at the chain center and the cavity potential depth are kept constant. (b) Average cooling rates (in ${\mathrm{s}}^{-1}$) of the generalized modes as a function of the chain size (in logarithmic scale for the vertical axis). (c) Steady-state occupation of the vibrational modes for $N=51$, and (d) corresponding cooling rates of the 51 eigenmodes which are dominantly vibrational (in logarithmic scale). We note that for studying the scaling behavior we fix $\eta$ and $\kappa$ so that the results are not optimized for the larger system size (see the text for parameters and discussion).

Figure 6

(a) Generalized eigenfrequencies ${\omega }_{\tilde{\alpha }}$ in units of $\kappa$, when fixing ${\mathrm{\Delta }}_c=-1.8\kappa$ so that the cavity is close to resonance with the kink mode at $\eta \simeq 200\kappa$. The blue dashed line is the effective cavity detuning $-{\mathrm{\Delta }}_{\mathrm{eff}}/\kappa$. (b) Asymptotic occupation numbers of the vibrational modes; only modes with nonzero coupling to the cavity are shown. The kink mode is shown in black while the highest frequency mode is shown in yellow (light gray). (c) Entanglement between the cavity fluctuations and all the modes (black solid line) and between the cavity and the kink mode only (red dashed line), quantified by the logarithmic negativity. Parameters are given in the text.

Figure 7

(a) Mean value of number of photon fluctuations as a function of the cavity detuning and (b) the mean occupation of the kink mode, which is minimum at resonance; the line is interrupted in the range in which the kink mode is not coupled to the cavity. (c) Spectrum at the cavity output (in units of $1/\mathrm{Hz}$) as a function of the cavity detuning and the photon frequency shift $\nu$ (in units of $\kappa$). The peak due to the kink mode vanishes near $\nu =0$ at ${\mathrm{\Delta }}_c\approx -1.9\kappa$ and is highlighted by the red dashed line. There is a cutoff in the color scale at ${10}^{-12}{\mathrm{Hz}}^{-1}$ to enhance the spectrum of the kink mode. (d) Output spectrum at ${\mathrm{\Delta }}_c=-1.8\kappa$, when the kink is resonantly coupled to the cavity mode; the inset shows a magnification of the kink mode, which is spectrally separated by a gap.