Quantum Algorithmic Readout in Multi-Ion Clocks

Optical clocks based on ensembles of trapped ions promise record frequency accuracy with good short-term stability. Most suitable ion species lack closed transitions, so the clock signal must be read out indirectly by transferring the quantum state of the clock ions to cotrapped logic ions of a different species. Existing methods of quantum logic readout require a linear overhead in either time or the number of logic ions. Here we describe a quantum algorithmic readout whose overhead scales logarithmically with the number of clock ions in both of these respects. The scheme allows a quantum nondemolition readout of the number of excited clock ions using a single multispecies gate operation which can also be used in other areas of ion trap technology such as quantum information processing, quantum simulations, metrology, and precision spectroscopy.

Figure 1

(a) In an atomic clock based on $N_C$ trapped ions the frequency of a laser is locked to an atomic transition, e.g., in a Ramsey sequence of two $\pi /2$ pulses (Hadamard gates $H$) enclosing a free evolution time $T$. Measurement of the number $\widehat{N}$ of clock ions in $|1⟩$ yields an error signal for the deviation of the laser from resonance. (b) For ion species lacking the cycling transition needed for direct state detection, a quantum algorithmic readout can be used to map $\widehat{N}$ onto $N_L$ cotrapped logic ions of a different species whose state can be detected efficiently.

Figure 2

Quantum algorithmic readout illustrated for $N_C=3$ clock and $N_L=2$ logic ions: after a Ramsey sequency (blue, dash-dotted box) the clock ions are in a superposition of states $|N{⟩}_C$, where $N\in [0,N_C]$ is the number of ions in excited states $|1⟩$ (their Hamming weight). A quantum algorithmic readout $\mathcal{U}$ (red, dashed box) maps $N$ in binary representation onto a number $N_L=\lceil {\log }_2(N_C+1)\rceil$ of logic ions. Detection of the logic ions in the $\{|0⟩,|1⟩\}$ basis provides the number $N$. (a) Quantum algorithm for the indirect measurement of the Hamming weight, taken from Refs. [22, 23]. (1) Logic ions are initialized in $|\underset{\_}{0}{⟩}_L=|00{⟩}_L$ (states $|\underset{\_}{N}{⟩}_L$ denote the binary representation of $N$). (2) The state is quantum Fourier transformed into $|\underset{\_}{\mathbf{0}}{⟩}_L$. (3) Controlled-phase gates rotate the state in Fourier space to $|\underset{\_}{\mathbf{N}}{⟩}_L$ for a state $|N{⟩}_C$ of clock ions. (4) An inverse Fourier transform yields the state $|\underset{\_}{N}{⟩}_L$ of logic ions. (b) The same algorithm decomposed in terms of multi-ion Mølmer-Sørensen (MS) gates. The Hadamard gates shown in gray need not be executed. Removing Hadamard gates displayed with dashed boxes merges the algorithmic readout with the Ramsey sequence, and requires the laser fields in the MS gates to be phase coherent with the Hadamard gates in the Ramsey sequence. Explicit, general forms of all gates are given in Ref. [39].

Figure 3

(a) Schematic level diagram with relevant frequencies during MS gates: atomic transition ${\omega }_0$, bichromatic driving fields at ${\omega }_0\pm \delta$, detuning ${\delta }_k$ from sideband ${\nu }_k$ due to collective mode $k$. (b) Average readout error probability $P_{\mathrm{err}}$ (red, solid line) and multispecies MS gate duration $\tau$ (blue, dashed line) versus detuning ${\delta }_1$.