Phonon-to-spin mapping in a system of a trapped ion via optimal control

We propose a protocol for measurement of the phonon number distribution of a harmonic oscillator based on selective mapping to a discrete spin-1/2 degree of freedom. We consider a system of a harmonically trapped ion, where a transition between two long-lived states can be driven with resolved motional sidebands. The required unitary transforms are generated by amplitude-modulated polychromatic radiation fields, where the time-domain ramps are obtained from numerical optimization by application of the chopped random basis algorithm (CRAB). We provide a detailed analysis of the scaling behavior of the attainable fidelities and required times for the mapping transform with respect to the size of the Hilbert space. As one application we show how the mapping can be employed as a building block for experiments which require measurement of the work distribution of a quantum process.

Figure 1

Schematic depiction of the spin mapping $U_m$ for the case of $m=2$ (left, initial state; right, final state): Starting from a product state ${\sum }_n{c}_n|\downarrow ,n\rangle$ with spin down and occupation in different number states of the harmonic oscillator (the circle diagrams show $|c_n{|}^2)$, only the occupation in the number state $|n=m=2\rangle$ $(|c_2{|}^2$, yellow) is excited to $|\uparrow \rangle$ while the occupation initially in number states $|n\ne 2\rangle$ $({\sum }_{n\ne 2}{\left|c_n\right|}^2$, gray) ends up in $|\downarrow \rangle$. As shown on the right, after the mapping, $|c_2{|}^2$ (yellow) is distributed over the spin-up states while ${\sum }_{n\ne 2}{\left|c_n\right|}^2$ (gray) is distributed over the spin-down states. Note that only the spin population of the final state matters while the state of the motional mode is not specified.

Figure 2

Performance of $U_m\left(T\right)$ for different $m$ as it acts on the states $|\downarrow ,n\rangle$: (a) $m=0$, (b) $m=1$, (c) $m=2$, and (d) $m=3$. The columns show the resulting population in the upper spin state $p_{\uparrow }={\sum }_k|\langle \uparrow ,k|U_m{\left(T\right)|n,\downarrow \rangle |}^2$ depending on the initial phonon-number state $|n\rangle$: The yellow columns show the population for $m$, where the spin should be flipped, and the gray columns show the spin-up population for phonon numbers $n$, where we want to keep the state in spin down.

Figure 3

Employing unitary phonon to spin mapping for a QND filter: The panels show a sequence of mapping and spectroscopic decoupling operations which enables measurement of the work distribution of a quantum process. (a) A state prepared in $|\downarrow \rangle$ with an arbitrary state of the vibrational mode. (b) Population from $|m=2\rangle$ is transferred to $|\uparrow \rangle$ by means of $U_2$. The vibrational populations are reshuffled in an unspecified way. (c) Population from $|\downarrow \rangle$ is shelved to a metastable auxiliary state $|a_1\rangle$ for spectroscopic decoupling. (d) The population which was not shelved is transferred to $|\downarrow ,2\rangle$ by applying the time-reversed mapping $U_2^{†}$. (e) A general quantum process $\mathcal{L}$ (unitary and/or dissipative) takes place which performs work on or transfers heat to the vibrational mode. (f) The population of $|m^{\prime }=0\rangle$ is mapped to $|\uparrow \rangle$ by means of $U_0$, and the remaining population in $|\downarrow \rangle$ is shelved to another metastable state $|a_2\rangle$ in (g). (h) Readout takes place by cycling to population of the ground state through an excited state $|e\rangle$ and detecting the resonance fluorescence.

Figure 4

(a) Energy-level scheme of a single trapped ion. The relevant levels are product states of the spin state and the harmonic oscillator modes given by the phonon number states $|n\rangle$. Lasers drive transitions on the carrier as well as on the blue and red sidebands. (b) Relative Rabi frequencies [given by the matrix elements $M_{n,n+\delta n}={\mathrm{\Omega }}_{n,n+\delta n}^{\left(\alpha \right)}\left(t\right)/{\mathrm{\Omega }}^{\left(\alpha \right)}\left(t\right)]$ for $\eta =0.25$. On resonance this is the relative coupling strength of the carrier transition (black circles, $\delta n=0)$, the blue one for the blue sideband (stars, $\delta n=1)$, and the red one for the red sideband (crosses, $\delta n=-1)$.

Figure 5

Pulse shapes after optimization. Pulse height in units of ${\mathrm{\Omega }}_0=2\pi \times 50$ kHz. (a) The optimal pulse for $m=1,T=400$ $\mu \mathrm{s}$, where three radiation fields (black, carrier; red, red sideband; blue, blue sideband) are applied and all of them are modulated continuously in time. The transformation works on the harmonic oscillator levels $n=0,\cdots ,9$. (b) Two fields (black, carrier; red, red sideband) are applied and the control is limited to phase switching by $\pi$ while the power is kept constant $[{\mathrm{\Omega }}^{\left(\alpha \right)}\left(t\right)={\mathrm{\Omega }}_0$ or ${\mathrm{\Omega }}^{\left(\alpha \right)}\left(t\right)=-{\mathrm{\Omega }}_0]$. The transformation operates on the harmonic oscillator levels $n=0,\cdots ,3$ for $m=0,T=300$ $\mu \mathrm{s}$.

Figure 6

Optimization results for different control scenarios: fidelity $F(m,10)$ for three driving fields (black squares), driving only the carrier and the blue sideband transition (blue circles) and using only the carrier and the red sideband transition (red crosses) as a function of $m$.

Figure 7

Operation error $1-F$ as a function of $N$, the number of basis states considered. The results are for realizing $U_0^N$ in an operation time of $T=300$ $\mu \mathrm{s}$ using carrier and red sideband. The black squares show the results allowing continuous modulation of the pulse power over time. The gray circles show the results obtained when only phase flips at constant power are allowed (compare Fig. 5 for the pulse shape).

Figure 8

Waiting times in units of $1/{\mathrm{\Omega }}_0^{\left(b\right)}$. Theoretical Poincaré recurrence times $T_P$ (red crosses) and required waiting times found by numerical simulations (black crosses and blue squares) are plotted for different numbers $N$ of included phonon excitations. The results were obtained for ${\mathrm{\Omega }}_0^{\left(b\right)}=2\pi \times 50$ kHz and $\varepsilon =0.02$. Blue squares are waiting times $T_F$ for crossing the error threshold [Eq. (C5)]; black crosses are waiting times $T_{\mathrm{\Delta }\varphi }$ for constant $\mathrm{\Delta }\varphi$ [Eq. (C6)]. The gray circles show $T_{\mathrm{opt}}$, the operation time needed for the optimized maps $U_0\left(N\right)$ with carrier and red sideband presented in Fig. 7.