Continuous quantum clock with high precision and long recurrence time

Continuous clocks, i.e., clocks that measure time in a continuous manner, are regarded as an essential component of sensing technology. Precision and recurrence time are two basic features of continuous clocks. In this paper, in the framework of quantum estimation theory various models for continuous quantum clocks are proposed, and all tools of quantum estimation theory are employed to seek the characteristics of clocks with high precision and long recurrence time. Then, in a resource-based approach, the performances of the proposed models are compared. It is shown that quantum clocks based on an $n$ two-qubit system not only can have better precision than quantum clocks based on a $2n$ one-qubit system but also support long recurrence time. Finally, it is shown that while employing $n$ entangled qubits improves the precision of clocks by a factor of $1/\sqrt{n}$, it inevitably worsens the recurrence time of the clock.

Figure 1

(a) Energy levels of a system with Hamiltonian $H=\frac{\omega }{2}{\sigma }_z$ are considered a qubit system, where the states $|0\rangle$ and $|1\rangle$ are eigenstates of the Pauli matrix ${\sigma }_z$. (b) When the system initially is prepared in the state $|+\rangle$, it starts to oscillate between the states $|+\rangle$ and $|-\rangle$ with frequency $\omega$, where the states $|\pm \rangle$ are eigenstates of the Pauli matrix ${\sigma }_x$. This dynamics is used to estimate the time with optimal precision.

Figure 2

Schematic model of a quantum clock based on the dynamics of $n$ one-qubit systems. All qubits are initially prepared in the state $|+\rangle$ and then evolve with Hamiltonian $H=\frac{\omega }{2}{\sigma }_z$. At finite time $t$ a measurement in the basis $\left\{|\pm \rangle \right\}$ is carried out on each qubit. From the statistics of the measurement outputs the time $t$ can be estimated, where the minimum error of the estimation scales as $1/\left(\sqrt{n}\omega \right)$.

Figure 3

The error of estimation versus time for different values of $n$ and $\omega$. Left: $\omega =0.5\frac{\text{rad}}{\text{s}}$. Right: $\omega =1.0\frac{\text{rad}}{\text{s}}$.

Figure 4

The dynamics of a two-qubit system is employed to propose a quantum clock with high precision and long RT.

Figure 5

Schematic model of a quantum clock based on the dynamics of $n$ two-qubit systems. All two-qubit systems are initially prepared in the state $|+\rangle |+\rangle$ and then evolve with the Hamiltonian given in Eq. (31). At finite time $t$ a measurement in the basis $\left\{|\pm \rangle |\pm \rangle \right\}$ is carried out on each two-qubit system. From the statistics of the measurement outputs the time $t$ can be estimated, where the minimum error of the estimation scales as $\frac{1}{\sqrt{n\left(\frac{{\omega }^2+{\mathrm{\Omega }}^2}{2}\right)}}$. In this model the RT scales as $1/min[\omega ,\mathrm{\Omega }]$.

Figure 6

Average of estimators ${\widehat{t}}_m\left(\mathbf{k}\right)$ with all $l_m=0$.

Figure 7

The average of the estimator given in Eq. (54) for time interval $[0,2\pi ]$.

Figure 8

The error of the estimator given in Eq. (54) for time interval $[0,2\pi ]$ and various values of $n$. The horizontal parts of the curves touch the corresponding Cramer-Rao lower bound. The green curve touches this bound for a longer time than the other curves.

Figure 9

Schematic model of a quantum clock based on the dynamics of $n$ entangled qubits. The Entangled $n$-qubit system is initially prepared in the state $\left(\right|0\cdots 0\rangle +|1\cdots 1\rangle )/\sqrt{2}$ and then evolves with the Hamiltonian given in Eq. (58). At finite time $t$ a measurement in the basis $\left\{|\pm \rangle \cdots |\pm \rangle \right\}$ is carried out on the system. From the statistics of the measurement outputs the time $t$ can be estimated, where the minimum error of the estimation scales as $1/n\omega$. Although the model based on entanglement resources improves the precision of quantum clocks, it cannot support long RT (see the text).