Supraclassical measurement using single-atom control of an atomic ensemble

We analyze the operation of a sensor based on atom interferometry, which can achieve supraclassical sensitivity by exploiting quantum correlations in mixed states of many qubits. The interferometer is based on quantum gates which use coherently controlled Rydberg interactions between a single atom (which acts as a control qubit) and an atomic ensemble (which provides register qubits). In principle, our scheme can achieve precision scaling with the size of the ensemble—which can extend to large numbers of atoms—while using only single-qubit operations on the control and bulk operations on the ensemble. We investigate realistic implementation of the interferometer, and our main aim is to develop an approach to quantum metrology that can achieve quantum-enhanced measurement precision by exploiting coherent operations on large impure quantum states. We propose an experiment to demonstrate the enhanced sensitivity of the protocol and to investigate a transition from classical to supraclassical sensitivity which occurs when using highly mixed probe states.

Figure 1

(a) The protocol: The control qubit is prepared at a high purity in state $|0\rangle$ followed by a Hadamard gate ($H_C$), and a $cnot$ gate prepares the probe state. Next the interaction $u_{\omega }$ is applied, driving the evolution of the probe state. The $cnot$ is reapplied, and finally, the control atom's state is measured. (b) Level schemes: Four-level schemes for the control and register atoms. The control atom is coupled to the Rydberg state $|R_C\rangle$ via a laser pulse. Register atoms must be coupled by three lasers, far detuned from the intermediate $|e\rangle$ but Raman resonant with the (unperturbed) Rydberg levels. $|0\rangle \to |e\rangle$ and $|1\rangle \to |e\rangle$ drive a $\pi$ pulse between the $|0\rangle$ and the $|1\rangle$ states only when the electromagnetically induced transparency condition is lifted by the strong Rydberg-Rydberg interaction due to the control atom in state $|R_C\rangle$. (c) Implementation: In the experiment, the tight control trap will contain only a single atom, while the larger register trap will be populated by an average of 25 atoms. The atomic positions sample the density distribution for the traps as described in the text. (d) Gravity measurement: Implementation of a gravity measurement may be accomplished by using independently controlled and state-dependent optical potentials, which allow the register trap to be divided along the $z$ axis ($\mathbf{g}$ lies along $z$).

Figure 2

Dephasing of interference fringes due to the reduced register purity and register-size fluctuations. (a) Interference fringes in ${\sigma }_Z$ as $t$ is scanned with $N_R=25,p_C=1$, and $p_R=0.95$ (dashed black line) and $p_R=0$ [dot-dashed (red) line]. The solid black line represents the fringe for each of 25 perfectly pure qubits (for comparison to a perfect classical interferometer). (b) Sensitivity obtained via the Fisher information for the fringes obtained in (a), using the same colors. (c) Interference fringes corresponding to the inclusion of Poissonian fluctuations in $N_R$, which lead to further dephasing of the fringes. (d) Sensitivity obtained from the fringes in (c).

Figure 3

Results of the full density-matrix model of the protocol with $N_R=1$, 2, or 3 atoms. Interferometer “output” with a finite laser line width, for $N_R=1$ atoms (a), $N_R=2$ atoms (b), and $N_R=3$ atoms (c). As described in the text, we obtained each point by sampling the atomic density distribution for the position of control and ensemble atoms $\nu =49$ times, yielding $P_{0,1}\left(\omega \right)$ and hence $\langle {\sigma }_Z\rangle$ for each run. Due to the finite range of the Rydberg-Rydberg interaction, the gates have a range of fidelities, which means that there is a spread of $P_{0,1}\left(\omega \right)$ obtained at any given value of $\omega$. In each plot the purity of the control and register qubits is $p_C=p_R=0.95$ and we plot the average of the expectation values $\langle {\sigma }_Z\rangle$ obtained as a function of $\delta \omega /{\omega }_0$. Lines are fits to the output, from which we find that increasing the line width from 10 to 100 kHz degrades the interferometer contrast from 0.928 to 0.908 in (a), from 0.896 to 0.886 in (b), and from 0.865 to 0.834 in (c).

Figure 4

(a) Simulated effects of gate imperfections due to strong register-register interactions or weak control-register interactions. Open circles represent the falling gate contrast as more register atoms are added for a 10-kHz laser line width; filled diamonds, for a 100-kHz laser line width. An exponential fit allows us to extrapolate to a worst-case scenario, where for $N_R=25$ atoms the contrast has fallen to 0.75. Register-register interactions degrade the fidelity of the gate. The dashed black line shows that the fidelity falls when ${\mathrm{\Omega }}_3/{\mathrm{\Omega }}_2=10$, and the dot-dashed line, when ${\mathrm{\Omega }}_3/{\mathrm{\Omega }}_2=15$, as calculated using the analytic form given in [13]. (b) Interferometer output with $N_R=25$ atoms when $\omega$ is scanned. The purity of the control qubit and register qubits is 0.95, $t=375\mu \mathrm{s}$, and we scan the frequency over the range $\pm \delta \omega =0.{15}^{-3}{\omega }_0$, where ${\omega }_0=2\pi \times 5326\mathrm{Hz}$. Points show the results of $\nu =49$ repetitions at each value of $\delta \omega /{\omega }_0$ obtained by including the falling gate fidelity in (a) and including the atom losses due to light-induced collisions. The solid black line shows the fringes in the limit $\nu ⟶\infty$, but not taking into account atom losses, and the dashed line shows the fringes produced with perfect gate fidelity.