Single-mode displacement sensor

We show that one can determine both parameters of a displacement acting on an oscillator with an accuracy which scales inversely with the square root of the number of photons in the oscillator. Our results are obtained by using a grid state as a sensor state for detecting small translations in phase space (displacements). Grid states were first proposed [D. Gottesman et al., Phys. Rev. A 64, 012310 (2001)] for encoding a qubit into an oscillator: an efficient preparation protocol of such states, using a coupling to a qubit, was later developed [B. M. Terhal and D. Weigand, Phys. Rev. A 93, 012315 (2016)]. We compare the performance of the grid state with the quantum compass or cat code state and place our results in the context of the two-parameter quantum Cramér-Rao lower bound on the variances of the displacement parameters. We show that the accessible information about the displacement for a grid state increases with the number of photons in the state when we measure and prepare the state using a phase estimation protocol. This is in contrast with the accessible information in the quantum compass state which we show is always upper bounded by a constant, independent of the number of photons. We present numerical simulations of a phase estimation based preparation protocol of a grid state in the presence of photon loss, nonlinearities, and qubit measurement, using no post-selection, showing how the two effective squeezing parameters which characterize the grid state change during the preparation. The idea behind the phase estimation protocol is a simple maximal-information gain strategy.

Figure 1

Wigner function of a grid state (top) and a quantum compass state (bottom), both with $\overline{n}\approx 12$. The grid state has been generated by the protocol in [16] using adaptive phase estimation (corresponding to $l=1$ and an adaptively varying $\phi$ in Fig. 2) in $M=8$ rounds, both for $S_p$ and $S_q$ applied to the vacuum state. The distance between the center of the Wigner function peaks is $\sqrt{2\pi }$ in both the $p$ and $q$ directions, thus the unit cell has area $2\pi$. Since the displacement can shift the grid in both positive and negative $p$ and $q$ directions, only displacements which can add at most $\pi /2$ photons to a vacuum state can be distinguished.

Figure 2

Phase estimation for a unitary operator $U$, for example, $U=S_p=e^{i\sqrt{2\pi }p}$ or $S_q$, consists of applying the “Ramsey” circuit of this figure for $M$ rounds, with possibly varying $l$ and phases $\phi$ repeatedly on one oscillator input state. If the input state is an eigenstate of $U$ with eigenvalue $e^{i\theta }$, then the probability for outcome 0 for the qubit measurement is given by $\mathbb{P}\left(0\right|\theta )=\frac{1}{2}[1+cos(l\theta +\phi )]$. In textbook phase estimation with $M$ ancilla qubits, one takes $l=2^k$ with $k=M-1,...,0$ starting at $k=M-1$. The rotation $R_z\left(\phi \right)$ depends on the outcomes of previous qubit measurements (feedback) so the circuits effectively implement the textbook quantum Fourier transform in sequential manner. In the phase estimation used in Sec. 5 we use $l=1$ and the feedback phases $\phi$ are chosen to maximize the information gain about the phase.

Figure 3

Top: the growth of the von Neumann entropy $S\left(\rho \right)$ with the number of photons $\overline{n}$ in a quantum compass state which is displaced by a small random amount and a constant, photon number independent, Wehrl entropy upper bound for $S\left(\rho \right)$ derived in Sec. 4. Bottom: the growth of the accessible information in the grid state as a function of $2M\approx {log}_2\left(\overline{n}\right)$.

Figure 4

Phase estimation circuit for preparing a grid state. One has $\alpha =\sqrt{\pi }$ for a measurement of $S_p$ and $\alpha =i\sqrt{\pi }$ for $S_q$.

Figure 5

Equivalent circuit for $D(-Z\alpha /2)$, here $R(-Z\pi /2)=exp(iZ{a}^{†}a\pi /2)$.

Figure 6

Effective squeezing after $M$ rounds, starting with a squeezed state $\mathrm{\Delta }=0.2$. Solid lines: ${\mathrm{\Delta }}_p$. Dashed lines: ${\mathrm{\Delta }}_q$. Top: $H=\chi a^{†}aZ+H_{Kc}$. Bottom: $H=\chi a^{†}aZ+H_{Kcq}$.

Figure 7

Effective squeezing after $M$ rounds of interleaved phase estimation with Kerr interaction, starting with the vacuum state. Solid lines: ${\mathrm{\Delta }}_p$. Dashed lines: $S_q$. Odd round numbers represent a measurement of $S_p$, even rounds a measurement of $S_q$.

Figure 8

Effective squeezing after $M$ rounds, starting with a squeezed state $\mathrm{\Delta }=0.2$. Solid lines: ${\mathrm{\Delta }}_p$. Dashed lines : ${\mathrm{\Delta }}_q$. Top: photon loss. Bottom: amplitude damping.

Figure 9

Effective squeezing after $M$ rounds, starting with a squeezed state $\mathrm{\Delta }=0.2$. Solid lines: ${\mathrm{\Delta }}_p$. Dashed lines: ${\mathrm{\Delta }}_q$. Top: imperfect projection. Bottom: readout errors.

Figure 10

Circuits used in the sequential implementation of textbook phase estimation for the unitary operator $S_p$. This sequential realization of phase estimation is identical to normal phase estimation as it merely uses a semiclassical realization of the quantum Fourier transform so that a single qubit is used in each round (see, for example, [31]). Assume that phase estimation uses $M$ ancillas prepared in $|00...0\rangle$. The top circuit in this figure is repeatedly executed for $k=M-1,...,1$ starting at $k=M-1$, each circuit using one qubit. For $k=0$, we use the circuit at the bottom, i.e., we do not apply the unconditional displacement $S_p^{-1/2}$: this has no effect on the phase estimation protocol, but ensures that the measurement for $S_p$ and $S_q$ fully commute. The only reason to have the unconditional displacement in the top circuit is that one minimizes the overall number of photons used in the protocol by doing so. The phase $\phi$ in $R_z\left(\phi \right)$ will depend on the outcomes of all previously measured qubits in accordance with the description of the semiclassical Fourier transform.

Figure 11

The unit $S_p$ Meas (resp. $S_q$ Meas) with outcomes $x_p$ or $x_p^{\prime }$ (resp. $x_q$ and $x_q^{\prime }$) represents the approximate measurement of the eigenvalue of $S_p$ (resp. $S_q$) by repeatedly using the circuits in Fig. 10. The circuit identity follows from the fact that the $S_p$ measurement and the $S_q$ measurement commute as (powers of) $S_p$ and $S_q$ commute.