Heisenberg-Limited Qubit Read-Out with Two-Mode Squeezed Light

We show how to use two-mode squeezed light to exponentially enhance cavity-based dispersive qubit measurement. Our scheme enables true Heisenberg-limited scaling of the measurement, and crucially, it is not restricted to small dispersive couplings or unrealistically long measurement times. It involves coupling a qubit dispersively to two cavities and making use of a symmetry in the dynamics of joint cavity quadratures (a so-called quantum-mechanics-free subsystem). We discuss the basic scaling of the scheme and its robustness against imperfections, as well as a realistic implementation in circuit quantum electrodynamics.

Figure 1

Phase-space representation of dispersive qubit read-out for different input states: (a) coherent state, (b) single-mode phase-squeezed state, (c) amplitude-squeezed state, (d) two-mode squeezed state in the QMFS $(X_{-},Y_{+})$. The purple dashed lines represent the input state and the blobs represent the output fields. The input state is displaced along the $X$ axis and the signal is encoded in the quadrature corresponding to the $Y$ axis with homodyne detection; as depicted in the leftmost panel, the read-out error corresponds to the overlap of the two marginals. Dispersive interaction with the qubit rotates the output field by the angle ${\phi }_{\mathrm{qb}}$ for the ground state $|0⟩$ (in blue) and $-{\phi }_{\mathrm{qb}}$ for the excited state $|1⟩$ (in red). Ideally, one wants the output state to be phase squeezed regardless of qubit state [the dotted “desiderata” states in (b)]; this is not possible when using single-mode squeezing due to the qubit-induced rotation. Our new QMFS scheme [panel (d)] does not suffer from this problem.

Figure 2

Circuit QED implementation of QMFS dispersive qubit read-out with two-mode-squeezed states produced by a nondegenerate paramp (NDPA). The signal is encoded in the joint quadrature $Y_{+,\mathrm{out}}\propto Y_{1,\mathrm{out}}+Y_{2,\mathrm{out}}$; see the text for details.

Figure 3

(a) SNR as a function of integration time $\tau$ for different protocols: coherent state drive (the black line), displaced single-mode squeeze state (the blue line), two-mode squeezed QMFS setup (the red line). We assume an optimal dispersive shift $\chi =\kappa /2$; for the QMFS setup, the cavity is presqueezed ($t_0\ll -1/\kappa$) with squeeze strength $e^{2r}=100$. The coherent drive is turned on at $\tau =0$. For the single-mode case, for each $\tau$ we optimize the squeeze strength $e^{2r}\in [1,100]$ and angle (see Ref. [25]). The QMFS scheme gives an exponential SNR enhancement, especially in the most interesting regime where $\tau \sim 1/\kappa$ (the shaded region). (b) Integration time $\tau$ required to achieve a fidelity $F=99.99\%$, as a function of $e^{2r}$; parameters as in (a), except that ${\overline{n}}_0=100$. Black lines correspond to an unsqueezed drive, where the drive strength is increased such that the intracavity photon number is the same as in the QMFS scheme, i.e., ${\overline{n}}_0\to {\overline{n}}_0+4{\sinh }^{2}r$. The solid curves correspond to the case of no photon losses (efficiency $\eta =1$), while the dashed curves correspond to $\eta =0.9$. (c) Total intracavity photon number needed to achieve $F=99.99\%$ in a measurement time $\tau$. Even with nonzero photon losses, the use of squeezing can dramatically reduce the number of intracavity photons.

Figure 4

(a) SNR enhancement as a function of the dispersive shift asymmetry (${\chi }_{1,2}=\delta \chi \pm \overline{\chi }$) for different resonator linewidth asymmetries (${\kappa }_{1,2}=\overline{\kappa }\pm \delta \kappa$) calculated for $\overline{\chi }=\overline{\kappa }/2$, $\kappa \tau =10$ and $e^{2r}=100$. The dashed line is the maximal SNR obtained by optimizing $\delta \kappa$. (b) Calculated dispersive shifts as a function of transmon anharmonicity $E_C$ from a numerical diagonalization of a transmon-resonator system for each of the resonators. The parameters are $E_J/h=25\mathrm{GHz}$, ${\omega }_1/2\pi =7.6\mathrm{GHz}$, ${\omega }_2/2\pi =7.9\mathrm{GHz}$, $g_1/2\pi =8\mathrm{MHz}$ and $g_2/2\pi =15\mathrm{MHz}$. The vertical dashed line shows a typical value of $E_C$ that leads to equal and opposite dispersive shifts.