Realizing modular quadrature measurements via a tunable photon-pressure coupling in circuit QED

One of the most direct preparations of a Gottesman-Kitaev-Preskill (GKP) qubit in an oscillator uses a tunable photon-pressure (also called optomechanical) coupling of the form $\widehat{q}{\widehat{b}}^{†}\widehat{b}$, enabling us to imprint the modular value of the position $\widehat{q}$ of one oscillator onto the state of an ancilla oscillator. We analyze the practical feasibility of executing such modular quadrature measurements in a parametric circuit-QED realization of this coupling. We provide estimates for the expected GKP squeezing induced by the protocol and discuss the effect of photon loss and other errors on the resulting squeezing.

Figure 1

Timeline of the measurement protocol. First, the ancilla oscillator is initialized to a coherent state $|\alpha \rangle$. Then, the parametric drive is turned on for time $t_{\mathrm{coupl}}$, coupling the target and ancilla oscillators with the unitary $U_{\mathrm{PP}}$. Finally, the parametric drive is turned off and the ancilla oscillator is coupled to a lossy oscillator. From this lossy oscillator, the state is released into a transmission line, where it is amplified and measured.

Figure 2

Wigner functions of states in the target and ancilla oscillators and probability distribution $\mathbb{P}\left(\beta \right)$ over measurement results of the heterodyne measurement of the ancilla oscillator mode. The initial state of the ancilla oscillator is the coherent state $|\alpha =\sqrt{3}\rangle$. The measurement result is the one with maximum likelihood with respect to $\mathbb{P}\left(\beta \right)$ (marked by a yellow cross). Top row: Starting with a vacuum state (${\mathrm{\Delta }}_q={\mathrm{\Delta }}_p=1$) in the target mode, a measurement of $S_q$ results in an effective squeezing of the final state of ${\mathrm{\Delta }}_q=0.18$, while ${\mathrm{\Delta }}_p=1$ is unchanged. The final state is most like the GKP $|-\rangle$ state for the following reason: We start with a vacuum state—which is closest to the $+1$ eigenstate of $X$. Besides, the measurement result gives an eigenvalue of $S_q$ close to $+1$ so we are in the GKP code space. In order to center the outgoing state, we apply an additional unconditional displacement equal to $Z^{-3}$ which changes the initial eigenvalue $+1$ of $X$ to $-1$. Bottom row: the initial state in the target mode is a squeezed vacuum state with ${\mathrm{\Delta }}_q=3$ and ${\mathrm{\Delta }}_p=1/3$. The effective squeezing of the final state ${\mathrm{\Delta }}_p=1/3$ is again unchanged, while ${\mathrm{\Delta }}_q=0.18$ for the outgoing state. The resulting state is squeezed with respect to both quadratures. Now the final state is close to a GKP $|-\rangle$ displaced by half a logical, i.e., $X^{-1/2}$, for the following reason. Again, we started with a squeezed vacuum state, which is closest to the $+1$ eigenstate of $X$ and the unconditional displacement is $Z^{-3}$, which changes the eigenvalue to $-1$. However, the measurement result now gives an eigenvalue of $S_q$ close to $-1$, indicating that the state is shifted out of the code space, by half a logical $X$.

Figure 3

Electric circuit realizing the photon-pressure coupling. The target oscillator (label $T$, right) is coupled to the ancilla oscillator (label $A$, left) via a Josephson junction. The coupling between the ancilla oscillator and the readout line is tunable, and only turned on during readout. The loop formed by the Josephson junction and the inductances $L_A,L_T$ is threaded by an external flux ${\mathrm{\Phi }}_{\mathrm{ext}}\left(t\right)$, which is a classical, time-dependent variable. Possible implementations of the readout switch are discussed in Sec. 2c.

Figure 4

Purple: We numerically simulate the average amount of squeezing ${\mathrm{\Delta }}_q$ [see Eqs. (1)] obtained using a coherent state $|\alpha \rangle$ with $\overline{n}={\left|\alpha \right|}^2$ photons to measure $S_q$ on a vacuum input state. In more detail, we generate a $\beta$ and ${\rho }_{\beta }$ and calculate ${\mathrm{\Delta }}_q\left({\rho }_{\beta }\right)$; the error bars indicate the standard deviation over different measurement results $\beta$. Blue: Mean effective squeezing estimate according to Eq. (13), using the Villain approximation to evaluate the expectation value for the sharpness on a vacuum state. Green: A simple approximate expression for the mean effective squeezing is $\langle {\mathrm{\Delta }}_q\rangle \approx 1/\sqrt{{4\pi \left|\alpha \right|}^2}$. Yellow: A lower bound on the green curve which replaces the average value $\langle \left|\beta \right|\rangle$ by $\sqrt{\langle |\beta {|}^2\rangle }$. Overall, the mean squeezing parameter goes down as $1/\sqrt{\overline{n}}$, where $\overline{n}$ is the average number of photons in the ancilla state used to implement the modular $q$ measurement.

Figure 5

Wigner functions and probability distribution $\mathbb{P}\left(\beta \right)$ over measurement results using the heterodyne measurement of the ancilla oscillator, including the leading nonlinear term. The initial states are a squeezed vacuum state with ${\mathrm{\Delta }}_q=3$ and ${\mathrm{\Delta }}_p=1/3$ in the target oscillator and the coherent state $|\alpha =\sqrt{3}\rangle$ in the ancilla oscillator. The measurement result is the one with the maximum likelihood with respect to $\mathbb{P}\left(\beta \right)$. The strength of the third-order term is set to $\frac{{\xi }_T^2}{{\xi }_A^2}={10}^{-3}$; compare Table 1. Top: Original third-order nonlinearity according to Eq. (17). The effective squeezing of the final state is ${\mathrm{\Delta }}_p=0.42,{\mathrm{\Delta }}_q=0.2$. Bottom: Third-order nonlinearity with a modified drive; see Eq. (19). The effective squeezing of the final state is ${\mathrm{\Delta }}_p=0.41,{\mathrm{\Delta }}_q=0.18$, demonstrating that ${\mathrm{\Delta }}_q$ is unchanged compared to the ideal measurement in Fig. 2.

Figure 6

Top: The external drive $x_{\mathrm{ext}}\left(t\right)$ required to obtain a photon-pressure coupling in the rotating frame; see Eq. (C1). As can be seen, the lowest frequency component of $x_{\mathrm{ext}}\left(t\right)$ is ${\omega }_T/2$. The starting point of $x_{\mathrm{ext}}(t=0)=\pi /2$ corresponds to the “off” setting where the target and ancilla oscillators are completely decoupled. The solid and dashed lines correspond to choosing the drive with a positive or negative sign, $x_{\mathrm{ext},\pm }\left(t\right)$, respectively. Bottom: The prefactor $cos\left[x_{\mathrm{ext}}\left(t\right)\right]$ of the self- and cross-Kerr terms in Eq. (4). The function is periodic and changes sign with frequency ${\omega }_T$. Purple: Drive required to obtain the maximum coupling strength i.e., $\delta =1$. The drive corresponds to a triangular wave. Green: the coupling strength is reduced to $\delta =0.5$. In this case, the drive is close to a simple cosine.

Figure 7

Amplitude $|b_n|$ of the first four harmonics with frequency ${\omega }_n$ of the Fourier series of either drive $x_{\mathrm{ext},\pm }\left(t\right)$; see Eq. (C1). Purple: Maximal coupling strength ($\delta =1$). Blue: $\delta =0.5$. Green: $\delta =0.1$. For $\delta =0.5$, it is sufficient to use only two harmonics in order to achieve a relative error below $1\%$.

Figure 8

Box-and-whisker plot of the expected number of photons (left) and the effective squeezing ${\mathrm{\Delta }}_p$ (right) of the final state after a measurement of $S_q$. The target oscillator starts in the vacuum state, and the ancilla oscillator with the coherent state $|\alpha =\sqrt{\overline{n}}\rangle$ (compare Fig. 4). For every $\alpha$, a total of 200 samples was simulated. The orange line indicates the median, the box indicates the 25 and 75 percentiles, the whiskers indicate the 5 and 95 percentiles, and events above or below these thresholds are shown individually. For some events, the final state of the target oscillator has a mean photon number exceeding 100, and therefore a large Hilbert space is required to faithfully represent those states. From analytical considerations, we know that ${\mathrm{\Delta }}_p=1$ should be constant, independent of the measurement results. As can be seen, errors are negligible up to $\overline{n}=3.5$; for $\overline{n}=4$, the relative error for most events is still less than 1%.