Dispersive qubit measurement by interferometry with parametric amplifiers

We perform a detailed analysis of how an amplified interferometer can be used to enhance the quality of a dispersive qubit measurement, such as one performed on a superconducting transmon qubit, using homodyne detection on an amplified microwave signal. Our modeling makes a realistic assessment of what is possible in current circuit-QED experiments; in particular, we take into account the frequency dependence of the qubit-induced phase shift for short microwaves pulses. We compare the possible signal-to-noise ratios obtainable with (single-mode) SU(1,1) interferometers with the current coherent measurement and find a considerable reduction in measurement error probability in an experimentally accessible range of parameters.

Figure 1

Schematic of the experimental setup for implementing one version of the SU(1,1) interferometer (using nondegenerate parametric amplifiers) in present-day microwave components, which could be either lumped “3D” structures or on-chip integrated devices. Many additional components of such an actual setup are omitted (e.g., attenuators, additional amplifiers, mixer local oscillators, data acquisition hardware); only the parts that are essential to our scheme are shown. A pump is to be distributed by a power splitter to the nondegenerate two parametric devices (PA) (for example, the “Josephson parametric converter” [13]). The PAs work in reflection, requiring circulators to separate input from output modes. It is assumed that the cavity containing the qubit is probed in reflection. The isolator/circulator at the bottom of the figure serves both to define the cold vacuum input to PA1 and to provide the necessary beam path from PA1 to PA2 for the reference mode of the SU(1,1) interferometer. Relative phases of the pump beams, and of the two interferometer arms, must be precisely set, requiring careful choice of the propagation lengths along all these paths. We show the HEMT amplifier (but not other amplifiers that would be involved in this setup) since it is necessary to consider whether the amplification provided by PA2 is large enough to overcome the nonideal noise characteristics of the HEMT.

Figure 2

Possible scenarios of a dispersive qubit measurement in a circuit-QED setting where $\phi$ is the qubit-state-dependent phase shift. The pump beams of the parametric (PA) and degenerate parametric amplifiers (DPA) are not explicitly depicted. (a) The (two-mode) SU(1,1) interferometer. When both input modes are prepared in the vacuum state and the parametric amplifiers are chosen such that $r=r_1=r_2$, ${\theta }_1-{\theta }_2=\pi$, the total number of photons at output $N_{\text{out}}=N_{1,\text{out}}+N_{2,\text{out}}$ is a sensitive probe for the phase $\phi$, i.e., ${\left(\Delta \phi \right)}^2=\frac{1}{{sinh}^2\left(r\right)}$ at $\phi =0$ [16]. In our scenario, we consider a coherent pulse in mode $a_{1,\text{in}}$ and a homodyne measurement is done on the outgoing mode $a_{1,\text{out}}$. (b) The squeeze scenario in which a coherent input pulse is first squeezed by a degenerate parametric amplifier before picking up a to-be-determined phase shift. A phase-insensitive amplifier subsequently amplifies the signal so that homodyne measurement is possible. (c) The standard coherent dispersive measurement scenario in which a coherent pulse, after having picked up a phase shift at the cavity, is amplified. (d) The single-mode SU(1,1) interferometer in which two degenerate parametric amplifiers sharing the same pump are used. The relative difference of the phases ${\theta }_1$ and ${\theta }_2$ is determined by the pump beam.

Figure 3

Phase-space sketches. In all three figures, we start with a coherent or squeezed state with $\langle p\rangle =\mathrm{Im}\left(\alpha \right)=0$ and the state picks up a phase shift ${\phi }^{\pm }$. (a) A coherent state picks up one of two phases depending on the state of a qubit $|\alpha \rangle \to |\alpha e^{\pm i\phi }\rangle$. Values $|{\phi }^{+}-{\phi }^{-}|>\pi$ do not give rise to greater distinguishability when measuring $\mathrm{Im}\left(\alpha \right)$, but as we will find in this paper modeling the input as a multimode signal and constraining the number of photons in the cavity will lead to a better SNR for larger phase shifts. (b), (c) The optimal direction of squeezing depends on the phase shift which is $\pi /2$ in (b) and close to 0 in (c). It is clearly not possible to minimize the noise for both $\pm$ signals when ${\phi }^{\pm }$ is not close to the points 0 or $\pi /2$.

Figure 4

Frequency dependence of phase shifts ${\phi }^{\pm }(\omega -{\omega }_r)$ for $\kappa =52$ MHz and $\chi =36$ MHz. The shift between the two $arctan$ functions equals $2\chi$.

Figure 5

Example of the number of photons in the cavity $\overline{n}\left(t\right)$ versus time $t$ for some representative choice of parameters $\Delta ,g,\kappa ,T_{\text{pulse}},n_{\text{pulse}}$. The cavity initially is excited by a Gaussian pulse and the maximum number of photons inside the cavity $n_c$ stays well below the critical value set by the detuning $\Delta$ and the coupling $g$ at $n_{\mathrm{crit}}\simeq 41.53$.

Figure 6

(a) The probability of error $P_{\mathrm{error}}$ of the standard coherent measurement scheme versus $\frac{2\chi }{\kappa }$. Other parameters in this plot (which do not directly affect where the error probability is minimal) are ${\omega }_r/2\pi =6.789$ GHz, ${\omega }_q/2\pi =5.5$ GHz, and $T_m=1.2T_{\text{pulse}}$, $T_{\mathrm{pulse}}=2/W$, $n_{\text{pulse}}=9$. (b) $P_{\mathrm{error}}$ versus normalized pulse width $W/\kappa$ for different values of phase shift (determined by $2\chi /\kappa$).

Figure 7

(a) Comparison between ${\mathrm{SNR}}_{\mathrm{SU}(1,1)+\mathrm{PA}}$ and ${\mathrm{SNR}}_{\mathrm{coherent}+\mathrm{PA}}$ for two different relative phases ${\theta }_2-{\theta }_1=\pi$ and ${\theta }_2={\theta }_1$. (b) Comparison between ${\mathrm{SNR}}_{\mathrm{SU}(1,1)+\mathrm{DPA}}$ and ${\mathrm{SNR}}_{\mathrm{coherent}+\mathrm{DPA}}$ for the optimal phase choice ${\theta }_2-{\theta }_1=\pi$. (c) Comparison between ${\mathrm{SNR}}_{\mathrm{SU}(1,1)+\mathrm{DPA}}$ (with the optimal phase choice ${\theta }_2-{\theta }_1=\pi$) and ${\mathrm{SNR}}_{\mathrm{SU}(1,1)+\mathrm{PA}}$ (with the optimal phase choice ${\theta }_2={\theta }_1)$. In all scenarios, we assume the optimal $G_1=3.12$ dB, $G_2=20$ dB (a practical maximum amplifier gain), and the number of photons in the cavity is at most 5. In the case of PA-based SU(1,1) interferometer, the photon flux [see Eq. (18) and Appendix] before the first PA is $31.23\kappa$ and the photon flux after the first PA but before the cavity is $F_t=32.5\kappa$ which is equal to the photon flux before the cavity for the coherent schemes. For the DPA-based interferometer, the photon flux before the cavity is $21.7\kappa$ which is amplified to $32.5\kappa$ after the first DPA.

Figure 8

Central result of the paper: The probability of error versus measurement time $T_m$ (a) for a pulse with time duration $T_{\text{pulse}}=160$ ns where the optimum total number of photons in the pulse before first PA is $n_{\mathrm{pulse}}=58.98$. (b) For a pulse with time duration $T_{\text{pulse}}=60$ ns where the optimum total number of photons in the pulse before the first PA is $n_{\mathrm{pulse}}=19.36$.

Figure 9

Same as Fig. 8, except that the assumed noisy post amplification (with HEMT) is replaced by ideal amplification. This quantifies the (small) amount by which the HEMT noise increases the error rate; at the minimum, the change is from 0.0007 to 0.0003. The relative performance of the coherent state vs SU(1,1) schemes is essentially unchanged.