Theory of Josephson photomultipliers: Optimal working conditions and back action

We describe the back action of microwave-photon detection via a Josephson photomultiplier (JPM), a superconducting qubit coupled strongly to a high-quality microwave cavity. The back-action operator depends qualitatively on the duration of the measurement interval, resembling the regular photon annihilation operator at short interaction times and approaching a variant of the photon subtraction operator at long times. The optimal operating conditions of the JPM differ from those considered optimal for processing and storing of quantum information, in that a short $T_2$ of the JPM suppresses the cavity dephasing incurred during measurement. Understanding this back action opens the possibility of performing multiple JPM measurements on the same state, hence performing efficient state tomography.

Figure 1

This figure is a diagrammatic representation of the potential energy of a JPM as a function of the superconducting phase difference, and shows the interaction with a microwave cavity.

Figure 2

(a) Diagonal and (b) off-diagonal ${\chi }^1$-matrix elements for a bare JPM as a function of time, where ${\alpha }_j={\beta }_{jj}$ as defined in Eq. (20).

Figure 3

Comparison of the (a) diagonal and (b) off-diagonal ${\chi }^1$-matrix elements of a bare JPM with one experiencing additional pure dephasing $1/T_2=10{\gamma }_1$ (marked with circles). In both plots, each curve's color indicates its row in the ${\chi }^1$ matrix, and in (b) distance from the diagonal is indicated by line style.

Figure 4

(a) Shows the asymptotic limit of the off-diagonal ${\chi }^1$-matrix elements as a function of $T_2$. (b) Shows the time scale over which these asymptotic limits are reached; (b) is not monotonic due to the coherent oscillations present for long $T_2$.

Figure 5

(a) Shows the detection probability for a bare JPM and a JPM experiencing pure dephasing for a coherent state input (as described in Sec. 5a with $\left|\alpha \right|=0.5$). (b) Shows the detection probability for a bare JPM and a JPM experiencing pure dephasing for a one-photon Fock state input.

Figure 6

This figure shows the diagonal ${\chi }^1$-matrix elements shared by a bare JPM and a JPM experiencing energy relaxation. Energy relaxation matrix elements are represented by circles on the plots.

Figure 7

Diagonal ${\chi }^1$-matrix elements ${\alpha }_j^{\left(r\right)}\equiv {\chi }_{j-1j+rj-1j+r}^1$ corresponding to photon detection after loss of $r$ photons. These elements are zero unless an energy dissipation mechanism is present.

Figure 8

(a) Shows the detection probability for a bare JPM and a JPM experiencing energy relaxation for a coherent state input [same as in Fig. 5a]. (b) Shows the detection probability for a bare JPM and a JPM experiencing energy relaxation for a one-photon Fock state input [same as in Fig. 5b].

Figure 9

This figure shows the asymptotic value of detection probability for a JPM experiencing energy relaxation as a function of the energy relaxation rate $T_1^{-1}$. The black circles represent numerically simulated data points, while the curve is a linear fit between the simulated data points. As expected, it is a monotonically decreasing function.

Figure 10

Detection probabilities for a bare JPM and one experiencing dark counts for (a) a one-photon Fock state input state and (b) an $\alpha =1$ coherent state input state.

Figure 11

These figures show the detection probability of a JPM experiencing pure dephasing in the low and intermediate $T_2$ regimes for one- and two-photon Fock state inputs. The detection probability obtained from the analytical solution described in this section is compared to a numerical simulation (via the Liouville supermatrix approach) of the detection probability. (a) represents the low $T_2$ regime, with $\frac{1}{T_2}=10000{\gamma }_1$ and (b) is the intermediate $T_2$ regime, with $\frac{1}{T_2}={\gamma }_1$.

Figure 12

For a bare JPM, the detection probability is shown for a measurement occurring at $t_{m}g$ values of $0.126$ in (a), $1.26$ in (b), $2.52$ in (c), and $12.6$ in (d). $\alpha$ runs from $\alpha =0.01$ to $\alpha =1$ in 0.01 intervals, and in each figure, all three curves are scaled to be equal at $\alpha =0.01$. The time in (d) is chosen such that it is representative of the long-time steady state of the back action.

Figure 13

For a JPM experiencing energy relaxation (with $T_1=\frac{1}{{\gamma }_1}$ as before), the detection probability is shown for a measurement occurring at $t_{m}g$ values of $0.126$ in (a), $1.26$ in (b), $2.52$ in (c), and $12.6$ in (d). $\alpha$ runs from $\alpha =0.01$ to $\alpha =1$ in 0.01 intervals, and in each figure, all three curves are scaled to be equal at $\alpha =0.01$.

Figure 14

For a JPM experiencing pure dephasing (with $T_2=\frac{10}{{\gamma }_1}$ as before), the detection probability is shown for a measurement occurring at $t_{m}g$ values of $0.126$ in (a), $1.26$ in (b), $2.52$ in (c), and $12.6$ in (d). $\alpha$ runs from $\alpha =0.01$ to $\alpha =1$ in 0.01 intervals, and in each figure, all three curves are scaled to be equal at $\alpha =0.01$.