Amplification of realistic Schrödinger-cat-state-like states by homodyne heralding

We present a scheme for the amplification of Schrödinger cat states that collapses two smaller states onto their constructive interference via a homodyne projection. We analyze the performance of the amplification in terms of fidelity and success rate when the input consists of either exact coherent state superpositions or of photon-subtracted squeezed vacua. The impact of imprecise homodyne detection and of impure squeezing is quantified. We also assess the scalability of iterated amplifications.

Figure 1

Setup for the generation of approximate small-amplitude odd cat states. A squeezed vacuum, represented here by the pumping of a ${\chi }^2$ nonlinear medium, is partially reflected onto an “on-off” photon detector such as an avalanche photodiode (APD). Upon the reflection of at least one photon from the squeezed vacuum, a state that likens the squeezed single photon is prepared in the limit of unit transmission $T\to 1$.

Figure 2

Top: Contour plot of the fidelity of the PSSV state with $|{\kappa }_{-}\left(\alpha \right)\rangle$ given an input squeezing ${\xi }_x$ between 0 and 15 dB. Bottom: Maximum achievable fidelity of the PSSV with $|{\kappa }_{-}\left(\alpha \right)\rangle$ (black) and corresponding success probability as a function of $\alpha$ (gray). In both cases, the transmission of the subtraction beam splitter is 99$\%$.

Figure 3

Fidelity between an ideal cat state $|{\kappa }_{-}\left(\alpha \right)\rangle$ and the small-amplitude cat states obtained fromthe squeezed vacuum ${\xi }_p=-3$ dB at four squeezing purities $\mathcal{P}$ (marked in decreasing darkness of gray): 100$\%$, 90$\%$, 80$\%$, and 70$\%$. The Wigner functions of the PSSVs corresponding to these four purities are shown at the top. As a reference for what the “ideal” output ought to look like, the Wigner function of the state $|{\kappa }_{-}\left(1.06\right)\rangle$, which has the highest fidelity with the PSSV obtained from pure squeezing, is shown at the bottom. The transmission of the subtraction beam splitter is set at 99$\%$.

Figure 4

Setup for the amplification of two ideal Schrödinger cat states into a larger even cat state. The two inputs are mixed on a symmetric beam splitter and one of the outputs is projected onto an $x$ quadrature window of width $\Delta Q$ around $x_0=0$ by an otherwise ideal homodyne detector. Inset: Wave functions $\langle x|0\rangle$ (solid curve) and $\langle x|{\kappa }_{+}\left(2\right)\rangle$ (dashed curve) of the vacuum and of an even cat state of size $\alpha =2$, respectively. The two functions are best distinguished at $x=0$ where their overlap is minimized.

Figure 5

Contour plot of the fidelity between an even cat of size $\beta$ and the amplification obtained from two odd cats of size $\alpha$. The post-selection window is set to 1 SNU.

Figure 6

Top: Maximum fidelity (black) and corresponding success probability (gray) of the amplified output with respect to an even cat state of size $\beta$. Bottom: Amplitude $\alpha$ of the input required to obtain the maximum fidelity of the output with an even cat state of size $\beta$. The dotted line marks the $\sqrt{2}$ amplification factor. The homodyne postselection window is 1 SNU wide.

Figure 7

Solid curves: Fidelity between an even cat state of size $\beta =\sqrt{2}\alpha$ and the output of mixing two odd cat states of size $\alpha$ for five homodyning widths $\Delta Q$ (in SNUs). Dashed curves: Corresponding success probabilities. (The curve for $\Delta Q\to 0$ is zero throughout since the probability of picking out the exact $x=0$ quadrature is vanishingly small.) The shading of the curves is lightened with larger homodyning windows.

Figure 8

Fidelity of the state amplified from two cat states of size $\alpha =1$ with an even cat of size $\beta =\sqrt{2}$ for a homodyning window $\Delta Q\in$ $[0,15]$ SNUs. The decrease in fidelity for wider homodyning windows is understandable from the increased overlap between the vacuum and the cat state, as illustrated in the inset of Fig. 4. The Wigner profile of the output is shown for three sample values of $\Delta Q$ at 1, 5, and 10 SNUs, respectively.

Figure 9

Contour plot of the fidelity between an even cat state $|{\kappa }_{-}\left(\beta \right)\rangle$ and the amplified PSSV obtained from an antisqueezing ${\xi }_x$ between 0 and 15 dB. The transmission of the subtraction beam splitter is 95$\%$. The contour lines of PSSV generation fidelity are overlaid to better visualize the shift in target amplitude $\beta$ resulting from the amplification. The blank vertical stripe for ${\xi }_x\le 0.8$ dB is a region where numeric underflow is too frequent to produce reliable data. (This is because the normalization factor which enters in the fidelity is itself inversely proportional to the success probability. The latter tends to negligible values for small squeezing.)

Figure 10

Top: Dependence of the effective size $\alpha$ of the PSSV-state input on the target size $\beta$ of the output such that the fidelity of the output with $|{\kappa }_{+}\left(\beta \right)\rangle$ is maximized. The dotted line marks the $\alpha /\beta =1/\sqrt{2}$ amplification ratio. The subtraction beam splitter for the PSSV generation is set to 95$\%$ and the homodyning window to 1 SNU. Middle: Pure squeezing required of the input in order to achieve the maximal fidelity with an even cat state of size $\beta$. Bottom: Maximum fidelity obtainable at the output with an even cat state of size $\beta$ (dashed black), along with the corresponding success probability of the amplification (dashed gray). Also shown is the maximum fidelity of the required input PSSV with an odd cat state of size $\alpha$ (solid black), and the success probability of the PSSV generation (solid gray). (In all three graphs, the finer dash-dotted lines refer to the results of Lund et al. in [30], with, in the bottom graph, the black and gray shadings representing fidelity and success probability, respectively.)

Figure 11

Robustness of the PSSV amplification to squeezing impurity (top) and homodyning width (bottom). The fidelity with an even cat state of size $\beta =1.5$ and the success probability of the amplification (assuming offline PSSVs) are plotted in black and gray, respectively. The transmission of the subtraction beam splitter from which the small-amplitude cat states are generated is 95$\%$ and the antisqueezing in $p$ is fixed, ${\xi }_p=-2.9$. [This squeezing is chosen such that the fidelity with $|{\kappa }_{+}(\beta =1.5)\rangle$ is maximized in the case of pure squeezing and $\Delta Q=1$ SNU—cf. the middle plot of Fig. 10.]

Figure 12

Maximum fidelity (squares) and corresponding effective size $\beta$ (disks) of the output as a function of the number of iterations. The inputs to the first iteration are an ideal odd cat state of size $\alpha =1$ (black) or a PSSV squeezed by 3 dB (gray).

Figure 13

Success probability of iterated amplifications for an input cat state $|{\kappa }_{-}(\alpha =1)\rangle$ as well as for various input PSSVs with squeezing $\xi \in \{0.5,1,2,3,6,9\}$ dB.

Figure 14

Probability $P_{\text{err}}$ of mistaking an even cat state of size $\beta$ for a vacuum state as a function of the quadrature acceptance window $\Delta Q$.

Figure 15

Sketch of a black box quantum circuit made up of $N=5$ input modes, $M=4$ measurement modes, and $N-M=1$ output modes. Each state occupying mode $k$ is represented by a Wigner function $W^{\left(k\right)}$. The input ${\prod }_n^N{W}_{\text{in}}^{\left(n\right)}$ is mapped according to (A15) into a linearly transformed state ${\prod }_n^N{\tilde{W}}_{\text{in}}^{\left(n\right)}$ which is then projected onto a “measurement state” ${\prod }_m^M{W}_{\widehat{\Pi }}^{\left(m\right)}$.