Quantum nondemolition gate operations and measurements in real time on fluctuating signals

We demonstrate an optical quantum nondemolition (QND) interaction gate with a bandwidth of about 100 MHz. Employing this gate, we are able to perform QND measurements in real time on randomly fluctuating signals. Our QND gate relies on linear optics and offline-prepared squeezed states. In contrast to previous demonstrations on narrow sideband modes, our gate is compatible with quantum states temporally localized in a wave-packet mode including non-Gaussian quantum states. This is the cornerstone of realizing quantum error correction and universal gate operations.

Figure 1

Experimental setup. (a) Decomposition of a QND gate. (b) Peripheral systems for the input signal preparation and the output measurements. Input optical signal is sent to either Input 1 or Input 2. (c) Experimental setup of the QND gate. Squeezing gates A and B share an optical delay line in orthogonal polarizations. “SHG” labels a second-harmonic generator; “HD” labels a homodyne detector.

Figure 2

Time-domain traces for 300 ns. (a) The filtered input white signals. (b) The filtered output homodyne signals: red (gray) is ${\widehat{x}}_1^{\mathrm{out}}$; cyan (light gray) is ${\widehat{x}}_2^{\mathrm{out}}$. (c) Results of the cancellation: red (gray) is ${\widehat{x}}_1^{\mathrm{out}}$; cyan (light gray) is ${\widehat{x}}_2^{\mathrm{out}}$. (d) The filtered output homodyne signals for vacuum inputs: red (gray) is ${\widehat{x}}_1^{\mathrm{out}}$; cyan (light gray) is ${\widehat{x}}_2^{\mathrm{out}}$. (e) The filtered output homodyne signals for vacuum inputs: red (gray) is ${\widehat{p}}_1^{\mathrm{out}}$; cyan (light gray) is ${\widehat{p}}_2^{\mathrm{out}}$.

Figure 3

Power spectra of output quadratures when a random signal is added to ${\widehat{x}}_1^{\mathrm{in}}$. Black (bottom) shows shot noise. Red shows results of vacuum-state input. Dotted magenta shows results of coherent-state input. Dashed green shows cancellation of the random signal.

Figure 4

Spectra of the QND quantities. (a), (b) Transfer coefficients when the random signal is added to ${\widehat{x}}_1^{\mathrm{in}}$ or ${\widehat{p}}_2^{\mathrm{in}}$, respectively. Red (top), blue (middle), and green (bottom) traces are $T_{\mathrm{S}}+T_{\mathrm{P}}, T_{\mathrm{S}}$, and $T_{\mathrm{P}}$ respectively. (c), (d) Conditional variances for vacuum inputs, i.e., the variances of ${\widehat{x}}_1^{\mathrm{out}}-g_x{\widehat{x}}_2^{\mathrm{out}}$ and $g_p{\widehat{p}}_1^{\mathrm{out}}+{\widehat{p}}_2^{\mathrm{out}}$ at the gain $g_x=0.41$ and $g_p=0.39$, respectively (see Appendix pp4), normalized by the shot noise spectrum. Blue (bottom) and green (top) traces are for the cases with and without the ancillary squeezed vacua, respectively.

Figure 5

Frequency characteristics of the LPFs, obtained by a network analyzer (Keysight, E5061B). Blue is gain, dotted green is phase.

Figure 6

Frequency characteristics of the HPFs, obtained by a network analyzer (Keysight, E5061B). Blue is gain, dotted green is phase.

Figure 7

Obtained response function of the LPF. Blue (top) shows LPF-1, green (middle) shows LPF-2, red (bottom) shows LPF-3.

Figure 8

Power spectrum of the input white signal. Green (top) shows unfiltered signal, blue (middle) shows filtered signal, red (bottom) shows oscilloscope noise floor.

Figure 9

Noise power spectra of four homodyne detectors. (i) Green shows optical shot noise spectra without the LPF. (ii) Blue shows optical shot noise spectra with the LPF. (iii) Cyan shows detector dark noise spectra without the LPF. (iv) Red shows detector dark noise spectra with the LPF. (v) Magenta shows oscilloscope noise floor.

Figure 10

Noise power spectra of ancillary squeezed vacua, normalized by the shot-noise spectrum. Green (upper) shows antisqueezed quadrature. Blue (lower) shows squeezed quadrature.

Figure 11

Experimental setup for input signal preparation. Beam 2 is sent to either Input 1 or Input 2.

Figure 12

Cancellation of the modulated signal by the feed-forward in the squeezing gate A. (a), (b) Gain and phase of the reference beam in the squeezing gate A, used for calibration of the other traces. (c), (d) Gain and phase of the feed-forward beam in the squeezing gate A. (e), (f) The results of the cancellation of the modulated signal.

Figure 13

Cancellation of the modulated signal by the feed-forward in the squeezing gate B. (a), (b) Gain and phase of the reference beam in the squeezing gate B, used for calibration of the other traces. (c), (d) Gain and phase of the feed-forward beam in the squeezing gate B. (e), (f) The results of the cancellation of the modulated signal.

Figure 14

Experimental input autocorrelation, input-output cross correlation, and response function $(f_{\text{in}\to 1}*f_{1\to 1}^x*f_{1\to \text{out}})\left(t\right)$ in the time domain (top panels) and in the frequency domain (bottom panels).

Figure 15

Estimated response functions, used for the cancellation in Fig. 16. (i) $(f_{\text{in}\to 1}*f_{1\to 1}^x*f_{1\to \text{out}})\left(t\right)$. (ii) $(f_{\text{in}\to 1}*f_{1\to 2}^x*f_{2\to \text{out}})\left(t\right)$. (iii) $(f_{\text{in}\to 2}*f_{2\to 2}^x*f_{2\to \text{out}})\left(t\right)$. (iv) $(f_{\text{in}\to 1}*f_{1\to 1}^p*f_{1\to \text{out}})\left(t\right)$. (v) $(f_{\text{in}\to 2}*f_{2\to 1}^p*f_{1\to \text{out}})\left(t\right)$. (vi) $(f_{\text{in}\to 2}*f_{2\to 2}^p*f_{2\to \text{out}})\left(t\right)$. (vii) $(f_{\text{in}\to 1}*f_{1\to \text{out}})\left(t\right)$.

Figure 16

Power spectra of all the cases where a random signal is added to one of the input quadratures ${\widehat{x}}_1^{\text{in}}, {\widehat{x}}_2^{\text{in}}, {\widehat{p}}_1^{\text{in}}$, and ${\widehat{p}}_2^{\text{in}}$. Black (bottom) shows shot noise. Red shows the QND outputs with vacuum-state inputs. Cyan (upper in left-end row) shows optical random signal at the input. Dotted magenta shows the QND outputs with the random signal input. Dashed green shows cancellation of the random signal without the response functions. Blue shows cancellation of the random signal with the response functions, which almost overlaps with red.

Figure 17

Response functions of the QND gate in the frequency domain. Blue shows response function $|F_{\text{in}\to 1}\left(\omega \right)F_{1\to \text{out}}\left(\omega \right)|$. Dotted green shows response functions $|F_{\text{in}\to 1}\left(\omega \right)F_{k\to l}^{x,p}\left(\omega \right)F_{1\to \text{out}}\left(\omega \right)|$. Red (bottom) shows response functions $|F_{k\to l}^{x,p}\left(\omega \right)|$.

Figure 18

Normalized variances (dimensionless) of (a) ${\widehat{x}}_1^{\mathrm{out}}-g_x{\widehat{x}}_2^{\mathrm{out}}$ and (b) $g_p{\widehat{p}}_1^{\mathrm{out}}+{\widehat{p}}_2^{\mathrm{out}}$. Cyan (upper) markers show experimental results when ancillary squeezed vacua are not used. Red (lower) markers show experimental results when ancillary squeezed vacua are used. Green (top) curves show theoretical variance when ancillary squeezed vacua are not used. Magenta (middle) curves show theoretical variance when ancillary squeezed vacua with $-2.8$ dB of squeezing are used. Blue (bottom) curves show theoretical variance for the ideal QND interaction with $G=1$. Gray lines show entangled criterion.