Quantum Linear Time-Translation-Invariant Systems: Conjugate Symplectic Structure, Uncertainty Bounds, and Tomography

Abstract

Linear time-translation-invariant (LTI) models offer simple, yet powerful, abstractions of complex classical dynamical systems. Quantum versions of such models have so far relied on assumptions of Markovianity or an internal state-space description. We develop a general quantization scheme for multimode classical LTI systems that reveals their fundamental quantum noise, is applicable to non-Markovian scenarios, and does not require knowledge of an internal description. The resulting model is that of an open quantum LTI system whose dilation to a closed system is characterized by elements of the conjugate symplectic group. Using Lie group techniques, we show that such systems can be synthesized using frequency-dependent interferometers and squeezers. We derive tighter Heisenberg uncertainty bounds, which constrain the ultimate performance of any LTI system, and obtain an invariant representation of their output noise covariance matrix that reveals the ubiquity of "complex squeezing" in lossy systems. This frequency-dependent quantum resource can be hidden to homodyne and heterodyne detection and can only be revealed with more general "symplectodyne" detection. These results establish a complete and systematic framework for the analysis, synthesis, and measurement of arbitrary quantum LTI systems.

Figures (6)

• Figure 1: Quantum LTI System. A quantum LTI system is described by $n$ input quantum noises, which are mapped to $m$ output quantum noises. (a) In general, a quantum LTI system will map $n$ accessible inputs and $k$ inaccessible inputs to its $m$ outputs through a map $\mathbfcal{A}$. For a linear system, the maps from accessible and inaccessible inputs to outputs, $\mathbfcal{G}$ and $\mathbfcal{N}$, can be considered separately as in (b) (see \ref{['main:eq:xoutMN']}). (c) By adding $n+k-m$ ancillary output modes (see \ref{['main:subsec:quantumNoiseInLtiSystems']}) and the maps $\mathbfcal{K}$ and $\mathbfcal{L}$ from accessible and inaccessible inputs to ancillary outputs, the map $\mathbfcal{A}$ can be dilated to a closed system (see \ref{['main:subsec:LTIdilated']}).
• Figure 2: Optical decomposition (\ref{['main:thm:opt_decomp']}) of a frequency-dependent LTI quantum system. While formally similar to the Bloch-Messiah decomposition on the real symplectic group $\text{Sp}(2n, \mathbb{R})$Bloch62, note that each component is a function of frequency and the beam-splitter component is complex-valued.
• Figure 3: Two-mode general symplectic transformation acting on vacuum. Appropriately choosing the frequency-dependent coefficients $\alpha_1,\alpha_2,\beta_1,\beta_2,\theta_1$ and $\theta_2$ enables the generation of the $\{\Sigma, \Delta\}$ form of \ref{['main:eq:sigma_delta_SDM']}.
• Figure 4: Symplectodyne detector. (a) Shows the implementation of a single-mode symplectodyne detector. The local oscillator $\alpha(t)$ with an arbitrary waveform is coupled to a signal field $\hat{a}(t)$ with a balanced coupler. The spectrum of the differential photocurrent $\hat{I}(t)$ is used to recover the spectral density matrix of the signal. Depending on the local oscillator spectrum, symplectodyne detection subsumes homodyne, heterodyne, or synodyne detection. The relationship between the local oscillator tone (red arrows) and carrier frequency (dashed blue arrows) is indicated for each detection scheme. (b) Multimode SDM tomography can be performed by subjecting the output fields of a quantum LTI system to symplectodyne detection and correlating their output photocurrents.
• Figure 5: (a) General scheme for production of complex squeezing, and its detection. (b) Disappearance of squeezing in a homodyne detector due to frequency-dependent loss. The plot shows the limiting value of input squeezing ($r_\text{lim}$ in \ref{['main:eq:rlim']}) in dB units at which a homodyne detector fails to diagnose the presence of squeezing due to frequency-dependent losses $F_+^2$ and $F_-^2$. (c) Comparison of the optimal quadrature measurement for a lossy system described by \ref{['main:eq:lossy_SDM_out']}, using homodyne ($\Lambda_-^\mathbb{R}$, red) or synodyne detection ($\Lambda_-^\mathbb{C}$, blue). The sideband efficiencies are taken to be $F_+^2 = 0.99$ and $F_-^2 = 0.3$. The dashed lines are the asymptotic expressions in \ref{['main:eq:Lambda_lim']}. The green vertical line shows $r_\text{lim}$, above which the measured squeezing becomes hidden.

Theorems & Definitions (22)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Definition 4
• Definition 5
• Theorem 2: Optical decomposition of $\mathrm{Sp}^\dagger(2n)$
• Definition 6
• Theorem 3
• Theorem 4