Photon catalysis acting as noiseless linear amplification and its application in coherence enhancement

Photon catalysis is an intriguing quantum mechanical operation during which no photon is added to or subtracted from the relevant optical system. However, we prove that photon catalysis is in essence equivalent to the simpler but more efficient noiseless linear amplifier. This provides a simple and zero-energy-input method for enhancing quantum coherence. We show that the coherence enhancement holds both for a coherent state and a two-mode squeezed vacuum (TMSV) state. For the TMSV state, biside photon catalysis is shown to be equivalent to two times the single-side photon catalysis, and two times the photon catalysis does not provide a substantial enhancement of quantum coherence compared with single-side catalysis. We further extend our investigation to the performance of coherence enhancement with a more realistic photon catalysis scheme where a heralded approximated single-photon state and an on-off detector are exploited. Moreover, we investigate the influence of an imperfect photon detector and the result shows that the amplification effect of photon catalysis is insensitive to the detector inefficiency. Finally, we apply the coherence measure to quantum illumination and see the same trend of performance improvement as coherence enhancement is identified in practical quantum target detection.

Figure 1

Schemes of photon catalysis (a) for the single-mode Fock state $|n\rangle$ and (b) for the coherent state and (c, d) single-side catalysis (biside catalysis) for the two-mode optical entanglement state. $m=1$ represents single-photon catalysis; $m>1$, multiphoton catalysis. ${\mathrm{BS}}_1$ and ${\mathrm{BS}}_2$ denote the beam splitters, with transmittance $T$.

Figure 2

Coherence before and after single-photon catalysis is performed on coherent state $|\alpha \rangle$: (a) $\alpha =0.5$; (c) $\alpha =0.1$. (b, d) Corresponding probability of single-photon catalysis for a coherent state with $\alpha =0.5$ and $\alpha =0.1$, respectively. In (c) and (d), lines with squares and lines with asterisks show the validity of the approximation. Photons in each mode are truncated within the $\left\{\right|0\rangle ,|1\rangle ,\cdots ,|19\rangle \}$ subspace to maintain numerical convergence and computation feasibility.

Figure 3

Coherence before and after single-side photon catalysis (lines with triangles) and biside photon catalysis (lines with asterisks) is performed on a TMSV state ($\lambda =0.5$). (b) Corresponding probability of single-side and biside photon catalysis for TMSV with $\lambda =0.5$. Photons in each mode are truncated within the $\left\{\right|0\rangle ,|1\rangle ,\cdots ,|11\rangle \}$ subspace to maintain numerical convergence and computation feasibility.

Figure 4

Coherence before and after single-side photon catalysis (lines with triangles) and biside photon catalysis (lines with asterisks) is performed on a weak TMSV state: (a) $\lambda =0.1$. (b) Corresponding probability of single-side and biside photon catalysis. Lines with squares and downward triangles are to check the validity of the approximation in Eq. (17) and Eq. (18). Photons in each mode are truncated within the $\left\{\right|0\rangle ,|1\rangle ,\cdots ,|11\rangle \}$ subspace to maintain numerical convergence and computation feasibility.

Figure 5

Realistic schemes of photon catalysis for (a) a single-mode coherent state and (b) a TMSV state. The approximated single-photon state in C mode is heralded when detector ${\mathrm{D}}_{\mathrm{D}}$ registers ‘on’ results. All detectors are on-off photon detectors.

Figure 6

Coherence before and after photon catalysis in (a, b) a weak coherent state ($\alpha =0.20$) and (c, d) a TMSV state ($r=0.40$). Parameter for the heralded single-photon state $r^{\prime }=0.10$. All optical states are truncated within the eight-dimensional subspace spanned by $|0\rangle ,|1\rangle ,\cdots ,|7\rangle$.

Figure 7

(a) Scheme of amplification with photon catalysis and (b) original Ralph and Lund's scheme with detectors of limited efficiency $\eta$. The transmittance of ${\mathrm{BS}}_1$ and ${\mathrm{BS}}_2$ is 0.5 and $T$, according to the Ralph and Lund's scheme [1]. The transmittances of beam splitters ${\mathrm{BS}}^{\prime }$, ${\mathrm{BS}}_3$, and ${\mathrm{BS}}_4$ are all set to $\eta$ to simulate the nonunit detection efficiency. All inputs in optical modes E, ${\mathrm{E}}_1$, and ${\mathrm{E}}_2$ are vacuum state and all photon detectors ${\mathrm{D}}_{\mathrm{D}}$, ${\mathrm{D}}_1$, and ${\mathrm{D}}_2$ are ideal unit-efficiency detectors. The trash bins denote the discarding operation. Amplification of (c1, c2) coherent state $|\alpha \rangle$ and (d1, d2) TMSV state with realistic photon catalysis and NLA scheme.

Figure 8

Coherence and modified success probability as a function of detector efficiency $\eta$ for amplification of (a, b) weak ($\alpha =0.20$) and (c, d) strong ($\alpha =0.50$) coherent states. The transmittances for beam splitters BS and ${\mathrm{BS}}_2$ are set to be unequal to obtain the same modified success probability. The squeezing parameters ${\lambda }^{\prime }=tanh\left(0.1\right)$ for all numerical evaluations in (a)–(d) and all optical states are truncated within the 10-dimensional subspace spanned by $|0\rangle ,|1\rangle ,\cdots ,|9\rangle$.

Figure 9

Coherence and modified success probability as a function of detector efficiency $\eta$ for amplification of (a, b) weak ($\lambda =0.10$) and (c, d) strong ($\lambda =0.50$) TMSV states. The transmittances for beam splitters BS and ${\mathrm{BS}}_2$ are set to be unequal to obtain the same modified success probability. The squeezing parameter is ${\lambda }^{\prime }=tanh\left(0.1\right)$ for all numerical evaluations in (a)–(d) and all optical states are truncated within the eight-dimensional subspace spanned by $|0\rangle ,|1\rangle ,\cdots ,|9\rangle$.

Figure 10

(a) Quantum target detection with $M$ copies of the TMSV state. A beam splitter with transmittance $T_0=1-\kappa$ is used to model the target and joint quantum measurement is exploited to extract the target information. (b) Basic scheme for practical quantum illumination with mode-by-mode measurement of each copy and optical parametric amplification.

Figure 11

(a) Error probability and (b) coherence in practical quantum illumination with mode-by-mode measurement. Other parameters: $N_s=0.01$, $\kappa ={10}^{-3}$, $N_B=0.95$, $M=2\times {10}^5$. All optical states are truncated within the 16-dimensional subspace spanned by $|0\rangle ,|1\rangle ,\cdots ,|15\rangle$.