Complete temporal mode characterization of non-Gaussian states by a dual homodyne measurement

Optical quantum states defined in temporal modes, especially non-Gaussian states, such as photon-number states, play an important role in quantum computing schemes. In general, the temporal mode structures of these states are characterized by one or more complex functions called temporal mode functions (TMFs). Although we can calculate the TMF theoretically in some cases, experimental estimation of a TMF is more advantageous to utilize the states with high purity. In this paper, we propose a method to estimate complex TMFs. This method can be applied not only to arbitrary single temporal mode non-Gaussian states, but also to two temporal mode states containing two photons. This method is implemented by continuous-wave dual homodyne measurement and does not need prior information of the target states nor state reconstruction procedure. We demonstrate this method by analyzing several experimentally created non-Gaussian states.

Figure 1

Conceptual diagram. The input state is a non-Gaussian state in unknown temporal modes. We estimate the temporal mode structure by measuring conjugate quadratures $\widehat{X},\widehat{P}$ by dual homodyne measurement and processing the data by the CPCA.

Figure 2

Schematic of the CPCA. For every single-shot measurement, we take the quadrature values $M$ times in $[0,T]$. We convert the complex variable ${\widehat{\beta }}_{t_j}$ into uncorrelated variables by the PCA where we utilize the correlation $\langle {\widehat{\beta }}_{t_j}^{†}{\widehat{\beta }}_{t_k}\rangle$ calculated from $N$-frame data. From the eigenfunctions and eigenvalues, we can estimate the TMFs of the input states.

Figure 3

Experimental setup for the heralded creation of time-bin qubits $p_1|1_{w_1},0_{w_2}\rangle +p_2|0_{w_1},1_{w_2}\rangle$ and time-bin qutrits $q_1|2_{w_1},0_{w_2}\rangle +q_2|1_{w_1},1_{w_2}\rangle +q_2|0_{w_1},2_{w_2}\rangle$. Ti:sapphire denotes a titanium sapphire laser, second-harmonic generator (SHG), mode cleaning cavity (MCC), acousto-optical modulator (AOM), nondegenerate optical prametric oscillator (NOPO), splitting cavity (SC), filter cavity (FC), avalanche photodiode (APD), LO, and homodyne detector (HD). In the photon-subtraction experiments, the AOM is removed, and the SC is replaced by a 97% reflection beam splitter in the time-bin qubit generation setup.

Figure 4

Analysis results of $|{\varphi }_1\rangle$ to $|{\varphi }_4\rangle$. Left: First 50 eigenvalues of matrix $C_t$ are shown in the red (dark gray) bar, and the vacuum state is shown in light blue (light gray). Middle: The first eigenfunctions $e_1\left(t\right)$ are shown in real lines. The blue (dark gray) and orange (light gray) lines show the real and imaginary parts of $e_1\left(t\right)$, respectively. As for $|{\varphi }_1\rangle$ and $|{\varphi }_2\rangle$, theoretical predictions are shown in the broken lines. Right: Wigner functions and photon-number distributions of $e_1$.

Figure 5

(a) Second eigenfunction $e_2\left(t\right)$ of $|{\varphi }_3\rangle$ and $|{\varphi }_4\rangle$. (b) Two-mode photon-number distribution about $e_1$ and $e_2$. Red (dark gray) bars show expected photon-number correlation of $|{\varphi }_4\rangle$.

Figure 6

Estimated $f_1\left(t\right),f_2\left(t\right)$ of time-bin qutrits $|{\varphi }_5\rangle$ and $|{\varphi }_6\rangle$. Wigner function and photon-number distribution of each modes are also shown.