Experimental demonstration of a time-domain multidimensional quantum channel

We present the first experimental realization of a flexible multidimensional quantum channel where the Hilbert space dimensionality can be controlled electronically. Using electro-optical modulators (EOM) and narrow-band optical filters, quantum information is encoded and decoded in the temporal degrees of freedom of photons from a long-coherence-time single-photon source. Our results demonstrate the feasibility of a generic scheme for encoding and transmitting multidimensional quantum information over the existing fiber-optical telecommunications infrastructure.

The transmission of this state through a narrow band filter is given by which is proportional to the overlap of the k-th and j-th temporal patterns, with the proportionality constant 2 0 filter T  , which is the product of transmission of the filter and its time constant.
We have previously shown theoretically 18 that one can achieve very low error rates based on linear phase modulation, where a linear temporal phase profile is applied to the single-photon pulse, resulting in a frequency shift  in the spectrum, If the frequency shift due to the linear phase pattern is large enough so that the overlap between the modulated spectrum and the un-modulated one is negligible, the two states can be considered practically orthogonal. This approach can be used to encode and decode multidimensional quantum information in the frequency basis -the computational basis -as well as in the basis of frequency superpositions. In this approach, the ability to write and read superpositions of quantum states, which is a crucial property of a quantum In our experimental implementation, in order to encode quantum information in the time domain of a single photon, a single-photon source with long lifetime is required.
To achieve this long coherence, we use a heralded single-photon source ( Fig. 1)  The encoding and decoding of the information in temporal wavepackets of the single photons (Eq. 3) is done with fast EOMs. The computational basis for our highdimensional quantum system 18 is defined by a set of discrete frequency bins. The first non-trivial multidimensional quantum system is a qutrita three-level quantum system.
Our method of encoding quantum information is not limited to qutrits; however, we focus on the three-dimensional Hilbert space of a qutrit to demonstrate the basic concept experimentally. For the computational basis, the three eigenstates of the system can be encoded in the frequency basis as Restricting ourselves to phase modulation, we can minimize such loss. In particular, we studied the case of a specific set of sinusoidal phase modulations, with the state after modulation given by where i J is the i-th order Bessel function of the first kind,  is the modulation index and  is the modulation angle. We have kept up to the first order in Eq. 5 and omitted higherorder contributions. We wish to find values of  and  so that the two states given by The state preparation of the photon is completed with one EOM, and the state projection in the receiver is performed with another EOM and a filter cavitythe detection cavity. There is a trade-off in the detection cavity bandwidth between error rate and detection efficiency. The optimal bandwidth for maximum information transfer was found to be roughly 1.5 photon , with  photon the single photon bandwidth 18 . For easier alignment, this cavity is designed to have a confocal geometry (Fig.1), that is, the separation of the cavity mirrors is set to be the same as their radius of curvature: R 1 = R 2 = L = 100 mm. The free spectral range of this cavity is thus FSR det = 1.5 GHz, which is much broader than the frequency range of the EOM encoding. The measured linewidth of the cavity is 7.0 0.4  MHz, which agrees qualitatively with the linewidth estimation from the FSR det and the specified reflectivity of the mirrors (99%). An arbitrary waveform generator is used to provide phase-locked driving signals of EOMs for both the modulation and demodulation, and the resulting spectra were first characterized with classical light. As shown in Table 1 are then recorded with a photodiode as shown in Fig. 2. We then performed a truth table measurement, yielding the pair-wise overlap of the five quantum states (see Table 2 and    In the two ( 33  and 22  ) diagonal blocks of the measurement matrix (Table 2 and Fig.3), where the measurements are done within the same basis, the off-diagonal elements in the measurement matrix (i.e. rates when the encoding and decoding profiles are mismatched) are much lower than the diagonal elements (i.e. rates when encoding and decoding profiles are correctly matched). The probability should ideally be zero for the off-diagonal elements in the measurement matrix; however, there is a finite probability of cross-talk induced detection, which is measured to be less than 12.3% in total. We attribute the finite cross-talk for different vectors in the same basis to the finite bandwidth of EOM driving electronics, EOM polarization alignment, single-photon state purity and cavity fluctuation. Furthermore, due to finite bandwidth in the electronics, the photons prepared in state 0 and 2 get extra spectral components, resulting in additional losses in these states, thus the variation in the diagonal elements of the measurement matrix. We verify that the coherence between computational basis states is preserved, by confirming the orthogonality between the superposition states S  and S  . To further investigate the coherent behavior of the superposition vectors, we varied the phase of the state preparation (Eq. 5), and then projected the photon onto the S  state.
The result is shown in Fig. 4 where one can clearly see the signature of the coherence when the phase of the state is varied, in good agreement with the quantum optical calculations, performed with no free parameters (Fig. 4). Such coherence cannot be explained by a classical model with probabilistic preparation of the different states. In conclusion, we have experimentally demonstrated fiber-optical multidimensional quantum channel based on temporal manipulation of single photons.
Our controllable-dimension temporal encoding scheme paves the way to incorporate multi-dimensional quantum information into the existing fiber optical telecommunication infrastructure.