Increasing the dimensionality of quantum entanglement is a key enabler for high-capacity quantum communications and key distribution [1, 2], quantum computation [3] and information processing [4, 5], imaging [6], and enhanced quantum phase measurement [7,8]. A large Hilbert space can be achieved through entanglement in more than one degree of freedom (known as hyperentanglement [2,7,9]), where each degree of freedom can also be expanded to more than two dimensions (known as high-dimensional entanglement). The high-dimensional entanglement can be prepared in several physical attributes, for example, in orbital angular momentum [1,[10][11][12] and other spatial modes [13][14][15]. The drawback of these high-dimensional spatial states is complicated beam-shaping for entanglement generation and detection, which reduces the brightness of the sources as the dimension scales up, and complicates their use in optical-fiber-based communications systems. In contrast, the continuous-variable energy-time entanglement [16][17][18][19][20][21][22] is intrinsically suitable for high-dimensional coding and, if successful, can potentially be generated and be communicated in the telecommunication network. However, most studies focus on time-bin entanglement, which is discrete-variable entanglement with typical dimensionality of two [23][24][25]. Difficulties in pump-pulse shaping and phase control limit the dimensionality of the time-bin entanglement [26], and high-dimensional time-bin entanglement has not been fully characterized because of the overwhelmingly complicated analyzing interferometers. On the other hand, a biphoton state with a comb-like spectrum could potentially serve for high-dimensional entanglement generation and take full advantage of the continuous-variable energy-time subspace. Based on this state, promising applications have been proposed for quantum computing, secure wavelength-division multiplexing, and dense quantum key distribution [3,27,28]. A phase-coherent biphoton frequency comb (BFC) is also known for 3 its mode-locked behavior in its second-order correlation. Unlike classical frequency combs, where mode-locking directly relies on phase coherence over individual comb lines, the mode-locked behavior of a BFC is the representation of the phase coherence of a biphoton wavepacket over comb-line pairs, and results in periodic recurrent correlation at different time-bins [29, 30]. This time correlation feature can be characterized through quantum interference when passing the BFC through an unbalanced Hong-Ou-Mandel (HOM)-type interferometer [31]. A surprising revival of the correlation dips can be observed at time-bins with half the period of the BFC revival time.However, because of the limited type-I collinear spontaneous parametric downconversion (SPDC) configuration in the prior studies [29], post-selection was necessary for the BFC generation where the signal and idler photons are indistinguishable, limiting the maximum two-photon interference to 50 %.Here we achieve high-dimensional hyperentangle...

Graphene, a unique two-dimensional material of carbon in a honeycomb lattice [1], has brought remarkable breakthroughs across the domains of electronics, mechanics, and thermal transport, driven by the quasiparticle Dirac fermions obeying a linear dispersion [2-3]. Here we demonstrate a counter-pumped all-optical difference frequency process to coherently generate and control THz plasmons in atomic layer graphene with an octave tunability and high efficiency. We leverage the inherent surface asymmetry of graphene for a strong second-order nonlinear polarizability  (2) [4-5], which together with tight plasmon field confinement, enables a robust difference frequency signal at THz frequencies. The counter-pumped resonant process on graphene uniquely achieves both energy and momentum conservation. Consequently we demonstrate a dual-layer graphene heterostructure that achieves the charge-and gate-tunability of the THz plasmons over an octave, from 9.4 THz to 4.7 THz, bounded only by the pump amplifier optical bandwidth. Theoretical modeling supports our single-volt-level gate tuning and optical-bandwidth-bounded 4.7 THz phase-matching measurements, through the random phase approximation with phonon coupling, saturable absorption, and below the Landau damping, to predict and understand the graphene carrier plasmon physics. 2The discovery of graphene spurred dramatic advances ranging from condensed matter physics, materials science to physical electronics, mechanics, and thermal processes. In optics [6][7], the additional chiral symmetry of the Dirac fermion quasiparticles of graphene [8] enables an optical conductivity defined only by the fine structure constant  [9], one that is remarkably charge-density tunable [10][11] and with broadband nonlinearities [12][13][14][15]. The collective oscillations of the two-dimensional correlated quasiparticles in graphene [16] naturally make for a fascinating cross-disciplinary field in graphene plasmonics [17], with applications ranging from tight-field-enhanced modulators, detectors, lasers, polarizers, to biochemical sensors [18][19][20][21][22]. Different from conventional noble metal plasmons, graphene plasmons are dominant in the terahertz and far-infrared frequencies [23]. To excite and detect these plasmons, specialized techniques such as resonant scattering nanoscale antennae near-field microscopy or micro-and nano-scale scattering arrays have been pursued, albeit still using terahertz/far-infrared sources [24][25][26][27][28]. Recently nonlinear optical processes, only with free-space experiments, have proven especially effective in generating graphene plasmons with efficiencies up to 10 -5 [4][5]. However, to date, it is challenging to generate, detect, and control on-chip graphene plasmons all-optically, a key step towards planar integration and next-generation high-density optoelectronics.Concurrently THz generation has recently been revisited by a number of studies for imaging, spectroscopy, and communications [29]. While a wide tunability in THz can provide new g...