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Multi-Color Continuous-Variable Quantum Entanglement in Dissipative Kerr Solitons

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Multi-Color Continuous-Variable Quantum Entanglement in Dissipative Kerr Solitons Multi-color continuous-variable quantum entanglement in dissipative Kerr solitons Ming Li,1, 2 Yan-Lei Zhang,1, 2 Xin-Biao Xu,1, 2 Chun-Hua Dong,1, 2 Hong X. Tang,3 Guang-Can Guo,1, 2 and Chang-Ling Zou1, 2, ∗ 1Key Laboratory of Quantum Information, CAS, University of Science and Technology of China, Hefei 230026, China 2CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China. 3Department of Electrical Engineering, Yale University, New Haven, CT 06511, USA (Dated: January 20, 2021) In a traveling wave microresonator, the cascaded four-wave mixing between optical modes allows the gener- ation of frequency combs, including the intriguing dissipative Kerr solitons (DKS). Here, we theoretically in- vestigate the quantum fluctuations of the comb and reveal the quantum feature of the soliton. It is demonstrated that the fluctuations of Kerr frequency comb lines are correlated, leading to multi-color continuous-variable entanglement. In particular, in the DKS state, the coherent comb lines stimulate photon-pair generation and also coherent photon conversion between all optical modes, and exhibit all-to-all connection of quantum en- tanglement. The broadband multi-color entanglement is not only universal, but also is robust against practical imperfections, such as extra optical loss or extraordinary frequency shift of a few modes. Our work reveals the prominent quantum nature of DKSs, which is of fundamental interest in quantum optics and also holds potential for quantum network and distributed quantum sensing applications. Introduction.- Over the past decades, nonlinear optics tech- tum network [18]. For the case above the threshold, although nologies become the backbone of modern optics, enabling previous studies of c(2) optical combs indicate the potential frequency conversion, optical parametric oscillation, ultrafast applications of comb in the multipartite cluster-state genera- modulation, and non-classical quantum sources. Especially, tion for quantum computing [7, 19, 20], the quantum nature various nonlinear optics mechanisms enabled the generation of DKS have not been studied excepting one pioneer work by of the optical frequency comb, which has attracted lots of re- Chembo [21], who pointed out the mode-pairs locating sym- search interests due to its revolutionary applications in pre- metrically in two sides of the pump mode exhibit significant cision spectroscopy, astronomy detection, optical clock, and squeezing. optical communication [1–7]. Recently, by harnessing the In this Letter, we theoretically investigated the CV entan- enhanced nonlinear optical effect in a microresonator, the glement between comb lines in a microcavity and compare Kerr frequency comb, in particular, the dissipative Kerr soli- the quantum features of comb in different states, i.e. the state ton (DKS) as phase-locked frequency comb, has been exten- below the threshold, primary comb, and DKS. We showed that sively studied [8]. Such DKS have been realized on photonic the DKS could stimulate pair-generation and coherent conver- (3) chips with various c materials, showing advances in com- sion interactions between optical modes and thus produces a pact size, scalability, low power consumption, great spectral complex network links all optical modes. Evaluated by the range and large repetition rate [9–12]. entanglement logarithm negativity, it is demonstrated that a Therefore, the DKS is appealing for photonic quantum in- large number of comb lines with different colors are entan- formation science. From one aspect, the DKS produces an gled together when the cavity is prepared at the soliton state. array of stable and locked coherent laser sources with equally In particular, a group of modes show all-to-all quantum en- spaced frequencies, which provides a coherent laser source tanglement, indicating that arbitrary two modes over a large for driving nonlinear light-matter interaction as well as stable bandwidth could be entangled. The all-to-all entanglement frequency reference of local oscillators for heterodyne mea- persists even some modes in this complex network are elim- surements via frequency multiplexing. By taking advantage inated. This multi-color quantum-entangled state [22] holds of the coherence over a large frequency band, the quantum great potential in application in multi-user quantum communi- key distribution utilizing DKS has been experimentally stud- cation, quantum teleportation network [23, 24] and quantum- enhanced measurements [25–28]. arXiv:2101.07734v1 [physics.optics] 16 Jan 2021 ied [13], which promises commercialized high-speed quan- tum communication. From another aspect, the Kerr nonlinear Model and principle.- Figure1(a) illustrates an optical mi- interaction is inherently a quantum parametric process that croring cavity sided-coupled to a waveguide for generating describes the annihilation of two photons and simultaneous dissipative Kerr solitons [8]. By injecting a pump laser into generation of signal-idler photon pairs, which imply quan- a resonance of the cavity, the intracavity optical field builds tum correlations among comb lines. When working below up and the photons are converted between optical modes via the threshold, the DKS devices have been exploited to gen- cascaded four-wave mixing (FWM) due to the Kerr nonlin- erate both discrete- and continuous-variable (CV) quantum earity, leading to a broad comb spectrum at the output. For entangled states [14–17], holding the potential for one-way investigating the DKS in the microcavity, we consider a group quantum computing and high-dimensional entanglement be- of 2N + 1 modes belongs to the same mode family, with the tween different colors that could be distributed over the quan- spatial distributions described by Y(−!r ;q) = f (−!r )eimq [29]. 2 the x to classify different FWM terms, as all the mode-pairs of (a) x could interact with each other. As shown by Fig.1(b), each horizontal line of x represents the group of signal-idler mode- pairs which are labeled by arrows with the same color and ori- entation. The dashed orange arrow represents the degenerate Pump case i = j. It is anticipated that the photon-pair generation for mode-pair (i; j) would be stimulated by all the coherent light fields belongs to the same x, which might lead to the bipartite (b) † † CV entanglement by the effective interaction ai a j + aia j 휉 =4 for all pairs in x. Meanwhile, a single mode also connects to multiple x. As an example, for mode m = −3 in Fig.1(b), it 휉 = 0 could be entangled with mode m = 3 with x = 0, and m = 2 with x = −1, as well as m = 7 with x = 4. In addition, for the 휉 = −1 phase-matching i + j = k + l, there are also coherent photon conversion processes between mode (i;l) simulated by mode ( j;k) and vice verse, which also distributes the quantum cor- relations between different modes. Consequently, the FWM -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 in the microresonator results in a complex network, with op- tical modes serving as nodes, and the links corresponding to FIG. 1. (a) Schematic of dissipative Kerr soliton generation in an photon-pair generation and coherent photon conversion. optical microring cavity with a continuous-wave laser drive. (b) The complex network of FWM in a microcavity (Fig.1(b)) Four-wave mixing processes classified by the conserved total angular implies complicated dynamics of optical fields, which can be momentum x, with photon-pairs generated or annihilated simultane- ously in mode-pairs (i; j) with x = i + j. The paired modes are la- numerically evaluated by solving the quantum dynamics of beled by the same color and orientation, and the index x can be taken bosonic modes according to the Hamiltonian [Eq. (1)]. In- from −2N to 2N. Here we only show x = −1;0;4 as an illustration. stead of solving the intractable many-body nonlinear equa- Blue arrow: coherent photon conversion between modes m = −4;0 tions via full quantum theory, we adopt the mean-field treat- stimulated by m = −1;3, with x = −1. ment of the strongly pumped system, i.e. the mean and fluctu- ation of optical field in each mode are solved separately, due to the weak nonlinearity g =k 1 in practice [32]. The op- Here, m 2 m denotes the mode index, which corresponds 0 erator of the cavity mode a is approximated by the sum of to the angular momentum of the resonant modes and m = i a classical field described by a amplitude a and a fluctua- f−N;−N − 1;:::;−1;0;1;:::;N − 1;Ng. Due to the disper- i tion described by a bosonic operator da . According to the sion, the resonant frequencies could be expanded with respect i m2 Heisenberg equation and discarding the high-order terms of to the index as wm = w0 +mD1 + 2! D2 +:::: [30]. The Hamil- the fluctuation operators, the dynamics of the classical fields tonian describing this multimode system reads [31] and their fluctuations follow N † † † d ∗ H = ∑ diai ai + g0 ∑ d (i + j − k − l)ai a j akal (1) ai = biai − ig0 ∑a j akal + epd (i); (2) j=−N i jkl dt jkl N † d p in in the rotating frame of ∑ (wp + jD1)a a j. Here, d j = da = b da + 2k a j=−N j dt i i i a i w − w + j2D =2 is the mode detuning by neglecting higher 0 p 2 † ∗ ∗ † −ig0 akalda + a aldak + a akdal ; (3) order dispersion terms, a (ai) is the photon creation (anni- ∑ j j j i jkl hilation) operator, g0 is the vaccum coupling strength of the Kerr nonlinearity, d(·) is the Kronecker delta function and re- respectively. Here, the summation takes over all possible flects the phase-matching condition i + j = k + l according to permutation ( j;k;l) with i = k + l − j, bi = −idi − ki, k j is the angular momentum conservation [29, 31].
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