Disorder-protected quantum state transmission through helical coupled-resonator waveguides Download: 579次
1. INTRODUCTION
Topological photonics is emerging as a way to create disorder-immune waveguides for light using edge modes of media with nontrivial topological properties [13" target="_self" style="display: inline;">–
One promising application of topological photonics is the robust generation and transport of quantum state of light [12–
Fig. 1. Temporally indistinguishable photons within the temporal resolution propagating through different delay lines can be temporally distinguishable given the delay provided by the ring resonator waveguides is sensitive to disorder, i.e., random red or blue shifts of the individual resonators. Insets below illustrate various possible effects of disorder: (a) phase shift via the difference in phase velocities, (b) difference in arrival times due to variation of the group velocities, and (c) wavepacket distortion due to higher-order dispersion and wavelength-dependent reflection.
A potential solution is to use the robustness of topological edge states to protect quantum states of light [13,20–
Thus far, topologically protected waveguiding has been largely demonstrated in two or higher dimensional photonic systems, while 1D topologically protected transport has required synthetic dimensions [30] or time modulation such as adiabatic pumping [31]. There is another approach to achieve disorder-resistant waveguiding in 1D without a topological bandgap, based on directly implementing a spin-momentum locked dispersion by breaking time-reversal symmetry, which has been demonstrated using 1D electronic quantum wires [32]. Recently, we proposed a model of a quasi-1D coupled-resonator optical waveguide (CROW) exhibiting a similar helical spin-momentum locked dispersion (H-CROW), by combining circulation direction and sublattice spin-like degrees of freedom. The former effectively breaks time-reversal symmetry, while the coupling between different sublattices can be tuned to create a sublattice-momentum locked dispersion relation [33]. This results in a suppression of backscattering and enhancement of localization length compared with the regular CROW model as well as preservation of phase information. It is noteworthy, as it provides the way to miniaturize a disorder-resistant waveguide by inducing helical transport.
In this paper, we study the propagation of quantum states of light through H-CROWs and demonstrate that temporal indistinguishability can be robust against moderate disorder, thereby enabling the protection of entangled states. First, in Section
2. MODEL OF H-CROWS WITH DISORDER-ROBUST TRANSPORT
The helical-CROW (H-CROW) consists of two legs of resonant ring cavities coupled via off-resonant link rings, as illustrated in Fig.
Fig. 2. Schematic of the helical coupled-resonator optical waveguide (H-CROW). Pseudospin-momentum locking is achieved after a certain propagation distance, where each sublattice exhibits definite momentum for designated circulation mode, thereby facilitating a disorder-resistant transport. As opposite circulations exhibit opposite helicity, two co-propagating channels can be realized.
The tight-binding Hamiltonian governing time evolution of one specific circulation direction (counterclockwise) in the absence of disorder reads [24,25,33,36] where is the hopping strength, and field operators represent the annihilation operators at the sites in the th unit cell, while their conjugates are associated with the corresponding creation operators. The complex hopping terms arise due to the specially introduced asymmetry of the link rings [33]. Note that we measure frequencies with respect to a resonance of a single isolated ring, such that the eigenvalues of are modal frequency detunings with respect to this resonance.
Let us now consider a periodic lattice, which enables us to compactly write the Hamiltonian in -space as where is the crystal momentum, , , are Pauli matrices, and we now interpret the upper () and lower () layers as corresponding to up and down pseudospin degrees of freedom, respectively. The eigenvalues of are . The first component, , describes the symmetric part of the intra-leg coupling, which vanishes under our choice of hopping phase [33], is analogous to a Zeeman field, and and resemble intrinsic and Rashba-like spin-orbit couplings, respectively.
For the opposite excitation (clockwise), propagation is governed by the time-reversed Hamiltonian , which exhibits opposite hopping phases due to time-reversal symmetry [37]. We introduce a total Hamiltonian composed of both circulations given in the direct product form, . We obtain pseudospin-momentum locking in the center of the passband (), where , , and the wave group velocity becomes . It supports the most resistant light propagation against disorder, since for the small momentum deviations has the form where Eq. (
At the band center, H-CROWs show the maximum Anderson localization length and most resistant temporal pulse propagation, since the most significant disorder is misalignment of the rings’ resonant frequencies, which is diagonal in the sublattice basis and does not flip the pseudospin [12,16,25,26,33,38].
Importantly, we can employ the circulation degree of freedom to use H-CROWs as two-mode delay lines (see Fig.
We first formulate equations for the field operators to calculate the transmission at a given frequency ; the equation reads [25,39] where (ccw) and (cw) index counterclockwise and clockwise circulation modes, respectively, is the intrinsic scattering losses of each cavity, is coupling strength to the input/output leads at each edge of the array, and is the array length. The input field entering the first unit cell is , and we assume wave packets with identical temporal distributions but opposite circulations and sublattices, which excite the two helical modes.
The reflection () and transmission () amplitudes can be expressed for the two inputs as follows [24,25]: where operators without a “hat” refer to their corresponding field components of sublattices , [40]. The derivation is given in Appendix
We present in Fig.
Fig. 3. Classical wave transport through H-CROWs and CROWs in the presence of moderate disorder and intrinsic losses . (a), (b) Disorder-averaged field intensity profiles at in the first 10 rings of an (a) H-CROW and (b) CROW. (c) Frequency-dependent transmission spectra. Solid lines indicate the disorder average; shaded regions represent 65% confidence interval. Maximum of average is and at for the H-CROW and CROW, respectively. (d) Dependence of the transmission at on the disorder strength . (e) Wave packet delay time as a function of the input frequency. (f) Distribution of delay times at , where is the root mean square delay.
We plot in Fig.
The effects of disorder on the group delay time versus frequency detuning are presented in Fig.
3. PRESERVATION OF PHOTON INDISTINGUISHABILITY
In this section, we consider the transmission of two identical photons forming a separable quantum state at the input and analyze the degree of temporal photon indistinguishability at the output of the H-CROW and CROW. Specifically, we consider an input state , with one photon in the clockwise mode and a temporally identical photon in the counterclockwise mode. The output state in the frequency domain has the form where the subscripts and indicate the Hilbert space corresponding to upper/lower part of output port with the field operators composed of field creation operators of each output port . Note that we are working with scalar fields, assuming a fixed polarization state.
We compare a degree of the temporal overlap of the two photons after each one propagates through a different part of the device by calculating the coincidence probability. It was the first experimental witness of quantum property, as Hong et al. showed quantumness by generating entangled photons and measuring their coincidence counts versus the controlled delay to one of the paths [34]. When the total time delay is zero, coincidence rates after a 50:50 beam splitter reach a minimum and vanish due to the quantum interference when photons are temporally indistinguishable. Accordingly, we analyze the photon interference at the output with a tunable temporal delay, as illustrated in Figs.
Fig. 4. (a), (b) Schematics of coincidence measurement using tight-binding models of an H-CROW and a pair of regular CROWs. We consider measurements for two photons exhibiting opposite helicity with controlled delay time before a 50:50 beam splitter (BS) and resulting coincidence probability of two photons to produce simultaneous “clicks” of single-photon detectors. (c) Coincidence versus controlled delay time for 20-site long CROW structures. (d) Minimum coincidence values with respect to the number of sites. Blue solid line and dots represent the average for H-CROWs and red for CROWs. Error bars indicate 65% confidence interval for 500 disorder realizations.
We perform a comparison against regular CROWs with the same input and same propagation length, as illustrated in Figs.
4. PROTECTION OF PHOTON ENTANGLEMENT
We now aim to show that H-CROWs can preserve a peculiar quantum property of transmitted photons, entanglement inherently originating from the quantum coherence. Let us consider an N00N state as an input, . Hereafter we omit the ‘ab’ notation. Such states are strongly sensitive to all effects of the disorder, as sketched in Fig.
Next, we determine the coincidence probability in the general case, considering all the effects due to disorder. We compute the density matrix of the output state using the projection operators respectively. Again, denote annihilation operators on upper and lower output legs, respectively. The coincidence probability with the normalized output is given by In agreement with the expression written in Fock basis in Eq. (
To quantify the mixture of the output state induced by disorder, we also analyze another quantity, the purity , which is bounded by , where is the dimension of Hilbert space, i.e., for the two-photon case. The maximum value corresponds to pure states and the minimum to fully mixed states. The state purity after passing the controlled delay is where we omit the integral variable to simplify notation.
The form of Eq. (
To verify this reasoning, we plot in Figs.
Fig. 5. Disorder-robust transmission of N00N states using the H-CROW. (a), (b) Statistics of the output coincidence probability for the (a) H-CROW and (b) regular CROWs. (c) Output state purity versus controlled delay time for the H-CROW (blue) and regular CROW (red), with error bars indicating 65% confidence interval. (d) Exponentiated entanglement entropy of the upper output port, , as a function of the photon number . distinguishes maximally entangled states from separable states . We use an ensemble of 500 disorder realizations and disorder strength .
We additionally show in Fig.
To further quantify the effect of disorder, we consider the entanglement entropy , which indicates the capacity for encoding quantum information [51], where . Note that, as coherence terms vanish by partial trace, the controlled delay does not affect the entropy. We compute the entanglement entropy in Appendix
We show in Fig.
5. CONCLUSION
In this paper, we have studied the propagation of quantum states of light through helical coupled-resonator waveguides (H-CROWs). Regular CROWs can serve as delay lines in integrated photonic circuits; however, they exhibit strong sensitivity to fabrication disorder preventing reliable transmission of wave packets. H-CROWs exploit an additional sublattice degree of freedom to achieve disorder-resistant transport, which arises due to one-way modes at the center of their transmission band, whose propagation direction is fixed by the excited sublattice (known as pseudospin-momentum locking). Using numerical solutions of tight-binding models describing H-CROWs and regular CROWs, we have shown that the former can be used to more reliably transport quantum states of light in the presence of disorder.
We first showed that transmission probability and wave packet delay times have narrower fluctuations and provide more ballistic-like transport compared with regular CROWs. Next, we showed that two identical photons transmitted through an H-CROW can preserve the indistinguishability of their temporal wave packets and, accordingly, demonstrate the quantum Hong–Ou–Mandel interference. Finally, we showed that path-entangled two-photon N00N states are preserved as pure entangled states, while the effect of disorder is only expressed through the accumulation of a relative phase between the photon pairs. We note that this relative phase fluctuation can be compensated by simply placing a single tunable phase shifter at one of the output ports [38]. The H-CROWs perform better at reliably transporting both types of quantum states, while quantum features are strongly suppressed by disorder in regular CROWs.
In the future, it will be interesting to generalize our findings to multimode entangled states and multimode H-CROWs. Moreover, in this work we considered weakly coupled resonators described by the tight binding approximation, similar to the experiments in Refs. [25,26]. In order to increase the operating bandwidth, it would be essential to consider more strongly coupled lattices, which must be modeled using the transfer matrix method [33]. We expect that our results can provide a practical way to create robust integrated photonic delay lines, which can serve as essential components facilitating reliable generation and guiding of the quantum state of light for multiple applications, including scalable quantum information processing.
[2] L. Lu, J. D. Joannopoulos, M. Soljačić. Topological photonics. Nat. Photonics, 2014, 8: 821-829.
[18]
[34]
[37]
[40]
[41]
[42]
[44]
[51]
[53]
Article Outline
JungYun Han, Andrey A. Sukhorukov, Daniel Leykam. Disorder-protected quantum state transmission through helical coupled-resonator waveguides[J]. Photonics Research, 2020, 8(10): 10000B15.