Engineering quantum diode in one-dimensional time-varying superconducting circuits

Abstract

A potential application, quantum diode based on the adiabatic pumping between two specific left and right edge modes, is explored in a one-dimensional cyclically modulated circuit quantum electrodynamic dimer mapped successfully to the paradigmatic Su-Schrieffer-Heeger model. The quantum diode is characterized by the presence of nonreciprocity in transport, which describes the one-way transfer between excitations at both boundary resonators of the lattice. We find that the quality of the quantum diode defined by fidelity can be improved by increasing the modulation amplitude, i.e., the one-way excitation transfer process becomes more and more pronounced with the increase of the modulation amplitude. By further modifying the cyclical modulation and optimizing the control function, we also realize a much faster one-way excitation transfer to accelerate the nonreciprocal transport in the quantum diode, where almost a threefold reduction in time spent can be achieved. Our work provides a distinct idea and insight for the application of the quantum transport in topological systems.

Introduction

The discovery of topological phases of matter^1,2 is an important milestone in the development of condensed matter physics, which could be first observed by topological insulators^3,4,5, e.g., the integer quantum Hall effect on a two-dimensional square lattice⁶. Over the past decades, topological insulators have been a long-standing research interest and have attracted extensive attention due to the existence of edge modes^7,8 insensitive to local perturbations and disorder (such as thermal fluctuations and defects), one simple and celebrated example is described by the one-dimensional Su-Schrieffer-Heeger (SSH) model⁹ of polyacetylene with spontaneous dimerization. Benefiting from these extraordinary properties of edge modes^{3,10,11,12,13,14,15,16}, they can be potentially exploited in quantum information processing^17,18 and quantum computation¹⁹. For instance, the topological pumping²⁰ between edge modes^{21,22,23,24,25,26,27,28,29,30} at both boundaries of the SSH model, which depends on the adiabatic modulation of the temporal parameter of system and further yields the excitation transport across bulk, enables the ability in robust quantum state transfer^{31,32,33,34,35,36,37,38,39}.

On the other hand, with the rapid developments and significant advances in micronano manufacturing and material processing technology, circuit quantum electrodynamic (QED) systems^40,41,42 have recently become one of the promising candidates for implementing quantum optics⁴³, quantum computation⁴⁴, quantum information processing^45,46, and quantum simulation^47,48,49, which is attributed to flexibility in structural designability and parameter adjustability on a single-site level. As an example, easy construction and high controllability make them a powerful tool to realize lattice models and further to map and investigate topological insulators⁵⁰. Moreover, based on different variations of the SSH model simulated by the circuit-QED lattice, a number of robust and fast quantum state transfer schemes have been proposed via topological pumping of edge modes^{51,52,53,54,55}. However, although quantum state transfer is a straightforward application of topological pumping of edge modes, it is also meaningful to try to engineer a feasible quantum optical device in terms of such a pumping process.

In this paper, a 1D dimerized circuit-QED lattice with cyclical modulation is investigated and is mapped to the paradigmatic SSH model. We find that the transfer between excitations at both boundary resonators of the lattice is unidirectional, which is attributed to the nonreciprocity of transport emerging in the adiabatic pumping between two specific left and right edge modes and allows us to engineer a quantum diode based on the one-way excitation transfer. We also employ fidelity to quantitatively reflect the quality of the quantum diode and further address the effect of the modulation amplitude on the one-way excitation transfer outcome. Additionally, in order to accelerate the nonreciprocal transport in the quantum diode, an appropriate cyclical modulation is applied to realize a much faster one-way excitation transfer, where the optimization of the control function indicates that a threefold reduction in time spent can be achieved.

Results

Mapping from circuit-QED lattice to the paradigmatic SSH model

The system considered here is a circuit-QED lattice made up of a 1D chain of transmission line resonators and superconducting qubits, as shown in Fig. 1. Each unit cell contains two different resonators respectively labeled by A_m and B_m ($m\in \left[1,M\right]$ is the index of unit cell with M giving the number of unit cells). Moreover, every two intracell (intercell) resonators are indirectly coupled by a two-level flux qubit ${Q}_{m}^{1}$ (${Q}_{m}^{2}$). The whole system is thus governed by the following Hamiltonian

$$\begin{array}{ll}{H}_{f}=&\mathop{\sum}\limits_{m}\left[\left({\omega }_{a}{a}_{m}^{{\dagger} }{a}_{m}+{\omega }_{b}{b}_{m}^{{\dagger} }{b}_{m}\right)+\left(\frac{{\omega }_{1}}{2}{{\sigma }^{z}}_{m,1}+\frac{{\omega }_{2}}{2}{\sigma }_{m,2}^{z}\right)\right],\\ {H}_{h}=&\mathop{\sum}\limits_{m}\left[{g}_{1}{\sigma }_{m,1}^{-}\left({a}_{m}^{{\dagger} }+{b}_{m}^{{\dagger} }\right)+{g}_{2}{\sigma }_{m,2}^{-}\left({b}_{m}^{{\dagger} }+{a}_{m+1}^{{\dagger} }\right)\right.\\ &\left.+\,{g}_{1}{\sigma }_{m,1}^{+}\left({a}_{m}+{b}_{m}\right)+{g}_{2}{\sigma }_{m,2}^{+}\left({b}_{m}+{a}_{m+1}\right)\right],\end{array}$$

(1)

where H_f represents the free Hamiltonian of all the resonators and qubits. ${a}_{m}^{{\dagger} }$ (${b}_{m}^{{\dagger} }$) and ω_a (ω_b) are the photonic creation operator and frequency of resonator A_m (B_m), respectively. ω_k (k = 1, 2) is the tunable level spacing of flux qubit ${Q}_{m}^{k}$, which can be modulated by the magnetic field provided by a flux-bias line^48,56,57,58, and ${\sigma }_{m,k}^{z}={\left\vert e\right\rangle }_{m,k}\left\langle e\right\vert -{\left\vert g\right\rangle }_{m,k}\left\langle g\right\vert$ denotes the Pauli z operator of qubit ${Q}_{m}^{k}$ with $\left\vert g\right\rangle$ and $\left\vert e\right\rangle$, respectively, being the corresponding ground and excited levels. H_h describes the photonic hopping between two adjacent resonators assisted by qubit. g_k is the coupling strength between resonator and qubit and ${\sigma }_{m,k}^{-}={\left\vert g\right\rangle }_{m,k}\left\langle e\right\vert$ (${\sigma }_{m,k}^{+}={\left\vert e\right\rangle }_{m,k}\left\langle g\right\vert$) is the lowering (raising) operator of qubit ${Q}_{m}^{k}$.

Fig. 1: Schematic diagram of the circuit-QED lattice.

The circuit-QED lattice consists of a 1D array of transmission line resonators and superconducting qubits. The number of unit cells is given by M and each unit cell (the dashed rectangle) contains two different resonators A_m and B_m, except that there is only the resonator A_M in the last unit cell. Moreover, two intracell and intercell resonators are coupled by two-level flux qubits ${Q}_{m}^{1}$ and ${Q}_{m}^{2}$, respectively.

For clarity, we now turn to the rotating frame with respect to an external driving frequency ω_d and to the interaction picture with respect to the level spacings ω₁ and ω₂. Moreover, when all the qubits are prepared in the ground state, the effective Hamiltonian of the whole system reads

$$\begin{array}{lll}{H}_{{{{\rm{eff}}}}}\,=\,\mathop{\sum}\limits_{m}\left[\left({{{\Delta }}}_{a}-\frac{{g}_{1}^{2}}{{{{\Delta }}}_{1}}-\frac{{g}_{2}^{2}}{{{{\Delta }}}_{2}}\right){a}_{m}^{{\dagger} }{a}_{m}\right.\\ \qquad\quad +\,\left({{{\Delta }}}_{b}-\frac{{g}_{1}^{2}}{{{{\Delta }}}_{1}}-\frac{{g}_{2}^{2}}{{{{\Delta }}}_{2}}\right){b}_{m}^{{\dagger} }{b}_{m}\\ \qquad\quad -\,\frac{{g}_{1}^{2}}{{{{\Delta }}}_{1}}\left({a}_{m}^{{\dagger} }{b}_{m}+{b}_{m}^{{\dagger} }{a}_{m}\right)\\ \qquad \quad \left.-\,\frac{{g}_{2}^{2}}{{{{\Delta }}}_{2}}\left({b}_{m}^{{\dagger} }{a}_{m+1}+{a}_{m+1}^{{\dagger} }{b}_{m}\right)\right]\end{array}$$

(2)

here Δ_o = ω_o − ω_d (o = a, b) and Δ_k = ω_k − ω_d are the detunings of resonator and qubit with respect to the external driving field, respectively. For simplicity, we further focus on the following parameter regimes

$$\begin{array}{lll}{{{\Delta }}}_{a}-\frac{{g}_{1}^{2}}{{{{\Delta }}}_{1}}-\frac{{g}_{2}^{2}}{{{{\Delta }}}_{2}}&=&0,\\ {{{\Delta }}}_{b}-\frac{{g}_{1}^{2}}{{{{\Delta }}}_{1}}-\frac{{g}_{2}^{2}}{{{{\Delta }}}_{2}}&=&0,\\ \frac{{g}_{1}^{2}}{{{{\Delta }}}_{1}}=J\left(1-\lambda \cos \phi \right)&=&{J}_{1},\\ \frac{{g}_{2}^{2}}{{{{\Delta }}}_{2}}=J\left(1+\lambda \cos \phi \right)&=&{J}_{2},\end{array}$$

(3)

where $\lambda \in \left[0,1\right]$ and $\phi \in \left[0,2\pi \right]$ are the amplitude and phase of cosine modulation, respectively, and J is referred to as the energy unit throughout the paper. In the current superconducting circuit experiments, g_k can be tuned in the range of 0-400 MHz⁵⁹ and ω_k can be modified from 100 MHz to 15 GHz^60,61, which thus provides a considerably large modulation range for J_k. Accordingly, the final total effective Hamiltonian can be rewritten as follows

$${H}_{{{{\rm{tot}}}}}=\mathop{\sum}\limits_{m}-\left({J}_{1}{a}_{m}^{{\dagger} }{b}_{m}+{J}_{2}{a}_{m+1}^{{\dagger} }{b}_{m}+{{{\rm{H.c.}}}}\right).$$

(4)

Obviously, Eq. (4) shows a nearest-neighbor tight-binding dimer with cyclical modulation, which reflects the most typical feature of the paradigmatic SSH model.

Quantum diode in the paradigmatic SSH model

Before proceeding, let us start by presenting the energy spectrum of the circuit-QED lattice. In this paper, we are interested in the lattice with an odd number of resonators, i.e., there is only the resonator A_M in the last unit cell. Figure 2 displays the energy spectrum of the system with λ = 0.5 as a function of ϕ, we can observe that a gap separating the upper and lower bands closes and reopens at ϕ = 0.5π and 1.5π. Furthermore, different from the common even-sized chain in which ϕ = 0.5π and 1.5π are the topological transition points and a pair of zero-energy edge modes respectively occupying opposite boundaries of the lattice emerges only if $\phi \in \left[0,0.5\pi \right)\cup \left(1.5\pi ,2\pi \right]$ (topologically nontrivial phase) and disappears once $\phi \in \left(0.5\pi ,1.5\pi \right)$ (topologically trivial phase), there is always a single zero-energy mode in the whole region of $\phi \in \left[0,2\pi \right]$, which reproduces the celebrated even-odd effect of the paradigmatic SSH model due to the chiral symmetry of the system⁷. The single zero-energy mode can be derived as

$$\left\vert {{{\Theta }}}_{0}\right\rangle =\tilde{N}\mathop{\sum }\limits_{m=1}^{M}{\left(-\frac{{J}_{1}}{{J}_{2}}\right)}^{m-1}{a}_{m}^{{\dagger} }\left\vert {{{\bf{0}}}}\right\rangle ,$$

(5)

with $\tilde{N}=\sqrt{\frac{1-{\left(\frac{{J}_{1}}{{J}_{2}}\right)}^{2}}{1-{\left(\frac{{J}_{1}}{{J}_{2}}\right)}^{2M}}}$ and $\left\vert {{{\bf{0}}}}\right\rangle$ being normalized coefficient and vacuum state, respectively. It can be easily demonstrated that when $\phi \in \left[0,0.5\pi \right)\cup \left(1.5\pi ,2\pi \right]$, the single zero-energy mode is mainly distributed at the left boundary of the lattice, whereas it is mainly concentrated on the right boundary of the lattice for $\phi \in \left(0.5\pi ,1.5\pi \right)$.

Fig. 2: Energy spectrum of the circuit-QED lattice as a function of the modulation phase ϕ in the scheme of cosine modulation.

The number of unit cells is M = 11 and the modulation amplitude is λ = 0.5. There is a single zero-energy mode throughout the modulation phase $\phi \in \left[0,2\pi \right]$. The red and green lines delineate the left and right edge modes, respectively, and these blue curves delineate the gapped bulk modes.

As mentioned above, we can find that with ϕ continuously varying from 0 to π, the single left edge mode will gradually convert into the single right edge mode. Therefore, according to the theory of adiabatic temporal evolution, when the system is initialized by the left edge mode at ϕ = 0 and ϕ is further scanned sufficiently slowly from 0 to π, the instantaneous eigenmode of the system can always remain the zero-energy mode during evolution and in the end, the left edge mode will be dynamically pumped to the right edge mode at ϕ = π, which leads to excitation transport across the bulk and can be potentially exploited in quantum state transfer. In practice, the adiabatic sweep of ϕ can be achieved by the external control. For instance, we can adiabatically control the external magnetic field to modulate Δ_k such that J_k is effectively tuned. While there is an intensive investigation on topological pumping scenarios of edge modes or quantum state transfer proposals in different generalized counterparts of the paradigmatic SSH model, here, we aim to reveal another interesting application in the paradigmatic SSH model, i.e., quantum diode based on the adiabatic pumping between the two edge modes at ϕ = 0 and π. Different from the conventional electronic diode⁶², the so-called quantum diode is elucidated in the following description: if we initially inject an excitation into the leftmost resonator A₁, as ϕ is slowly swept from 0 to π, the excitation will be transited to the rightmost resonator A_M and be output from it. On the contrary, when an excitation is initially injected into the rightmost resonator A_M, despite still slowly ramping ϕ from 0 to π, there will be no output from the leftmost resonator A₁. It turns out that a nonreciprocal transport behavior characterized by the one-way excitation transfer occurs and if we treat the leftmost and rightmost resonators A₁ and A_M as two ports, the whole system can be used for engineering and preparing a quantum diode.

To this end, we first set $\phi =\frac{t}{\tau }\pi$ with $\frac{t}{\tau }\in \left[0,1\right]$, where t stands for the evolution time and τ dominates the evolution speed. Furthermore, the corresponding transport process can be captured by solving the time-dependent Schrödinger equation numerically (ℏ = 1)

$$\begin{array}{lll}i{\partial }_{t}{\psi }_{m,A}\left(t\right)&=&-{J}_{2}\left(t\right){\psi }_{m-1,B}\left(t\right)-{J}_{1}\left(t\right){\psi }_{m,B}\left(t\right),\\ i{\partial }_{t}{\psi }_{m,B}\left(t\right)&=&-{J}_{1}\left(t\right){\psi }_{m,A}\left(t\right)-{J}_{2}\left(t\right){\psi }_{m+1,A}\left(t\right),\end{array}$$

(6)

where ${\psi }_{m,A}\left(t\right)$ (${\psi }_{m,B}\left(t\right)$) is the probability amplitude on resonator A_m (B_m) at time t. At the same time, in order to quantitatively evaluate the quality of the quantum diode, we introduce fidelity to directly measure the transport outcome, which is defined as

$$\begin{array}{lll}{F}_{l}&=&| \langle {{{\Psi }}}_{g,M}| {{{\Psi }}}_{f,M}\rangle | ,\\ {F}_{r}&=&| \langle {{{\Psi }}}_{g,1}| {{{\Psi }}}_{f,1}\rangle | ,\end{array}$$

(7)

where the fidelity F_l (F_r) corresponds to the case of an excitation initially injected into the leftmost (rightmost) resonator A₁ (A_M) with $\left\vert {{{\Psi }}}_{f,M}\right\rangle$ ($\left\vert {{{\Psi }}}_{f,1}\right\rangle$) and $\left\vert {{{\Psi }}}_{g,M}\right\rangle$ ($\left\vert {{{\Psi }}}_{g,1}\right\rangle$) being the normalized final state when temporal evolution is finished and the target state for an excitation injected into the rightmost (leftmost) resonator A_M (A₁), respectively. It is clear that ${F}_{l}\left({F}_{r}\right)\in \left[0,1\right]$ and ${F}_{l}\left({F}_{r}\right)\to 1$ indicates that the initial leftmost (rightmost) excitation is completely transited to the opposite boundary resonator A_M (A₁) and is only output from it, however, there will be no output at all from the opposite boundary resonator A_M (A₁) if ${F}_{l}\left({F}_{r}\right)\to 0$. For the quantum diode here, the most perfect quality is characterized by F_l → 1 and F_r → 0.

Figure 3a and b displays the fidelities F_l and F_r versus λ and τ, respectively. We can observe that as λ increases and τ decreases, F_l rises, but F_r is nearly vanishing all the time, which implies that the increase of λ can improve the quality of the quantum diode as long as τ is sufficiently small. The underlying mechanism behind this phenomenon can be interpreted as follows: on the one hand, the excitation initially injected into the leftmost resonator A₁ mostly contains a superposition of the left edge mode at ϕ = 0, on the contrary, there is almost no superposition of the left edge mode at ϕ = 0 in the excitation initially injected into the rightmost resonator A_M. On the other hand, the excitation transfer process purely depends on the adiabatic pumping from the left edge mode at ϕ = 0 to the right edge mode at ϕ = π. Hence, when we sufficiently slowly ramp ϕ from 0 to π, the initial leftmost excitation will be transfered to the rightmost resonator A_M, whereas there is a large possibility that the initial rightmost excitation will not be transferred to the leftmost resonator A₁. Additionally, as clarified by Eq. (5), the increasing λ gradually enhances the localization of these zero-energy edge modes on the corresponding boundary resonator, including the left edge mode at ϕ = 0 and the right edge mode at ϕ = π, so that the weight of the left edge mode at ϕ = 0 in the eigenmode superposition of the initial leftmost (rightmost) excitation is elevated (reduced). As a result, the adiabatic pumping from the left edge mode at ϕ = 0 to the right edge mode at ϕ = π is more and more dominant in the excitation transfer process from the leftmost resonator A₁ to the rightmost resonator A_M, which contributes to obtaining a final state that is more localized on the rightmost resonator A_M and thus to a higher F_l.

Fig. 3: Fidelity of the excitation transfer in the scheme of cosine modulation.

a Fidelity F_l versus the modulation amplitude λ and the total evolution time τ. b Fidelity F_r versus the modulation amplitude λ and the total evolution time τ. The modulation phase ϕ is scanned from 0 to π.

Fast one-way excitation transfer in the quantum diode

Preceding analysis and discussion illustrate that the optimal selection of λ for the perfect excitation transfer F_l → 1 in the circuit-QED lattice is λ = 1, since the two edge modes at ϕ = 0 and π with λ = 1 are entirely localized on the corresponding boundary resonators A₁ and A_M respectively so that the eigenmode superposition of the initial leftmost excitation only incorporates the left edge mode at ϕ = 0, moreover, as ϕ is sufficiently slowly swept from 0 to π, the excitation transfer process from the leftmost resonator A₁ to the rightmost resonator A_M is exactly equivalent to the adiabatic pumping from the left edge mode at ϕ = 0 to the right edge mode at ϕ = π actually and the final state will be the desired right edge mode at ϕ = π, in other words, the initial leftmost excitation will be completely transferred to the rightmost resonator A_M, thereby yielding the perfect excitation transfer F_l = 1. However, the adiabatic temporal evolution is an ideal limit, i.e., the pumping can instantaneously remain in the zero-energy mode all the time over the whole evolution unless τ → ∞, if τ is finite, there will be typically some (even very small) nonadiabatic excitations to instantaneous bulk modes, which causes the leak into the space of other eigenmodes. Therefore, it is impossible to have the perfect excitation transfer F_l = 1 due to the existence of finite-time nonadiabatic excitation and we thus need to set a lower bound for the most perfect quality of the quantum diode. Here, two alternative standards are considered: (i) one of the standards is chosen as F_l ⩾ 99% and F_r ⩽ 0.1%, (ii) the other is chosen as F_l ⩾ 99.9% and F_r ⩽ 0.1%. Next, we will hunt for the minimum τ required for the two standards, respectively.

When λ = 1, we present in Fig. 4a, c the instantaneous change of J₁ and J₂ and the instantaneous energy spectrum of the system within a total evolution time τ, respectively, Fig. 4e further shows F_l and F_r against τ. We find that standards (i) and (ii) can be achieved respectively at τ = 312 and 406, where F_l = 99.3207% and F_r = 0.0157% for standard (i) and F_l = 99.9001% and F_r = 0.0297% for standard (ii). Experimentally, for a typical choice of $\frac{J}{2\pi }=250\,{{{\rm{MHz}}}}$, τ = 312(406) corresponds to a real duration of $t=\frac{\tau }{J}\approx 0.20\,(0.26)\,{{{\rm{\mu s}}}}$, which is much less than the decoherence time 30–140 μs^48,63 of qubit.

Fig. 4: Instantaneous change of the nearest-neighbor coupling, instantaneous energy spectrum of the circuit-QED lattice, and fidelity of the excitation transfer in different modulation schemes when the modulation amplitude λ = 1.

For the schemes of (a) cosine modulation and (b) tangent modulation with the free parameter $\xi =\frac{\pi }{2}+0.4$, instantaneous change of the nearest-neighbor couplings J₁ and J₂ within a total evolution time τ. For the schemes of (c) cosine modulation and (d) tangent modulation with the free parameter $\xi =\frac{\pi }{2}+0.4$, instantaneous energy spectrum of the circuit-QED lattice within a total evolution time τ, where the red and green lines delineate the instantaneous left and right edge modes, respectively, and these blue curves delineate the instantaneous gapped bulk modes. For the schemes of (e) cosine modulation and (f) tangent modulation with the free parameter $\xi =\frac{\pi }{2}+0.4$, fidelities F_l and F_r against the total evolution time τ.

In fact, it is necessary to realize a much faster one-way excitation transfer, which is conducive to accelerate the nonreciprocal transport in the quantum diode. Therefore, we first discuss what determines adiabatic condition and analyze why the cosine modulation in J₁ and J₂ is not appropriate for a fast pumping between the two edge modes at ϕ = 0 and π briefly. Adiabatic temporal evolution is required to satisfy adiabatic condition, meaning that the changing velocity in time of J₁ and J₂ need to be much less than the magnitude of the corresponding instantaneous gap. Furthermore, within a total evolution time τ, while the instantaneous gap between the zero-energy mode and the bulk mode on either side shown in Fig. 4c decreases as t increases and reaches its minimum at t = 0.5τ, the cosine modulation shown in Fig. 4a indicates that the changing velocity in time of J₁ and J₂ is the slowest at the beginning and becomes faster and faster when it approaches t = 0.5τ, which makes a relatively long total evolution time τ is necessary to avoid nonadiabatic excitations and to further attain a high F_l. Accordingly, it will be instructive for us to seek out a control function^64,65,66,67 that makes the changing velocity in time of J₁ and J₂ matches well with the magnitude of the corresponding instantaneous gap.

Let us now replace the cosine modulation in J₁ and J₂ with the following control function

$$\begin{array}{lll}{J}_{1}\left(t\right)&=&J\left\{1-\lambda \frac{\tan \left[\frac{t}{\tau }\left(2\pi \,-\,2\xi \right)\,+\,\xi \right]}{\tan \xi }\right\},\\ {J}_{2}\left(t\right)&=&J\left\{1+\lambda \frac{\tan \left[\frac{t}{\tau }\left(2\pi \,-\,2\xi \right)\,+\,\xi \right]}{\tan \xi }\right\},\end{array}$$

(8)

with free parameter $\xi \in \left(\frac{\pi }{2},\pi \right)$. When λ = 1 and $\xi =\frac{\pi }{2}+0.4$, the instantaneous change of J₁ and J₂ and the instantaneous energy spectrum of the system within a total evolution time τ are also shown in Fig. 4b, d, respectively. We can observe that in contrast to the scheme of cosine modulation, the changing velocity in time of J₁ and J₂ decreases as t increases and reaches its minimum at t = 0.5τ, in the meanwhile, the instantaneous gap between the zero-energy mode and the bulk mode on either side is also the largest at the beginning and still becomes smaller and smaller when it approaches t = 0.5τ, which indicates that the changing velocity in time of J₁ and J₂ may coincide with the magnitude of the corresponding instantaneous gap via an optimization search in the range of ξ. After undergoing a simple search, as shown in Fig. 4f, we find that for $\xi =\frac{\pi }{2}+0.4$ chosen above, the one-way excitation transfer can be achieved respectively at τ = 95 with F_l = 99.2910% and F_r = 0.0472% for standard (i) and τ = 135 with F_l = 99.9132% and F_r = 0.0387% for standard (ii), and the operation times for τ = 95 and 135 with $\frac{J}{2\pi }=250\,{{{\rm{MHz}}}}$ are t ≈ 0.06 μs and 0.09 μs, respectively, both are also much shorter compared with the typical decoherence time of qubit. Furthermore, as an example, we also clearly present in Fig. 5 the one-way excitation transfer process when τ = 135. As a result, in the quantum diode, the improvement in the time spent of the nonreciprocal transport is pronounced, which is almost 300% shorter compared with the scheme of cosine modulation, and we can indeed realize a much faster one-way excitation transfer by employing the control function in Eq. (8) with $\xi =\frac{\pi }{2}+0.4$.

Fig. 5: One-way excitation transfer process.

a Excitation transfer process when an excitation is initially injected into the leftmost resonator A₁. b Excitation transfer process when an excitation is initially injected into the rightmost resonator A_M. The nearest-neighbor couplings J₁ and J₂ are governed by the scheme of tangent modulation with the modulation amplitude λ = 1, the free parameter $\xi =\frac{\pi }{2}+0.4$, and the total evolution time τ = 135. The inset in (b) shows the finial excitation population for the initial rightmost excitation and it is obvious that there is almost no output from the leftmost resonator A₁.

Note that disorder is naturally unavoidable in practical experimental device. In the circuit-QED lattice, the imperfection mainly resides in the nearest-neighbor couplings J₁ and J₂ and can be described by the following Hamiltonian

$${H}_{d}=\mathop{\sum}\limits_{m}-\left(d{J}_{1}^{m}{a}_{m}^{{\dagger} }{b}_{m}+d{J}_{2}^{m}{a}_{m+1}^{{\dagger} }{b}_{m}+{{{\rm{H.c.}}}}\right),$$

(9)

where $d{J}_{1}^{m}$, $d{J}_{2}^{m}=W{{\Xi }}$ with W being the disorder strength and ${{\equiv }}\in [-0.5,0.5]$ being uniformly and randomly distributed. In order to find out whether the quantum diode is immune to the imperfection, Fig. 6 displays F_l and F_r as a function of W when τ = 95 and 135, respectively. For each $d{J}_{1}^{m}$ and $d{J}_{2}^{m}$, we choose 1000 disorder samples and average over the results of all of these samples to obtain the corresponding F_l and F_r. We can observe that as W increases, F_l shows a wide plateau with F_l > 99% when τ = 95 and F_l > 99.9% when τ = 135, which is ascribed to the nontrivial topological property of the system endowed by the chiral symmetry of the circuit-QED lattice and further reflects the robustness of the excitation transfer from the leftmost resonator A₁ to the rightmost resonator A_M against the imperfection, furthermore, it drops quickly as disorder continues to enhance. However, compared with F_l, F_r is more sensitive to the imperfection and can sustain a very low value only when W is relatively small. Therefore, the quantum diode is robust against moderate disorder.

Fig. 6: Fidelity of the excitation transfer as a function of the disorder strength W in the scheme of tangent modulation.

The modulation amplitude is λ = 1, the free parameter is $\xi =\frac{\pi }{2}+0.4$, and the total evolution time is τ = 95 and 135, respectively. The fidelities F_l and F_r are placed on the left and right axises, respectively, averaged over 1000 disorder samples.

Discussion

In conclusion, we have revealed that the adiabatic pumping between two specific left and right edge modes in a 1D cyclically modulated circuit-QED lattice with dimerization, which is the photonic analog of the paradigmatic SSH model, showing a nonreciprocity of transport and producing the one-way transfer between excitations at both boundary resonators of the lattice such that we can engineer a quantum diode based on the one-way excitation transfer. We also demonstrate that the quality of the quantum diode defined by fidelity can be improved by increasing the modulation amplitude, in other words, the one-way excitation transfer process becomes more and more pronounced as the modulation amplitude increases. Furthermore, we find an appropriate cyclical modulation to realize a much faster one-way excitation transfer and to further accelerate the nonreciprocal transport in the quantum diode, where there is almost a threefold reduction in time spent via the optimization of the control function.

Methods

Energy spectrum of the circuit-QED lattice

In the scheme of cosine modulation, in order to evaluate the energy spectrum of the circuit-QED lattice, in the basis of $\{\left\vert j\right\rangle ={c}_{j}^{{\dagger} }\left\vert {{{\bf{0}}}}\right\rangle ,1\,\leqslant\, j\,\leqslant\, 2M-1\}$ (c = a for odd j and c = b for even j), equation (4) can be written as a $\left(2M-1\right)\times \left(2M-1\right)$ matrix, which takes the form

$$\begin{array}{r}{H}_{{{{\rm{tot}}}}}=-\left(\begin{array}{ccccccc}0\,&\,{J}_{1}\left(\lambda ,\phi \right)\,&\,0\,&\,\cdots \,&\,0\,&\,0\,&\,0\\ {J}_{1}\left(\lambda ,\phi \right)\,&\,0\,&\,{J}_{2}\left(\lambda ,\phi \right)\,&\,\cdots \,&\,0\,&\,0\,&\,0\\ 0\,&\,{J}_{2}\left(\lambda ,\phi \right)\,&\,0\,&\,\cdots \,&\,0\,&\,0\,&\,0\\ \cdots \,&\,\cdots \,&\,\cdots \,&\,\cdots \,&\,\cdots \,&\,\cdots \,&\,\cdots \\ 0\,&\,0\,&\,0\,&\,\cdots \,&\,0\,&\,{J}_{1}\left(\lambda ,\phi \right)\,&\,0\\ 0\,&\,0\,&\,0\,&\,\cdots \,&\,{J}_{1}\left(\lambda ,\phi \right)\,&\,0\,&\,{J}_{2}\left(\lambda ,\phi \right)\\ 0\,&\,0\,&\,0\,&\,\cdots \,&\,0\,&\,{J}_{2}\left(\lambda ,\phi \right)\,&\,0\\ \end{array}\right),\end{array}$$

(10)

where the nonzero matrix elements ${H}_{j,j+1}=\left\langle j\right\vert {H}_{{{{\rm{tot}}}}}\left\vert j+1\right\rangle ={H}_{j+1,j}=\left\langle j+1\right\vert {H}_{{{{\rm{tot}}}}}\left\vert j\right\rangle ={J}_{1}\left(\lambda ,\phi \right)\left({J}_{2}\left(\lambda ,\phi \right)\right)$ for odd (even) j with j ∈ [1, 2M − 2]. Therefore, when λ is given and ϕ varies, we can obtain the energy spectrum of the circuit-QED lattice by diagonalizing Eq. (10) numerically. Moreover, the energy spectrum of the system for the scheme of tangent modulation can be obtained in the same way.

Calculation of fidelity

In the scheme of cosine modulation, regarding the fidelity calculation, a necessary procedure is that we need to capture the final state when the adiabatic sweep of ϕ is finished. For an initial excitation, after substituting $\phi =\frac{t}{\tau }\pi$, we can capture the state $\left\vert {{\Psi }}\left(t\right)\right\rangle$ at any time t during evolution by solving the time-dependent Schrödinger equation of the system numerically (ℏ = 1),

$$i\frac{\partial }{\partial t}\left\vert {{\Psi }}(t)\right\rangle ={H}_{{{{\rm{tot}}}}}(t)\left\vert {{\Psi }}(t)\right\rangle .$$

(11)

Due to the fact that $\left\vert {{\Psi }}\left(t\right)\right\rangle$ can always be spanned by the basis $\{\left\vert j\right\rangle ,1\,\leqslant\, j\,\leqslant\, 2M-1\}$, i.e., $\left\vert {{\Psi }}\left(t\right)\right\rangle ={\sum }_{m}{\psi }_{m,A}\left(t\right){a}_{m}^{{\dagger} }\left\vert {{{\bf{0}}}}\right\rangle +{\psi }_{m,B}\left(t\right){b}_{m}^{{\dagger} }\left\vert {{{\bf{0}}}}\right\rangle$ with ${\psi }_{m,A}\left(t\right)$ and ${\psi }_{m,B}\left(t\right)$ being the probability amplitudes on resonators A_m and B_m at time t, this thus leads to the establishment of the following system of differential equations for ${\psi }_{m,A}\left(t\right)$ and ${\psi }_{m,B}\left(t\right)$ when λ is given,

$$\begin{array}{r}i\frac{\partial }{\partial t}\left(\begin{array}{c}{\psi }_{1,A}\left(t\right)\\ {\psi }_{1,B}\left(t\right)\\ {\psi }_{2,A}\left(t\right)\\ \cdots \\ {\psi }_{M-1,A}\left(t\right)\\ {\psi }_{M-1,B}\left(t\right)\\ {\psi }_{M,A}\left(t\right)\\ \end{array}\right)=-\left(\begin{array}{ccccc}0\,&\,{J}_{1}\left(t\right)\,&\,\cdots \,&\,0\,&\,0\\ {J}_{1}\left(t\right)\,&\,0\,&\,\cdots \,&\,0\,&\,0\\ 0\,&\,{J}_{2}\left(t\right)\,&\,\cdots \,&\,0\,&\,0\\ \cdots \,&\,\cdots \,&\,\cdots \,&\,\cdots \,&\,\cdots \\ 0\,&\,0\,&\,\cdots \,&\,{J}_{1}\left(t\right)\,&\,0\\ 0\,&\,0\,&\,\cdots \,&\,0\,&\,{J}_{2}\left(t\right)\\ 0\,&\,0\,&\,\cdots \,&\,{J}_{2}\left(t\right)\,&\,0\\ \end{array}\right)\left(\begin{array}{c}{\psi }_{1,A}\left(t\right)\\ {\psi }_{1,B}\left(t\right)\\ {\psi }_{2,A}\left(t\right)\\ \cdots \\ {\psi }_{M-1,A}\left(t\right)\\ {\psi }_{M-1,B}\left(t\right)\\ {\psi }_{M,A}\left(t\right)\\ \end{array}\right).\end{array}$$

(12)

It can be further expressed as a compact form, i.e., equations (6), where ${J}_{1}\left(t\right)=J\left(1-\lambda \cos \frac{t}{\tau }\pi \right)$ and ${J}_{2}\left(t\right)=J\left(1+\lambda \cos \frac{t}{\tau }\pi \right)$, together with the two initial conditions ${\psi }_{m,A}\left(0\right)={\delta }_{m,1}$ and ${\psi }_{m,B}\left(0\right)=0$ for the leftmost excitation and ${\psi }_{m,A}\left(0\right)={\delta }_{m,M}$ and ${\psi }_{m,B}\left(0\right)=0$ for the rightmost excitation (δ is the Kronecker function). According to ϕ being scanned from 0 to π and $\phi =\frac{t}{\tau }\pi$, evolution is terminated at t = τ, i.e., $\frac{t}{\tau }\in \left[0,1\right]$. As a result, when λ and τ is given, for both the initial leftmost and rightmost excitations, the corresponding transport process and finial state can be captured successively, which allows us to further calculate the fidelities F_l and F_r. Moreover, for the scheme of tangent modulation, the fidelities F_l and F_r and the excitation transport process can also be obtained in the same way. In the numerical simulation, we solve Eq. (12) by resorting to the fourth-order variable-step Runge-Kutta method.