Non-reciprocal population dynamics in a quantum trimer

royalsocietypublishing.org/journal/rspa Research Cite this article: Downing CA, Zueco D. 2021 Non-reciprocal population dynamics in a quantum trimer. Proc.R.Soc.A477:20210507. https://doi.org/10.1098/rspa.2021.0507 Received: 22 June 2021 Accepted: 20 October 2021 Subject Areas: optics, solid-state physics, quantum physics Keywords: chirality, non-reciprocity, two-level systems, open quantum systems Author for correspondence: C. A. Downing e-mail: [email protected] Electronic supplementary material is available online at https://doi.org/10.6084/m9.figshare. c.5705312. Non-reciprocal population dynamics in a quantum trimer C. A. Downing1and D. Zueco2 1Department of Physics and Astronomy, University of Exeter, Exeter EX4 4QL, UK 2Instituto de Nanociencia y Materiales de Aragón (INMA), CSIC-Universidad de Zaragoza, Zaragoza 50009, Spain CAD, 0000-0002-0058-9746;DZ,0000-0003-4478-1948 We study a quantum trimer of coupled two-level systems beyond the single-excitation sector, where the coherent coupling constants are ornamented by a complex phase. Accounting for losses and gain in an open quantum systems approach, we show how the mean populations of the states in the system crucially depend on the accumulated phase in the trimer. Namely, for non-trivial accumulated phases, the population dynamics and the steady states display remarkable non-reciprocal behaviour in both the singly and doubly excited manifolds. Furthermore, while the directionality of the resultant chiral current is primarily determined by the accumulated phase in the loop, the sign of the flow may also change depending on the coupling strength and the amount of gain in the system. This directionality paves the way for experimental studies of chiral currents at the nanoscale, where the phases of the complex hopping parameters are modulated by magnetic or synthetic magnetic fields. 1. Introduction Reciprocity in the animal kingdom is manifested by the evolution of reciprocal altruism: ‘you scratch my back, and I will scratch yours’ [1]. Aside from mere grooming, the consequences of reciprocity for the sharing of food, medicine and knowledge are profound. However, the breakdown of reciprocity, perhaps fuelled by a lack of affinity or obligation, can also lead to certain benefits for the non-reciprocator, who can profit from the nonreciprocal interaction [2]. 2021 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/ by/4.0/, which permits unrestricted use, provided the original author and source are credited. Downloaded from https://royalsocietypublishing.org/ on 01 April 2022 2 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 477:20210507 .......................................................... In condensed matter physics, there is currently a revolution in the fabrication and mastery of nanostructures which can exploit quantum mechanics [3,4]. This progress promises a new paradigm of quantum technologies which seek to transform the modern world [5,6]. In particular, the field of quantum optics provides the ideal framework to describe light–matter interactions and the quantum aspects of the latest metamaterials, which are commonly built from nanoscopic lattices of meta-atoms [7–10]. Recently, it was noticed that the introduction of the concept of non-reciprocity into nanophotonic systems will have sweeping implications for the control of light–matter coupling [11–14], and hence for future quantum technology. Non-reciprocal interactions between meta-atoms in metamaterials can immediately be seen to be advantageous for future chiral devices, such as circulators and isolators, which rely on the directional transfer of energy and information at the nanoscale [15–22]. In 2017, Roushan et al.[23] reported the directional circulation of photons in a triangular loop of superconducting qubits. In a pioneering experiment for chiral quantum optics, the team observed chiral ground-state currents and probed the unusual quantum phases of strongly interacting photons (for a review of strongly interacting photons, see [24]). The required synthetic magnetic fields were realized by sinusoidally modulating their qubit–qubit couplings, which led to the necessary complex phases attached to the coherent coupling constants [25]. Such complex phases can appear in various ways; for example: in a real magnetic field through the Peierls substitution [26,27], via a Peierls tunnelling phase even in the absence of an external magnetic field [28], using a time-dependent coupling Hamiltonian [29,30], constructing synthetic gauge fields using synthetic lattices [31], using light-induced gauge potentials [32–34], designing inductor–capacitor circuits [35], by considering circularly polarized dipoles [36] or by careful pumping, which gives rise to complex potentials [37]. Inspired by the landmark experiment of Roushan et al. [23], who modelled their photonic system as harmonic oscillators, in this work we study a trimer of two-level systems (2LSs) in order to probe the whole energy ladder, including the effects of saturation due to the strong interactions. The 2LS approximation may be realized in an abundance of physical systems, as catalogued in [38], including superconducting qubits [39,40], cold atoms [41] and plasmons in metallic nanoparticles [42]. We consider our 2LS trimer in a triangular geometry (figure 1a), in order to form a loop which may enclose a non-trivial accumulated phase (depending on the phases of the complex hopping parameters), which is akin to an Aharonov–Bohm ring [43]. Importantly, we go beyond the single-excitation limit, which allows us to study the circulation of multiple excitations in our system as we modulate the amount of gain and loss in the trimer. Prior studies of trimers have primarily focused on including losses in a non-Hermitian Hamiltonian approach [44–52] (for a review of non-Hermitian classical and quantum physics, see [53]; for a review of non-Hermitian systems and topology, see [54]), while other investigations have employed an open quantum systems approach [55–57]. Here we employ a quantum master equation so that the dynamics is both stable and regular by construction, and in doing so we go beyond models restricted to strictly obeying non-Hermitian or PT symmetric Hamiltonians [58,59]. The rest of this work is organized as follows: in §2, we introduce our model; we reveal chiral steady states in §3; we present instances of non-reciprocal dynamics in §4; and in §5 we draw some conclusions. In addition, the electronic supplementary material provides some calculational details and figures. 2. Model We consider a trimer of 2LSs, which interact via coherent qubit–qubit coupling. Importantly, we allow for the coupling constants to have non-zero complex phases, which is the key ingredient that allows non-reciprocity to emerge [60–64]. Effectively, we study the Aharonov–Bohm effect [43] in a tight-binding quantum ring with three sites, in an open quantum systems approach. The generated phase φis both gauge invariant (the energies and eigenstates become dependent on the phase) and physically consequential (non-reciprocity is induced in the quantum transport). In Downloaded from https://royalsocietypublishing.org/ on 01 April 2022 3 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 477:20210507 .......................................................... w0 w0 w0 ge±iq31 ge±iq23 ge±iq12 (a) w6 w5 w7 w8 w3 w2 w1 w4 3w0 2w0 w0 0 N = 3 N = 2 N = 1 N = 0 rung uncoupled coupled (b) Figure 1. (a) A sketch of the trimer system, where each 2LS is of resonance frequency ω0, and the magnitude of the three coupling constants is g. Each hopping is associated with a phase θnn+1.(b) The four-rung energy ladder of the trimer, codified bythenumberofexcitationsN,whenthesystemisintheuncoupled(left)andcoupled(right)regimes.(Onlineversionincolour.) §2a, where we introduce the Hamiltonian formulation, we show how the phase φgeneralizes the eigenfrequencies. We include dissipation in the system in §2b, where we introduce the quantum master equation and incoherent gain processes. (a) Hamiltonian The Hamiltonian operator ˆ Hfor the system reads (we take ¯ h=1 throughout) ˆ H=ω0(σ† 1σ1+σ† 2σ2+σ† 3σ3)+g(eiθ12 σ† 1σ2+eiθ23 σ† 2σ3+eiθ31 σ† 3σ1+h.c.), (2.1) where we have used cyclic boundary conditions, corresponding to the triangle geometry sketched in figure 1a. The transition frequency of each 2LS is ω0and the coherent coupling between 2LS-nand 2LS-(n+1) is of magnitude g≥0andphaseθnn+1. The raising (lowering) operator of the nth 2LS is σ† n(σn), which satisfy the algebra of two distinguishable systems, with the anticommutator relation {σn,σ† n}=1 and the commutator relations [σn,σ† m]=[σn,σm]=0, where n= m. The Hamiltonian ˆ Hof equation (2.1) defines four subspaces, spanned by the eigenstates corresponding to N={0, 1, 2, 3}excitations. Explicitly, the subspaces are given by {|0},N=0, (2.2a) {σ† 1|0,σ† 2|0,σ† 3|0},N=1, (2.2b) {σ† 2σ† 1|0,σ† 3σ† 1|0,σ† 3σ† 2|0},N=2 (2.2c) and {σ† 3σ† 2σ† 1|0},N=3, (2.2d) where the vacuum state, without any excitations, is |0=|0, 0, 0. The energy ladder defined by equation (2.2) is sketched in figure 1b, in the weak (left) and strong (right) coupling regimes. The ground state is defined by ˆ H|0=ω1|0, and has the eigenvalue ω1=0 (purple lines in figure 1b). The triply excited state is characterized by ˆ Hσ† 3σ† 2σ† 1|0=ω8σ† 3σ† 2σ† 1|0, and is associated with the maximal eigenvalue ω8=3ω0(pink lines). These two extreme rungs of the energy ladder are the same in the coupled and uncoupled regimes (left and right in figure 1b), because they are associated with the wholly unoccupied state and the wholly occupied state. However, for the intermediate rungs associated with N={1, 2}excitations the nature of the coherent coupling is important. In the basis {σ† 1|0,σ† 2|0,σ† 3|0}, the singly excited (N=1) subspace has the 3 ×3 matrix representation H1=⎛ ⎜ ⎝ ω0geiθ12 ge−iθ31 ge−iθ12 ω0geiθ23 geiθ31 ge−iθ23 ω0 ⎞ ⎟ ⎠, (2.3) Downloaded from https://royalsocietypublishing.org/ on 01 April 2022 4 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 477:20210507 .......................................................... and the eigenvalues readily follow from equation (2.3) as ω2=ω0+2gcos φ+2π 3, (2.4a) ω3=ω0+2gcos φ+4π 3(2.4b) and ω4=ω0+2gcos φ 3, (2.4c) where we have introduced the quantity φ=θ12 +θ23 +θ31, (2.5) which describes the accumulated phase φin the trimer and is tantamount to the Aharonov–Bohm phase of a quantum ring [43]. Clearly, equation (2.4) exposes the first ramification of including non-trivial phases, even at the bedrock level of the eigenfrequencies, where it precipitates degeneracies at the trivial phases φ={0, π,2π}and otherwise presents non-trivial splittings of the energy levels. In the basis {σ† 2σ† 1|0,σ† 3σ† 1|0,σ† 3σ† 2|0}, the doubly excited (N=2) subspace has the 3 ×3 matrix representation H2=⎛ ⎜ ⎝ 2ω0ge−iθ12 geiθ31 geiθ12 2ω0ge−iθ23 ge−iθ31 geiθ23 2ω0 ⎞ ⎟ ⎠, (2.6) such that the three eigenvalues of equation (2.6) are given by ω5=2ω0+2gcos φ+2π 3, (2.7a) ω6=2ω0+2gcos φ+4π 3(2.7b) and ω7=2ω0+2gcos φ 3, (2.7c) which are identical to equation (2.4) up to a constant shift in frequency of ω0. We plot in figure 2 the eigenfrequencies ωnof the energy ladder using equations (2.4) and (2.7), as a function of the accumulated phase φ(see equation (2.5)). Most notably, the accumulated phase φcrucially determines the magnitude, ordering and degeneracy of both the single-excitation subspace (red, blue and green lines) and double-excitation subspace (orange, cyan and lime lines) eigenfrequencies, in a manifestation of the Aharonov–Bohm effect [43] for a three-site quantum ring, going beyond the single-excitation sector. Notably, a triangular trimer is the most elementary system in which the phase of the coherent coupling is important at the simplest-level of the eigenfrequencies. In a two-site dimer, with Hamiltonian ˆ Hdi =ω0(σ† 1σ1+σ† 2σ2)+g(eiθ12 σ† 1σ2+h.c.), the single-excitation eigenfrequencies are unaffected by the phase θ12.Theysimplyreadω±=ω0±g, such that the energy ladder of the dimer is formed by {2ω0,ω+,ω−,0}[36,65]. Moreover, a linear trimer (or indeed a linear chain of any size) will not support a gauge-independent phase, since it is crucial to have a ring geometry in order to mimic Aharonov–Bohm-style physics. (b) Quantum master equation Upon assuming weak coupling to the environment and Markovian behaviour, and after discarding fast-oscillating (non-resonant) terms, the quantum master equation of the trimer Downloaded from https://royalsocietypublishing.org/ on 01 April 2022 5 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 477:20210507 .......................................................... w8 w7 w6 w5 w4 w3 w2 w1 0 1 2 3 eigenfrequencies ⁄w0 01 − 2p3 − 2p pp f Figure 2. The eight eigenfrequencies ωnof the trimer (in units of ω0) in the coupled regime, as a function of the accumulated phase φ(see equations (2.4), (2.5) and (2.7)). In the figure, the coherent coupling strength g=ω0/25. (Online version in colour.) system reads [66] ∂tρ=i[ρ,ˆ H]+ n=1,2,3 γn 2Lσn+ n=1,2,3 Pn 2(Lσn)†, (2.8) where the Hamiltonian operator ˆ His given by equation (2.1), and where we have used the following super-operators in Lindblad form: Lσn=2σnρσ† n−σ† nσnρ−ρσ† nσn(2.9) and (Lσn)†=2σ† nρσn−σnσ† nρ−ρσnσ† n. (2.10) Here γn≥0 is the damping decay rate of each individual 2LS, and Pn≥0 is the incoherent pumping rate into 2LS-n. In equation (2.8), the first term on the right-hand side is responsible for the unitary evolution (the von Neumann equation), and the second term accounts for losses into heat baths. The third term in equation (2.8) describes gain processes, so that the master equation can model both a normally ordered system and a variety of inverted systems. The formal structure of equation (2.8) is tantamount to the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation, which has remarkable utility across quantum optics and atomic and condensed matter physics, as reconfirmed by recent experiments. For example, Barredo and co-workers Downloaded from https://royalsocietypublishing.org/ on 01 April 2022 6 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 477:20210507 .......................................................... studied blockade-type phenomena in coupled Rydberg atoms [67], where dissipators in the form of equation (2.9) sufficiently captured the effects of atomic losses due to spontaneous emission (in this experiment, the coupling g≃5 MHz and the loss γn≃0.3 MHz). Furthermore, the quantum nature of evanescently coupled optical waveguides satisfying parity–time symmetry was investigated by Klauck and colleagues [68], who modelled waveguide loss well by a GKSL master equation (in this experiment, g≃49 GHz and γn≃38 GHz). The aforementioned experiment of Roushan and co-workers—that with a trio of superconducting qubits—may be characterized by the parameters g≃4MHzandγn≃0.1 MHz [23]. In what follows, we shall be interested in the interplay between non-reciprocity in transport, whose emergence has already been hinted at by the eigenfrequencies of equations (2.4) and (2.7) becoming sensitive to the gauge-independent phase φ, and the loss and gain in the open quantum system, which can be controlled through the parameters γnand Pn, respectively. 3. Chiral steady states The non-reciprocity of the trimer system first manifests itself at the level of the steady-state populations of the collection of 2LSs. In this section, we characterize the asymmetries in the steady-state populations and steady-state currents, as a function of the accumulated phase φ in the system (see equation (2.5)). We relegate the calculations to the electronic supplementary material. We consider the trimer in the set-up sketched in figure 3, with equal damping rates γ0 (γn=γ0,wheren={1, 2, 3}) (purple arrows in the figure), and of non-zero pumping rate P1 into 2LS-1 (yellow arrow), while the other pumping rates are zero (P2=P3=0). We show the resultant steady-state populations in figure 4 for the accumulated phase φ={0, π/4, π/2}in the {left, middle, right}panels. Therefore, we can see the standard situation when φ=0, and two example non-reciprocal cases when φ={π/4, π/2}. In the top (bottom) panels, the magnitude of the coherent coupling g=γ0(g=5γ0). The labelling of the mean population of the state |i,j,k is displayed in the legend of figure 4a, and states with N={0, 1, 2, 3}excitations are shown with increasingly thick lines. Let us start by considering figure 4a, where the phase φ=0. The fine purple line corresponds to the mean population of the ground state |0, 0, 0, which is the only possible state at vanishing pumping P1γ0, and it monotonically decreases with increasing pumping rate P1, since the nontrivial states become populated. The results for the set of single-excitation states are given by the thin lines, and comprise the mean populations of the states |1, 0, 0,|0, 1, 0and |0, 0, 1,which are denoted by green, blue and red lines, respectively. Since only 2LS-1 is being pumped, the |1, 0, 0population (green line) grows quickly with increasing pumping rate P1, and approaches unity within the large pump limit P1γ0. Meanwhile, the populations of the states |0, 1, 0and |0, 0, 1(blue and red lines, respectively) are identical owing to the absence of any accumulated phase φ, and they form a hump structure since they are not populated in the low or high pump limits. The results for the set of two-excitation states are given by the medium thickness lines, and comprise the mean populations of the states |1, 1, 0,|1, 0, 1and |0, 1, 1, which are denoted by orange, cyan and lime lines, respectively. The mean populations of the states |1, 1, 0and |1, 0, 1 (orange and cyan lines, respectively) are the same, forming a hump structure peaked at a higher pumping rate than the single-excitation populations of |0, 1, 0and |0, 0, 1. As only 2LS-1 is being fed with gain, the |0, 1, 1mean population (lime line) is negligible, as is the mean population of the triply excited state |1, 1, 1, which is represented by the thick pink line. This panel exemplifies the standard reciprocal situation, without any asymmetries or surprises. In figure 4b, we have a non-trivial accumulated phase φ=π/4. The effect is to break two symmetries in the steady-state populations. In the single-excitation subspace, the populations of the states |0, 1, 0and |0, 0, 1(blue and red lines, respectively) are no longer identical (see figure 4a). Similarly, in the two-excitation subspace, the populations of the states |1, 1, 0and |1, 0, 1(orange and cyan lines, respectively) are now noticeably different. These asymmetries are the hallmark of non-reciprocity in the trimer system, as caused by the directionality imposed by Downloaded from https://royalsocietypublishing.org/ on 01 April 2022 7 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 477:20210507 .......................................................... P1g0 g0g0 ge±iq31 ge±iq23 ge±iq12 Figure 3. A sketch of the trimer with specific parameter choices (see equations (2.1) and (2.8)). Each 2LS is of resonance frequency ω0and damping rate γ0(purple arrows). The 2LS-1 is subject to gain at a rate P1(yellow arrow), while P2=P3=0. The magnitude of the three coherent coupling constants is g, and the hopping between sites nand n+1 is augmented with the complex argument θnn+1. (Online version in colour.) the non-zero phase φ.Infigure 4c, the accumulated phase is increased to φ=π/2, showcasing further population imbalances in both the first and second rung of the energy ladder, in a manifestation of multi-excitation Aharonov–Bohm physics. Notably, if we were to further consider φ=3π/2, the result would effectively be the opposite of that in figure 4c,whereφ=π/2. That is, the populations of |0, 1, 0and |0, 0, 1would be reversed, and those of |1, 1, 0and |1, 0, 1 would also be reversed, with respect to figure 4c. In figure 4d–f, the magnitude of the coherent coupling is increased to g=5γ0(in figure 4a–c,g=γ0). This stronger coupling leads to a significantly richer structure of the mean populations of the system, since the doubly and triply excited states have more chances to be populated. Figure 4dshows the reciprocal case with φ=0, where there is a clear region of large population inversion. Indeed the triply excited state |1, 1, 1has the most chance of being excited approximately within 10γ0<P1<100γ0(thick pink line). Non-reciprocity appears in figure 4e,f,whereφ=π/4andφ=π/2, respectively, and where two population symmetries have been broken in the same manner as in figure 4b,c. That is, the N=1 excitation mean populations (red and blue lines) and the N=2 excitation mean populations (orange and cyan lines), which coincide in figure 4d, are now completely distinguishable in figure 4e,f. Perhaps surprisingly, figure 4e,falso showcases a region in which the population of 2LS-3 is greater than that of 2LS-2, an area which is bordered by the thin vertical lines. Primarily, this inversion is because of the population of |1, 0, 1(cyan lines) being greater than the population of |1, 1, 0(orange lines) for moderate ratios of P1/γ0, where the system is mostly in the twoexcitation sector. Outside of this moderate pumping region, one sees that for low pumping P1γ0, where the system is mostly in the one-excitation sector, 2LS-2 is more excited than 2LS-3, owing to the population of |0, 1, 0(blue lines) being greater than the population of |0, 0, 1(red lines). Similarly, for large pumping P1γ0the population imbalance is also in favour of 2LS-2, as guaranteed by the population of |1, 1, 0(orange lines) being greater than the population of |1, 0, 1(cyan lines). The populations of each individual 2LS, rather than those of the states |i,j,k, can be explicitly seen in the electronic supplementary material, figure S1. Most Downloaded from https://royalsocietypublishing.org/ on 01 April 2022 8 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 477:20210507 .......................................................... (b) 0 0.2 0.4 0.6 0.8 1.0 steady-state population f = 0 f = p/4 f = p/2 |0, 0, 0Ò |0, 0, 1Ò |1, 0, 0Ò |0, 1, 0Ò |1, 1, 0Ò |1, 1, 1Ò |1, 0, 1Ò |0, 1, 1Ò (a) g = g0 (c) 0 0.2 0.4 0.6 0.8 1.0 steady-state population 10–2 10–1 11010 2103 P1/g0 (d) 10–2 10–1 11010 2103 P1/g0 (e)4.37 88.3 10–2 10–1 11010 2103 g = 5g0 P1/g0 (f)3.72 88.33 Figure 4. Steady-state populations in the trimer as a function of the pumping rate P1into 2LS-1, in units of the common decay rateγ0(seetheconfigurationinfigure3).Theotherpumpingratesarezero(P2=P3=0).Weshowresultsfortheaccumulated phase φ={0, π/4, π/2}in the {left, middle, right}panels. Top (bottom) panels: the magnitude of the coherent coupling g=γ0(g=5γ0). The labelling of the mean population of the state |i,j,kis displayed in the legend in (a), and states with N={0, 1, 2, 3}excitations are shown with increasingly thick lines. Thin, vertical lines in (e,f): guides for the eye at the ratios of P1/γ0, which form a region in which 2LS-3 is more populated than 2LS-2. (Online version in colour.) notably, the region of inverted population imbalance only occurs within the thin vertical lines in figure 4e,f, since it requires both a non-trivial accumulated phase φand a sufficiently strong coupling g. An important observable to consider is the steady-state current across the three sites of the trimer. To do so, let us consider the continuity equation at each site n, ∂t(σ† nσn)=i[σ† nσn,ˆ H]=Inn+1−In−1n, (3.1) where the Hamiltonian operator ˆ His given by equation (2.1). In equation (3.1), we have introduced the local current operator Inn+1, describing the transfer of excitations between two neighbouring sites nand n+1 in the trimer (we assume modular arithmetic for the indices), as Inn+1=ig(eiθnn+1σ† nσn+1−e−iθnn+1σ† n+1σn). (3.2) The global current operator Inaturally follows as I=I12 +I23 +I31, (3.3) and we donate the mean versions of these quantities as J=I(3.4a) and Jnn+1=Inn+1. (3.4b) The steady-state (ss) versions of these quantities, Jss and Jss nn+1, portray how the excitations in the system are transferred at large time scales. The results are presented in figure 5a,b, as a function of the pumping rate P1into 2LS-1. We show results for the accumulated phase φ={0, π/4, π/2} with increasingly thin lines, and in the left (right)-hand panels the magnitude of the coherent coupling g=γ0(g=5γ0). Figure 5ahighlights the absence of a steady-state current when φ=0 (thick green line). When φ=π/4 (medium pink line), a non-zero steady-state current is able to be supported because of the population imbalance between 2LS-2 and 2LS-3, and it has a maximal Downloaded from https://royalsocietypublishing.org/ on 01 April 2022 9 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 477:20210507 .......................................................... –0.3 –0.2 –0.1 0 0.1 0.2 0.3 12 23 31 (e) –0.8 –0.4 0 0.4 0.8 4.37 88.3 (f) g = 5g0 –0.03 –0.02 –0.01 0 0.01 0.02 0.03 (b) g = g0 –0.05 –0.04 –0.03 –0.02 –0.01 0 j = 0 j = p/4 j = p/2 (a) –0.3 –0.2 –0.1 0 0.1 0.2 0.3 12 23 31 (c) –0.8 –0.4 0 0.4 0.8 (d) P1 = g0P1 = g0 Jss ⁄ g0 0 –0.3 –0.2 –0.1 0.1 0.2 0.3 10–2 10–1 11010 2103 12 23 31 (g) –0.8 –0.4 0 0.4 0.8 10–2 10–1 11010 2103 f = p/2 f = p/4 f = 0 3.72 88.33 (h) nn+1 Jss ⁄ g0 nn+1 Jss ⁄ g0 nn+1 Jss ⁄ g0 nn+1 Figure 5. (a,b) Global steady-state current Jss in the trimer, as a function of the pumping rate P1into 2LS-1, in units of the common decay rate γ0(see the configuration in figure 3). The other pumping rates are zero (P2=P3=0). We show results for the accumulated phases φ={0, π/4, π/2}with increasingly thin lines. (c–h) Local currents Jss nn+1for the three phases φcorresponding to (a,b) (see equation (3.4)). The dashed, solid and dotted lines represent Jss 12,Jss 23 and Jss 31, respectively. Thin vertical lines: guides for the eye at the ratio of P1/γ0corresponding to sign changes of the global steady-state current Jss.Left (right)-hand panels: the magnitude of the coherent coupling g=γ0(g=5γ0). (Online version in colour.) value around P1≃10γ0.Thecaseofφ=π/2 (thin cyan line) displays the greatest steady-state current, as follows from figure 4c,f, where the mean population asymmetries are also greatest. In figure 5b, the effect of increased coherent coupling gleads to a notably different behaviour. While the reciprocal case current remains zero (thick green line), and the currents in the nonreciprocal cases (thinner lines) remain zero in the limiting cases of vanishing pumping and large pumping (these asymptotics are guaranteed from figure 4, because of saturation), the intermediate behaviour is more interesting. The steady-state current Jss becomes a sign-changing Downloaded from https://royalsocietypublishing.org/ on 01 April 2022 16 royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 477:20210507 .......................................................... 40. Kjaergaard M, Schwartz ME, Braumuller J, Krantz P, Wang JIJ, Gustavsson S, Olive WD. 2020 Superconducting qubits: current state of play. Annu. Rev. Condens. Matter Phys. 11, 369–395. (doi:10.1146/annurev-conmatphys-031119-050605) 41. Cooper NR, Dalibard J, Spielman IB. 2019 Topological bands for ultracold atoms. Rev. Mod. 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