Abstract
The identification of fractionalized excitations, such as Majorana quasiparticles, would be a striking signal of the realization of exotic quantum states of matter. While the paramount demonstration of such excitations would be a probe of their nonAbelian statistics via controlled braiding operations, alternative proposals exist that may be easier to access experimentally. Here we identify a signature of Majorana quasiparticles, qualitatively different from the behaviour of a conventional superconductor, which can be detected in cold atom systems using alkalineearthlike atoms. The system studied is a Kitaev wire interrupted by an extra site, which gives rise to superexchange coupling between two Majoranabound states. We show that this system hosts a tunable, nonequilibrium Josephson effect with a characteristic 8π periodicity of the Josephson current. The visibility of the 8π periodicity of the Josephson current is then studied including the effects of dephasing and particle losses.
Introduction
The search for observable signatures that identify exotic states of quantum matter and their fractionalized excitations has become a main focus of research in quantum physics. A paradigmatic example is the hunt for Majorana quasiparticles (MQPs) that exist at the ends of topological superconductors^{1}. First experimental evidence^{2,3,4,5,6,7} consistent with the presence of MQPs has recently been reported in various superconducting hybrid systems^{8,9,10}. While the ultimate goal is to probe the existence of nonAbelian anyons such as MQPs by performing controlled braiding operations, several possible fingerprints have been proposed that may be easier to access experimentally.
A prominent example hallmarking MQPs is the fractionalization of the Josephson effect, which can exhibit a 4π (half frequency) period due to a nonequilibrium population of excited states that is protected by fermion parity conservation^{1,4}. However, a similar, though nonprotected, fractionalization is also known to occur in conventional Swave superconductors, due to the presence of accidental midgap states^{11,12}. As a new signature for MQPs, here we show how a dissipationless, nonequilibrium 8π periodic Josephson effect occurs when two MQPs are subject to a superexchange coupling via a controllable energy level interrupting a Kitaev chain, an effect that is not found in Swave superconductors. In addition, we show how our model can be realized in systems of cold atoms in optical lattices, where isolation from the environment creates an ideal platform for the study of such nonequilibrium phenomena.
Our proposal is motivated by remarkable recent experimental progress with cold atom systems, including the observation of the nonequilibrium Josephson effect^{13}, initially demonstrated with Bose–Einstein condensates^{14,15}, and later observed over the BoseEinstein Condensate (BEC)–BardeenCooperSchrieffer (BCS) crossover^{16,17}. These results demonstrate not only the ability to measure nonequilibrium signals, but in addition, this realization of the 2π Josephson effect^{17} will provide a crucial piece of our implementation. More concretely, in our proposal, the starting point is an atomic realization of the Kitaev wire^{18,19,20,21}, here using a system of alkaline earth atoms (AEAs) coupled to a BEC reservoir (Fig. 1b). AEAs allow the creation of a controllable extra site by means of speciesdependent potentials^{22}, while the reservoir allows both the implementation of the Kitaev wire and the modification of the Josephson phase via an underlying Josephson effect of the reservoir itself. In addition, we investigate the visibility of this effect by studying the transient dynamics of the Josephson current in the presence of imperfections, including various dissipation mechanisms (singleparticle losses and dephasing) captured by a quantum master equation. Our simulations support not only the observability of the 8π effect, but further underline how this signature is characteristic of MQPs: while 4π peaks in the Fourier signal cannot be distinguished from those arising from midgap states in an ordinary Swave SC, and peaks at 4π, 2π and zero frequency can be enhanced from dissipation, the 8π signal visible in our setup provides a signature that cannot be confused with these undesired effects.
Results
Model Hamiltonian
We consider spinless fermions with field operators , where j=0, … N−1 labels the sites of a onedimensional (1D) lattice in ring geometry. The model Hamiltonian reads as
which describes a proximityinduced Pwave superconductor^{1} with pairing Δ, interrupted by an extra site at j=0, which is assumed to be not affected by the pairing (Fig. 1a). The hopping strength is denoted by t and the chemical potential relative to halffilling by μ. The site at j=0 is connected to its neighbours by the hoppings t_{L} and t_{R}, respectively, and has an energy offset μ_{0}. The phase factor e^{iΦ/2} on the hopping between j=0 and j=1 models a flux that advances the phase of a Cooper pair by Φ when moving around the ring.
For μ<2t, Δ>0 and t_{L}=t_{R}=0, in the limit of large N the system hosts a single pair of zeroenergy MQPs^{1}, γ_{L} and γ_{R}, which are localized exponentially around j=N−1 and j=1, respectively. All other quasiparticles of the superconductor are gapped, such that along with γ_{L} and γ_{R} form a subspace that is energetically detached from the bulk spectrum. To understand the qualitative Φ dependence of equation (1) in the physically relevant regime t_{L}, t_{R}<<Δ, t, we hence consider a minimal model encompassing the dynamics within this lowenergy sector. Decomposing into the Majorana operators , and setting μ_{0}=0, the effective Hamiltonian then reads as
In Fig. 2, we compare the energy spectra of H_{J}(Φ) and H(Φ). The full qualitative agreement confirms that the effective Hamiltonian H_{J}(Φ) captures the basic Josephson physics of the full model H(Φ). To understand the various level (avoided) crossings in Fig. 2, we first focus on the symmetric case t_{L}=t_{R}. At Φ=0, we have , that is, the four Majorana operators form two disjoint pairs giving rise to two singleparticle (hole) excitations with energy . The four possible manybody states then have the energies (−t_{L}, 0, 0, t_{L}), which explains the twofold degeneracy at E=0. At Φ=π, we have , that is, γ_{L} and γ_{R} are coupled to the same Majorana operator γ_{x}. This gives rise to a zero mode in the singleparticle spectrum and the manybody energies are as reflected in the crossings at Φ=π in Fig. 2. At Φ=2π, we have , that is, the analogous situation to Φ=0, but with a sign change of a singleparticle excitation energy, reflecting the change of the fermion parity in the ground state^{1}. At Φ=3π, the situation is analogous to Φ=π with γ_{R}→−γ_{R}. As for Φ=4π, we note H_{J}(4π)=H_{J}(0). However, despite the 4π periodicity of H_{J}, adiabatically following the ground state in Fig. 2 through the various crossings leads to an 8π periodic pattern. This is a phenomenon of spectral flow, where the system is pumped to an excited state during one 4π cycle of the Hamiltonian, and only returns to the initial state after a second cycle.
We emphasize that the level crossings in Fig. 2 are of quite different physical nature. The crossings between states with different fermion parity at odd multiples of π are robust as long as the fermion parity is conserved. By contrast, the crossings at even multiples of π require left/right symmetry and a midgap state on the additional site: this is realized by tuning the junction parameters, namely, μ_{0}=0 and symmetric tunnelling t_{L}=t_{R}. However, tuning of the bulk parameters within the topological superconductor (TSC) phase supporting the MQPs γ_{L} and γ_{R} is not required as long as the bulk gap is much larger than t_{L} and t_{R}. In a solidstate setting, the decoherence due to the coupling to phonons implies that observing the nonequilibrium population of the unprotected excited state presents a serious challenge. In contrast, in the cold atom setting proposed here, such decoherence channels are not present, thus stabilizing these effects.
Below we describe how the model given in equation (1) can be realized in systems of AEAs trapped in optical lattices, before discussing in more detail the visibility of the 8π Josephson effect in the presence of various imperfections.
Experimental realization
There are three points required for the realization of our setup: the implementation of a 1D Kiteav chain, the addition of the single site separating the two ends of the wire, and the time control of the phase Φ. To address these points in a concrete setup, we consider a system of fermionic AEAs^{23,24,25,26,27,28,29,30,31,32,33}, trapped in their ^{1}S_{0} ground state in a 1D lattice. The choice of AEAs allows us to independently trap the ^{1}S_{0} ground state g〉 and the ^{3}P_{0} metastable excited state atoms e〉. While our model is for spinless (single species) fermions, the ability to trap two species independently will be, as discussed in more detail below, of crucial use to implement the junction architecture of equation (1). We also note that while AEAs are well known in the experiments for their additional SU(N) symmetry^{34}, here, the choice of AEAs rests on the above reason, and the SU(N) symmetry plays no role.
We first address the implementation of a 1D Kiteav chain. While the hopping terms (t) arise naturally in the lattice, pairing terms (Δ) can be induced by coupling the fermions in the lattice to a BEC reservoir, where a radiofrequency (RF) field is used to break up Cooper pairs directly into neighbouring sites in the lattice, as described in ref. 18.
Second, we address how we can interrupt the chain with a single site. First, at the position j=0, a barrier is engineered to inhibit g〉 atoms from being at this site, which splits the Kitaev wire into two. This can be done using a highly focused beam at the socalled antimagic wavelength, which acts as a sink for e〉, and oppositely on g〉^{22}, resulting in the e〉 atoms only being trapped at this site. Thus, the e〉 atom at site j=0 acts as the additional site coupling the two ends of the wire. While natural hopping into and out of this site is deterred by this barrier, the tunnelling (t_{L} and t_{R}) are then reintroduced with Raman processes involving a clock transition^{35,36,37}.
Last, we address the time control of the phase Φ. In fact, the barrier which inhibits g〉 atoms to be trapped at j=0 also acts as the mechanism that controls the phase Φ. This can be seen as follows. The barrier is turned on via a laser which is highly localized at the j=0 position in the optical lattice, but homogenous in the remaining directions and impacts the BEC reservoir, bisecting it into two regions. For a barrier that is only a few times larger than the coherence length of the system, it will act as a thin tunnelling barrier between the two regions. If the two regions have a different Cooper pair density, an ordinary a.c. Josephson effect will occur, giving rise to a relative phase Φ across the junction that oscillates in time^{17}. The Josephson frequency ω_{J} of this oscillation is proportional to the population imbalance, which constitutes the analogue of a bias voltage in the solidstate context. Due to the proximity effect, this timedependent phase is inherited by the 1D lattice system, giving rise to the model described in equation (1). Here ω_{J} is on the order of the bare trap frequency and can be controlled via the barrier and reservoir parameters.
Within this setup, there are two main ways to demonstrate the 8π periodicity of the Josephson effect by current measurements. First, it is possible to use local interferometric probes, as realized, for example, in ref. 38, or to infer the current behaviour from density measurements^{16,17}. Second, one can observe clear signatures of the 8π periodicity using the relation between the timedependent momentum distribution and the current operator^{39,40}. For the model defined in equation (1), the relevant current at the junction is defined by:
where denotes the real time on which the Hamiltonian is dependent via the modulation of the phase , with the Josephson frequency ω_{J}, such that Φ(0)=0. Since the system we investigate does not display translational invariance, the global current operators cannot be described solely in terms of momentum distribution (momentum is not a good quantum number). Indeed, the total current reads:
where the presence of the last two terms reflects the fact that momentum is not a conserved quantity. While these terms are not directly accessible in cold atom experiments, it is possible to identify signatures of the 8π periodicity via the first term.
In Fig. 3, we show the timedependent behaviour of the current (Fig. 3a) and the various components of the momentum distribution (Fig. 3c–d) as a function of time in different parameter regimes, for a system of N=10 sites. For system parameters where the current has a dominant 8π periodicity (the orange line in Fig. 3a), the momentum components n(k) individually mirror this. This is shown in the orange (triangle) line of Fig. 3c and d, where the identical parameters have been taken. However, when system parameters are such that the current has a dominant 4π periodicity (the blue dashed line in Fig. 3a), the momentum components reflect this. This is shown with the blue (circles) lines of Fig. 3c and d, again with the identical parameters.
We now address the question of the integrity of this 8π Josephson effect in our proposed setup subject to imperfections. First, we address the influence of Hamiltonian imperfections t_{L}≠t_{R} as well as μ_{0}≠0 leading to avoided crossings in the level spectrum at integer multiples of 2π (Fig. 2). We find that Landau–Zener processes restore the 8π periodicity of the current at finite bias voltage. Thereafter, we investigate the effect of singleparticle losses, induced threebody collisions between particles in the wire and pairs in the reservoir, and dephasing in the framework of a Markovian quantum master equation^{41,42}.
Transport dynamics and 8π Josephson effect
We study the current through the junction region at site j=0, as defined in equation (3). In the limit of perfect adiabatic evolution ω_{J}→0, at the symmetric parameter point t_{L}=t_{R}, μ_{0}=0, the current will be 8π periodic, as indicated in the dispersion relation (Fig. 2); any deviation from this finetuned parameter point will cause a gap to open and the adiabatic current will be 4π periodic. However, the 8π effect is restored at finite ω_{J} due to the Landau–Zener effect. This tradeoff between finite ω_{J} and finite imperfections is analysed within the coherent time evolution governed by equation (1) in Fig. 3, where we numerically calculate the current J(τ) as a function of time (Fig. 3a). For small ω_{J} and weak imperfections (orange, solid line), the current displays a clear 8π periodicity, while increasing imperfections at fixed ω_{J} is detrimental (blue, dashed line). However, larger ω_{J} allows the system to follow the avoided crossings due to Landau–Zener tunnelling, thus restoring the 8π periodicity (black, dotdashed line).
To provide a quantitative picture of the interplay between imperfections and ω_{J}, we extract the height of the 8π peak () and the 4π peak from the Fourier transform of the current over a total phase change of Φ_{T}=8π. The ratio of these two peaks is shown in Fig. 3b) as a function of ω_{J} and (t_{L}−t_{R}). At intermediate ω_{J}, the 8π peak dominates over a wide range of parameters: remarkably, even for imperfections of a few per cent, the 8π signal is still an order of magnitude stronger than that at 4π. This behaviour has been verified with Φ_{T}=32π. Figure 3 shows the data with Φ_{T}=8π to minimize the compound effect of several Landu–Zener crossings (a finite particle loss stabilizes this effect and data at Φ_{T}=32π is shown in these cases, as discussed in the next section).
Dissipation and opensystem dynamics
In addition to imperfections that cause the system to move away from the symmetric point t_{L}=t_{R}, μ_{0}=0, an experimentally relevant imperfection is due to the coupling of the system to its environment. To account for this, we consider two dissipative channels. The first is a singleparticle loss at the site j with the rate κ_{j}: in cold atom settings, this represents losses due to inelastic collisions with the background BEC reservoir. The second source of dissipation is dephasing due an effective measurement at rate γ_{j} of the local occupation number by the environment. This typically represents the effect of spontaneous emission in optical lattice settings. Assuming a weak coupling to a Markovian quantum bath, the time evolution of the system is then governed by the master equation
where ρ is the density matrix of the system and the superoperator is the Lindblad dissipator for an arbitrary Lindblad jump operator O. As long as γ_{j}=0, equation (5) is still quadratic in the field operators and can be solved numerically efficiently. By contrast, γ_{j} leads to quartic terms in the master equation (5), which we treat in an exact diagonalization analysis. In what follows, we present results for the full master equation in systems of N=10 sites.
To study the impact of a finite κ_{j}≡κ on the integrity of the 8π effect, we numerically solve the master equation (5) and calculate the current in the presence of finite loss. In such open settings, the system dynamics is now determined by the competition of three energy scales, corresponding to ω_{J}, the energy scale related to Hamiltonian imperfections and κ. At fixed κ, one expects a stronger 8π signal for intermediate ω_{J}, since both Landau–Zener tunnelling works at its best even in the presence of imperfections and dissipation becomes detrimental only after many oscillations periods.
A few examples of the current evolution as a function of time are depicted in Fig. 4a): the main effect of dissipation is to damp the current signal in the system, thus inhibiting transport. However, even for relatively large decay rates (black line, corresponding to decay collision rates of order κ≃1 Hz (ref. 18) to be compared with t_{L}≃200 Hz), the signal stays 8π periodic for intermediate timescales (combined with a exponentially decaying envelope).
Following the above analysis, again we quantify the 8π effect by extracting the ratio of the 8π and the 4π peaks from the Fourier spectrum. This ratio is shown for various system parameters and loss rates in Fig. 4b–d, and illustrates the regimes in which the 8π signal can be seen. In Fig. 4b), we plot the ratio at fixed κ: the best attainable regime is for intermediate values of the velocity, where imperfections are relatively harmless up to values on the order of a few per cent. In Fig. 4c, t_{L}−t_{R} is fixed: here again intermediate speeds work at best, and values of the dissipation of the order of 10^{−2} can be tolerated. Finally, in Fig. 4d, the speed of the ramp is fixed: the signal is solid in the regime of low losses, and, for intermediate values of imperfections, larger values of the losses, κ, can be tolerated. The strong signal at these intermediate values of t_{L}−t_{R} is consistent with what is expected from Landau–Zener theory, which predicts an optimal tunnelling rate at intermediate gap values in case of finite dissipation and finite speed.
We have repeated these calculations in the presence of a finite dephasing rate γ. In this case, the system dynamics is not quadratic in the fermions, so our study was limited to system sizes up to N=10 sites. A sample of the results is presented in Fig. 5a). Overall, we found that it has qualitatively the same effect as κ, which can be understood in terms of the protection of the nonequilibrium excited states. While the decay channel κ mixes states with different parity the decay channel γ mixes states within the same patrity, both contributing equally through the evolution from 0 to 8π. Finally, we have checked how the main effects discussed here are affected by finitesize effects. In the regimes of interest, those effects are negligible at N=10. For the γ=0 case, we have checked this explicitly for some sample points up to N=30, while for the γ≠0 case, we have systematically checked consistency with the N=8 case.
In summary, the 8π periodicity of the current profile is robust to both the Hamiltonian imperfections and the dissipation considered here. Monitoring the evolution for shorter time (for example, for a single 8π cycle) can also substantially improve the signal, as in that case the role of particle losses is less detrimental.
Manybody effects
Finally, we consider the effect of a finite interaction on the energy spectrum of the model. At a qualitative level, these interactions have a similar effect as the direct tunnelling between the two superconducting islands: in the presence of interactions, the MQPs are less localized, and the overlap of their wavefunctions can lead to a direct interaction between them. To quantitatively study the effect of interactions on our model, we consider a nearestneighbour interaction of the form
where . We show the impact of the interaction on both the current pattern (Fig. 5b) and the lowlying spectrum (Fig. 6) for various values of the interaction U/t. When the interaction strength remains the lowestenergy scale U<t_{L}<t (Fig. 6a), a gap will open in the spectrum at even multiples of π, and effects the current measurement similar to the case of other Hamiltonian imperfections t_{L}≠t_{R}. As the interaction strength increases, the size of this gap increases, until values of U∼t the assumption of four lowlying states separated by an energy gap is no longer valid (Fig. 5c). Moreover, we numerically verified that distinct interactions in the bulk and at the junction have similar consequences on the Josephson effect, in agreement with the discussion above (both interactions lead to a delocalization of the MQP wavefunctions). Typically, in systems of AEAs for lattice spacings on the order of 250–500 nm the ratio U/t∼10^{−3} (ref. 32), well within the regime where the 8π behaviour can be seen.
Discussion
The periodicity of the Josephson effect is closely related to the charge of the particles involved in the tunnelling processes. Intuitively, an 8π periodicity then corresponds to a fractional charge of , which is the physical picture behind the timereversal protected fractional Majorana fermions discussed in ref. 43. In contrast, our model does not involve fractional charges, and our effective Hamiltonian H_{J}(Φ) (equation (1)) is hence 4π periodic in Φ, in agreement with the Byers–Yang theorem^{44}. The 8π Josephson effect in our setup is a phenomenon of spectral flow: the system is pumped to an excited state after slowly increasing Φ by 4π, and returns to the ground state after a second 4π cycle. Our work thus shows that an 8π periodic signal can also emerge due to nonprotected crossings, analogue to what has been shown to occur for the 4π effect. However, in the latter case, the accidental 4π periodicity occurs when the underlying system is a conventional superconductor; here this 8π effect arises when the underlying system hosts ‘normal’ Majorana fermions.
We note that while a 12π periodic Josephson effect has been put forward in the context of two connected quantum wires^{45}, we emphasize that these effects are dissipationfull, as there is no controllable gap separating the crossing branches of the Josephson junction from the bulk states.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Additional information
How to cite this article: Laflamme, C. et al. Nonequilibrium 8π Josephson effect in atomic Kitaev wires. Nat. Commun. 7:12280 doi: 10.1038/ncomms12280 (2016).
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Acknowledgements
We acknowledge useful discussions with M. Baranov, S. Nascimbène, P. Recher and B. Trauzettel. This work was supported by the ERC Synergy Grant UQUAM, SIQS, SFB FoQus (4016N23), and the Austrian Ministry of Science BMWF as part of the UniInfrastrukturprogramm of the Focal Point Scientific Computing at the University of Innsbruck. C.L. is partially supported by NSERC.
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Laflamme, C., Budich, J., Zoller, P. et al. Nonequilibrium 8π Josephson effect in atomic Kitaev wires. Nat Commun 7, 12280 (2016). https://doi.org/10.1038/ncomms12280
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DOI: https://doi.org/10.1038/ncomms12280
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