Split-ring polariton condensates as macroscopic two-level quantum systems

Superposition states of circular currents of exciton-polaritons mimic the superconducting flux qubits. The current states are formed by a macroscopic number of bosonic quasiparticles that compose a single quantum state of a many-body condensate. The essential difference between a polariton fluid and a superconducting current comes from the fact that in contrast to Cooper pairs polaritons are electrically neutral, and the magnetic field would not have a significant effect on a polariton flow. Nevertheless, the phase of a polariton condensate must change by an integer number of 2$\pi$, when going around the ring. If one introduces a $\pi$-phase delay line in the ring, the system is obliged to propagate a clockwise or anticlockwise circular current to reduce the total phase gained over one round-trip to zero or to build it up to $2\pi$. We show that such a $\pi$-delay line can be provided by a dark soliton embedded into a ring condensate and pinned to a potential well created by a C-shape non-resonant pump-spot. The physics of resulting split-ring polariton condensates is essentially similar to the physics of flux qubits. In particular, they exhibit pronounced coherent oscillations passing periodically through clockwise and anticlockwise current states. We predict that these oscillations may persist far beyond the coherence time of polariton condensates. As a consequence, the qubits based on split-ring polariton condensates are expected to possess very high figures of merit that makes them a valuable alternative to superconducting qubits.

Superposition states of circular currents of exciton-polaritons mimic the superconducting flux qubits. The current states are formed by a macroscopic number of bosonic quasiparticles that compose a single quantum state of a manybody condensate. The essential difference between a polariton fluid and a superconducting current comes from the fact that in contrast to Cooper pairs polaritons are electrically neutral, and the magnetic field would not have a significant effect on a polariton flow. Nevertheless, the phase of a polariton condensate must change by an integer number of 2π, when going around the ring. If one introduces a π-phase delay line in the ring, the system is obliged to propagate a clockwise or anticlockwise circular current to reduce the total phase gained over one roundtrip to zero or to build it up to 2π. We show that such a π-delay line can be provided by a dark soliton embedded into a ring condensate and pinned to a potential well created by a C-shape non-resonant pump-spot. The physics of resulting split-ring polariton condensates is essentially similar to the physics of flux qubits. In particular, they exhibit pronounced coherent oscillations passing periodically through clockwise and anticlockwise current states. We predict that these oscillations may persist far beyond the coherence time of polariton condensates. As a consequence, the qubits based on split-ring polariton condensates are expected to possess very high figures of merit that makes them a valuable alternative to superconducting qubits.
Introduction. While a tremendous progress in the development of quantum technologies is apparent 1 , it is still unclear which material platform is the most suitable for the realisation of future quantum computers and simulators 2 . Among the leaders of the quest are superconducting circuits with Josephson junctions 3 , cold atoms in optical traps 4 , ions 5 , purely photonic systems 6 . The semiconductor platform legs slightly behind so far, while a lot of interesting fundamental works on twodimensional quantum systems based on semiconductor quantum dots 7 , rings 8 , as well as on the topologically protected 9 electronic systems in semiconductor nanostructures has been published. Recently, a series of papers demonstrated a high potentiality of semiconductor microcavities in the strong light-matter coupling regime for hosting ensembles of phase-locked bosonic condensates of half-light-half-matter quasiparticles: exciton polaritons (hereafter referred to as polaritons for brevity) 10,11 . It has been argued that the phase locking process in an array of polariton condensates may be used for the minimization of a classical many-body XY-Hamiltonian 11 . Polariton condensates may be formed at elevated temperatures, optically controlled and mutually phase-locked on a picosecond time scale. These features constitute their main potential advantages over other material platforms for realisation of quantum simulators. On the other hand, a polariton qubit has never been convincingly demonstrated till now, and it has been argued that the dissipative nature of exciton-polaritons characterised by ultrashort radiative lifetimes would prevent their use for implementations of quantum algorithms 12 .
Here we argue that a strong fundamental similarity of superfluid polariton flows 13 and superconducting electric currents may be exploited to build a polariton analogue of the superconducting flux qubit. Superconducting flux qubits are based on superpositions of clockwise and anti-clockwise currents formed by millions of Cooper pairs [14][15][16] , see Figs. 1a,b. In order to excite the system in a superposition state, the half-quantum flux of magnetic field is passed through the superconducting circuit containing one or several Josephson junctions. The system is forced to generate a circular current to either reduce the magnetic flux to zero or to build it up to a full-quantum flux.
While electrically neutral polaritons are much less sensitive to the external magnetic field 17 than Cooper pairs, the circular currents of superfluid polaritons 18 can be efficiently controlled by introducing a potential defect (a phase delay line) in a polariton ring. The defect couples counter-propagating polariton currents. This results in a formation of a two-level quantum system based on a splitring polariton condensate. One of the efficient methods for the realization of such a defect implies pinning a dark soliton 19,20 , that is characterised by a π-jump of phase of a superfluid, to the slot in the polariton ring. The πphase delay line embedded in a circle forces the superfluid to flow clockwise or anti-clockwise in order to either build up the phase variation along the loop to 2π or to reduce it to zero. We run numerical experiments showing the spontaneous excitation of robust coherent oscillations of the resulting polariton qubit state on the Bloch sphere.
Interestingly, the dephasing time in such a system non-resonantly pumped by a continuous-wave (cw) laser source appears to be orders of magnitude longer than the characteristic oscillation period (about 125 ps). This may result in very high figures of merit of qubits based on split-ring polariton condensates. This is because the phase gradient that governs the current states of polariton condensates is insensitive to of the overall time-dependent phase characterising the condensate as a whole object. The lifetime of circular polariton currents is much longer than the coherence time of a polariton condensate that sustains these currents. Our analysis shows a high potentiality of semiconductor microcavities as a platform for realisation of quantum information devices.

Results
The origin of the two-level quantum system. Let us consider a close circuit filled with a coherent quantum fluid. The phase of the many-body wave function ψ(t, s) of the fluid ϕ must obey the equality: D ∂ s ϕds = 2π , which is the quantisation condition for the topological invariant ∈ Z also known as the winding number 21,22 . Here s is the coordinate along the circuit of a total length D. If the circuit is subjected to some effective vector potential A, the quantisation condition becomes D ∂ s ϕds − θ = 2π . The phase delay θ = ΛΦ/ induced by the vector field is determined by its flux Φ and the constant Λ which defines pulse rescaling rule,p →p − ΛA. The effective flux governs the energy spectrum of a twolevel system based on counter-propagating currents with opposite winding numbers, as Fig. 1d shows. At the particular value θ = π, the states with l = 0 and l = 1 are degenerate in energy similar to the case of the superconducting flux qubit.
The energy gap between l = 0 and l = 1 states appears in the presence of a defect embedded in a circuit, see Fig. 1c,d. The defect causes back-scattering of the currents and mixes them. The eigenstates of this system mimic the linear superposition states of a superconducting flux qubit.
For electrically neutral particles the phase delay θ can be engineered either by the circular motion of the defect 23 or by exploiting the spin-orbit coupling in the presence of external magnetic fields 17 . However, the specific case of θ = π can be realised in a much simpler way. A properly designed defect is able to pin the dark soliton 24,25 state characterised by a π-jump of the phase. In the presence of a dark soliton, the current states with the phase changing by π and −π over the remaining part of the circuit form two superposition states |0 and |1 , which constitute a two-level quantum system or qubit.
The dynamics of a split-ring condensate.
To be specific, we consider the system shown schematically in Fig. 2a. A semiconductor microcavity formed by a couple of Bragg mirrors contains an ensemble of embedded quantum wells. The strong coupling of cavity photons and quantum-well excitons results in the appearance of new eigenmodes of the structure, the exciton polaritons 26 . We assume that the polaritons are created by the nonresonant cw optical pump of a C-shape (see Methods for details on the pump beam geometry). Obeying the bosonic statistics, polaritons form a Bose-Einstein condensate which remains localised under the pump spot due to the finite polariton lifetime.
In the case of a condensate confined in a thin ring of a large diameter, the local modulation of the pump shape (see Fig. 2b) leads to the appearance of a dark soliton 24 . In addition to the persistent current states with non-zero average momenta, 27 , the couple of closely spaced in energy states with zero average currents appears. These are symmetric and anti-symmetric superposition states similar to those presented in Fig. 1d. The value of the energy gap between these states is dependent on the parameters of the defect. The angular dependencies of the magnitudes and the phases of the corresponding wave functions are shown in Fig. 2c,d (see Methods for details). In what follows, we shall focus on the oscillatory regime of a 2D split-ring condensate. In this regime, 28,29 , the condensate is initially formed in a superposition of |0 and |1 eigenstates. The system exhibits long-living quantum beats whose frequency is governed by the energy splitting of |0 and |1 states. The split-ring condensate passes periodically through clockwise and anticlockwise current states. Its dynamics can be conveniently mapped to a Bloch sphere. To prepare the system in the superposition state and trigger the oscillations we introduce the spatial modulation of the pump beam intensity, as Fig. 2a shows. The numerical modelling is performed using the generalized Gross-Pitaevskii equation for the many-body wave function of the condensate ψ(r, t) coupled to the rate equation for the density of the reservoir of incoherent excitons n R (r, t). The details of the model and the chosen parameters are described in Methods. Figure 3 shows the oscillatory regime of the splitring polariton condensate.
We characterize nonstationary circular current states of the condensate by the normalized average angular momentum introduced as dr is the actual average angular momentum and N (t) = |ψ(r, t)| 2 dr is the number of polaritons in the condensate. In contrast to the winding number , the average angular momentum m(t) continuously varies in the course of the evolution of the condensate. The oscillations of the polariton state in The harmonic oscillations of the angular momentum of the condensate can be considered as a fingerprint of a many-body two-level quantum system. The spectral analysis of the oscillatory dynamics of the system reveals two sharp resonances appearing, split in energy by about 32 µeV (see Supplementary Figure 3) that corresponds to the period of the oscillations seen in Fig. 3a. Figure 4 shows the trajectory of the quantum state of the split-ring polariton condensate on the surface of the Bloch sphere based on |0 and |1 eigenstates (see de- tails of the mapping in Methods). Retaining only the classical fluctuations of the initial polariton field and neglecting the stochastic processes ( Fig. 3 and Fig. 4a), we obtain the stable oscillations that persist during over 10 ns (truncation time of this numerical simulation) and correspond to a circular trajectory close to the equator of the Bloch sphere. When including the quantum fluctuations described by Eq. (3) in Methods, we find that the uncertainty in the mapping procedure to the Bloch sphere becomes larger and the trajectory of our system on the surface of the Bloch sphere looks noisy. Remarkably, the noise does not affect the stability of oscillations that persist over the whole calculation period showing no apparent decay. We conclude that the harmonic oscillations in a two-level quantum system formed by a split-ring polariton condensate may persist over tens of nanoseconds or even longer. This is much beyond the single polariton lifetime (as short as 6 ps in our case) and the coherence time of the condensate as a whole (about 100 ps) 30 . The reason of the surprising stability of oscillations is in the topological protection of circular current states in splitring condensates. The superfluid currents are governed by the spatial distribution of the phase of the condensate. The localisation radius of the condensate (20 µm in our case) is much smaller than the coherence length in the system (over 100 µm) 31 , which is why the coherence of superfluid polariton condensate is preserved. is broken and the system exhibits a fast decay. Fig. 5 shows the variation of the dynamics of the system resulting from the variation of the value of P 1 : while t is shorter than 3000 ps, the system exhibits the same stable oscillations as those shown in Figs. 3 and 4. Next, as a result of the decrease of P 1 from 0.1 × P 0 to 0.09 × P 0 at t = 3000 ps, the fast decay of the oscillations is observed, so that, eventually, the system relaxes to one of the eigenstates, specifically, to the state |0 shown in Fig. 6a. We emphasize that the decay time of oscillations is still independent of the coherence time of the condensate in this regime. It is fully governed by P 1 parameter that controls the magnitude of the step potential. The trajectory of the system on the Bloch sphere that describes the decay of the oscillations has been smoothed with use of the Bezier function 33 for clarity.
One can see in Fig. 6a that the wave function of the split-ring condensate is anti-symmetric with respect to the horizontal axis (y = 0), which passes through the center of the slot. A π phase jump appears at y = 0. With the further decrease of P 1 down to 0.03 × P 0 , the system relaxes to the basis state |1 shown in Fig. 6b. It represents the symmetric pattern with a pair of π phase jumps close to the horizontal axis (y = 0). Note that both basis states are characterised by zero average polariton flow, m = 0, and represent 2D counterparts of the basis states shown in Fig. 2c,d. Besides, they are nearly perfectly orthogonal. Their orthogonality is essential for mapping the system to the Bloch sphere. It is important to note that, in a general case, the ring condensate is not expected to relax to the lowest energy state corresponding to the upper pole of the Bloch sphere. This is a characteristic feature of polariton lasers: out of all quantum states the system chooses one that maximises the occupation number of the condensate, but not necessarily one that is characterised by the lowest energy 10,11 . This is why incoherent processes of acousticphonon assistant energy relaxation are not expected to affect the dynamics of qubits based on split-ring polariton condensates.
C-shape potentials Till now we were considering the polariton condensates imprinted in a planar microcavity by means of the optical pumping. Their spatial localisation was imposed by the shape of the non-resonant pump used for their excitation. An alternative way to realize split-ring polariton condensates is by using laterally confined C-shape potentials produced by chemical etching of planar cavities. In this case, we expect a stronger confinement of polaritons and more control tools for shaping the condensates. The drawback of this system as compared to fully optically induced split-ring condensates is in its rigidity: each time, to change the geometry of an array of polariton condensates one would need to grow a new sample. We also consider the combined method of lateral confinement by using the etched micropillars where ring condensates are formed due to the repulsion of polaritons from the exciton reservoir formed in the center of the pillar by a non-resonant optical pumping. Persistent superfluid currents of exciton-polaritons were recently observed in such structures 32 . Shifting the pump spot from the center of the pillar one should be able to realize split-ring con-densates. The results of numerical simulations of the harmonic oscillations in polariton condensates confined to C-shape potentials are shown in the Supplementary materials.

Discussion
The simulations described above demonstrate that at certain conditions split-ring polariton condensates behave as two level quantum systems demonstrating longstanding coherent oscillations. Considering this system as a qubit, one should be able to estimate its figure of merit given by the ratio of the decoherence time to the characteristic single logic operation time. For the set of parameters used in our simulations that correspond to a conventional GaAs-based microcavity, the decoherence time of the qubit appears to be many orders of magnitude longer than the single polariton lifetime and than the coherence time of the condensate as a whole. Even accounting for quantum fluctuations, we do not observe any significant decay of oscillations over a time-scale of ten nanoseconds for a certain range of the control pump power P 1 . Estimating the single logic operation time by the period of the oscillations on the Bloch sphere that is of the order of 125 ps in our case, we end up with a figure of merit of more than 100, that matches those of best superconducting qubits 3 .
This high figure of merit can be achieved in a split-ring condensate because it is localised on a spot that is much smaller than the coherence length in the polariton system and because the overall phase of the condensate that is subject to a fast decoherence is fully decoupled from the superfluid phase current dynamics which defines the trajectory of the considered quantum system on a Bloch sphere. It is also important that in the stationary regime the energy relaxation of the condensate as a whole does not occur. The fluctuations of the number of particles in the condensate do not have any significant effect on the energy splitting between |0 and |1 states that controls the frequency of oscillations.
Implementation of quantum algorithms In order to fully characterize the applicability of polariton qubits for quantum information processing, one should address the issues of setting the initial quantum state of a polariton qubit, coupling between different qubits, elementary logic operations and read-out of the information from a set of polariton qubits. While discussing these subjects in their integrity is out of the scope of this work, we would like to briefly express our vision on the concept of quantum information processing with use of polariton qubits.
Setting a split-ring condensate into a given quantum state can be achieved with use of a resonant short pump pulse focused on a specific spot of the ring. A similar technique has been employed for setting the phase of polariton Rabi oscillations 34 . The read-out of a quantum state of the qubit can be done combining the time-and spatially-resolved photoluminescence and interferometry measurements. Note that this is a "weak measurement" method that does not fully destroy the measured quantum state, while it perturbs it to some extent. Conceptually, in a similar way, a SQUID-based read-out perturbs but does not fully destroy the quantum state of a superconducting flux qubit 3,35 . The proposed optical read-out technique is currently being used for studies of XY-simulators based on an array of exciton-polariton condensates 11 . Finally, the coherent coupling between qubits can be achieved by means of their exchange by polaritons propagating in the plane of a microcavity, in a full similarity to the coupling mechanism employed in XY-simulators 11 . This simple coupling scheme proved efficient for pairing of nearest neighbours in an array of polariton condensates. In order to achieve coupling of distant condensates one can use one-dimensional optical waveguides imprinted lithographically in the plane of a semiconductor microcavity 36 .
To summarize, we have demonstrated that a coherent many-body quantum system represented by a bosonic condensate of exciton-polaritons placed in a split-ring geometry sustains stable and long-living oscillations between two circular current states. The polariton system qualitatively reproduces the behaviour of a superconducting flux qubit. In a remarkable similarity to the flux qubit, in the considered split-ring polariton condensate a two-level quantum system is formed by superposition states of clockwise and anticlockwise circular currents. The size of the system is much less than the coherence length of a polariton condensate, which is why superfluid polariton currents are well preserved. The oscillatory dynamics of the system is sensitive neither to the overall coherence time of the condensate nor to the single polariton lifetime. This ensures a high figure of merit for qubits based on split-ring polariton condensates. The present analysis paves the way to the realisation of a new semiconductor platform for quantum information processing. The evident advantages of the considered quantum system are in its high scalability, full optical control, high operation temperature, ultrafast logic operations and potential integrability with classical semiconductor based nano-electronic devices.

Methods
The model. To predict the dynamics of a 2D splitring polariton condensate in a semiconductor microcavity, the driven-dissipative Gross-Pitaevskii equation coupled with the rate equation for the density of the exciton reservoir is used: where ψ(r, t) is the wave function of a polariton condensate subject to the boundary conditions 37 . n R (r, t) is the exciton reservoir density, m * = 10 −4 m e is the effective mass of polaritons on the lower branch (m e is the free electron mass), γ c = 1/6 ps −1 and γ R = 2γ c are the polariton and the reservoir decay rates, respectively. R = 0.01 ps −1 µm 2 is the rate of the stimulated scattering from the exciton reservoir to the polariton condensate. g c = 6 × 10 −3 meV µm 2 and g r = 2g c describe the interaction of polaritons between themselves and with the reservoir excitons, respectively. V (r) is the external potential. The cw nonresonant pump P (r) of C-shape, as shown in Fig. 2a, is characterised by the spatial distribution of the intensity given by: , x > 0, where the pump power variation factor P 1 is introduced to control the energy gap between |0 and |1 states. P 1 is a vital parameter for the realization of a polariton qubit. While the classical fluctuations are taken into account in the initial polariton field, it is also possible to include the effect of quantum fluctuations within the classical field approximation by adding a complex stochastic term in the truncated Wigner approximation 38 The basis states shown in Fig. 2c,d are obtained using the one-dimension equivalent of 2D model (1) which is relevant to the polariton condensate confined in a thin ring of a wide radius R 0 . The transformation to the 1D model is performed with a substitution ψ(r, t) = Φ(ρ)ψ(φ, t), where Φ(ρ) accounts for the radial distribution of the condensate, ρ and φ are polar coordinates. After integrating out the radial dependence Eq. (1) reduces to the 1D model with ∇ 2 → ∂ φφ and m * → R 2 0 m * . Figures 2c,d show the solutions of the 1D problem obtained with a pump distribution shown in panel (b). The width of the pump slot is 2 µm while the pump amplitude is 3.5 × P th .
Mapping the dynamics on the Bloch sphere. For a qubit based on a two-level system formed by the states |0 and |1 , any state |ψ on the surface of the Bloch sphere can be represented as a linear combination of two basis states: |ψ = α|0 + β|1 , where the normalization condition |α| 2 + |β| 2 = 1 is implied. The circular current states can be represented as: . For simplicity, we associate |m ±0.4 (points corresponding to the plots (b) and (d) in Fig. 3) with | and | and choose these states as the basis for our numerical fitting procedure. In this way, we obtain: The quantum state visited by the condensate can be characterized by a pseudo-spin vectorŜ = S xêx +S yêy +S zêz whose components are defined as: S x = 1 2 (αβ * + α * β), S y = i 2 (α * β − αβ * ), S z = 1 2 (|β| 2 − |α| 2 ). The mapping of the condensate dynamics to the Bloch sphere is realised using the method of Maximum Inherit Optimization.
Calculation of the angular momentum. We note that the current direction at the position of the slot is different for the inner part (r < r 0 ) and the outer part (r > r 0 ) of the polariton condensate, as shown in the right panels of Fig. 3b-d (r 0 = 8 µm). Fig. 3a shows the angular momentum for the outer part of the polariton condensate that manifests a pronounced superfluid phase current.
Supplementary material: Split-ring polariton condensates as macroscopic two-level quantum systems Half-quantum currents trapped in C-shape potentials Circular superfluid currents with fractional angular momenta demonstrating persistent coherent oscillations can also be found in the case of a condensate confined to a C-shape external potential created e.g. by etching of a planar microcavity sample. In order to describe the system in this case we intriduce the additional stationary potential V (r), see Eq. (1) in the main text. We consider the non-resonant excitation of the system by a broad pump as illustrated in Fig. S1. The considered potential contains a narrow barrier, which is different from the optically induced potential distribution shown in Fig. 2a in the main text. In the main text, the slot in the pumpring corresponds to a potential well for the polariton condensate. Under the excitation by a non-resonant broad pump with a relatively low intensity (P 0 = 1.3 × P th ), an oscillating state with its angular momentum varying between m ±0.5 is obtained, as shown in Figs. S1c-e. The tunneling of polaritons through the narrow barrier mimics the Josephson dynamics. It is important to underline that Josephson oscillations between two condensates observed in Ref. 1 decay on the timescale of the coherence time of polariton condensates because of the decoherence between two condensates. In contrast, in the ring geometry we work with a single condensate. The fluctuations of its overall phase do not affect the phase difference between its parts situated to the right and left sides of the potential barrier, which is why the oscillations persist on a much longer time-scale in our case. We estimate that the decoherence time in our system scales exponentially with the ratio of the coherence length to the diameter of the ring condensate. It may exceed several tens of nanoseconds in realistic GaAs-based microcavities.
As the pump intensity increases, the system achieves a steady state regime where the normalised angular momentum is fractional, as shown in Fig. S1f. Note that this state is different from those in Fig. 3b,d in the main text, since there is no clear π phase jump at the potential barrier here. Making the pump intensity much stronger, one can see clearly the tunneling of polaritons under the potential barrier, see Fig. S1g. The tunneling ensures a smooth phase variation in this case, so that the angular momentum of the condensate approaches m = 1, see Fig. S1c.
We note that the shape of the external potential strongly influences the spatial distribution of the polariton density in the condensate. If the potential width is increased (inset to Fig. S2a), the same broad pump as in Fig. S1a can create a C-shape solution with the angular momentum m ±0.5 (Fig. S2a), and a clear π phase jump 2 is observed at the potential barrier as shown in Fig. S2b. This solution is very similar to that of Fig. 3 in the main text. As in the previous case, whilst the pump intensity increases, the phase difference between both sides of the potential barrier changes from π to 0, leading to the formation of the current state with an integer angular momentum, see Figs. S2a,c.
Spectral analysis of the oscillations in a split-ring polariton condensate The strong evidence of the two-level nature of the oscillating split-ring polariton condensate is provided by its energy spectrum. The Fourier spectrum of the dynamics of the condensate wave-function (Fig. 3 in the main text) S(ω) = ψ(t, r)e −iωt dtd 2 r is shown in Fig. S3. Note that besides the pair of peaks corresponding to the eigenstates of the system |0 and |1 shown in Fig. 6, the comb of the side-band peaks of attenuated intensities also appears in agreement with the recent predictions 3,4 . These satellites result from the nonlinear processes in the driven-dissipative polariton system and are indicative of the deviation of the considered split-ring condensate from an ideal two-level linear quantum system. The intensities of the side peaks increase as the pump power increases and the interactions between counter-rotating polariton currents become important.