Introduction

Quantum simulation, that holds promises of solving particular problems exponentially faster than any classical computer, is a rapidly expanding field of research1,2,3. The information in quantum computers is stored in quantum bits or qubits, which have found several physical realizations; quantum simulators have been nowadays realized and/or proposed that employ trapped ions4, ultracold quantum gases5, photonic systems6, quantum dots7, and superconducting circuits1,8,9. Solid state devices, and in particular those relying on the Josephson effect10, are gaining ground as preferable elementary units (qubits) of quantum simulators since they exhibit relatively long coherence times and extremely low dissipation11. Several variants of Josephson qubits that utilize either charge or flux or phase degrees of freedom have been proposed for implementing a working quantum computer; the recently anounced, commercially available quantum computer with more than 1000 superconducting qubit CPU, known as D-Wave 2XTM (the upgrade of D-Wave TwoTM with 512 qubits CPU), is clearly a major advancement in this direction. A single superconducting charge qubit (SCQ)12 at milikelvin temperatures behaves effectively as an artificial two-level “atom” in which two states, the ground and the first excited ones, are coherently superposed by Josephson coupling. When coupled to an electromagnetic (EM) vector potential, a single SCQ does behave, with respect to the scattering of EM waves, as an atom in space. Indeed, a “single-atom laser” has been realized with an SCQ coupled to a transmission line resonator (“cavity”)13. Thus, it would be anticipated that a periodic structure of SCQs demonstrates the properties of a transparent material, at least in a particular frequency band. The idea of building materials comprising artificial “atoms” with engineered properties, i.e., metamaterials, and in particular superconducting ones14, is currently under active development. Superconducting quantum metamaterials (SCQMMs) comprising a large number of qubits could hopefully maintain quantum coherence for times long enough to reveal new, exotic collective properties. The first SCQMM that was only recently implemented comprises 20 flux qubits arranged in a double chain geometry15. Furthermore, lasing in the microwave range has been demonstrated theoretically to be triggered in an SCQMM initialized in an easily reachable factorized state16.

Results

Superconducting Quantum Metamaterial Model

Consider an infinite, one-dimensional (1D) periodic SCQ array placed in a transmission line (TL) consisting of two superconducting strips of infinite length17,18 (Fig. 1a,b); each SCQ, in the form of a tiny superconducting island, is connected to each bank of the TL by a Josephson junction (JJ). The control circuitry for each individual SCQ (Fig. 1c), consisting of a gate voltage source Vg coupled to it through a gate capacitor Cg, allows for local control of the SCQMM by altering independently the state of each SCQ19. The SCQs exploit the nonlinearity of the Josephson effect and the large charging energy resulting from nanofabrication to create artificial mesoscopic two-level systems. A propagating EM field in the superconducting TL gives rise to nontrivial interactions between the SCQs, that are mediated by its photons20. Those interactions are of fundamental importance in quantum optics, quantum simulations, and quantum information processing, as well. In what follows, it is demonstrated theoretically that self-induced transparency21 and Dicke-type superradiance (collective spontaneous emission)22 occur for weak EM fields in that SCQMM structure; the occurence of the former or the latter effect solely depends on the initial state of the SCQ subsystem. Most importantly, self-induced transparent (SIT) or superradiant (SRD) pulses induce quantum coherence effects in the qubit subsystem. In superradiance (resp. self-induced transparency), the initial conditions correspond to a state where the SCQs are all in their excited (resp. ground) state; an extended system exhibiting SRD or SIT effects is often called a coherent amlpifier or attenuator, respectively. These fundamental quantum coherent prosesses have been investigated extensively in connection to one- and two-photon resonant two-level systems. Superradiant effects have been actually observed recently in two-level systems formed by quantum dot arrays23 and spin-orbit coupled Bose-Einstein condensates24; the latter system features the coupling between momentum states and the collective atomic spin which is analogous to that between the EM field and the atomic spin in the original Dicke model. These results suggest that quantum dots and the atoms in the Bose-Einstein condensate can radiatively interact over long distances. The experimental confirmation of SIT and SRD in extended SCQMM structures may open a new pathway to potentially powerful quantum computing. As a consequence of these effects, the value of the speed of either an SIT or SRD propagating pulse in a SCQMM structure can in principle be engineered through the SCQ parameters25, which is not possible in ordinary resonant media. From a technological viewpoint, an EM (light) pulse can be regarded as a “bit” of optical information; its slowing down, or even its complete halting for a certain time interval, may be used for data storage in a quantum computer.

Figure 1: Schematic drawing of a charge qubit superconducting quantum metamaterial (SCQMM).
figure 1

(a) The SCQMM comprising an infinite chain of identical charge qubits in a superconducting transmission line. Each qubit consists of a superconducting island that is connected to the electrodes of the transmission line through two Josephson junctions, formed in the regions of the dielectric layers (blue). The propagating electromagnetic vector potential pulse is also shown schematically out of scale. (b) The side view of the SCQMM in which the relevant geometrical parameters and the field orientations are indicated. (c) A unit cell of the superconducting quantum metamaterial which also shows the control circuitry of the charge qubit, consisting of a gate potential Vg applied to it through the gate capacitor Cg.

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In the following, the essential building blocks of the SCQMM model are summarized in a self-contained manner, yet omitting unnecessary calculational details which are presented in the Supplementary Information. The energy per unit cell of the SCQMM structure lying along the x–direction, when coupled to an EM vector potential , can be readily written as17,18

in units of the Josephson energy EJ = Φ0Ic/(2πC), with Φ0, Ic and C being the magnetic flux quantum, the critical current of the JJ, and the capacitance of the JJ, respectively. In equation (1), φn is the superconducting phase on the nth island, β = (8πdEJ)−1/2(Φ0/2π), with d being the separation between the electrodes of the superconducting TL, and the overdots denote differentiation with respect to the temporal variable t. Assuming EM fields with wavelengths , with being the distance between neighboring qubits, the EM potential is approximately constant within a unit cell, so that in the centre of the nth unit cell . In terms of the discretized EM potential Az,n(t), the normalized gauge term is an = 2πdAx,n/Φ0. The classical energy expression equation (1) provides a minimal modelling approach for the system under consideration; the three angular brackets in that equation correspond to the energies of the SCQ subsystem, the EM field inside the TL electrodes, and their interaction, respectively. The latter results from the requirement for gauge-invariance of each Josephson phase.

Second Quantization and Reduction to Maxwell-Bloch Equations

The quantization of the SCQ subsystem requires the replacement of the classical variables φn and by the corresponding quantum operators and , respectively. While the EM field is treated classically, the SCQs are regarded as two-level systems, so that only the two lowest energy states are retained; under these considerations, the second-quantized Hamiltonian corresponding to equation (1) is

where p, p′ = 0, 1, E0 and E1 are the energy eigenvalues of the ground and the excited state, respectively, the operator (an,p) excites (de-excites) the nth SCQ from the ground to the excited (from the excited to the ground) state, and are the matrix elements of the effective SCQ-EM field interaction. The basis states Ξp can be obtained by solving the single-SCQ Schrödinger equation (−∂2/∂φ2 − Ep + 2 cos φ)Ξp = 0. In general, each SCQ is in a superposition state of the form . The substitution of |Ψn〉 into the Schrödinger equation with the second-quantized Hamiltonian equation (2), and the introduction of the Bloch variables , , Rz(n) = |Ψn,1|2 − |Ψn,0|2, provides the re-formulation of the problem into the Maxwell-Bloch (MB) equations

that are nonlinearly coupled to the resulting EM vector potential equation

where δαn = αn−1 − 2αn + αn+1, D = (V11 − V00)/(2χ), Ω2 = (V00 + V11)/2, μ = V10/χ = V01/χ, and Δ = ε1 − ε0 ≡ (E1 − E0)/χ, with χ = ωJ/EJ. In the earlier equations, the overdots denote differentiation with respect to the normalized time t → ωJt, in which ωJ = eIc/(C) is the Josephson frequency and e, are the electron charge and the Planck’s constant devided by 2π, respectively.

Approximations and Analytical Solutions

For weak EM fields, the approximation can be safely used. Then, by taking the continuum limit αn(t) → α(x, t) and Ri(n; t) → Ri(x; t) (i = x, y, z) of equations (3–, , 6), a set of simplified, yet still nonlinearly coupled equations is obtained, similar to those encountered in two-photon SIT in resonant media27. Further simplification can be achieved with the slowly varying envelope approximation (SVEA) by making for the EM vector potential the ansatz α(x, t) = ε(x, t)cos Ψ(x, t), where Ψ(x, t) = kx − ωt + ϕ(x, t) and ε(x, t), ϕ(x, t) are the slowly varying pulse envelope and phase, respectively, with ω and being the frequency of the carrier wave of the EM pulse and its wavenumber in the superconducting TL, respectively. In the absence of the SCQ chain the EM pulse is “free” to propagate in the TL with speed β. At the same time, equations (3, 4, 5) for the Bloch vector components are transformed according to Rx = rx cos (2Ψ) + ry sin (2Ψ), Ry = ry cos (2Ψ) − rx sin (2Ψ), and Rz = rz. Then, collecting the coefficients of sinΨ and cosΨ while neglecting the rapidly varying terms, and averaging over the phase Ψ, results in a set of truncated equations (see Supplementary Information). Further manipulation of the resulting equations and the enforcement of the two-photon resonance condition Δ = 2ω, results in

where c = β2k/ω = 2β2k/Δ, and the truncated MB equations

which obey the conservation law . In equation (9), the n–dependence of the ri (i = x, y, z) is suppressed, in accordance with common practices in quantum optics.

The ri can be written in terms of new Bloch vector components Si using the unitary transformation rx = Sx cos Φ − Sz sin Φ, ry = Sy, and rz = Sz cos Φ + Sx sin Φ, where Φ is a constant angle to be determined. Using a procedure similar to that for obtaining the ri, we get , , and , where and tan Φ ≡ γ = 4D/μ. The combined system of the equations for the Si and equations (7, 8) admits exact solutions of the form ε = ε(τ = t − x/v) and Si = Si(τ = t − x/v), where v is the pulse speed. For the slowly varying pulse envelop, we obtain

where is the pulse amplitude and τp = {χ(σμ/ω)[v/(c − v)]}−1 its duration, with . The decoherence factor γ can be expressed as a function of the matrix elements of the SCQ-EM field interaction, Vij, as γ = 2(V11 − V00)/V10 that can be calculated when the latter are known. Such Lorentzian propagating pulses have been obtained before in two-photon resonant media28,29; however, SIT in quantum systems has only been demonstrated in one-photon (absorbing) frequency gap media, in which solitonic pulses can propagate without dissipation30. The corresponding solution for the population inversion, Rz, reads

where , and the plus (minus) sign corresponds to absorbing (amplifying) SCQMMs; these are specified through the initial conditions as Rz(−∞) = −1, ε(−∞) = 0 and Rz(−∞ = +1), ε(−∞) = 0 for absorbing and amplifying SCQMMs, respectively (with Rx(−∞) = Ry(−∞) = 0 in both cases). The requirement for the wavenumber k being real, leads to the SCQ parameter-dependent condition 2χ2(V11 + V00) < (E1−E0)2 for pulse propagation in the SCQMM. Thus, beyond the obtained two-photon SIT or SRD, the propagating EM pulse plays a key role in the interaction processes in the qubit subsystem: it leads to collective behavior of the ensemble of SCQs in the form of quantum coherent probability pulses; such pulses are illustrated here through the population inversion Rz.

The corresponding velocity-amplitude relation of the propagating pulse reads

Equation (12) can be also written as a velocity-duration expression, since the pulse amplitude and its duration are related through . The duration of SRD pulses cannot exceed the limiting value of τM = ω(c − v)/(χμv). From equation (12), the existence of a critical velocity c, defined earlier, can be immediately identified; that velocity sets an upper (lower) bound on the pulse velocity in absorbing (amplifying) SCQMM structures. Thus, in absorbing (amplifying) SCQMM structures, pulses of higher intensity propagate faster (slower). That limiting velocity is generally lower than the corresponding one for two-photon SIT or SRD in ordinary media, β, which here coincides with the speed of the “free” pulse in the TL (Fig. 2). As can be inferred from Fig. 2, the increase of decoherence through γ makes the velocity to saturate at its limiting value c at lower amplitudes ε; that velocity can be reduced further with increasing the ratio of the TL to the pulse carrier wave frequency Ω/ω through proper parameter engineering. Moreover, effective control of v in SCQMMs could in principle be achieved by an external field31 or by real time tuning of the qubit parameters. That ability to control the flow of “optical”, in the broad sense, information may have technological relevance to quantum computing25. Note that total inversion, i.e. excitation or de-excitation of all qubits during pulse propagation is possible only if γ = 0, i.e., for V00 = V11; otherwise (V00 < V11) the energy levels of the qubit states are Stark-shifted, violating thus the resonance condition. Typical analytical profiles for the EM vector potential pulse ε(τ) and the population inversions Rz(τ) both for absorbing and amplifying SCQMMs are shown in the insets of Fig. 2. The maximum of ε(τ) reduces considerably with increasing γ, while at the same time the maximum (minimum) of Rz decreases (increases) at the same rate.

Figure 2: The velocity-amplitude relation in two-photon superradiant (TPSRD, amplifying) and two-photon self-induced transparent (TPSIT, absorbing) superconducting quantum metamaterials (SCQMMs) & quantum coherent pulse profiles.
figure 2

In all subfigures, the pulse velocity v in units of β as a function of the electromagnetic vector potential pulse amplitude ε0 is plotted and compared with the corresponding curves for ordinary (atomic) amplifying and absorbing media (brown- and green-dotted curves, respectively). The horizontal magenta-solid (resp. black-solid) lines indicate the limiting velocity in ordinary amplifying and absorbing media, v/β = 1 (resp. amplifying and absorbing SCQMMs, v = c < β). (a) V00 = V11 = 1, V01 = V10 = 0.8, χ = 1/5, E1 − E0 = 3 (γ = 0 and Ω/ω = 0.3). Left Inset: The electromagnetic vector potential pulse envelop (ε/εM)2 and the population inversion function Rz(n) profiles as a function of the slow variable (τ/τM in a frame of reference that is moving with velocity v, for TPSIT (absorbing) SCQMMs. Right Inset: Same as in the left inset for TPSRD (amplifying) SCQMMs. (b) V00 = 0.6, V11 = 1.4, V01 = V10 = 0.8, χ = 1/5, E1 − E0 = 3 (γ = 2 and Ω/ω = 0.3). Left Inset: The electromagnetic vector potential pulse envelop (ε/εM)2 and the population inversion function Rz(n) profiles as a function of the slow variable (τ/τM in a frame of reference that is moving with velocity v, for TPSIT (absorbing) SCQMMs in the presence of relatively strong decoherence (γ = 2). Right Inset: Same as in the left inset for TPSRD (amplifying) SCQMMs. (c) V00 = V11 = 3, V01 = V10 = 0.8, χ = 1/5, E1 − E0 = 3 (γ = 0 and Ω/ω = 0.52). (d) V00 = 3, V11 = 3.8, V01 = V10 = 0.8, χ = 1/5, E1 − E0 = 3 (γ = 2 and Ω/ω = 0.52). The effect of non-zero decoherence (γ ≠ 0) become apparent by direct comparison of (a) with (b,c) with (d). The pulse velocity v in SCQMMs saturates with increasing ε0 to vm/β, that can be significantly lower than that achieved in ordinary TPSIT and TPSRD media, i.e., β. The parameters of the SCQMM can be engineered to slow down the pulse velocity v at the desired level for high enough amplitudes ε0. Note that v is also the velocity of the coherent qubit pulse.

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The system of equations (7, –9) can be reduced to a single equation using the parametrization rx = R0γσ2[1 − cos θ], ry = −R0σ sin θ, and rz = R0{1 − σ2[1 − cos θ]}, of the Bloch vector components. Then, a relation between the Bloch angle θ ≡ θ(x, t) and the slow amplitude ε can be easily obtained, that leads straightforwardly to the equation . Time integration of that equation yields , that conforms with the famous area theorem: pulses with special values of “area” θ(x) = 2πn conserve that value during propagation.

Here we concentrate on the interaction of the SCQs with the EM wave and we are not concerned with decoherence effects in the SCQs due to dephasing and energy relaxation. This is clearly an idealization which is justified as long as the coherence time exceeds the wave propagation time across a relatively large number of unit cell periods. In a recent experiment26, a charge qubit coupled to a strip line had a dephasing time in excess of 200 ns, i.e., a dephasing rate of 5 MHz, and a photon loss rate from the cavity of 0.57 MHz. Those frequencies are very small compared with the transition frequency of the considered SCQs which is of the order of the Josephson energy (i.e., a few GHz)17,18. Therefore, we have neglected such decoherence effects in the present work. The decoherence factor γ, which in Fig. 2b,d has been chosen according to the parameter values in ref. 17, is not related to either dephasing or energy relaxation. That factor attains a non-zero value whenever the matrix elements of the effective SCQ-EM field interaction, V11 and V00, are not equal.

Numerical Simulations

In order to confirm numerically the obtained results, the equations (3–, , 6) are integrated in time using a fourth order Runge-Kutta algorithm with constant time-step. For pulse propagation in absorbing SCQMMs, all the qubits are initially set to their ground state while the vector potential pulse assumes its analytical form for the given set of parameters. A very fine time-step and very large qubit arrays are used to diminish the energy and/or probability loss and the effects of the boundaries during propagation, respectively. The subsequent temporal evolution in two-photon SIT SCQMM, as can be seen in Fig. 3a,b, in which several snapshots of the population inversion Rz(n; t) and the vector potential pulses an(t), respectively, are shown, reveals that the latter are indeed capable of inducing quantum coherent effects in the qubit subsystem in the form of population inversion pulses! In Fig. 3a, the amplitude of the Rz(n; t) pulse gradually grow to the expected maximum around unity in approximately 60 time units, and they continue its course almost coherently (although with fluctuating amplitude) for about 160 more time units, during which they move at the same speed as the vector potential pulse (Fig. 3b). However, due to the inherent discreteness in the qubit subsystem and the lack of inter-qubit coupling, the Rz(n; t) pulse splits at certain instants leaving behind small “probability bumps” that get pinned at particular qubits. After the end of the almost coherent propagation regime, the Rz(n; t) pulse broadens and slows-down until it stops completely. At the same time, the width of the an(t) pulse increases in the course of time due to discreteness-induced dispersion. A comparison with the corresponding analytical expressions reveals fair agreement during the almost coherent propagation regime, although both the Rz(n; t) and an(t) pulses travel slightly faster than expected from the analytical predictions. The temporal variable here is normalized to the inverse of the Josephson frequency ωJ which for typical parameter values is of the order of a few GHz17. Then, the almost coherent induced pulse regime in the particular case shown in Fig. 3 lasts for ~160 × 10−9 s, or ~160 ns, which is of the same order as the reported decoherence time for a charge qubit in ref. 26 (i.e., 200 ns).

Figure 3: Numerical validation of the analytical expressions for two-photon self-induced transparent (TPSIT) and superradiant (TPSRD) propagating pulses.
figure 3

(a) Snapshots of the population inversion pulse Rz(n; t), excited by the induced quantum coherence in the qubit subsystem by the electromagnetic vector potential pulse, in the absence of decoherence (γ = 0); the pulse propagates to the right (time increases downwards) in TPSIT (absorbing) superconducting quantum metamaterials (SCQMMs). The snapshots are taken at intervals of 20 time-units starting at t = 20 and they are displaced vertically to avoid overlapping (blue pulses). The corresponding pulses from the analytical expression equation (11) at the same time-instants are shown in red. (b) Snapshots for the corresponding evolution of the electromagnetic vector potential pulse an(t), that exhibits significant broadening as time passes by; the numerical and analytical pulses are shown in blue and red color, respectively. (c) The same as in a in TPSRD (amplifying) superconducting quantum metamaterials. The resulting propagation is not as simple as expected from the theoretical analysis; instead of a population inversion pulse, it is observed a rather kink-like front propagating to the the right (blue) with a velocity considerably less than that predicted analytically for the pulse, which analytical form is shown in red. (d) The same as in (b) in TPSRD (amplifying) superconducting quantum metamaterials. The velocity of the an(t) pulse (blue) is the same as that of the propagating population inversion front, Rz(n; t); however, it exhibits less broadening with time in comparison with the corresponding numerical an(t) pulse in b. The predicted analytical form is shown in red. Parameter values: χ = 1/5, β = 6, V00 = V11 = 1, V01 = V10 = 0.8, E1 − E0 = 3, and v/c = 0.7 (for (a,b)); v/c = 1.25 (for (c,d)).

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The situation seems to be different, however, in the case of two-photon SRD pulses, as can be observed in the snapshots shown in Fig. 3c,d for Rz(n; t) and an(t), respectively. Here, the lack of the inter-qubit interaction is crucial, since the SCQs that make a transition from the excited to their ground state as the peak of the an(t) pulse passes by their location, cannot return to their excited states after the an(t) pulse has gone away. It seems, thus, that the an(t) pulse creates a type of a kink-like front that propagates at the same velocity. It should be noted that the common velocity of the Rz(n; t) and an(t) pulses is considerably different (i.e., smaller) than the analytically predicted one, as it can be inferred by inspection of Fig. 3c,d. Even more complicated behavioral patterns of two-photon SRD propagating pulses and the effect of non-zero decoherence factor are discussed in the Supplementary Information.

Conclusion

An SCQMM comprising SCQs loaded periodically on a superconducting TL has been investigated theoretically using a minimalistic one-dimensional model following a semiclassical approach. While the SCQs are regarded as two-level quantum systems, the EM field is treated classically. Through analytical techniques it is demonstrated that the system allows self-induced transparent and superradiant pulse propagation given that a particular constraint is fulfilled. Most importanty, it is demonstrated that the propagating EM pulses may induce quantum coherent population inversion pulses in the SCQMM. Numerical simulation of the semiclassical equations confirms the excitation of population inversion pulses with significant coherence time in absorbing media. The situation is slightly different in amplifying media, in which the numerically obtained, induced population inversion excitations are kink-like propagating structures (although more complex behaviors discussed in the Supplementary Information also appear). Moreover, the limiting pulse velocity in both amplifying and absorbing SCQMMs is lower than the corresponding one in two-photon resonant amplifying and absorbing ordinanary (atomic) media. That limiting velocity in SCQMMs can in principle be engineered through the SCQ parameters.

Additional Information

How to cite this article: Ivić, Z. et al. Qubit lattice coherence induced by electromagnetic pulses in superconducting metamaterials. Sci. Rep. 6, 29374; doi: 10.1038/srep29374 (2016).