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Electromagnetic Waves in Josephson Qubit Lines

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Electromagnetic Waves in Josephson Qubit Lines Phys. Status Solidi B 246, No. 5, 955–960 (2009) / DOI 10.1002/pssb.200881568 p s sb solidi status www.pss-b.com physica Quantum metamaterials: basic solid state physics Electromagnetic waves in Josephson qubit lines A. M. Zagoskin1,2,3*, A. L. Rakhmanov1,4, Sergey Savel’ev1,2, and Franco Nori1,5 1 Frontier Research System, The Institute of Physical and Chemical Research (RIKEN), Wako-shi, Saitama, 351-0198, Japan 2 Department of Physics, Loughborough University, Loughborough LE11 3TU, United Kingdom 3 Physics and Astronomy Department, The University of British Columbia, Vancouver, B.C., V6T 1Z1, Canada 4 Institute for Theoretical and Applied Electrodynamics RAS, 125412 Moscow, Russia 5 Department of Physics, Center for Theoretical Physics, Applied Physics Program, Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109-1040, USA Received 15 October 2008, revised 4 February 2009, accepted 4 February 2009 Published online 3 April 2009 PACS 03.67.–a, 41.20.Jb, 82.25.Cp ∗ Corresponding author: e-mail [email protected] We consider the propagation of a classical electromagnetic oscillating bandgap. Similar behaviour is expected from a wave through a transmission line, formed by identical super- transmission line formed by flux qubits. The key ingredient conducting charge qubits inside a superconducting resonator. of these effects is that the optical properties of the Josephson Since the qubits can be in a coherent superposition of quan- transmission line are controlled by the quantum coherent state tum states, we show that such a system demonstrates interest- of the qubits. ing new effects, such as a “breathing” photonic crystal with an © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim The progress in experimental and theoretical investiga- we have already seen that putting qubits in resonators pro- tion of mesoscopic structures, in particular superconduct- vides several advantages, such as a convenient control and ing qubits [1], made it possible now to shift focus to a readout and increased decoherence time, and is therefore more interesting business of building an “artificial matter”, relevant from the experimental point of view. a structure containing a large enough number of qubits, The propagation of an electromagnetic wave along this which maintain quantum coherence for a long enough time sytem will be affected by the coherent quantum dynamics to demonstrate certain new properties. One can consider a of qubits. Should one expect anything unusual? working quantum computer an ultimate example of such Let us consider an infinite number of superconducting a structure. Nevertheless we don’t need to go that far. For charge qubits placed at equal intervals l between two mas- starters, we can consider the propagation of electromag- sive superconductors separated by a distance D of the same netic wave through a large set of qubits, considering the order (Fig. 1) [3]. The superconductors form a waveguide, latter as an effective medium. Such media built of classi- and qubits – an effective medium filling it. The magnetic cal artificial components are known as metamaterials. De- field of the electromagnetic wave propagating in this struc- pending on their structure and parameters, they can demon- ture must be parallel to the superconducting banks. If ne- strate some strikingly unusual properties (e.g., refraction of glect – as we do – the energy losses, it must be also nor- the electromagnetic wave in so-called left-handed materi- mal to the direction of the wave propagation. Therefore als [2]). We will therefore call a medium built of qubits a H = Hyey, and it is conveniently described by the vector quantum metamaterial. potential A = Azez. We will see that if we choose realistic A simple example of a quantum metamaterial if put a parameters, the wavelength of a propagating wave is much large number of qubits in a one-dimensional resonator. Of larger than interqubit distance, and therefore one can ne- course, a chain of interacting qubits would do as well, but glect all dependence on the transverse coordinates. More- © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim pssb solidi status physica 956 A. M. Zagoskin et al.: Electromagnetic waves in Josephson qubit lines though, we can treat the field classically, and therefore as- sume that its amplitude is small, i.e., αn 1. The physical sense of the latter inequality is that the magnetic flux per unit cell area HyD×l is much smaller than Φ0. Under such ... ... assumptions, the Hamiltonian for a single qubit is D jn l 2 ∂ 2 H = − − αn cos ϕn. (4) ∂ϕn z As usual, we restrict the states of each qubit to either its ground state |0, with energy E0, or excited state |1 A E y with energy 1. We can also introduce the Heisenberg ba- H sis states x {|0 exp (iεt/2), |1 exp (−iεt/2)} , (5) where ε = E1−E0 is the dimensionless qubit excitation Figure 1 An example of 1D quantum metematerial. Identical ω0 l energy. charge qubits are placed at equal intervals between bulk su- n Ψ perconductors separated by a distance D. Control circuits of the The wave function of the th qubit, n,isasum qubits are not shown. n iεt/2 n −iεt/2 Ψn = C0 |0 e + C1 |1 e . (6) Using the standard time-dependent perturbation theory, we over, the vector potential within a single cell (say, between see from Eq. (4) that in the presence of the field the nth and (n+1)st qubits) is approximately constant and A dCn can be denoted as zn. i k = α2 V n (t) Cn (t) By now we have learned enough about charge qubits to dt n km m (7) directly write the energy per unit cell m=1,2 ⎡ ⎤ 2 2 with some initial conditions Cn(t =0)=Cn0.Here E 2πDA˙ 2πDA˙ k k E = J ⎣ zn +˙ϕ + zn − ϕ˙ ⎦ 2 n n V n (t)=k| cos ϕ |m 2ωJ Φ0 Φ0 km n are matrix elements of the field–qubit interaction calcu- 2πDAzn − EJ cos ϕn + (1) lated in the Heisenberg basis. Φ0 Varying the energy (2), we obtain the equation for the 2 electromagnetic field in the linear approximation: 2πDAzn Dl Azn+1 −Azn +c o s ϕn − + . 2 Φ0 8π l α¨n −β (αn+1 + αn−1 − 2αn)+αnΨn| cos ϕn|Ψn =0. (8) Here the dot denotes the time derivative, and we assume This equation together with Eqs. (7) allows to determine that the charge qubit has two identical Josephson junctions the behaviour of both the field and the qubits, but it is not with ωJ = eIc/C each. We took into account that, in the yet what we need. Indeed, this equation still contains the presence of the vector potential, the superconducting phase states of individual qubits, while we strive to look at the big differences across the junctions of the nth qubit, ±ϕn, ac- picture. If, as we assumed, the field wavelength exceeds the quire a gauge term, αn =2πDAzn/Φ0. length of a single unit cell, it will not “see” separate qubits. E = El/E , Introducing the dimensionless units J and Then, instead of a difference equation (8) for αn(t) and t → ω t, J we can rewrite Eq. (1) as Ψn(t), we can write a differential equation for α(x, t) and Ψ(x, t) 2 2 2 2 : E =˙ϕn +˙αn − 2cosαn cos ϕn + β (αn+1 − αn) , (2) ∂2α α¨ − β2 + V α =0,V= Ψ(x)| cos ϕ(x)|Ψ(x). where 2 0 0 2 ∂x 2 1 Φ0 EEM (9) β = ≡ (3) n · l x 8πlDEJ 2π EJ Here was replaced by . Staying within the perturbation theory approach, we characterizes the ratio (EEM/EJ ) of electromagnetic and split the electromagnetic wave into the larger incident Josephson energies. wave, α0, and a smaller scattered wave α1. A quantum Eventually we will consider such properties of a quan- state of the system is now described by the wave function tum metamaterial as lasing, which would require a quan- iεt/2 −iεt/2 tum description of the electromagnetic field. At this point, Ψ(x, t)=C0(x, t)|0e + C1(x, t)|1e . (10) © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com Original Paper Phys. Status Solidi B 246, No. 5 (2009) 957 In the unperturbed state the coefficients in this equation are as the “ground state” plasma frequency of the medium. 0 Ci (x). Splitting, in their turn, the coefficients Ci(x, t) into From Eq. (11) we obtain 0 the unperturbed solution Ci (x) and a small perturbation 1 2 2 1 1 C0 (x, t) iA cos (kx) sin(2ωt) Ci (x, t), |Ci |1, and using Eq. (7), we derive = − t + V00 2 2ω t iC1 = dtα2 V C0 + V C0e−iεt 1 2 2 iεt 0 0 00 0 01 1 C1 (x, t) A cos (kx) e − 1 0 = − (15) t V01 2 ε iC1 = dtα2 V C0 + V ∗ C0eiεt . 1 0 11 1 10 0 (11) ε + eiεt [2iω sin(2ωt) − ε cos(2ωt)] 0 + 2 2 Here the matrix elements Vik = i| cos ϕ|k are calculated 4ω − ε using the unperturbed wave functions. Obviously V ∗ = 10 The initial disturbance of the wave function, in its turn, V01. produces a disturbance α1 in the propagating wave. For For the unperturbed EM wave α0, we obtain from this perturbation, using Eq. (9), we derive Eq. (9) a usual-looking wave equation 2 2 2 ∂ α1 ∂ α0 α¨ − β + V α + ΔV α =0, α¨ − β2 + V α =0. 1 ∂x2 00 1 0 0 (16) 0 ∂x2 0 0 (12) ΔV Nevertheless the specifics of a quantum metamaterial is 0 being the perturbation of the field–qubit coupling.
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