Electromagnetic Waves in Josephson Qubit Lines

Home , Quantum metamaterial

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

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-

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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. 1/2 By means of Eqs. (15) we find hidden in the quantity (V0) ΔV (t)=−|V |2A2 cos2(kx) 0 2 0 2 0 0∗ iεt 0 01 V0 = |C | V00 +|C | V11 +C C e V10 +h.c., (13) 0 1 0 1 1 2(2ω2 − ε2)cos(εt)+ε2 cos(2ωt) × − . which plays the role of the Josephson plasma frequency. ε ε (4ω2 − ε2) (17) Now we see directly that the “optics” of electromagnetic wave propagation through the system is determined by the We see that the electromagnetic wave is in resonance with quantum state and quantum dynamics of the qubits, which the qubit line if its frequency is half the inter-level distance, 2 are supposed to be under our direct control. ω = ε/2. This is due to the term proportional to α in the Before proceeding, it is wise to check whether the var- Hamiltonian (4). Obviously, this result does not hold near 1 ious assumptions we made are consistent. To begin with, the resonance, since the condition |Ci |1 is no longer we completely neglected both dephasing and relaxation in valid. qubits, and losses in the transmission line formed by the An almost identical result is obtained if all qubits are bulk superconductors. Of course, this only holds as long as initially in the excited state |1. We should only exchange the characteristic time scale of the effect we investigate is 0 ↔ 1 and ε ↔−ε in Eqs. (14)–(17), which leads to short enough. If use typical enough experimental data on 1 charge qubits in a superconducting resonator [4], the de- k(ω)= ω2 − V , β 11 (18) phasing rate of a qubit is ∼ 5 MHz, the photon loss rate from the resonator 0.57 MHz, while the Josephson energy C1(x, t) iA2 cos2(kx) sin(2ωt) (which provides the scale for the effects we are consider- 1 = − t + ing here) is ∼ 6 GHz, which is orders of magnitude higher. V 2 2ω 11 Therefore our assumptions hold water. Moreover, for this 1 2 2 −iεt C0 (x, t) A cos (kx) 1 − e choice of parameters and sensible dimensions of the sys- = − (19) tem D ∼ l ∼ 10μm the parameter β ∼ 30.Sinceβ is the V10 2 ε velocity in the dimensionless wave equation (12), i.e. the −ε + e−iεt [2iω sin(2ωt)+ε cos(2ωt)] number of unit cells per wavelength, our second assump- + . 2 2 tion, the one which allowed us to derive (12), is also valid. 4ω − ε Now we can safely consider some characteristic solu- For the electromagnetic wave one gets tions of the above equations. For simplicity, we assume that α0 is a standing wave, α0 = A cos(ωt)cos[k(ω)x]. ∂2α α¨ − β2 1 + V α + ΔV α =0, First, assume that all the qubits are initially in the 1 2 11 1 1 0 (20) 0 0 ∂x ground state |0, i.e., C0 =1and C1 =0. In this case V0 = V00 and the dispersion law where ΔV (t)=−ΔV (t), 1 1 0 k(ω)= ω2 − V . β 00 (14) and ΔV0(t) is given by Eq. (17). There is, though, an important difference. If all qubits Thus, the metamaterial is transparent for the waves with are initially in the state |1, the metamaterial is an ac- 1/2 frequencies exceeding (V00) , which can be interpreted tive medium, with the complete population inversion. One

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Phys. Status Solidi B 246, No. 5 (2009) 959

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0 0 0.5 1 1.5 2 2.5 3 w Figure 3 Flux qubits in a resonator: Representation with lumped circuit elements. The length of a single section is d, its self- inductance L, and capacitance C. The phase velocity in the un- Figure 2 Breathing photonic crystal: contour curves of the wave 2 perturbed line is thus s = dΩ,whereΩ =1/LC. vector k as a function of ω and εt in the situation described by Eq. (29). The parameters used here are V00 = V01 =1,V11 =2 (units of the qubit Josephson energy EJ ), β =0.5, L =2.The nodes n and n +1is given by (ϕn+1 − ϕn)/L, L be- time-dependent gaps in the spectrum are clearly seen. ing the self-inductance of the section between the nodes. The coupling to the qubits takes place due to the fluxes φ = MIˆ n Then Vγγ = V00 and n p sent by the -th qubit through the corresponding section of the line, M being their mutual induc- ˆ Vδδ(t)=[V00 + V11 +2V01 cos(εt)]/4. tance, and Ip the persistent current operator in the qubit loop. (We could easily include the direct qubit–qubit in- We see that the photonic crystal arises if any of the matrix teraction and take into account the inhomogeneity of the elements is of the order of unity. The frequency gap is mod- structure, but these are not important for the current con- ulated by the value of V01/2 with the period Δt =2π/ε. sideration). It is straightforward now to write the equations If V01 ∼ 1, then the modulation is significant. If of motion for the (classical) field in the line, 2 2 |γ = |0eiεt −|1e−iεt /2 ϕ¨n + Ω (2ϕn − ϕn+1 − ϕn)=Ω (φn − φn−1). (30) In the continuous limit, this equation becomes and iεt −iεt ∂2ϕ(x, t) ∂φ(x, t) |δ = |0e + |1e /2, ϕ¨(x, t) − s2 = sΩ . ∂x2 ∂x (31) then the qubit wave function, Eq. (6), determines the right-hand Vγγ(t)=[V00 + V11 − 2V01 cos(εt)]/4, side via

(29) φ(x, t)=MI0Ψ(x, t)|σˆz |Ψ(x, t). (32) Vδδ(t)=[V00 + V11 +2V01 cos(εt)]/4. This equation will be completed by Eqs. (7) for Ψ(x, t), If V01 ∼ 1, we obtain the gap which is strongly modu- but with a different set of Vkm. Note that unlike Eq. (9) lated in time from zero, at t =(2n +1)π/2ε, to its maxi- for charge qubits, the unperturbed spectrum of Eq. (31) is mum value, when t = nπ/ε (here n is an integer). This is gapless. a “breathing” photonic crystal (Fig. 2). What conclusions can we make from the above results? Is it possible to replace charge qubits with flux qubits? An extended structure built of qubits behaves with respect One can see no reason why not, but the equations are to the propagation of a classical electromagnetic wave in a slightly different. Consider a set of flux qubits inductively way, which reveals quantum coherence of its constituting coupled to a transmission line (i.e., placed inside a strup elements. Not only one can tweak the parameters of this line resonator). It is convenient to represent such a system medium by changing the qubit states – this is, arguably, no using the approximation of lumped elements (Fig. 3). Fol- different from controlling parameters in a classical trans- lowing the approach of Ref. [7], we assign to each node mission line. Of more fundamental importance is the re- afluxvariableϕn, such that, e.g., the current between the sult that the classical field propagation through a quantum

960 A. M. Zagoskin et al.: Electromagnetic waves in Josephson qubit lines

metamaterial will reveal such essentially quantum effects [3] A.L. Rakhmanov, A.M. Zagoskin, S. Savel’ev, and F. Nori, as quantum beats due to the system being prepared in a co- Phys. Rev. B 77, 144507 (2008). herent superposition of quantum states. One can think of [4] J. Gambetta et al., Phys. Rev. A 74, 042318 (2006). this as an inversion of the double-slit experiment. There a [5] B. Saleh and M. Teich, Photonics (Wiley-Interscience, quantum particle interacted with a classical scatterer. Here Hoboken, 2007), chap. 7. the situation is reversed. An exciting part is, that this al- [6] C. Kittel, Introduction to Solid State Physics (Wiley, New lows us a direct glimpse into the very boundary between York, 1996). quantum and classical worlds. [7] M. H. Devoret, Quantum ﬂuctuations in electrical circuits, in: Quantum ﬂuctuations (Les Houches, Session LXIII, References 1995), edited by S. Reynaud, E. Giacobino and J. Zinn- Justin (Elsevier-Science, 1997), p. 351. [1] J.Q. You and F. Nori, Phys. Today, p. 42 (November 2005). [2] B. Saleh and M. Teich, Photonics (Wiley-Interscience, Hoboken, 2007), chap. 5.