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Quasi-Superradiant Soliton State of Matter in Quantum Metamaterials?

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Quasi-Superradiant Soliton State of Matter in Quantum Metamaterials? Eur. Phys. J. B (2017) Vol: No DOI: 10.1140/epjb/e2017-80567-7 THE EUROPEAN PHYSICAL JOURNAL B Regular Article Quasi-superradiant soliton state of matter in quantum metamaterials? Hidehiro Asai1,2, a, Shiro Kawabata1, Sergey E. Savel'ev2, and Alexandre M. Zagoskin2,3 1 Nanoelectronics Research Institute (NeRI), National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8568, Japan 2 Department of Physics, Loughborough University, Leicestershire LE11 3TU, UK 3 Theoretical Physics and Quantum Technologies Department, Moscow Institute for Steel and Alloys, 119049 Moscow, Russia Received 8 October 2017 Published online (Inserted Later) { c EDP Sciences, Societ`aItaliana di Fisica, Springer-Verlag 2017 Abstract. Strong interaction of a system of quantum emitters (e.g., two-level atoms) with electromagnetic field induces specific correlations in the system accompanied by a drastic increase of emitted radiation (superradiation or superfluorescence). Despite the fact that since its prediction this phenomenon was subject to a vigorous experimental and theoretical research, there remain open question, in particular, concerning the possibility of a first order phase transition to the superradiant state from the vacuum state. In systems of natural and charge-based artificial atom this transition is prohibited by \no-go" theorems. Here we demonstrate numerically and confirm analytically a similar transition in a one-dimensional quan- tum metamaterial { a chain of artificial atoms (qubits) strongly interacting with classical electromagnetic fields in a transmission line. The system switches from vacuum state to the quasi-superradiant (QS) phase with one or several magnetic solitons and finite average occupation of qubit excited states along the trans- mission line. A quantum metamaterial in the QS phase circumvents the \no-go" restrictions by considerably decreasing its total energy relative to the vacuum state by exciting nonlinear electromagnetic solitons. 1 1 Introduction atom excitations) as the light-atom interaction increases. 23 This topic remained subject of vigorous investigation ever 24 2 The interaction of light and matter in artificial opti- since [27,28]. Soon after this prediction it was pointed 25 3 cal media is the focus of a significant research effort out that the predicted transition is \an interesting arte- 26 4 (see e.g., [1{6]). The strong light-matter interaction in fact" [22], caused by neglecting the term quadratic in 27 5 such systems make possible such effects as unusual pho- the electromagnetic vector potential in the light-matter 28 6 ton collapse-and-revivals [7], Schr¨odinger-catstates [8], interaction Hamiltonian. Taking it into account elimi- 29 7 non-classical radiation [9], unusual Casimir effect and nates the superradiant state. This \no-go" theorem (see 30 8 pseudo-vacuum states [10]. For a subclass of these media e.g., [22{24]) was claimed not to apply to both bosonic 31 9 with extended spatial quantum coherence (quantum meta- and fermionic artificial-atom arrays (in particular, in case 32 10 materials [11{19]) a number of novel phenomena are of circuit QED), at least when driven by a laser to a 33 11 predicted, which do not have place in natural materials non-equilibrium state [26,29{32]. Nevertheless in [25] the 34 12 and classical metamaterials. This adds a new dimension theorem was restated for charge-qubit based circuit-QED 35 13 to the long-standing discussion about the possibility of systems. The precise requirements to a system capable 36 14 a superradiant transition in the system of atoms strongly of undergoing the superradiant phase transition [25,26] 37 15 interacting with electromagnetic waves in a cavity [20{26]. and possible connections between the superradiance and 38 16 In particular, reference [25] extends the \no-go" theorem similar phenomena such as the dynamical Casimir effect 39 17 to circuit QED with charge (but not flux) qubits. [33,34] or the essentially non-classical spontaneous radi- 40 18 It was predicted [20,21] that in a cavity containing ation [9,10] are still being vigorously debated, and the 41 investigation of these and related phenomena in artificial 42 19 many two-level atoms there existsUncorrected a possibility of a tran- Proof 20 sition from the vacuum state (with no photons and all structures is highly relevant. 43 21 atoms being in the ground state) to the superradiant state A quantum metamaterial (QMM) is a globally quantum 44 22 (with nonzero photon occupation number accompanied by coherent array of artificial atoms with a limited control 45 of their quantum state [12]. In this article we consider 46 ? Supplementary material in the form of one pdf file from superradiant-like transition in the presence of external 47 the Journal web page at magnetic field in such a structure. Though the basic 48 https://doi.org/10.1140/epjb/e2017-80567-7. properties of a QMM are qualitatively independent on 49 a e-mail: hd-asai@aist.go.jp its specific realization [18,19], here we use a model of a 50 Page 2 of6 Eur. Phys. J. B (2017) Vol: No qubit level occupation which, in turn, is generated by the 78 solitons themselves. 79 2 Model 80 The system in question is a quantum counterpart of the 81 standard Josephson transmission line [35] where Joseph- 82 son junctions linking two superconducting banks are 83 replaced by charge qubits (Fig.1), i.e., small supercon- 84 ducting islands connected to both long superconduct- 85 ing banks with high-resistance tunneling junctions (R > 86 RQ ≈ 12 kΩ), and with controlled potential bias with 87 respect to the banks [36]. The quantum states of an island 88 differ by the number of extra Cooper pairs on them, and 89 are coupled to the electromagnetic field through their 90 electric charge. 91 As in [11] we describe qubits quantum-mechanically 92 while treating the electromagnetic fields in the transmis- 93 sion line classically (in line with the standard semiclassical 94 approach to atom-field interactions, valid in case of strong 95 enough fields [37]). We direct the vector potential A 96 Fig. 1. One-dimensional quantum metamaterial: (a) a chain z across the junctions (along z-axis) and assume it to be 97 of superconducting charge qubits playing the role of artificial atoms in a Josephson transmission line formed between two constant in each \cell" between two adjacent qubits. 98 superconducting banks connected via superconducting islands We start from the classical expression for the sys- 99 (charge qubits). The (green) wave represent distribution of tem's total (electromagnetic, electrostatic and Josephson) 100 energy as a function of the phases φn of the supercon- 101 the electromagnetic field in the structure. (b) The schematic P view of the structure. The magnetic field penetrates through ducting order parameters on the islands: Etotal = n En 102 the openings between the superconducting islands. The elec- with 103 tric and magnetic fields and the vector potential are assumed constant within each opening. 2 " 2 C ~ dφn πD dAz;n En = 2 + 2 4e dt Φ0 dt 2# dφn πD dAz;n 51 superconducting one-dimensional QMM essentially iden- + − 52 tical to the one of [11,16,17]: a one-dimensional chain of dt Φ0 dt 53 charge qubits between two superconducting banks and πDAz;n πDAz;n 54 interacting with electromagnetic fields, but not directly −EJ cos φn + + cos φn − Φ0 Φ0 55 with each other (Fig.1). As the result of our numer- 2 56 ical simulation applicable to strong field-qubit coupling 8π A − A + z;n+1 z;n : (1) 57 supported by an analytical calculation, we find a phase DL L 58 transition from the vacuum state to the new quasi- 59 superradiant phase, which appears when the field-qubit Here, the index n corresponds to the qubit number in 104 60 coupling induces the phase coherence of the QMM in the chain, C is the junction capacitance, Φ0 = hc=2e is 105 61 the presence of an external magnetic field. This effect IcΦ0 the flux quantum, E = is the Josephson coupling 106 62 is not necessarily limited to superconducting quantum J 2πc energy, I is the critical current of the Josephson junc- 107 63 metamaterials and may provide additional tools for the c tions linking the qubits to the superconducting busbars, D 108 64 investigation of meso- and macroscopic quantum coherent and L are distances between the neighbouring qubits and 109 65 systems. between the two superconducting banks, respectively. It is 110 66 As the field-qubit coupling increases, the vacuum state convenient to use dimensionless variables: the inter-qubit 111 67 becomes unstable, and the system undergoes the first distance l = L/λ, vector potential a = πDA /Φ , 112 68 order phase transition to the state with magnetic soli- z;n z;n 0 time τ = ! t and energy E = E =E , where the Joseph- 113 69 tons and a spatially varying occupation of the excited J n n J Uncorrectedson plasma frequency Proof!J = 2eIc=( C) and the effective 114 70 qubit states. The critical coupling strength can be reduced ~ spatial scale λ = c=! . Then the energy per unit is 115 71 by increasing the external magnetic field Hext, which J 72 also generates structural transitions between states with 2 2 73 different number of solitons. When cycling the external dφn daz;n 74 magnetic field around zero, the soliton number can be En = + − 2 cos φn cos az;n dτ dτ 75 either zero or nonzero at Hext = 0 resulting in a rich 2 76 variety of remagnetization loops associated with quan- a − a +β2 z;n+1 z;n : (2) 77 tum pinning of solitons on the spatial variations of the l Eur. Phys. J.
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