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arXiv:1606.09078 · Physical review. B./Physical review. B

Double-resonance response of a superconducting quantum metamaterial: Manifestation of nonclassical states of photons

M. A. Iontsev, Sergei I. Mukhin, M. V. Fistul

Published Nov 17, 2016• 8 citations
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Abstract

We report a theoretical study of the ac response of superconducting quantum metamaterials (SQMs), i.e., an array of qubits (two-level systems) embedded in a low-dissipative resonator. By making use of a particular example of a SQM, namely the array of charge qubits capacitively coupled to the resonator, we obtain a second-order phase transition between an incoherent (the high-temperature phase) and coherent (the low-temperatures phase) state of photons. This phase transition in many aspects resembles the paramagnetic-ferromagnetic phase transition. The critical temperature of the phase transition, ${T}^{\ensuremath{\star}}$, is determined by the energy splitting of two-level systems $\ensuremath{\delta}$, number of qubits in the array $N$, and the strength of the interaction $\ensuremath{\eta}$ between qubits and photons in the cavity. We obtain that the photon states manifest themselves by resonant drops in the frequency-dependent transmission $D(\ensuremath{\omega})$ of electromagnetic waves propagating through a transmission line weakly coupled to the SQM. At high temperatures the $D(\ensuremath{\omega})$ displays a single resonant drop, and at low temperatures a peculiar double-resonance response has to be observed. The physical origin of such a resonant splitting is the quantum oscillations between two coherent states of photons of different polarizations.

DOI
10.1103/physrevb.94.174510

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We report a theoretical study of the ac response of superconducting quantum metamaterials (SQMs), i.e., an array of qubits (two-level systems) embedded in a low-dissipative resonator. By making use of a particular example of a SQM, namely the array of charge qubits capacitively coupled to the resonator, we obtain a second-order phase transition between an incoherent (the high-temperature phase) and coherent (the low-temperatures phase) state of photons. This phase transition in many aspects resembles the paramagnetic-ferromagnetic phase transition. The critical temperature of the phase transition, T\ensuremath⋆{T}^{\ensuremath{\star}}T\ensuremath⋆, is determined by the energy splitting of two-level systems \ensuremathδ\ensuremath{\delta}\ensuremathδ, number of qubits in the array NNN, and the strength of the interaction \ensuremathη\ensuremath{\eta}\ensuremathη between qubits and photons in the cavity. We obtain that the photon states manifest themselves by resonant drops in the frequency-dependent transmission D(\ensuremathω)D(\ensuremath{\omega})D(\ensuremathω) of electromagnetic waves propagating through a transmission line weakly coupled to the SQM. At high temperatures the D(\ensuremathω)D(\ensuremath{\omega})D(\ensuremathω) displays a single resonant drop, and at low temperatures a peculiar double-resonance response has to be observed. The physical origin of such a resonant splitting is the quantum oscillations between two coherent states of photons of different polarizations.
We report a theoretical study of diverse non-classical photon states that can be realized in superconducting quantum metamaterials. As a particular example of superconducting quantum metamaterials, an array of SQUIDs incorporated in a low-dissipative transmission line (resonant cavity) will be studied. This system will be modeled as a set of two-level systems (qubits) strongly interacting with resonant cavity photons. We predict and analyze a second-order phase transition between incoherent (the high-temperature phase) and coherent (the low-temperatures phase) states of photons. In the equilibrium state the partition function Z of the electromagnetic field (EF) in the cavity is determined by the effective action S eff { P (τ)} that, in turn, depends on the imaginary time dependent momentum of the photon field P (τ). We show that the order parameter of this phase transition is the P 0 (τ) minimizing the effective action of the whole system. In the incoherent state, the order parameter P 0 (τ) = 0 but at low temperatures we obtain various coherent states characterized by non-zero values of P 0 (τ). This phase transition in many aspects resembles the Peierls metal–insulator and the metal–superconductor phase transitions. The critical temperature of such a phase transition T * is determined by the energy splitting of two-level systems Δ, the number of SQUIDs in the array N and the strength of the interaction η between SQUIDs and photons in the cavity.
Abstract I report a theoretical study of collective coherent quantum-mechanical oscillations in disordered superconducting quantum metamaterials (SQMs), i.e. artificial arrays of interacting qubits (two-levels system). An unavoidable disorder in qubits parameters results in a substantial spread of qubits frequencies, and in the absence of electromagnetic interaction between qubits these individual quantum-mechanical oscillations of single qubits manifest themselves by a large number of small resonant dips in the frequency dependent transmission of electromagnetic waves, |S21(ω)|2. We show that even a weak electromagnetic interaction between adjacent qubits can overcome the disorder and establish completely or partially synchronized quantum-mechanical dynamic state in the disordered SQM. In such a state a large amount of qubits displays the collective quantum mechanical oscillations, and this collective behavior manifests itself by a few giant resonant dips in the |S21(ω)|2 dependence. The size of a system r0 showing the collective (synchronized) quantum-mechanical behavior is determined in the one-dimensional SQMs as r0 ≃ a [K/δΔ]2, where K, δΔ, a are the effective energy of nearest-neighbor interaction, the spread of qubits energy splitting, and the distance between qubits, accordingly. We show that this phenomenon is mapped to the Anderson localization of spinon-type excitations arising in the SQM.
We study theoretically the collective quantum dynamics occurring in various interacting superconducting qubit arrays (SQAs) in the presence of a spread of individual qubit frequencies. The interaction is provided by mutual inductive coupling between adjacent qubits (short-range Ising interaction) or inductive coupling to a low-dissipative resonator (long-range exchange interaction). In the absence of interaction, the Fourier transform of the temporal correlation function of the total polarization (zzz projection of the total spin), i.e., the dynamic susceptibility $C(\ensuremath{\omega})$, demonstrates a set of sharp small magnitude resonances corresponding to the transitions of individual superconducting qubits. We show that even a weak interaction between qubits can overcome the disorder with a simultaneous formation of the collective excited states. This collective behavior manifests itself by a single large resonance in $C(\ensuremath{\omega})$. In the presence of a weak nonresonant microwave photon field in the low-dissipative resonator, the positions of dominant resonances depend on the number of photons, i.e., the collective ac Stark effect. Coupling of an SQA to the transmission line allows a straightforward experimental access of the collective states in microwave transmission experiments and, at the same time, to employ SQAs as sensitive single-photon detectors.
We report a theoretical study of the propagation of electromagnetic waves (EWs) through an array of superconducting qubits, i.e., coherent two-level systems, embedded in a low-dissipation transmission line. We focus on the near-resonant case as the frequency of EWs $\ensuremath{\omega}\ensuremath{\simeq}{\ensuremath{\omega}}_{q}$, where ${\ensuremath{\omega}}_{q}$ is the qubit frequency. In this limit we derive the effective dynamic nonlinear wave equation allowing one to obtain the frequency-dependent transmission coefficient of EWs, $D(\ensuremath{\omega})$. In the linear regime and a relatively wide frequency region we obtain a strong resonant suppression of $D(\ensuremath{\omega})$ in both cases of a single qubit and chains composed of a large number of densely arranged qubits. However, in narrow frequency regions a chain of qubits allows the resonant transmission of EWs with greatly enhanced $D(\ensuremath{\omega})$. In the nonlinear regime realized for a moderate power of applied microwave radiation, we predict and analyze various transitions between states characterized by high and low values of $D(\ensuremath{\omega})$. These transitions are manifestations of nonequilibrium steady states of an array of qubits achieved in this regime.
We report a theoretical study of coherent collective quantum dynamic effects in an array of NNN qubits (two-level systems) incorporated into a low-dissipation resonant cavity. Individual qubits are characterized by energy level differences ${\ensuremath{\Delta}}_{i}$ and a spread of ${\ensuremath{\Delta}}_{i}$ is taken into account. Noninteracting qubits display coherent quantum beatings with NNN different frequencies, i.e., ${\ensuremath{\omega}}_{i}={\ensuremath{\Delta}}_{i}/\ensuremath{\hbar}$. Virtual emission and absorption of cavity photons provides a long-range interaction between qubits. In the presence of such interaction we analyze quantum correlation functions of individual qubits Ci(t){C}_{i}(t)Ci​(t) to obtain two collective quantum-mechanical coherent oscillations, characterized by frequencies ${\ensuremath{\omega}}_{1}=\overline{\ensuremath{\Delta}}/\ensuremath{\hbar}$ and ${\ensuremath{\omega}}_{2}={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{R}$, where ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{R}$ is the resonant frequency of the cavity renormalized by interaction. The amplitude of these oscillations can be strongly enhanced in the resonant case when ${\ensuremath{\omega}}_{1}\ensuremath{\simeq}{\ensuremath{\omega}}_{2}$. These collective quantum oscillations can be directly observed, e.g., by measurements of frequency dependent transmission coefficient $D(\ensuremath{\omega})$ of electromagnetic field propagating in a transmission line coupled to the system.
We consider the propagation of a classical electromagnetic wave through a transmission line, formed by identical superconducting charge qubits inside a superconducting resonator. Since the qubits can be in a coherent superposition of quantum states, we show that such a system demonstrates interesting effects, such as a ``breathing'' photonic crystal with an oscillating band gap and a ``quantum Archimedean screw'' that transports, at an arbitrary controlled velocity, Josephson plasma waves through a transmission line. The key ingredient of these effects is that the optical properties of the Josephson transmission line are controlled by the quantum coherent state of the qubits.