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arXiv:1609.04625 · Scientific Reports

Quantum synchronization in disordered superconducting metamaterials

M. V. Fistul

Published Mar 2, 2017• 13 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1038/srep43657

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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 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 NNN, 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.
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 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 explore experimentally a quantum metamaterial based on a superconducting chip with 25 frequency-tunable transmon qubits coupled to a common coplanar resonator. The collective bright and dark modes are probed via the microwave response, i.e., by measuring the transmission amplitude of an external microwave signal. All qubits have individual control and readout lines. Their frequency tunability allows the number NNN of resonantly coupled qubits to change and a disorder in their excitation frequencies to be introduced with preassigned distributions. While increasing NNN, we demonstrate the expected N1/2{N}^{1/2}N1/2 scaling law for the energy gap (Rabi splitting) between bright modes around the cavity frequency. By introducing a controllable disorder and averaging the transmission amplitude over a large number of realizations, we demonstrate a decay of mesoscopic fluctuations which mimics an approach towards the thermodynamic limit. The collective bright states survive in the presence of disorder when the strength of individual qubit coupling to the cavity dominates over the disorder strength.
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.