Quantum synchronization in disordered superconducting metamaterials

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.

A key point allowing one to resolve such a puzzle is electromagnetic interactions between qubits. Two types of interactions can be realized in the SQMs: the nearest-neighbor interaction that is due to direct inductive or capacitive electromagnetic interaction between adjacent qubits 18 , and/or the long-range electromagnetic interaction arising due to consequent emission, propagation and absorbtion of virtual photons in the low-dissipative resonator coupled to the array of qubits 10,15,19 . Both interactions depend on the coupling coefficient g, where g is determined by mutual inductance between qubits or a single qubit and the resonator. The direct nearest-neighbor interaction is proportional to g, and the strength of long-range interaction is proportional to g 2 . Although the latter interaction is rather weak, the long-range character of such interaction allows to establish the collective effects. E.g. in ref. 15 the collective ac-Stark shift of qubit frequencies has been theoretically predicted. Moreover, this ac-Stark frequency shift is resonantly enhanced as the frequency of resonator ω r coincides with the average value of qubit frequencies ∆ = ∆ i .
In this Article we consider the case of inductive interaction between adjacent qubits, and we show that such interaction between qubits can overcome the disorder and establish collective synchronized quantum-mechanical oscillations characterized by just few frequencies. A weak inductive interaction results in a slight increase of energy of coupled qubits being in nonequivalent quantum states in respect to the energy of qubits being in the equivalent quantum states. Such energy difference, 4 K, is determined by the coupling coefficient g. By making use of the instanton method of analysis 15,20 we obtain that even in the case of a weak interaction, i.e. K ≪ Δ, a large amount of qubits δ∆  N K * ( / ) 2 are synchronized, and these qubits display collective quantum-mechanical oscillations of a single frequency. Correspondingly, N N / * is a total number of diverse quantum oscillations observed in disordered SQMs. Here, δΔ is the typical spread of energy level differences of individual qubits.
We show also that such quantum-mechanical synchronization phenomenon can be mapped to the Anderson localization 21 of spinon-type excitations occurring in the SQM. Indeed, the various disordered SQMs with nearest-neighbor interactions can be described by the quantum Izing model with a large transverse magnetic field 22,23 . In the limit of K < Δ the ground state of this model is the spin ordering in the transverse direction, and the low-lying excited states form the spinon band separated from the ground state by the energy gap of order Δ . In the presence of disorder all spinon states are localized, and a spread of localized wave function of spinons is of order r 0 . Therefore, the coherent quantum mechanical oscillations correspond to the resonant transitions between the ground state and the various excited states of localized spinons.
We anticipate that the fabrication of disordered SQMs with a tunable inductive coupling allows one to observe the crossover from a non-synchronized regime to the synchronized regime, and provides the direct evidence of synchronized quantum-mechanical oscillations. The realization of synchronized regime in intrinsically disordered SQMs will result in a substantial simplification of the process of qubits addressing. Since arrays of interacting qubits are exactly mapped on various quantum spin models, our method of instanton analysis can be used for a quantitative theoretical study of low-lying quantum-mechanical states in diverse interacting disordered quantum systems under equilibrium and non-equilibrium conditions [22][23][24][25] .

Superconducting Quantum Metamaterial Model and Hamiltonian of disordered SQM. Let us
to consider a one-dimensional array of N flux qubits modeled as two-level systems, with nearest-neighbor inductive electromagnetic interactions between adjacent qubits. In a particular case of flux qubits (3-Josephson junction SQUID) the two states correspond to the clockwise and anticlockwise flowing currents, and the energy level difference, Δ i , is determined by quantum tunneling between these states 12 . The frequencies of individual coherent quantum-mechanical oscillations that can be excited in a single qubit, are ω i = Δ i /ħ. The SQM is coupled to the linear transmission line, and this setup allows one to study the propagation and reflection of electromagnetic waves through the SQM. The schematic of such setup is shown in Fig. 1. Notice here, that similar setup has been used in ref. 1 in order to measure macroscopic quantum-mechanical oscillations excited in the disordered SQM.
The quantum dynamics of each qubit is characterized by the imaginary-time dependent degree of freedom, ϕ i (τ), and the qubits Hamiltonian has a form: where both the parameter m * and the potentials U i [ϕ] determine completely quantum-mechanical dynamics and the energy levels of isolated qubits. Moreover, the double-well potential results in small quantum-mechanical energy level differences Δ i of single qubits. Here, ± ϕ 0 are the values of ϕ in the classical stable states. In this model the energy level differences Δ i are expressed as For flux qubits the parameters m * , ϕ 0 and U i0 are determined by the capacitance and Josephson critical currents of Josephson junctions of qubits 12 . The unavoidable disorder in qubit parameters (U i0 ) leads to a spread of Δ i in the SQM. The Hamiltonian of inductive electromagnetic interaction between adjacent qubits is written as where the parameter g characterizing the strength of nearest-neighbor interaction is determined by mutual inductance between qubits. Thus, one can see that in the quantum regime the inductive interaction results in an increase of energy of qubits being in nonequivalent quantum states, 4 K, where ϕ = K g /2 0 2 is the effective strength of interaction between adjacent qubits. Partition Function and Instanton Analysis. The thermodynamic properties of disordered SQMs are determined by the partition function Z expressed through the Feynman path integral in the imaginary-time τ representation as We consider disordered SQMs with a weak electromagnetic interaction, i.e. K ≪ Δ i , and in this case the optimal path configurations are series of alternating instanton (anti-instanton) solutions uncorrelated in imaginary time interval [0, ħ/(k B T)] 15,20 (schematic of a typical path configuration is shown in Fig. 2). For a single i-th qubit the optimal path configuration ϕ(τ) has a form: Since the probability density to obtain a single instanton in the interval dτ is determined by Δ i the partition function Z i of a single qubit is written as . As the electromagnetic interaction between qubits is absent, i.e. K = 0, one can 20 .

and on the time interval [0, ħ/(k B T)] the average quantity of instantons and anti-instantons for an
To analyze the collective behavior of disordered SQM in the presence of interaction we will characterize each qubit by random value of N i , and the probability P K=0 {N i } shows sharp peaks on the values Moreover, in such description the frequencies ω i of quantum-mechanical oscillations occurring in the SQM, are determined by random values of N i as ω i = k B TN i /ħ. Notice here, that in the presence of a weak (K ≪ Δ i ) interaction between qubits the qubit frequencies ω i can differ from bare frequencies, Δ i /ħ. Indeed, substituting instanton-(anti)instanton solutions (5) in the Eqs (3) and (4) we obtain the probability P K {ω i } in the following form: Here, ∆ = ∆ i is the average value of energy splitting of qubits Δ i , and ϕ = K g /2 0 2 is the effective strength of electromagnetic interaction between instantons (anti-instantons) of adjacent qubits. We have taken into account that the spread of qubit splitting is small, i.e. δΔ ≪ Δ . The physical meaning of different terms in Eq. (10) is rather transparent: the disorder in bare frequencies Δ i /ħ results in a spread of instanton quantities, N i along the array, but the electromagnetic interaction allows to equalize the quantity of instantons on different qubits. Since the frequencies of quantum-mechanical oscillations are directly related to the quantities of instantons, one expect that the electromagnetic interaction results in the synchronization phenomena.

Correlations of Qubits Frequencies.
To analyze this effect quantitatively we consider the continuous limit as N ≫ 1, and using the periodic boundary conditions we expand ω(x) and Δ (x) (x is the coordinate along the array of qubits) in Fourier series as: Here, L = Na is the length of the SQM, and a is the distance between qubits. Taking into account that the values of Δ i are not correlated for different qubits, and the characteristic spread of bare energy level differences is δΔ we obtain |b n | 2 = 2(δΔ ) 2 a. The coordinate dependent correlations in qubit frequencies are characterized by the correlation function 〈 ω(x 1 )ω(x 2 )〉 : Thus, the distribution of qubit frequencies ω(x) is determined by the Fourier coefficients a n , which, in turn, can be obtained from the analysis of the partition function of interacting instanton liquid, Q. Substituting (11) in (10) the expression for Q is written in the following form: The optimal path a n in the Eq. (13) is obtained by following procedure. By making use of the substitution = ∑ π ( ) X a n n n L 2 2 2 and the equality and calculating the Gaussian integrals over a n explicitly we obtain The maximum of the exponent in (15) is reached for optimal values of X 0 and Y 0 , which are solutions of the system of equations  (14) occurs from particular values of a n which are determined by random values of qubit splitting b n . Explicitly the amplitudes a n are obtained by maximizing the exponent in (14) over the variables a n as Substituting in this expression Y = Y 0 the amplitudes a n are written as Here, we introduce the correlation radius r 0 characterizing the area where the quantum-mechanical oscillations are correlated, i.e.  = ∆ r k T aiY 2 / B 0 2 0 2 . In spite of the presence of uncorrelated disorder in qubits splitting, Δ i , the electromagnetic interaction between qubits can lead in long-range correlations (r 0 ≫ a) of frequencies of quantum-mechanical oscillations excited in the disordered SQMs. These correlations are described quantitatively by coordinate-dependent correlation function of qubit frequencies R(x) (Equation (12)). Substituting the amplitudes a n in (12) and calculating the sum over n, the correlation function R(x) is written explicitly as where the correlation radius r 0 can greatly exceed the distance between qubits a. A peculiar dependence of correlation radius on the strength of disorder has an origin in a sub-diffusive interacting term ∝ |ω i − ω i−1 | in the exponent of (10). It differs such a problem from e.g. fluctuation induced bending of strings and superconducting vortex lines 26 . Next, we notice that this analysis can be extended to the two-dimensional lattice of interacting qubits. Indeed, the Eqs (16) are valid for 2d square lattice with the substitution: (1/L 2 )∑ n → ∫ qdq. Calculating all integrals we obtain that in a 2d case the correlation radius is exponentially large, i.e. where the parameter α ≪ γ 2 characterizes the coupling between the transmission line and SQM. Thus, as the electromagnetic interaction between adjacent qubits are small, i.e. K ≪ δΔ , and a substantial spread of qubit splitting δΔ ≫ γ, the disordered SQM supports N non-synchronized coherent quantum oscillations of different frequencies. These non-synchronized coherent quantum-mechanical oscillations manifest themselves by N small resonant dips of magnitude α γ  / 2 in the |S 21 (ω)| 2 (see, Eq. (21)). However, the crossover to partially synchronized regime occurs as the inductive interaction between nearest-neighbors qubits overcomes the disorder in qubits energy level differences, i.e. K ≥ δΔ . In this regime the correlation radius r 0 exceeds the distance a between qubits, and a large amount of qubits, N r a * / 1 0 displays collective synchronized behavior characterized by a single frequency. In this limit the |S 21 | 2 displays N * dips of large magnitude α γ N * / 2 . As the interaction K goes over the value (δΔ )N 1/2 all qubits become synchronized, the collective quantum mechanical oscillations are established in a whole SQM, and a single large dip occurs in the |S 21 | 2 . Notice here, that main assumption of our analysis is the absence of correlations of instanton (anti-instanton) positions on the τ-axis, and this assumption is valid as K < Δ .

Discussion
In conclusion we have theoretically analyzed the excitation of coherent quantum-mechanical oscillations in disordered SQMs. Our analysis is based on the mapping of coherent quantum-mechanical oscillations to series of alternating instanton (anti-instanton) solutions in the path-integral approach. In this model the frequencies of quantum-mechanical oscillations excited in the SQMs, ω i , are directly related to the quantity of instantons on different qubits, N i . The disorder in qubits parameters results in a spread of N i along the array of qubits, and a weak electromagnetic interaction between adjacent qubits, K, leads to the partial alignment of these quantities, N i . Thus, we have obtained that a large amount of qubits can display synchronized collective behavior characterized , where E tot is the total energy of the spin chain, and N is the total number of spins. The x is the coordinate along the spin chain, and a is the distance between spins. The ground state energy, E gr tot = − NΔ /2, the effective potential for the low-lying excited states (the spinon band of width K) are shown. The wave functions of excited states (blue lines) are localized, and a spread of spinon wave functions characterizes the number of quantum-mechanical oscillations in the SQM. These oscillations correspond to the resonant transitions (green lines) between the ground state and the excited states of localized spinons.
Scientific RepoRts | 7:43657 | DOI: 10.1038/srep43657 by a single frequency. The size of the region showing synchronized behavior is determined by the ratio of the strength of interaction K to the typical spread of energy splitting of qubits, δΔ .
The fingerprints of synchronized regimes of coherent quantum oscillations are a few number of giant resonant dips in the frequency dependent electromagnetic waves transmission, |S 21 (ω)| 2 (see Fig. 1, and Eq. (21)). In the synchronize d quantum-me chanica l dynamic st ate a numb er of such res onant dips is δ = = ∆ N N L r N K / * / ( /2)[ / ] 0 2 . In ref. 1 the propagation of electromagnetic waves through the SQM containing of 20 qubits has been experimentally studied. In these experiments one or two giant resonant dips in the frequency dependent transmission coefficient, |S 21 (ω)| 2 , have been observed. These results indicate the excitation of synchronized coherent quantum mechanical oscillations in the SQM, and a plausible qualitative explanation of these results is the presence of substantial effective inductive coupling between adjacent qubits, i.e. δ∆ . −  K/( ) 2 5 3. However, one can not exclude an alternative explanation that a long-range interaction in an array of qubits also can induce the synchronized quantum-mechanical dynamic state. Such an interaction originates from the emission (absorption) of virtual photons of a low-dissipative resonator 2,15,19 . This type of interaction leads to the interaction between instantons (anti-instantons) of well separated qubits, and it also allows to equalize quantities of instantons (anti-instantons) on different qubits, and therefore, to establish a synchronized regime. We will address the quantitative analysis of synchronized quantum-mechanical oscillations induced by long-range interaction, elsewhere.