Dispersive Response of a Disordered Superconducting Quantum Metamaterial
Dmitriy S. Shapiro, Pascal Macha, Alexey N. Rubtsov, Alexey V. Ustinov
DDispersive Response of a DisorderedSuperconducting Quantum Metamaterial
Dmitriy S. Shapiro , , , Pascal Macha , , Alexey N. Rubtsov , and Alexey V. Ustinov , , Russian Quantum Center, Novaya St. 100, Skolkovo, Moscow region, 143025, Russia Kotel’nikov Institute of Radio Engineering and Electronics of the RussianAcademie of Sciences, Mokhovaya 11/7, Moscow, 125009, Russia Center of Fundamental and Applied Research, N.L. Dukhov All-Russia Institute ofScience and Research, Sushchevskaya 22, Moscow, 123055, Russia Physikalisches Institut, Karlsruhe Institute of Technology, D-76128 Karlsruhe,Germany ARC Centre for Engineered Quantum Systems, University of Queensland, Brisbane,Queensland 4072, Australia National University of Science and Technology MISIS, Leninsky prosp. 4, Moscow,119049, RussiaE-mail: [email protected]
Abstract.
We consider a disordered quantum metamaterial formed by an array ofsuperconducting flux qubits coupled to microwave photons in a cavity. We map thesystem on the Tavis-Cummings model accounting for the disorder in frequencies ofthe qubits. The complex transmittance is calculated with the parameters taken fromstate-of-the-art experiments. We demonstrate that photon phase shift measurementsallow to distinguish individual resonances in the metamaterial with up to 100 qubits,in spite of the decoherence spectral width being remarkably larger than the effectivecoupling constant. Our simulations are in agreement with the results of the recentlyreported experiment [1].
Keywords : quantum metamaterial; superconducting qubits; cavity QED; simulation a r X i v : . [ c ond - m a t . s up r- c on ] A p r
1. Introduction
The novel type of quantum metamaterials [2, 3] composed of arrays of superconductingtwo-level systems (qubits) offers a platform for quantum simulators and quantummemories [4, 5]. Quantum metamaterials can be employed as test bench for studiesof fundamental phenomena as ensemble quantum electrodynamics and spin resonancephysics on macroscopic level [6, 7]. The flux qubit [8, 9], which can be viewed as aartificial atom [10], is a µ m-sized superconducting loop with several Josephson junctionsacting as nonlinear circuit elements. The strongly anharmonic potential of the fluxqubit results in an effective two-level structure of the lowest pair of energy levels of thesystem, where the excitation frequencies fall into GHz range. Ground and excited levelsof the qubit correspond to quantum superpositions of states with opposite directions ofmacroscopic persistent currents in the qubit loop. When coupled to a photon field in asuperconducting cavity, the qubit becomes ”dressed” with photons. The experimentalstudies of superconducting qubit-cavity quantum electrodynamics have shown vacuumRabi oscillations [11, 12], spin echo and Ramsey fringes [13, 14], emission of singlemicrowave photons [15], and Lamb shift [16].While the artificially made ensembles of superconducting qubits have anunavoidable spread of excitation frequencies, in contrast to identical natural atoms,they are easily tunable by varying an external magnetic flux. This fact makes possibleobservation of the fundamental phenomena as dynamical Casimir effect by applying ofa non-stationary external drive [17, 18], under which GHz photons are created fromcavity vacuum fluctuations. In case of the multi-qubit system, a cm-long cavity isused as a transmission line where the virtual photon exchange provides a long-rangequbit-qubit interaction in a sub-wavelength metamaterial. It was shown by Tavis andCummings [19] that N identical two-level systems coupled to the single photon modegenerate a collective enhancement of the coupling constant proportional to √ N . In aqubit array with disordered values of excitation frequencies photons are also coupledto a collective superradiant mode and decoupled from other N − N = 100 qubits. Our simulationsclosely reproduce the experimental observations.
2. Model
Considering the interaction of N qubits with a photon in a resonator (see Fig. 1),we start from the Tavis-Cummings Hamiltonian, which includes qubit-qubit interactionterm H = ω r a + a + N (cid:88) i =1 (cid:15) i σ + i σ − i + N (cid:88) i =1 g i ( σ + i a + a + σ − i ) + N − (cid:88) i =1 g qq,i σ + i +1 σ − i (1)Bosonic operators a + , a in (1) correspond to the photon mode ω r in the cavity andthe single i -th qubit is described in terms of uppering and lowering Pauli operators σ + i = | e (cid:105)(cid:104) g | , σ − i = | g (cid:105)(cid:104) e | acting on its ground and excited states. The first and thesecond terms in Eq. (1) correspond to photon and qubit excitations of frequencies ω r and (cid:15) i , the third one is qubit-cavity coupling written in rotating wave approximation.The last term is an effective direct nearest neighbor qubit-qubit interaction.Figure 1: Sketch of quantum metamaterial formed by an array of flux qubits embeddedinto a microwave resonator.The qubit excitation frequency (cid:15) follows from microscopic Josephson and chargingenergies, E J and E C , which imply E J > E C in the operating regime of flux qubit. Whenthe external magnetic flux threading the qubit loop is equal to half of the magnetic fluxquantum Φ = Φ /
2, with Φ = h/ (2 e ), the total energy of the qubit has two symmetricminima related to the opposite circulations of persistent currents |↓(cid:105) and |↑(cid:105) where thetunneling rate ∆ between two wells depends on E J and E C . The condition that ∆ ishigher than dephasing rate Γ ϕ allows for quantum superpositions of |↓(cid:105) and |↑(cid:105) resultingin two non-degenerate states | g (cid:105) , | e (cid:105) being the basis of (1). By detuning the magneticflux from Φ = Φ / ε i (Φ) = 2 I p,i (cid:126) (cid:18) Φ − Φ (cid:19) , (2)where I p,i is the i -th qubit nominal current [8]. The excitation frequency of i -th qubitis given by the Floquet relation (cid:15) i (Φ) = (cid:113) ∆ i + ε i (Φ) . (3)The qubit-cavity coupling constants in Eq. (1) are renormalized bare constants g i = ∆ i (cid:15) i (Φ) g barei written in the rotated basis | g (cid:105) , | e (cid:105) . Everywhere below we work withthe effective constants. Under uniform flux biasing conditions, the spread in (cid:15) i dependsmainly on qubit excitation gaps ∆ i , rather than I p,i , due to exponential dependence of∆ i ∝ (cid:112) E J /E C exp( − α (cid:112) E J /E C ) on the E J /E C ratio which fluctuates from one fluxqubit to another. The system under consideration [1] is strongly disordered becausethe spread of excitation gaps ∆ i is estimated as high as σ ∆ ≤ ϕ but larger than relaxation rate κ of the cavityΓ ϕ > Ω > κ, Ω = (cid:115) (cid:88) i =1 ...N g i . In the experiment by Macha et al. [1], the external magnetic flux Φ tunes qubitexcitation frequencies in resonance with the cavity mode providing the shift of the phase ϕ (Φ) of a weak external probe signal at cavity mode frequency ω r / π = 7 .
78 GHz. Fromthe measured ϕ (Φ) it was found that number of qubits in an ensemble, which form acollective mode, corresponds to N = 8. Relevant parameters of the studied metamaterialare the following: qubit-cavity coupling strength g i / π = (1 . ± .
1) MHz, dephasingΓ ϕ / π = 55 MHz, persistent current I p = (74 ±
1) nA and average value of the excitationgap ∆ / π = 5 . g qq,i / π < ϕ (Φ). The interaction only slightly shifts energies of collective statesand does not contribute directly to the collective qubit-cavity coupling strength Ω, whichis the most relevant for the phase shift. The qubit dephasing in the system under consideration leads to smearing of the photondensity of states which becomes not informative. However, the experimentally measuredphase locking effect is quite prominent [1]. In our work here, we focus on the influenceof relatively large diagonal disorder in the qubit excitation frequencies and their number N on the phase shift ϕ (Φ) of the transmitted photons. We calculate the phase shiftfrom a complex phase of the photon Green function ϕ (Φ) = arg D ω , where ω is theprobe frequency and Φ is the external flux, counted from the symmetry point valueΦ /
2, which shifts all excitation energies in the qubit ensemble.Considering approximately resonant regime where the excitation frequencies ofqubits and resonator mode are close to each other | (cid:15) i − ω r | < ω r (4)we find a spectrum of the system in a regime of single excitation by means ofexact diagonalization of the Hamiltonian. This regime is realized under experimentalconditions due to small microwave probe power and low temperature of the system.Our solution corresponds to the fully quantum regime where all the variables σ − i , a arequantum fields.Below we introduce the basis of N + 1 states of approximately equal energies whichare related to single excitation either in the photon cavity or in one of N qubits. Namely,the state where a single photon is excited and all qubits are in the ground state reads | (cid:105) = |
1; 0 , . . . , (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) N +1 The excitation in 1-st, i -th or N -th qubits without the photon are given by | (cid:105) = |
0; 1 , , . . . , (cid:105) , | i + (cid:105) = |
0; 0 , . . . , , . . . , (cid:105) and | N + (cid:105) = |
0; 0 , . . . , , (cid:105) . The Hamiltonian (1) in a matrix form H i,j = (cid:104) i | H | j (cid:105) in terms of this single excitationbasis | i (cid:105) reads H i,j = ω r g g g g g . . .g (cid:15) g qq, . . .g g qq, (cid:15) g qq, . . .g g qq, (cid:15) g qq, . . .g g qq, (cid:15) g qq, . . .g g qq, (cid:15) . . . ... ... ... ... ... ... . . . . The photon Green function D ω is given by the first diagonal { , } element of theresolvent matrix GG i,j = [ ωδ i,j − H i,j − diag( iκ, i Γ ϕ , ... )] − . (5)This is the matrix of retarded Green functions, where the last term diag( iκ, i Γ ϕ , ... )introduced in order to take into account the damping in the cavity κ/ π = 715 kHz anddephasing of qubits. We note that the above expression for the photon Green function D ω and G i,j allowing us to calculate the phase shift coincides with the generic equationsobtained earlier by Volkov and Fistul [3].
3. Results N In our numerical studies, we model the system with large amount of qubits and inthe presence of disorder between them. Namely, we fix metamaterial parameters andplot the phase shift increasing qubit number N , but keeping spread of ∆ i constant,see Fig. 2, and vice versa, see Fig. 3. The reported results for the ϕ (Φ) areobtained at the experimental values ω/ π = ω r / π = 7 .
78 GHz, I p = 74 nA andthe following parameters: decoherence rate Γ ϕ / π = 33 MHz, average qubit excitationgaps ∆ / π = 5 . σ ∆ = 3 . g/ π = 1 . σ g = 1% and N = 7 qubitsin the ensemble. These parameters give best fit in the experimental regime at small N discussed in the next section 3.2.Figure 2 shows plots of the phase shift dependence on magnetic flux for N =20 , , , ,
500 qubits in the system and fixed spread σ ∆ = 3 .
6% of the normaldistribution of ∆ i . One can see that for N <
100 the phase shift fluctuates and dependson a particular realization of the qubit frequencies, while for
N >
100 it appears rathersmooth and regular, meaning that the array of qubits can be treated as a continuoussystem at these parameters. In Fig. 3, we show results for different values of disorderwith the following dispersions σ ∆ = 5 , . , , , N = 250fixed. One can see from the Fig. 3 that, with increasing disorder, the phase shift picturesstart to reveal fluctuations. For the realistic spread σ ∆ = 10% the system remains farfrom the ensemble limit, even for this relatively large number of qubits. N Next, we compare experimental [1] (Fig. 4, left panel) and numerical (Fig. 4, rightpanel) results for the phase shift ϕ (Φ). The resonator frequency ω/ π = ω r / π = 7 . I p = 74 nA are fixed. In our numerical method, we assumea normal distribution of the qubit excitation gaps and effective coupling. We selected aparameter distribution with dispersions σ ∆ = 3 .
6% and σ g = 1% for a system containing N = 7 qubits with an average excitation gap ∆ / π = 5 . ϕ (Φ) at qubit energy gap disorder σ ∆ = 3 .
6% of N = 20 , , , ,
500 qubits.Figure 3: Numerical results for the phase shift ϕ (Φ) of N = 250 qubits using fivedifferent values of disorder σ ∆ = 5 , . , , , ω/ π = ω r / π = 7 .
78 GHz and relaxation rate κ/ π = 715 kHz, averageexcitation gap ∆ / π = 5 . σ ∆ = 3 . g/ π = 1 . σ g = 1%, decoherenceΓ ϕ / π = 33 MHz, persistent current I p = 74 nA.the experimental data. Subsequently, we fitted the decoherence rate and effective qubitcoupling and found Γ ϕ / π = 33 MHz and g/ π = 1 . σ exp ∆ < exp / π = 5 . g exp / π = 1 . expϕ / π = 55 MHz and N exp = 8. The values for qubit number anddephasing reported in Ref. [1] were found under the assumption of identical qubits. Herewe show, that the experimental data can be reproduced fairly well under the assumptionof randomly distributed qubit parameters. In the experiment it was observed, thatan ensemble of N exp = 8 qubits, resonantly interacting with the cavity mode andmonitored over long time, spontaneously dissolved into two sub-ensembles of 4 qubitseach, resulting into two jumps in Fig. 4 (left panel). In our exact diagonalizationprocedure we do not find the formation of sub-ensembles but arrive at correspondencewith the experimental curves if parameters of the metamaterial and disorder are closeto the experimental values.Figure 4: Experimental (left panel) and numerical (right panel) results for the photonphase shift ϕ (Φ) as a function of external flux bias Φ, calculated for the system of N = 7qubits.
4. Conclusions
We theoretically studied the model of a flux qubit array coupled to a cavity, withdisorder in qubit excitation frequencies. The system under consideration contains afinite number of qubits and operates in the intermediate regime where disorder rangeand decoherence rate exceed the effective qubit-cavity coupling. We calculated thephoton Green function through the exact diagonalization of the Hamiltonian in the singleexcitation regime, assuming low power of external microwave driving. We found that theresonant phase shift of a transmitted probe signal shows quantitative correspondencewith the experimental data [1]. Variations in phase shift characteristics at differentvalues of the disorder and number of qubits in the system were presented.
5. Acknowledgments
This work was supported by the Ministry for Education and Science of RussianFederation under contract no. 11.G34.31.0062 and in the framework of IncreaseCompetitiveness Program of the National University of Science and Technology MISISunder contract no. K2-2014-025.
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