arXiv:1201.5299v2 [cond-mat.mes-hall] 8 Sep 2012 erdc h ekpwrdpnec bevdexperi- observed to dependence failed power field weak ge- electric the the applied reproduce include the which of TLS dependence of ometric theory conventional the of a rate, cay eoao soeae nargm flwtemperature low of regime a CPW in as- the ( quantum operated from applications, circuit is ranging these to resonator In fields, 2] [3–6]. diverse [1, electrodynamics of detection photon number tronomical a in used efrac fteedvcsi ietyrltdt the to related directly factor, is quality devices resonator these of performance htn.Ti eut nasrn oe eedneof dependence power by strong dissipated factor: a quality be in the can results This that thereby power saturated, photons. maximal get the dielectrics is the resonator limiting to also in applied expects TLS radiation the one increased, of theory, power this the the to as in that, According by TLS 9]. predicted of [8, one absorption glasses microwave the of with theory ob- agrees conventional the closely by that supported frequency shift strongly resonance is res- temperature-dependent of belief the servation in This surrounding located surfaces) [7]. 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2 by the dimensionless parameter λ = νp0/ǫ, here ǫ is the dielectric constant of the medium, it can also be viewed as coming from TLS coupling to elastic strain [18]. Straight- forward analysis shows that the same parameter also con- trols the phonon mean free path at low temperatures [19]. −3 The direct measurement gives values of λ 10 in bulk !t materials, so the interaction between TLS≈ is usually as- sumed to be irrelevant. The intrinsic microwave loss is due to the coupling of the electric dipole moment pˆ of the TLS to the applied Figure 1: Drift of the energy of a given TLS in the field of a electric field Ecosωt of the resonator. The resonator qual- few fluctuators. ity factor is related to the imaginary part of the dielectric function, ǫ (ω), i.e. Q−1 ǫ . At microwave frequen- ∝ |ℑ | cies and low temperatures, only the resonant contribution cated on the interface oxide surfaces (metal-air, metal- to the electric susceptibility tensor is relevant. One can substrate or substrate-air) [10, 12–14, 16], even though it compute the change in the dielectric function due to the remains unclear which of the surfaces is more relevant[20]. resonant response of an ensemble of TLS by assuming The estimates show [21] that the concentration of TLS that TLS are relaxed by phonons and using the Bloch in these thin surfaces is higher than in the bulk and con- equations that neglect the interaction between TLS [18]. sequently the average interaction between TLS is larger, In fact, one can get the power dependence of quality fac- implying that one has to develop the theory of microwave tor Q by following a more general, qualitative argument. absorption in interacting TLS in these resonators. In In the steady state driven by sinusoidal electric field, the contrast, the resonators that show Q √P and low in- power dissipation density is given by the time average trinsic loss in single photon regime are∝ made of Nb on SiO2/Si [10] and AlOx coated Nb on Si substrate [11] ∂D ω 2 P = E = E ǫ (1) and thus contain a significant amount of bulk amorphous h di  · ∂t  2 | | |ℑ | dielectrics (SiO2 or AlOx). It is natural to assume that where D= ǫE is the displacement vector. In the absence the large intrinsic loss in these resonators is due to TLS of TLS-TLS interactions, each TLS can decay from its located in the dielectric bulk which is described well by excited state only by emitting a phonon; the rate of this the conventional theory of independent TLS. emission gives the decay rate, Γ(ε), of its excited state. Developing the full theory of interacting TLS is a very The maximal power that a given TLS can absorb from difficult problem that was first discussed by Yu et al. [22] the photons in the resonator is εΓ. At small applied pow- and still remains controversial [23–26]. Fortunately, as we ers, the TLS that absorb photons have energies close to shall see below, one does not need to solve the full theory the frequency, ω0, of the photon field, i.e. ε ω0 . Γ. of interacting TLS in order to estimate the internal loss It is convenient to characterize the effect of| the− electric| of superconducting resonators due to interacting TLS. 1 field on TLS by their Rabi frequency Ω = 2 p E sin θ; for In our approach, we assume that the effective degrees small Ω < Γ TLS at resonance are excited with· proba- of freedom that remain active in the amorphous oxides bility (Ω/Γ)2. Since their density is νΓ, the total power surrounding the superconductors are described by fluc- 2 dissipation density reads Pd = νεΩ /Γ. Different is the tuating dipoles: some of these dipoles are characterized situation at larger appliedh powers.i There, even the TLS by fast transitions between their states (ε sin θ ω) and ∼ with energies further away from the frequency of the res- relatively small decoherence rates, i.e. Γ ω, and are ≪ onator, ω0, get excited. For TLS exactly at resonance, effectively coherent TLS; other dipoles are instead slow the electric field E cos ω0t results in oscillations with fre- and characterized by decoherence times shorter than the quency Ω. When Ω > Γ, TLS with energies ε ω0 . Ω typical time between the transitions, we shall call them are excited. Since their density is νΩ, the total| − dissipated| fluctuators. Due to the interaction between fluctuators power density reads Pd = νεΓΩ. By substituting it into and TLS, the frequency of the TLS jumps when the fluc- Eq. 1, one recovers immediatelyh i the quality factor power tuators in its vicinity change its state. This translates dependence Q √P. into the fact that instead of staying constant the energy The generality∝ of these qualitative arguments implies of a given TLS drifts with time (see Fig. 1). Notice that that, in the resonators where a much weaker power de- this phenomenological model is in a full agreement with pendence of Q has been observed, the dipoles that absorb experiments on charge noise performed in superconduct- the radiation do not get saturated as the applied power P ing SET and qubits [27]. is increased. This is possible if the interactions between We now use this phenomenological model to estimate dipoles are not negligible and lead to energy diffusion. the internal loss of the CPW resonators. We begin with The bulk of evidence indicates that the loss in high- qualitative arguments. Due to the interaction with fluc- Q superconducting CPW resonators is due to TLS lo- tuators, the TLS with energy level ε in resonance with the 3 applied electric field stays effectively in resonance only for these processes on a single TLS can be described by a short dwell time τ = γ−1 , where γ is the combined re- the Bloch equations: d S(t) = [ S(t) B(t)] R(t), dt h i h i× − laxation rate of the fluctuators that affect it (see Fig.1). where R(t)=(Γ2 Sx , Γ2 Sy , Γ1( Sz m)) is the re- If the time τ is short, i.e. τΩ 1, the TLS gets into laxation matrix, Bh (ti) ish thei totalh fieldi− acting on the ≪ 2 1 the excited state with probability e (Ωτ) and then TLS and m = tanh [ε/2T ] denotes the equilibrium pop- dissipates the energy away from theP resonance.∼ The av- 2 2 ulation of the TLS levels and T is the temperature. erage dissipation rate of this process is Γ=¯ γ e Ω /γ. Only TLS with energy level located withinP energy∝ γ In the conventional theory of TLS, this field has two B B0 B1 B0 from the resonator frequency ω contribute to this pro- components: = + , with = (0, 0,ε) and B1 ′ ′ 1 pE cess: their density is νγ and the total dissipated power = (Ω, 0, Ω )cos ωt , where Ω = 2 cos θ. The pres- 2 γmax ence of the fluctuators results in the additional time de- density reads Pd = νεΩ ´Ω (γ)dγ , where (γ)= h i P P pendence of the energy levels, εt = ε + ξ(t). Here ξ(t) P0/γ is the probability distribution of the fluctuator re- laxation rates [28]. After integration, we obtain that denotes a multi-level telegraph noise signal characterized 2 by switching rate γ. Pd = νP0εΩ ln (γmax/Ω) and by inserting it into Eq.1 h i The nonlinear Bloch equations are dramatically sim- we find that in this case Q ln (Ω/γmax) ln P , i.e. there is only a slow, logarithmic∝ dependence∝ of the qual- plified in realistic conditions, because the feasible elec- ity factor with the applied power P . tric field acting on the TLS has very little effect on them away from the resonance. Formally, in the sta- These qualitative arguments can be confirmed by more tionary solution of the Bloch equations driven with quantitative analytical computations. Before proceeding E cos ωt, all spin components oscillate with frequencies ∗ with the analytic derivation, we stress that the essen- nω: S = S exp( inωt) with S = S− . Relax- n n − n n tial ingredient of the phenomenological model is the large ation to theP stationary solution is determined by the de- value of the jumps in the TLS frequency, δε Γ caused t ≫ cay rate Γ1 Γ2 ω which sets the longest time scale by fluctuators. This implies that the interaction between in the problem.≪ It≪ can be described by allowing slow time the high frequency dipole and the fluctuators in its vicin- 2 dependence of Sn components. Finally, introducing the p0 ± x y ity is large: V (r) = 3 Γ. Because the typical dis- operators S = S iS and leaving only the leading ǫr ≫ ± tance between thermally activated TLS isr ¯ = ρ−1/3, with resonant terms the Bloch equations reduce to: ρ = νT , the typical interaction is V (¯r) = λT. Compar- + 6 −1 dS1 z + ing it with the relaxation rates of Γ 10 s observed i =ΩS (ω ε + iΓ2)S (2) ∼ dt 0 − − t 1 experimentally at frequencies ω 10 GHz (and expected z theoretically for phonon emissions)∼ we see that in typi- dS0 + 0 =Ω S1 Γ1(Sz m) (3) cal experimental conditions δε Γ for λ 10−3. The dt ℑ − − ∼ ∼ condition δε Γ, however, can be satisfied for lower fre- The macroscopic response is given by the average po- ≫ quencies or in materials with larger λ. The estimates of larization, Pω of TLS at frequency ω: the dissipation per unit volume of surface oxides show 1 that there their density of TLS is at least 10 times larger P = p sin θS+ (4) ω 2 1 than in the bulk amorphous materials [21]. This trans- lates into 10 times larger value of the parameter λ and where the average is taken over the distribution of TLS a completely different physics of microwave dissipation and their dipole moments. The quality factor is directly described above. related to P : Q−1 P /E. ω ∝ |ℑ ω| We now give the details for the analytical derivation If the jumps ξ(t) induced by the fluctuators are small, of the Q √P and its generalization for the model that i.e. ξ < Γ2, the stationary solution of Eqs.(2,3) gives the takes into∝ account energy drifts of TLS caused by fluc- spin component: tuators. The collection of TLS is characterized by the Ω(ω ε iΓ2)m z 1 + Hamiltonian Hint = i εiSi + 2 i,j Vab(ri rj )papb. S1 = 2 − 2− 2 −1 . (5) − (ω ε) +Γ2 +Ω Γ2Γ1 Here Vab(r) is the interactionP betweenP TLS that is due to − their dipole moments or virtual phonon exchange. In ei- At small fields, one can neglect the last term in the de- ther case, in the static limit, the interaction scales as nominator of Eq.(5). By substituting Eq.(5) into Eq.(4) 3 V (r) 1/r . At zero temperatures and for a small and then averaging over the distribution of TLS, one gets: ∼ value of the parameter λ 1, the Hamiltonian Hint ≪ πm 2 gives a coherent dynamics of the individual TLS. In this P = sin θ λE (6) |ℑ ω| 6 limit, the levels at high frequencies are broadened by phonon emission [9] that leads to the relaxation rate and consequently a Q factor that does not depend on the 3 Γ1 ε . At non-zero temperatures the levels are fur- field strength E. ∝ 2 ther broadened by dephasing caused by thermal phonons At large fields (Ω > Γ1Γ2), the third term in denomi- and other TLS [29, 30]. The combined effect of all nator of Eq.(5) dominates the second and this results in 4 a rapid decrease of the response. By averaging over the exponentially with their number this condition is satisfied distribution of TLS, one gets: even for a moderate number of fluctuators that affect a given TLS. In this situation the dissipation is dominated πm √Γ1Γ2 2 P = sin2 θ λE (7) by TLS with γ > Ξ Ω /Γ2 : |ℑ ω| 6  Ω  ∼ πm 2 γmaxΓ2 which translates into the usual Q √P dependence. Pω = sin θ λP0 ln 2 E (10) ∼ |ℑ | 6  Ω  In the opposite case of large jumps ξ Γ2 one should ≫ solve the full dynamical Eqs.(2,3) with noise ξ(t). The Both Eqs(8,10) lead to the weak, logarithmic depen- 2 details of the solution depend on the relation between the dence of Q at large electric fields (Ω > Γ2Γ1), while at relaxation rates Γ1, Γ2 and the jump rate γ. We discuss smaller fields, the imaginary part of the average polar- first the simplest case of large γ in which a given TLS ization is given by Eq.(6) and Q is constant. spends most of its time away from the resonance, so that As a final remark let us notice that in this model the di- when it moves back into the resonance its magnetization mensionless coupling between TLS remains small, λ 1. z + z ≪ S0 and S1 have their equilibrium values: S0 = m and This implies that thermodynamics properties and the real + S1 = 0. In this case the only quantity that controls the part of the dielectric constant are not affected by the in- response is the rate, γ, of incoming and outgoing jumps teraction. Therefore we expect the same temperature- into the resonance. The stochastic nature of this process dependent resonance frequency shift as predicted by the + implies that, in order to find the average value of S1 conventional theory of TLS in agreement with experi- that determines the response to electric field, we have ments [8, 9]. to solve the equation for the probability, ̺(s), to find In conclusion, we have shown that the resonance ab- the resonant TLS characterized by the three dimensional sorption from TLS is strongly affected by their interac- vector s = (x, y, z), where x and y are the real and the tion with classical fluctuators; this does not change the + z imaginary parts of S1 and z = S0 . This probability absorption at low powers but changes the square root obeys the evolution equation: dependence of the absorption into the logarithmic one ∂̺ d ds at high powers. The important condition is that each + ̺ = γ [δ(z m)δ(x)δ(y) ̺] fluctuator should move the TLS frequency more than its ∂t ds  dt  − − width due to the relaxation rate. This translates into s a higher ( 10 times larger) concentration of TLS than where d /dt is given by Eqs.(2,3) with constant ε. The ∼ equation is further simplified in the limit of large γ & the typical density in amorphous bulk materials.. We in- Ω Γ1, Γ2 where the analytic solution gives (see Supple- terpret the results of recent experiments displaying slow mentary≫ material), after averaging over the distributions power dependence of the quality factor in high-Q super- of ε and γ, (ε,γ)= νP0/γ: conducting CPW resonators as the evidence for a large P concentration of TLS located in the interface oxide sur- πm 2 γmax P = sin θ λP0 ln E (8) faces of the resonator. Very likely it implies that these |ℑ ω| 6 Ω   TLS have a different nature, e.g. localized electron states which results in a weak, logarithmic dependence of the at the superconductor oxide boundary. quality factor Q on the strength of the electric field. Note added: recent experiment directly observed the A similar logarithmic dependence occurs in the cases energy drift of TLS which is the basis of our model [31] when the dephasing rate is larger than the jump rates The research was supported by ARO W911NF-09- as well. In this case, one can neglect the time derivative 1-0395, DARPA HR0011-09-1-0009 and NIRT ECS- + z terms in Eq. (2) and express S1 through S0 and get the 0608842. general evolution equation:

∂̺k ∂ + [(Γ1(z m) Ξ z) ̺ ]= γ ̺ (9) ∂t ∂z − − k k kn n [1] T. Klapwijk, in 100 Years of Superconductivity, edited by where ̺k(z,t) is the probability for a given TLS to have H. Rogalla and P. H. Kes (CRC Press, 2011). z value S0 = z while subjected to the effective driving field [2] P. Day et al., Nature 425, 817 (2003). 2 2 2 Ξk =Ω Γ2/[(ω εk) +Γ2] in the state k of the fluctua- [3] A. Wallraff et al., Nature 431, 162 (2004). − 460 tors. The matrix γkl is the matrix of the transition rates [4] L. DiCarlo et al., et al., Nature , 240 (2009). between the states of the fluctuators. Similarly to the [5] M. Ansmann et al., et al., Nature 461, 504 (2009). 22 case of the small dephasing rate the saturation of TLS [6] M. Steffen et al., J. Phys. Condens. Matter. , 053201 (2010). does not happen if they stay mostly away from the res- [7] J. Martinis et al., Phys. Rev. Lett. 95, 210503 (2005). onance, so that the probability to find a given TLS in [8] P. Anderson et al. Philos. Mag. 25, 1 (1972). resonance n < Γ1/γ (see Supplementary material). Be- [9] J. Black and B. Halperin, Phys. Rev. B 16, 2879 (1977). cause the number of states of classical fluctuators grows [10] T. Lindstrom et al., Phys. Rev. B 80, 132501 (2009). 5

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