Relaxation in quantum dots due to evanescent-wave Johnson noise Amrit Poudel, Luke S. Langsjoen, Maxim G. Vavilov, and Robert Joynt Department of Physics, University of Wisconsin-Madison, Wisconsin 53706, USA (Dated: January 25, 2013) We present our study of decoherence in charge (spin) qubits due to evanescent-wave Johnson noise (EWJN) in a laterally coupled double quantum dot (single quantum dot). The high density of evanescent modes in the vicinity of metallic gates causes energy relaxation and a loss of phase coherence of electrons trapped in quantum dots. We derive expressions for the resultant energy 3 relaxation ratesof chargeandspinqubitsinavarietyofdot geometries, andEWJN isshown tobe 1 adominantsourceofdecoherenceforspinqubitsheldatlowmagneticfields. Previousstudiesinthis 0 fieldapproximatedthechargeorspinqubitasapointdipole. Ignoringthefinitesizeofthequantum 2 dot in this way leads to a spurious divergence in the relaxation rate as the qubit approaches the n metal. Ourapproachgoesbeyondthedipoleapproximationandremediesthisunphysicaldivergence a by taking into account the finite size of the quantum dot. Additionally, we derive an enhancement J of EWJN that occurs outside a thin metallic film, relative to the field surrounding a conducting 4 half-space. 2 PACSnumbers: 03.67.-a,03.65.Yz,42.50.Lc,73.21.-b ] l l a I. INTRODUCTION atomic12,13 and quantum dot based qubits.14 Our previ- h ouswork14 aswellasothertheoreticalestimates12 ofthe - s effectofJohnsonnoiseinatomicandquantumdotbased e Semiconducting quantum dots are promising candi- qubits usethe dipole approximation,whichis avalidap- m datesforscalablequantuminformationprocessing1. Sev- proximationif the distance from the metallic gate to the eral experiments performed on laterally coupled double at. quantum dots (DQDs) have demonstrated precise and qubitismuchlargerthanthe sizeofthe qubit. However, it may be necessary to go beyond the dipole approxima- m rapid control of the coupling between electronic charge tion in the case of EWJN in a quantum dot. statesandcoherentmanipulationoftrappedelectrons,2–4 - d leadingtorealizationofaDQDasaqubit. Quantumdots n arerealizedinavarietyofexperimentalsetups,including o aSiandGaAstwo-dimensionalelectrongas,2–4 semicon- c ductor nanowires5 and carbon nanotubes.6 In almost all [ oftheseimplementations,confinementandmanipulation In this work, we present our study of the energy re- 3 of an electron in a quantum dot is achieved by applying laxation of a single electron charge qubit in a DQD sys- v an electrostatic potential through metallic gates. While 3 tem and a single electronspin qubit in a single quantum the metallic gates are crucial for qubit control, they can 8 dot. We assume that the primary source of field fluc- also act as a source of decoherence during qubit opera- 9 tuations are the metallic top gates of the quantum dot 3 tions. architecture. Back gates are typically a distance on the 1. Several decoherence mechanisms, such as hyperfine order of a micron from the qubits, which is too far to 1 coupling of the trapped electron spin to host lattice nu- experience significant EWJN enhancement. We consider 2 clearspinsinspin-basedqubits7 andelectroncouplingto the detailedspatialvariationofthe electromagneticfield 1 phononmodes8–10 incharge-basedqubits,havebeenpre- fluctuations and present results beyond the dipole ap- v: viously studied in an effort to identify the major source proximationwhichtakeintoaccountthefinitesizeofthe i of decoherence in semiconductor qubits. A more recent quantum dot. We show that this extension of the dipole X study investigated decoherence due to voltage fluctua- approximationremoves the unphysical divergence in the ar tionsinthemetallicgatesusingthelumpedcircuitmodel relaxation rate at the metallic surface. This paper is or- of a DQD charge qubit.11 In almost all of these stud- ganized as follows: In Section II we present our formal- ies,8–11 the estimated energy relaxation rate is at least ism for calculating the relaxation rate of a charge qubit. anorderofmagnitudesmallerthantherateobservedex- Results are presented for a DQD geometry. Section III perimentally,2,3 suggesting that a different decoherence presentstheformalismandresultsfortherelaxationrate mechanism is dominant in current experimental setups of a spin QD. In Section IV we derive an enhancement for charge qubits. ofthe noise spectrum that results asthe thickness of the Here we present our study of decoherence in a quan- metallic gate is decreased. Finally, Section V summa- tumdotduetoelectromagneticfieldfluctuationsnearthe rizes our results. Our results indicate that EWJN is the metallicgates. Wefocusprimarilyonnoisefromthehigh dominant cause of energy relaxation in some spin qubit density of evanescent modes in the vicinity of metallic experiments, particularly those performed in a small ex- gates. This evanescent-wave Johnson noise (EWJN) has ternal magnetic field, and is comparable in effect with been identified as an important source of decoherence in previously studied noise sources in charge qubits. 2 II. CHARGE QUBIT following expression, which follows directly from Fermi’s golden rule: We consider a charge qubit realized in a gated lateral 1 DQD in an AlGaAs/GaAs heterostructure where elec- Γ = d3~r d3~r′M∗i(~r)Mj(~r′)χ (~r,~r′,ω). 1,c ~2 r r ij tron confinement along the z direction is much smaller ij Z Z X than in the x or y directions, so that we can safely de- (6) couple the dynamics along x and y directions from the z direction23. The total Hamiltonian of the chargequbit At finite temperature, the emission (transition from ex- anditsinteractionwiththeelectromagneticenvironment cited to ground state) and absorption (transition from is given by ground to excited state) rates are given by H =Hq +Hint (1) Γe1,c =(1+N(ω,T))Γ1,c , Γa1,c =N(ω,T)Γ1,c (7) where Hq is the Hamiltonian of the charge qubit in The Planck function N(ω,T) = 1/[exp(~ω/kBT) − 1] a DQD, which we model in the basis of the localized gives the average occupation number of environment charge states L , R as H = ε/2(L L R R)+ modes with frequency ω at temperature T. The spec- q {| i | i} | ih |−| ih | ∆/2(L R + R L). ε is the bias energy between the tral density of the vector potential χij is related to the two d|otish, a|nd|∆iihs t|he tunneling amplitude. In the en- retarded photon Green’s function Dij by15 ergy eigenbasis this Hamiltonian reduces to ~ω χij(~r,~r′,ω)≡ dτexp−iωτh[Ai(~r,t),Aj(~r′,t+τ)]i Hq = σz (2) Z 2 1 = ImD (~r,~r′,ω) , ij whereσ isthePaulimatrix,and~ω =√ε2+∆2. Forall −ǫ0 z our calculations except those in Fig. 2, we will set ε=0. where i,j are Cartesian indices that run over x,y,z and The interaction Hamiltonian Hint may be expressed in the square brackets denote the commutator. Dij is ob- this same basis as tained by solving ω2ǫ(~r,ω) δ 2+ +∂ ∂ D (~r,~r′,ω) Hint =− d~r σˆxM~r(~r)+σˆzM~φ(~r) ·A~(~r,t) , (3) (cid:20)− ij(cid:18)∇ c2 (cid:19) i j(cid:21) ik Z h i = 4π~ δ3(~r ~r′)δjk. (8) − − where A~(~r,t) is the vector potential of the fluctuating field. M~ and M~ are associated with energy relaxation Here the relative permittivity ǫ(~r,ω) characterizes the r φ geometry of a particular problem. In this section, we andpuredephasinginthe chargequbit,respectivelyand shall limit ourselves to the case where the metallic top are defined as gate of the lateral DQD is approximated by the half- e ie~ space,z <0. Thenwecanderiveananalyticalexpression M~ (~r) ψ∗(~r)p~ψ (~r) ψ∗(~r)ψ (~r) , r ≡mc + − − 2mc + − ∇ for Dij15,16 e M~φ(~r)≡2ime~chψ+∗(~r)p~ψ+(~r)−ψ−∗(~r)~pψ−(~r)i Dij(~r,~r′,ω)=4π12 ei~k·~rkD˜ij(~k,z,z′,ω)d~k, (9) −4mc ψ+∗(~r)ψ+(~r)∇−ψ−∗(~r)ψ−(~r)∇ (4) D˜ (~k,z,z′,ω)=2πi~Zeiq(z+z′) h i xx q Here m is the effective mass and p~ is the momentum op- eratorofthe trappedelectron. Becausewe areoperating q2c2 r (k,ω)cos2θ r (k,ω)sin2θ , within the weak field limit, the term proportional to A~2 ×" s − ω2 p # has been dropped from the interactionHamiltonian. We (10) choose the gauge where the scalar potential φ = 0 so thatE~ = ∂ A~. The expressionforH derivesfroman where r and r are Fresnel’s reflection coefficients given t int s p − interaction in terms of operator quantities of the form by ǫq q q q 1 1 Hint = e A~(~r,t) p~+~p A~(~r,t) . (5) rp(k,ω)= ǫq−+q1 , rs(k,ω)= q+−q1 . −2mc · · (cid:16) (cid:17) Here ~k (k ,k ), ~r (x x′,y y′), θ is the angle Thissymmetrizedversionofthevectorpotentialisnot x y k ≡ ≡ − − strictlynecessaryinourcasesinceourqubitresidesinthe between ~k and the x-axis, and q = ω2/c2 k2 and − vacuum where A~ =0, but we included it to keep our q = ǫω2/c2 k2 are the z-components of the photon ∇· 1 − p results more generally applicable. The zero temperature wavevector in the vacuum and the metal, respectively. relaxationrate Γ =1/T can be calculated using the All otpher components of D˜ can be derived from D˜ .15 1,c 1,c ij xx 3 In this work we consider ǫ iσ/ωǫ , where σ = 6 107 ≈ 0 × S/m is the conductivity of the copper gate. l = 30 nm 105 Wepausebrieflytomentionthattypicalmodelsofthe interaction of a DQD with the electromagnetic field use 104 l = 60 nm the dipole interaction Hamiltonian H = E~(~r) d~, (11) s) 103 2 l int n − · which will result in a relaxation rate of (1 102 T d d2ω 101 Γ = (12) 1,c 4~z3σ 100 in the quasistatic approximation, where d~ is the dipole moment of the qubit and E~(~r) is the strength of the 10−1 10 30 50 70 90 110 130 150 170 190 fluctuating electric field evaluated at the location of the z (nm) qubit. This expression approximates that the electric fieldisuniformoverthespatialextentofthequbit,which is equivalent to treating the qubit as a point dipole. As FIG. 1: Energy relaxation time T1 vs. the distance from the such,thequbitisabletocoupletoarbitrarilysmallwave- metallic gate to the DQDs z for dot geometry of d = 30 nm lengths of the electromagnetic spectrum, and the relax- and l = 30 nm (dashed and dash-dotted blue lines) and 60 ationrateisseentodivergeatshorterdistancesas 1/z3 nm (solid and dashed black lines) at 0 K, ω/2π = 1GHz, if the conductor is modeled with a local dielectric∼func- and ε = 0. Solid and dash-dotted lines are T1 times for the exact form of the interaction Hamiltonian, whereas dashed tion12,14. Usingthecompleteelectromagneticinteraction lines are for the dipole form of the interaction. The inset in Hamiltonian(5)accountsforfluctuationsofthefieldover the figure displays confining potential of a typical DQD in a the spatial extent of the qubit. If the wavelength of a one dimensional nanowire and the corresponding symmetric particular Fourier component of the field fluctuations is ground state and antisymmetric first excited state. smallerthanthe lengthofthe qubit inthatdirection,its influence on the electron will average out and it will not contribute to qubit relaxation. The exactand dipole ap- 102 proximation forms of the interaction Hamiltonian, Eqs. (5) and (11), converge when the distance from the gate becomes larger than the spatial extent of the qubit. ) We present calculations of the relaxation time for 0 ( chargequbitsthathighlightthedifferencesbetweenthese 1 z = 1 nm T twoformsoftheinteraction. FirstweconsiderDQDsina / 101 z = 10 nm one dimensional nanowire, which are realizable in semi- ) z = 50 nm ε conducting nanowires6 or carbon nanotubes5. In such (1 T a geometry, the wave functions of trapped electrons in quantumdotshaveappreciablespatialextentinonlyone direction. We model the confining potential of the DQD as a symmetric double square well potential and com- 100 0 10 20 30 40 50 putethelowesttwoeigenenergiesandwavefunctions. We ε/∆ thencomputethe relaxationratebetweenthesetwolow- est states which are separated by a fixed transition fre- quency ω/2π=1GHz. A plotof the wavefunctions and FIG. 2: Ratio of the T1 time for finite bias ε to the T1 time the shape of the potentialis shownin the inset ofFig. 1. at zero bias vs. ε/∆ for three values of distances z from the WeplottheenergyrelaxationtimeT1vs.thedistancez metallicgate: 1nm(solid blueline),10nm(dashedredline) fromthemetallicgatetotheDQDinFig.1. Inthisplot, and 50 nm (dash-dotted black line), for a dot geometry of we choose the size of the dot in the x-direction d = 30 d=30 nm and l=30 nm at 0 K temperature. nm and half the separation between the dots l = 30 nm (dash-dotted and dashed blue lines) and 60 nm (solid and dashed black lines). The curves that are shown in In Fig. 2, we presentthe ratio of T1 for a charge DQD solid and dash-dotted lines are relaxation times for the qubitatbiasǫtotheT1obtainedatǫ=0versustheratio exactformoftheinteractionHamiltonian,whereasthose ǫ/∆. An increaseinbiasincreasesthe levelsplitting and shownindashedlines areobtainedusingthe dipole form decreasesthedipolemomentoftheDQD.Sincetherelax- of the interaction. The curves show significant deviation ation time T1 1/ωd2, where d = 2lsin(arctan(∆/ε)) ∼ of the exact relaxation rate from the dipole relaxation is the dipole moment of the quantum dot, T1 increases rate at shorter distances and convergence of the two re- for larger bias. sults at longer distances. Next, we present results from the relaxation rate cal- 4 quantum dot. Here the system H and the interaction 104 s l = 30 nm Hint Hamiltonians are given by 103 H = gµ ~σ B~ /2 (13) s B 0 − · 102 H = gµ ~σ B~(~r,t) (14) l = 60 nm int B ) − · s (n 101 where ~σ is the vector of Pauli matrices, g is the gyro- T1 metric factor of the trapped electron in a quantum dot, 100 µ is the Bohr magneton, B~ is the externally applied B 0 magnetic field andB~(~r,t) is the fluctuating EWJN field. 10−1 The rate of spin flip from excited to ground at | ↑i | ↓i T =0K can be obtained from Fermi’s golden rule 10−2 10 30 50 70 90 110 130 150 170 190 1 z (nm) Γ = d3~r d3~r′M (~r)M (~r′) 1,s ~2 r,s r,s Z Z ǫ ǫ χB (~r,~r′,ω)n n × ijk ij′k′ kk′ j j′ FIG. 3: Energy relaxation time T1 vs. the distance from the (15) metallic gate to the DQDs z for dot geometry of d=f =30 nmandl=30nm(dashedanddash-dottedbluelines)and60 where repeated indices are summed over, and n are the nm(solidanddashedblacklines)at0K,ω/2π=1GHz,and j componentsofaunitvectornˆ inthedirectionofB~ . The ε=0. Solid and dash-dotted lines are T1 times for theexact 0 formoftheinteractionHamiltonian,whereasdashedlinesare effect of finite temperature on the transition rates is the for the dipole form of theinteraction. same as for charge qubits, as shown in Eq. (7). The magnetic spectral density χB and the matrix element ij M (~r) are r,s culation for a DQD in a two-dimensional quantum well. ε ε In this treatment we label the z-axis as the vertical con- χB(~r,~r′,ω)= ikm jnp∂ ∂ Im D (~r,~r′,ω) (16) ij ǫ c2 k n mp finement direction and do not consider excitations along 0 the z-direction. We model the confining potential by a Mr,s(~r) gµB ψ0(~r)2. (17) ≡ | | symmetric double rectangular well in 2D and numeri- Herethespinqubitfrequencyω =gµ B /~andψ (~r)is cally compute the lowest two eigenenergies and wave- B 0 0 the spatial part of the the groundstate wave function of functions. We then compute the electron relaxationrate the spin qubit. Equation (15) is a generalizationbeyond betweenthese two loweststates. The resultsarequalita- the dipole approximationof the simpler expression14: tively similar to the one-dimensionalcase and are shown in Fig. 3, where we plot the energy relaxation time T 1 g3µ3σB vs. the distance z from the metallic gate to the DQD. Γ = B 0, (18) 1,s 8~2ǫ c4z In this plot, we choose the size of the dot in the x- 0 direction d = 30 nm, the size in the y-direction f = 30 which has been obtained by using the quasistatic limit nm and half the separation between the dots l to be 30 fortheGreen’sfunction(16),andassumingitisconstant nm(dashedanddash-dottedbluelines)and60nm(solid over the spatial extent of the qubit. Equation (18) also and dashed black lines). We find that for l = 30 nm assumes the external magnetic field B~ points in the z- and z = 90 nm, T is 4 µs while for l = 60 nm, T 0 1 1 direction. is 1.6 µs. These relaxation times are somewhat longer A plot of energy relaxation time T vs. the distance than the experimentally reported value of T = 20 ns 1 1 from the metallic gate z for a spin qubit is displayed in in DQD-based charge qubits.3 We note that the relax- Fig. 4. Here we consider a single quantum dot of diam- ation rate for a two-dimensional DQD is shorter than eter d = 60 nm and approximate the ground state spa- for a one-dimensional DQD of comparable geometry by tial wave function of the spin qubit by the ground state about a factor of 5. A two-dimensional DQD is able to wave function of a harmonic potential. We assume the couple to obliquely oriented wavevectors in addition to Zeeman splitting between spin states is 50GHz, typical those which point in the direction of separationbetween of experiments in spin qubits.17 The solid line is the T thedots,andthiscanbereasonablyexpectedtoenhance 1 timeobtainedusingthenon-localmagneticspectralden- relaxation by a geometric factor of order unity. sity while the dotted line is obtained for a local spectral density, which diverges as 1/z as one approaches the ∼ metallicgate. ThereasonforsaturationoftheT timeat 1 III. SPIN QUBIT smaller distances is similar to the case for charge qubits. Thereisaslightdistinctioninthatthespincaseinvolves We now focus onthe calculationofthe relaxationrate a spatially extended dipole interaction, as opposed to for a single electron in a spin qubit realized in a single the charge case which involves qenuine quadrupole and 5 far considered the simpler top gate geometry of a con- 10−1 ducting half-space rather than the thin layer of finger gates used in these experiments. In the next section we d = 60 nm address modifications to our calculations that we expect from more realistic gate geometry. 10−2 ) s ( 1 IV. THIN METALLIC GATES T 10−3 A conducting half-space is an analytically convenient gategeometry,buta poorapproximationto the thintop gates commonly used in semiconductor devices. In this 10−4 section we present an exact treatment of the behavior of 0 50 100 150 200 EWJNinthevicinityofametallicfilmoffinitethickness. z (nm) Changing the half-space to a thin film affects EWJN by modifyingthereflectioncoefficientsr andr . Thepower s p spectrumoftheresultantEWJNisobtainedbysubstitut- FIG. 4: Energy relaxation time T1 vs. the distance from the metallic gate to quantum dot z for spin qubit single dot ge- ing these modified reflection coefficients into the photon ometry of d=60 nm at 0 K temperature and gµBB0/2π~= Green’s function (10), and the relaxation time of, e.g. a 50GHz. The solid line represent T1 time for the exact form charge qubit, is obtained by plugging Eq. (10) into Eqs. oftheinteractionHamiltonian,whereasthedashedlineisfor (6)and(9). Themodifiedreflectioncoefficientsforafilm thedipole form of theinteraction. of thickness a take the form ǫ2q2 q2 r (k,ω,a)= − 1 (19) highermultipolecontributions. Thisdistinctionislargely p q12+ǫ2q2+2iqq1ǫcot(q1a) technical,however,andasaturationofT1 asz →0isob- r (k,ω,a)= q2−q12 . (20) servedin both cases. We find thatthe T1 time for a spin s q2+q2+2iqq cot(q a) 1 1 1 qubit inaGaAs quantumdotwith anexternalmagnetic fieldof10T andz =90nmis150mswhichislargerthan They differ significantly from the half-space result only the experimentally reportedvalue17 of0.55ms,andgen- when the thickness a is of the order or smaller than the erallyEWJNdoesnotseemtobethedominantsourceof skindepthδ, andthey reduceto the half-spaceresultfor decoherence for semiconductor devices in large magnetic a δ. A derivation of Eqs. (19) and (20) is given in fields. GaAs has a strong spin-orbit interaction (SOI), the≫Appendix. Equations (19) and (20) are exact, but which mixes the Zeeman-split spin states with orbitally for a good conductor they can be cast into a simpler excited states. Spin relaxation can then occur via cou- approximate form pling of the qubit to piezoelectric phonon noise in the 2DEG layer. The relaxation rate from this mechanism 2q −1 1 scales as B5 and is the dominant pathway for spin re- rp(k,ω,a) 1+ cot(q1a) (21) ≈ ǫk laxation at large external magnetic fields B > 1T19,20. (cid:18) (cid:19) Additionally, Marquardtand Abalmassov21 calculate re- 2c2q1k −1 r (k,ω,a) 1 cot(q a) . (22) laxation of spin qubits from electric EWJN via the SOI. s ≈− − ǫω2 1 (cid:18) (cid:19) Again, mixing of the charge and spin states via the SOI allows spin relaxation to be induced from electric field These expressions have been obtained by expanding fluctuations. They estimate the power spectrum of the Eqs. (19) and (20) for large imaginary ǫ and then tak- Johnsonnoise using a lumped circuit model and found a ing the quasistatic approximation q ik. The first ap- B3 dependence ofthe relaxationrate. Our treatmentin- proximation is extremely accurate fo→r copper near zero volves a direct coupling of the fluctuating magnetic field temperature and the second is accurate for all distances from the top gates with the spin states, and our rate z suchthat EWJN is appreciablyenhanced aboveblack- scales linearly with the magnetic field. We therefore ex- body radiation14. The remarkable feature of Eqs. (21) pectourrelaxationpathwaytodominateatlowmagnetic and (22) is that they show the strength of the fluctu- fields, and indeed while we predict a much slower relax- ating fields outside the film are actually amplified rela- ationratethanmeasuredbyAmashaetal19 forB 7T, tive to the half-space result. This can be understood by ∼ atB =1T ourresultspredictT 1.5swhichiscompa- analogy to the behavior of a particle trapped in a finite 1 ∼ rable to their measured value of T = 1 s. Additionally, one-dimensional potential well. For a given width of the 1 in Si quantum dots with a 2T external magnetic field well, the wavefunction will have an exponentially decay- and z = 50 nm, we predict a T time of 6 ms which ing tail in the forbidden region. As the confinement is 1 is smaller than the experimentally reported value of 40 increased,theparticlewillbesqueezedanditswavefunc- ms.18 However,it must be kept in mind that we have so tionwillleakfartherinto theforbiddenregion. Itwillbe 6 1 V. DISCUSSION AND CONCLUSIONS In conclusion, we have presented a detailed study of 0.8 ce the effect of evanescent-wave Johnson noise on energy a p relaxation of quantum dots beyond the dipole approxi- s − 0.6 mation. We have noted that previous studies of charge alf and spin qubits which use the dipole approximation al- h1 T / 0.4 low contribution from infinitely large components of the m photon wavevector leading to overestimation and diver- filT1 gence of the energy relaxation rate as z 0. We have 0.2 → demonstrated that it is possible to remedy this spurious divergence by taking into account the finite size of the 0 quantum dot. While a non-local permittivity of the sur- 50 100 150 200 facemetalwillremovethedivergenceinthefieldfluctua- a (nm) tions atthe boundary,wehaveshownthat the finite size of the dot provides an alternative normalization mecha- nism by enforcing a finite cutoff in the magnitude of the FIG. 5: Ratio of energy relaxation time T1 from conducting film to T1 time from half-space vs. thickness of the film a contributing wavevector. In addition, we have derived a for a DQD charge qubit in onedimension with dot geometry novel enhancement of the EWJN field fluctuations that d= 30 nm and l = 60 nm at 0 K temperature. We take the occursoutsideametallicfilm,relativetothefieldoutside exact form of the interaction Hamiltonian. The distance z a metallic half-space. from the film or half-space is chosen as follows: z = 10 nm Thismanuscripthasfocusedexclusivelyonrelaxation, (black dash-dotted line), z = 50 nm (blue dashed line) and though we expect dephasing times from EWJN to be of z = 150 nm (solid red line). Other parameters are the same comparablemagnitude. ThepowerspectrumofEWJNis as in Fig. 1. linear in ω, and this will suppress contribution from the small frequency part of the electromagnetic spectrum, which typically enhances dephasing rates. While the temperature dependence of the relaxation rate is simply given by the Planck function, we do expect a more non- interestingtoseeifthisenhancementisobservableinthe trivial temperature dependence of the dephasing rate. Casimir attraction between 2 thin conducting plates. Of particular interest are experimental signatures of EWJN-inducedrelaxation. Notably,atzerotemperature Using the modified expression for the reflection coeffi- the charge relaxation rate scales linearly with the qubit cients,wecomputetheT timeofaDQDchargequbitin 1 transitionfrequencyandastheinversecubicpowerofthe onedimensionduetothemetallicfilm. InFig.5,weplot distance between the qubit and the metallic top gates. the ratio ofthe T time obtained for the film to the time 1 The zero temperature spinrelaxationrate scaleslinearly computed for the metallic half-space as a function of the with the external magnetic field and inversely with the film thickness. We takethe exactformofthe interaction distance to the gates. Hamiltonianforavarietyofdistancesfromthe gate. We Our results indicate that EWJNfrom the metallic top find that for distance z > a, the relaxation time due to gate is not a dominant source of relaxation in charge the film can be reduced by over an order of magnitude qubits, but can be the dominant noise source for energy relative to the half-space. It converges to the half-space relaxation in spin qubits held at low external magnetic resultaszbecomessmallerthanthethicknessofthefilm. field. Common semiconductor qubit architectures employ thin finger-shaped top gates which are more sparse than Acknowledgments the films considered here. An exact treatment of EWJN from a detailed finger gate geometry would be pro- hibitively difficult, but we expect to a reasonable ap- We thank M.A. Eriksson for useful discussions. This proximation that EWJN from finger gates will be re- work was supported by ARO and LPS grant no. duced by a factor of the fraction of the top gate layer W911NF-11-1-0030and NSF grant DMR 0955500. that is not composed of metal. Our results should then overestimate the relaxation rate by a geometric factor. APPENDIX We note however that newer accumulation-mode archi- tecturesemployasecondtopgateabovetheconfinement top gates22. These accumulation gates are solid sheets Derivation of Green’s tensor for a thin film and are typically around 100 nm from the qubit, so our treatment should accurately describe their contribution Here we present the calculation for the retarded pho- to relaxation. ton Green’s tensor outside of a thin conducting sheet of 7 permittivity ǫ. TheGreen’s functionwillsatisfyEq. (8). so that Here ~r′ is simply a parameter for the purposes of solv- ing this set of equations, and we take it to lie in the vacuum outside the conducting sheet. We will suppress 2πi~ the dependence of Dik(~r,~r′,ω) on ~r′ and ω to simplify D˜yy,kx(k,z)= q rs(k,ω,a)eiq(z+z′)+eiq|z−z′| the notation. The geometry of the problem is contained (cid:16) (A(cid:17)-5) entirely in the permittivity function ε(~r,ω). We take Thetermproportionaltoexp(iq z z′)isthefreephoton the boundaries of the conducting sheet to be located at | − | contributiontothepowerspectrum. Itwillhaveanimag- z = a and z = 0, with vacuum outside. Because the − inary component and thus contribute to relaxation only geometry is still translationally invariant in the x and y in the radiative regime, k ω/c. Within a skin depth directions, we employ the same Fourier expansion (9) as ≤ of separation from the metal, evanescent waves are or- in Section II. Solving Eq. (8) for a problem with planar ders ofmagnitude largerin field strengththan these free symmetry is greatly simplified by separately considering photons. They may be safely ignored in this context. A theFouriercomponentsof(9)thatarepolarizedinthex similar calculation yields the result for D˜ : and y directions. D˜ (~r) may then be reconstructed as yy,ky yy Dyy(~r)=Z (2dπ2~k)2ei~k·~rk D˜yy,ky(k,z)=−2πiω~2c2q rp(k,ω,a)eiq(z+z′)−eiq|z−z′| , × D˜yy,kx(k,z)cos2θ+D˜yy,ky(k,z)sin2θ (cid:16) (A-6(cid:17)) (cid:16) (A(cid:17)-1) where where D˜ = D˜ when k = 0, and D˜ = yy,kx yy y yy,ky DD˜˜yyyy,kwx(hke,nz)ktxhe=n b0e.coTmhees boundary value problem for rp(k,ω,a)≡ q12+ǫ2q(cid:0)2ǫ2sqi2n−(q1qa12)(cid:1)+sin2(iqq1qa1)ǫcos(q1a) D˜yy,kx(k,z)= CBe1iqez−+iq1z2π+~eBAiq2|eze−−iqizq1′zz| ,,, zz−<a≤−0az <0 =2(cid:0)isinq1a(cid:18)(cid:1)eiq1aǫǫqq+−qq11 −e−iq1aǫǫqq+−qq11(A(cid:19)-−71)  iq ≥ (A-2)  Ourinterestliesin the behaviorofthe fieldsfor z >0, A Taylor expansion of Eqs. (A-4) and (A-7) for large a soweneedonlytocalculateC. EnforcingthatD and yy,kx in the evanescent range of wavevectors,i.e., a Taylor ex- ∂D /∂z are continuous across the boundaries results yy,kx pansioninpowersofexp( 2q a),givesamonotonically in − | 1| increasingfunctionoffilmthickness, a. 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