Relaxation in quantum dots due to evanescent-wave Johnson noise from a metallic backgate

0.23 MB·English

by  Amrit Poudel

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. However,amore
2πi~ carefultreatmentrevealsthatthis isanerror. The naive
C = r (k,ω,a)eiqz′ (A-3)
s expansions of (A-4) and (A-7) for large a neglect an en-
q
hancement of the field spectrum that occurs for small k.
where In fact, EWJN is enhanced as the thickness is decreased
for anygoodconductor. Specifically, the enhancementis
q2 q2 sin(q a)
r (k,ω,a) − 1 1 preserved for a particular spatial Fourier component of
s ≡ q2+q2 sin(q a)+2iqq cos(q a) the Green’s function as long as 2q1 < 1. EWJN will
1 (cid:0) 1 (cid:1) 1 1 | ǫk |
−1 eventuallyvanish asa 0, but this does notoccur until
=2(cid:0)isinq a(cid:1) eiq1aq−q1 e−iq1aq+q1 an unphysically small →thickness is reached, on the order
1
(cid:18) q+q1 − q−q1(cid:19) of10−14 mforcopperatT =0K whichiswellbelowthe
(A-4)
applicability of the local permittivity model.
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