Quantum Photovoltaic Effect in Double Quantum Dots

0.21 MB·English

by  Canran Xu

Quantum Photovoltaic Effect in Double Quantum Dots
Canran Xu and Maxim G. Vavilov
Department of Physics, University of Wisconsin-Madison, Wisconsin 53706, USA
(Dated: January 16, 2013)
We analyze the photovoltaic current through a double quantum dot system coupled to a high-
qualitydrivenmicrowaveresonator. Theconversionofphotonsintheresonatortoelectronicexcita-
tions producesa current flow even at zero bias across theleads of thedoublequantum dot system.
We demonstrate that due to the quantum nature of the electromagnetic field in the resonator, the
3
photovoltaiccurrentexhibitsadoublepeakdependenceonthefrequencyωofanexternalmicrowave
1
source. Thedistancebetweenthepeaksisdeterminedbythestrengthofinteractionbetweenphotons
0
in the resonator and electrons in the double quantum dot. The double peak structure disappears
2
as strengths of relaxation processes increases, recovering a simple classical condition for maximal

n current when themicrowave frequency is equal to theresonator frequency.
a
J PACSnumbers: 73.23.-b,73.63.Kv,42.50.Hz,73.50.Pz

6
1
I. INTRODUCTION (a) µw resonator (c) Γe,L e Γe,R
]
l µw
l
a Theinteractionofelectronsinconductorswithelectro-
h magnetic fields has long been considered within a classi-
- calpictureofelectromagnetic(EM)radiation. Awidely– ε ℏΩ
s
e known example is the photon assisted tunneling (PAT) (b) ΓL,g ΓR,g
m in double quantum dot (DQD) systems,1 when the EM
S L R D
. fieldbringsanelectrontrappedatthegroundstatetoan g
t
a excited state and facilitates electron transfer. This clas-
BL PL BM PR BR
m sical description of the EM field breaks in high-quality
- microwave resonators based on superconducting trans- FIG. 1: (Color online) (a) An illustration of a DQD and a
d missionlinegeometry.2InteractionofsuchEMfieldswith
transmission lineresonatorcoupledtoanexternalmicrowave
n
electronicdevicesrequireaquantumtreatmentknownas source µw. (b) A schematic view of the DQD. Electrons are
o
the circuit quantum electrodynamics (cQED).3,4 confined to the left (L) and right (R) dots by barrier gates
c
[ Recently, severalexperimentalgroups studied systems BL, BM, and BR that also control electron tunneling rates
consisting of a superconducting high quality resonator betweenthesource,S,andtheleftdot,theleftandrightdots,
1 andaDQD5–9oravoltagebiasedCooperpairbox.10The andtherightdotandthedrain,D,respectively. Electrostatic
v
energiesoftwoquantumdotsaredefinedbytheplungergates,
coupling strength between a resonatorphoton mode and
8
PLandPR,andthePLgateisalsoconnectedtoanantinode
8 electronstates in a DQD is characterizedby the vacuum
oftheresonator,seee.g. Refs.7,9. (c)Electronicstatesofthe
7 Rabi frequency g with reported values in the range of
DQD are presented in both the eigenstate basis (solid lines)
3 g/2π 108Hz. These systems call for re-examination of
and the left–right basis (dashed lines). Tunneling from the
1. thePA∼Tbytakingintoaccountaquantumdescriptionof excitedstate, |ei,totheleft/right lead, with rateΓ and
e,L/R
0 theEMfieldintermsofphotonexcitations. Onemayex- from the left/right lead to the ground state, |gi, with rate
3 pectatleasttwoimportantdistinctionsfromtheclassical Γ are illustrated by arrows.
L/R,g
1 treatment: (1)theLambshiftthatrenormalizesquantum
:
v states of electrons and photons; (2) spontaneous photon
i emission that breaks symmetry between absorption and and reflects the Lamb shift of electronic energy states.
X
emission processes and is important in systems with ei- We also demonstrate that the interaction-induced split-
r ther a finite voltage bias between the leads11–13 or an ting is sensitive to the energy and phase relaxationrates
a
inhomogeneous temperature distribution.14 in the DQD.
In this paper we study the photovoltaic current Wenotethatthephotovoltaiceffectdiscussedhereisa
throughaDQDcoupledtoahigh-qualitymicrowaveres- common phenomenon when the current in an electronic
onator at zero bias across the DQD. The resonator is circuit is generated by out-of-equilibrium EM environ-
driven by external microwave source that populates a ment. Examples of this phenomenon include the current
photon mode of the resonator, see Fig. 1. The pho- response of a DQD in the vicinity of the biased quan-
tons excite electrons in the DQD and produce electric tum point contact17 or another circuit element out of
current even at zero bias, similar to the classical PAT equilibrium18 with the electronic system. However, be-
case.1,15,16 Weshowthatduetothecouplingofelectrons causeout-of-equilibriumphotonsoftheenvironmenthave
and photons, the current as a function of the source fre- abroadspectrum,thegeneratedcurrentdoesnotexhibit
quency has a multiple peak structure with splitting be- a resonantdependence onparametersofthe systemthat
tween the peaks determined by the coupling strength g weobserveinasystemofa singlemode highqualityres-
2
onator and a DQD. utilize the rotating frame approximation to obtain11,13
H 1 ∂ † Ω ω
= †H˜ i U = − σ (5)
II. MODEL ~ ~U U − ∂t U 2 z
+(ω ω)a†a+g(aσ++a†σ−)+F(a†+a),
0
−
We consider a DQD system with each dot connected
whereg =g sinθcharacterizestheactualstrengthofthe
0
to its individual electron reservoir at zero temperature
coupling between the microwave field and DQD states
and at zero bias between the reservoirs, see Fig. 1(b,c).
responsible for photon absorption or emission, the Pauli
The gate voltages of the DQD are adjusted near a triple matrices σ = e e g g , σ− = g e and σ+ =
point of its stability diagram.1 To be specific, we choose z | ih |− | ih | | ih |
e g are defined in terms of eigenstates of the electron
a triple degeneracy point between (N ,N ), (N +1,N ) | ih |
l r l r Hamiltonian, Eq. (1).
and (N ,N +1) electron states in the DQD and denote
l r We analyze the behavior of the system with Hamil-
these states as 0 , L and R , respectively. We model
tonian Eq. (5) in the presence of relaxation in electron
the system by t|heiH|amiiltoni|aniH˜ =H +H +H ,
DQD r int and photon degrees of freedom by employing the Born-
where H describes states with an extra electron in
DQD Markov master equation for the full density matrix
the left or right dot, L or R :
| i | i i
ρ˙ = ρ= [H,ρ]+ ρ. (6)
1 Ltot −~ Ldiss
H = ετ + τ , (1)
DQD z x
2 T
The first term on the r.h.s. of Eq.(6) describes the uni-
with ε being the electrostatic energy difference between tary evolution of the system and the second term ac-
the two states, and being the tunnel matrix element counts for the dissipative processes in the resonator and
of an electron betweeTn the dots. The Pauli matrices are DQD systems17
defined in the subspace of states L and R as τ = γ
| i | i x ρ κ (a)ρ+γ (σ−)ρ+ φ (σ )ρ
|RihL|+|LihR| and τz =|RihR|−|LihL|. A resonator Ldiss ≡ D D 2 D z (7)
driven by an external microwave source is described by +(Γ +Γ ) (c†)ρ+(Γ +Γ ) (c )ρ,
the Hamiltonian L,g R,g D g e,L e,R D e
where (x)ρ = 2xρx† x†xρ ρx†x /2 is the Lind-
Hr =~ω0a†a+2~F(a†+a)cosωt (2) blad suDperoperato(cid:0)r. The−relaxa−tion of(cid:1)the photon field
intheresonatorwithrateκisrepresentedbyκ (a)ρand
with a (a†) denoting the annihilation (creation) opera- D
the electron relaxation from the excited state e to the
tors for microwave photons in the resonator, ~F being ground state g with rate γ is represented by γ| i(σ−)ρ.
the amplitude of the external drive of the resonator and | i D
The last two Lindblad superoperators account for the
ω (ω) being frequency of the resonator (source). The
0 loading of the ground state g and unloading of the ex-
interaction between the microwave field and the DQD | i
citedstate e ofthedoublequantumdotviaelectrontun-
system is represented by11 neling in te|rmi s of operators c = 0 e and c† = g 0,
e | ih | g | ih |
respectively. The tunneling rates Γ = Γ cos2(θ/2),
H =~g (a†+a)τ . (3) L,g l
int 0 z Γ = Γ sin2(θ/2), Γ = Γ sin2(θ/2) and Γ =
R,g r e,L l e,R
Γ cos2(θ/2) are written in terms of tunneling rates Γ
This interaction describes the shift of energy difference r l/r
in the basis of L and R states.
betweenstates R and L duetotheelectricpotentialof | i | i
| i | i Note that in Eq.(6), the dynamics of state 0 only
the plunger gatesdefined by the microwavephotonfield. | i
appears via the tunneling terms involving D(c )ρ and
We assume that the photon field is distributed between e
(c†)ρ. These terms can be categorized by whether the
the left and right plunger gates, see Fig. 1(b) and does D g
empty state is loadedfrom the left or rightlead with co-
not influence the source and drain voltage to avoid the
rectification effects.19–21 efficientsdepending onprojectionofthe eigenstatesonto
theleft/rightstates,asshowninFig. 1. Inthispicture,17
Further calculations are more convenient in the basis
the photovoltaic current is given by
of the ground, g , and excited, e states of the Hamil-
| i | i
tonian, Eq.(1): θ θ
I =eΓ cos2 e ρ¯ e sin2 0 ρ¯ 0 . (8)
r
(cid:18) 2h | | i− 2h | | i(cid:19)
e =cos(θ/2) L +sin(θ/2) R ,
| i | i | i
g = sin(θ/2) L +cos(θ/2) R . (4) in terms of the reduced density matrix ρ¯ = Trph ρ ,
| i − | i | i where we traced out photon degrees of freedom of{th}e
Hereθ =arctan(2 /ε)characterizesthehybridizationof resonator. We alsoanalyzethe numberofphotonsinthe
the L or R staTtes. The energy splitting between the resonator,
| i | i
eigenstates~Ω=√ε2+4T2 canbetunedindependently N¯ =Tr a†aρ , (9)
byvaryingεand viadcgatevoltages. Wefurtherelim-
T (cid:8) (cid:9)
inate the time-dependence in Hamiltonian Eq.(2) by ap- where we trace out both photon and electron degrees of
plying unitary operator =exp( iωt(a†a+σ /2)) and freedom.
z
U −
3
4 0.2 γ =0
φ
ber3 γφ/2π=10MHz
Num ber γφ/2π=20MHz
2 m
n u
hoto1 n N0.1
P o
ot
h
0 P
ber1.5
1.5 m
u 1
N
1 on 0.5 0
pA 0.5 Phot 0 −8 −ε7/h.8, −G7H.6z−7.4−7.2 1.5
nt, A
e 0 p
urr nt,
C e 1
dc −0.5 γφ=0 Curr
−1 γ/2π=10MHz c
φ d0.5
−1.5 γφ/2π=20MHz
−2 0
−12 −8 −4 0 4 8 12 −0.2 −0.1 0 0.1 0.2
ε/h, GHz
(ω−ω )/2π, GHz
0
FIG. 2: (Color online) The average number of photons in
the resonator and the photovoltaic current as functions of FIG. 3: (Color online) The average number of photons in
level bias ε for T/2π = 1 GHz, F = 50 µs−1 and ω0/2π = the resonator and the photovoltaic current as a function of
8 GHz. Thecurrentis generated neartheresonant condition the frequency ω of the microwave drive for T/2π = 1 GHz,
when ε = ± ~2ω2−4T2 (vertical lines). The three curves F = 50 µs−1 and Ω = ω0 = 2π×8 GHz, g/2π = 48.5 MHz.
0
represent diffperent dephasingrates γφ theDQD. Forγφ =0,bothaveragenumberofphotonsN¯ andthephoto-
voltaiccurrentshowlocalminimaatω=ω0andlocalmaxima
near ω =(E1,±−E0)/~, shown by vertical lines. As the de-
phasingrateγ increases, thedoublepeaksmergetoasingle
φ
III. RESULTS
peak at ω=ω0.
The average number of photons in the resonator, N¯,
andthedccomponentofphotocurrentcanbefoundusing on the separation between energy levels in the double
thesteadystatesolutionofthemasterequation,(6),with quantum dot, controlled by the electrostatic energy dif-
ρ˙ =0. Wenumericallyfindthefulldensitymatrixρfora ference ε. We take frequency ω of microwave source to
double quantumdot andphotonfield of the resonatorin be equal to the resonator frequency, ω =ω , and fix the
0
the Fock’s space using Quantum Optics Toolbox22 and driveamplitudeF =50 µs−1. Dependenceoftheaverage
QuTiP23, both of which provide consistent results. The numberofphotonsintheresonatorandthephotocurrent
steady state solutionfor the density matrixρ defines the on energy ε is presented in Fig. 2 for three values of the
average number of photons N¯, Eq. (9), and the pho- dephasing rate γ /2π = 0, 10, 20MHz. As the energy
φ
tocurrent, Eq.(8). difference between the excited and ground states of the
Our choice of parameters is motivated by Ref. [7]. We quantumdotgoesthroughthe resonanceΩ=ω ,we ob-
0
choose the relaxation rate γ = 2π 25 MHz, the res- serve a significant suppression of the photon number in
×
onator relaxation rate κ/2π = 8 MHz, tunneling ampli- the resonator, see the top panel and the inset in Fig. 2.
tude between the individual dots /2π = 1 GHz, the This is expected behavior because the DQD system en-
T
atunndntehliengrersaotneaftroormfraeqduoetntcoyaωl0e/a2dπΓ=l/r8=G2Hπz.×3W0eMnHotze, hAabnscoersbepdhopthoontoanbssocarputsieontrainnstithieonrsesboentawteoernattheΩg≃rouωn0d.
thattokeepthecouplingconstantfinite,wehavetotake and excited electronic states. These electrons tunnel to
~Ω, since g = g0sinθ, Eq.(5), vanishes for = 0. the leads andgenerateelectric currentthough the DQD.
T ∼ T
Below we fix g0/2π = 200MHz. This current is shown in the lower panel of Fig. 2 and is
First, we investigate dependence of the photocurrent peaked at ε = ~2ω2 4 2 or ε/(2π~) 7.75GHz,
± 0− T ≃ ±
p
4
second effect and dephasing increases the magnitude of
1 photocurrent for the case of fixed ω =ω .
γ =0 0
φ Next, we consider the case when the frequency of the
γ /2π=10MHz
0.8 φ microwave source, ω, is varied while the energy splitting
er ~ΩoftheDQDandtheresonatorfrequencyω0 arefixed.
b
m0.6 Themicrowaveradiationismostlyreflectedwhenits fre-
u
N quency does not match the difference between energies
on 0.4 En,± of the resonator and DQD system defined by the
ot Jaynes-Cummings spectrum:
h
P
0.2 ~ ~∆
E =n~ω 4g2n+∆2, E = , (10)
n,± 0 0
± 2 2
0 p
where∆=ω Ωisthe detuningbetweentheDQDand
0
−
the resonator. We demonstrate that for DQD with weak
3
energy and phase relaxations, this resonant admittance
A
p of the microwave source to the resonator results in the
nt, peak structure of the average photon number and the
e 2
rr photocurrent.
u
C In Fig. 3, we plot the average number of photons in
c
d 1 the resonator and the photocurrent as a function of the
drive frequency ω for ω =Ω and for the choice of other
0
systemparametersidenticaltothoseforcurvesinFig.2.
0 Atvanishingdephasingrate,γ =0,weobserveadouble
−0.2 −0.1 0 0.1 0.2 φ
(ω−ω )/2π, GHz peak feature in both photon number and photocurrent
0 curves, see Fig. 3. These peaks at ω = (E E )/~
± 1,± 0
−
aredefinedby the levelspacing ofthe Jaynes-Cummings
Hamiltonian and are shown by vertical dashed lines in
Fig. 3. The two peaks merge at ω = ω as the dephas-
0
ing rate increases and destroys quantum entanglement
FIG.4: (Coloronline)Theaveragenumberofphotonsinthe
between photons and DQD states.
resonator and the photovoltaic current as a function of the
frequency ω of the microwave drive for detuned DQD and AtfinitedetuningbetweentheresonatorandtheDQD,
resonator system with Ω/2π = 8.1 GHz, ω0/2π = 8.0 GHz, ∆&g =2π 48.5 MHz,theeigenstatesofthesystembe-
×
the intradot tunneling T/2π = 1 GHz, and the drive ampli- come dominantly photon states or electron states of the
tudeF =50MHz. Thephotonaveragenumberhasapeakat DQD. As a result, the microwave source increases the
ω = (E1,−−E0)/~, Eq. (10), while the photovoltaic current number of photon excitations in the resonator when the
exhibitsadoublepeakfeatureatω=(E1,±−E0)/~(vertical microwave frequency is in resonance with the transition
lines). between the photon–like states, ω = (E E )/~.
1,− 1,− 0
−
Butthesourcehasaweakeffectattheresonancewiththe
electron-like states at frequency ω = (E E )/~.
1,+ 1,+ 0
indicated by dashed vertical lines. We present the corresponding dependence of −the pho-
OnefeatureinFig.2isthatthephotonnumberisalso ton number and the photocurrent in Fig. 4 for ω /2π =
0
reduced at zero bias ε, when the photovoltaic current 8 GHz, Ω/2π = 8.1 GHz (∆ = 100 MHz) and other pa-
vanishes. This suppression is a result of strong enhance- rameters identical to those for in Figs. 2 and 3. We in-
ment of the coupling constant g =g0 at ε =0, resulting deed observe one large peak in the photon number near
in stronger dissipation in the resonator and increase of the resonant condition for the dominantly photon state
off-resonant absorption rate. At the same time the pho- withenergyE whilethephotonnumberdoesnotshow
1,−
tovoltaic current vanishes at ε = 0 due to cancellation significant enhancement near the second resonance, cor-
between the two terms in Eq.(8). respondingtothetransitiontothedominantlyelectronic
The curves for the photon number and the current do state with energy E . The photocurrent still exhibits
1,+
not significantly change after the dephasing rate γ is double peak feature, but the peak corresponding to the
φ
introduced in addition to the energy relaxation rate γ. photon resonance is higher, when the microwave drive
Dephasing smears the resonant condition for the pho- produces a higher photon population.
ton absorption by the DQD and has two effects: (1) the We now consider a more idealistic regime of signifi-
number of photons increases a little near the resonance cantly reduced tunneling and relaxationrates Γ =Γ =
l r
Ω ω , see the inset in Fig. 2; (2) the resonant absorp- γ =2π 100 kHz, the drive amplitude F/2π =30 MHz
0
≃ ×
tion of photons by electrons is suppressed resulting in and ω = Ω = 2π 8 GHz. In this case additional reso-
0
×
reduction of the photocurrent. We note that in the case nances develop, see Fig. 5. These resonances correspond
presented in Fig. 2 the first effect is stronger than the to excitations ofseveralphotons in the cavityby the mi-
5
2
r (a) F/2π = 5 MHz
be0.2 F/2π = 10MHz
m
u er1.5 F/2π = 15MHz
N b
hoton 0.1 n Num 1
P 0 hoto
P0.5
A15
f
ent, 10 0 δ (GHz)
r
ur 5
C
dc 5 nt, pA 4
e
0 r 3
−0.1 −0.05 0 0.05 0.1 ur
C
(ω−ω )/2π, GHz c 2
0 d
(b) 1 (c)
1
|2,g>, |1,e>
n0.5 0
P −0.2 −0.1 0 0.1 0.2
|1,g>, |0,e> (ω−ω )/2π, GHz
0
0
0 1 2 3 ω1,−ω2,−
n |0,g>
FIG. 6: (Color online) Dependence of the average photon
numberandthecurrentonmicrowavefrequencyω forseveral
values of the drive amplitude F/2π =5, 10, 15 MHz, at zero
FIG. 5: (Color online) (a) The average number of photons
dephasing γ = 0 and other parameters are the same as in
in the resonator and the photovoltaic current as a function φ
data in Fig. 3. As the amplitude of the drive increases, the
of the frequency ω of the microwave drive for Ω = ω0 =
2π×8 GHz, the intradot tunneling T/2π =1 GHz, and the two peaks merge together to a single peak at ω=ω0.
driveamplitudeF/2π =30MHzandextremelylowtunneling
rates to the leads and the energy relaxation rate, Γ =Γ =
l r
γ = 2π × 100 kHz. The photon average number and the matchtheresonatortoproduceatwophotonoccupation
photocurrent have several peaks at ωn,± = (En,±−E0)/n~ of the resonator at ω =ω , but it matches the resonator
1
with n = 1, 2, 3, these frequencies, calculated from Eq. (10)
to populate the state with the energy E , which then
are shown by vertical lines). (b) The histogram presents the 2,±
decays to the lower energy states with n=1,0.
probabilities P to have n photons in the resonator steady
n
stateatdrivefrequencyω1(darkbars)andω2(greybars). (c) Next,weinvestigatedependenceofthephotonnumber
ThediagramisaschematicpicturefortheJaynes-Cummings in the resonator and the magnitude of the photovoltaic
energy levels showing single and two photon excitations. currentfordifferentamplitudesF ofthedrive. Theabove
discussion was mostly focused on a resonator containing
less than one photon. As the drive increases, the double
crowave source. When the frequency of the source satis- peak feature evolves to a single peak at the drive fre-
fies ~ωn =E E , the DQD-resonator system expe- quency equal to the frequency of the resonator, ω = ω .
n,± 0 0
−
riences transitions from the ground state to the energy We interpret this cross-over as a signature of changed
state E , cf. Ref. 24. These multiphoton transitions hierarchy of the terms in the system Hamiltonian. At
n,±
result in peaks of the average photon number and the weak drive, we have a JC Hamiltonian with its peculiar
magnitude of the photocurrent. Curves in Fig. 5 have energy levels, Eq.(10), and the drive can be viewed as a
three pairs of peaks at frequencies ω = ω g/√n weak probe testing the spectral structure of the coupled
n,± 0
±
marked by vertical dashed lines for n = 1,2,3. We no- resonator and DQD system. Once the drive reaches the
tice that for ω = ω the average photon number is strength of the g coupling, g 2π 50 MHz, a proper
1,2
≃ ×
nearly the same, see the top panel in Fig. 5(a), while waytotreatthesystemistostartwiththeFloquet–type
the photon distribution function is different, Fig. 5(b): states16,25,26 ofthedrivenresonatorandthentotakeinto
at ω = ω a non-zero P develops for a probability accounttheinteractionofthesestateswiththeDQDsys-
2,− 2
that the resonatorcontains two photons. This difference tem as a perturbation. In this picture, the photon reso-
in P indicates that the microwave drive line does not nance happens at ω =ω . The coupling g is responsible
n 0
6
for the formation of the broader “wings” in curves for reminiscentoftheLambshiftbyasingleelectromagnetic
theaveragephotonnumberandthephotocurrent. These mode. We also found that in the limit if extremely weak
wings are more pronounced in the photovoltaic current, relaxationratesoftheDQD,multiphotonresonancesde-
which is entirely due to the coupling between resonator velop when the energy difference between the states of
andDQD. This broadstructure ofthe generatedcurrent the coupled system is a multiple of ~ω.
asafunctionofthesourcefrequencyispreservedevenat As energy and phase relaxation rates of the DQD in-
strongerdrive. Thus,theshapeofthephotovoltaiccurve crease,thepeaksinthephotonnumberandthephotocur-
might provide an experimental approachto quantify the rent broaden and eventually merge in a single resonance
strength of the JC coupling constant. peak at the frequency ω of the resonator. In this limit,
0
the resonator photon mode and the DQD are no longer
describedasanentangledquantumsystemandthe reso-
IV. DISCUSSION AND CONCLUSION nantconditionfortheinteractionofthemicrowavesource
with the system corresponds to equal frequencies of the
We analyzed the photovoltaic current through a DQD source and the resonator mode, ω =ω .
0
system at zero voltage bias between the leads. The dou- At stronger microwavedrive,frequency dependence of
ble quantum dot interacts through its dipole moment to the averagephoton number in the resonator evolve from
a quantized electromagnetic field of a high quality mi- the Jaynes-Cummings double peaks at ω = ω g to a
0
±
crowave resonator. The interaction is described by the single peak at the resonator frequency ω . The single
0
Jaynes–Cummings Hamiltonian of a quantized electro- peak at ω = ω is a result of multi-photon transitions
0
magnetic field and a two level quantum system, repre- at strong drive by the microwave source that all merge
sented by ground and excited electronic states of the together due to finite width of multi-photon resonances.
double quantum dot. When a weak microwaveradiation Similar evolution to a single peak occurs for the pho-
is applied to the resonator, the source acts as a spec- tocurrentresponse,althoughthe photocurrentcurve has
tral probe that causes excitation of the system when the a broader width as a function of the source frequency ω,
energy difference between its eigenstates is equal to the this width corresponds to the strength of the coupling g
photon energy ~ω of the source. If this resonance con- betweenthe photonmode ofthe resonatorandthe DQD
dition is satisfied, the microwave source populates the andmaybeusedtocharacterizethestrengthofthiscou-
photonmodeoftheresonatorandgeneratesadirectcur- pling in experiments.
rent though the double dot system even at zero bias.
We demonstrated that at finite, but still low energy
andphaserelaxationratesoftheDQD,boththeaverage
Acknowledgments
number of photons in the resonator and the photocur-
rent through the DQD have a double-peak structure as
functions ofthe frequencyofthe microwavesource. This We thank R. McDermott and J. Petta for fruitful dis-
double peak structure reflects an avoided crossing of the cussions. The work was supported by NSF Grant No.
energystatesoftheDQDandthe resonatorphotonsdue DMR-1105178andbytheDonorsoftheAmericanChem-
to the interaction between the two subsystems and is ical Society Petroleum Research Fund.
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