Photon-mediated interaction between distant

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Received 13 Jul 2012 | Accepted 20 Dec 2012 | Published 29 Jan 2013

Engineering the interaction between light and matter is an important goal in the emerging eld of quantum opto-electronics. Thanks to the use of cavity quantum electrodynamics architectures, one can envision a fully hybrid multiplexing of quantum conductors. Here we use such an architecture to couple two quantum dot circuits. Our quantum dots are separated by 200 times their own size, with no direct tunnel and electrostatic couplings between them. We demonstrate their interaction, mediated by the cavity photons. This could be used to scale up quantum bit architectures based on quantum dot circuits or simulate on-chip phonon-mediated interactions between strongly correlated electrons.

DOI: 10.1038/ncomms2407

Photon-mediated interaction between distant quantum dot circuits

M.R. Delbecq1, L.E. Bruhat1, J.J. Viennot1, S. Datta1, A. Cottet1 & T. Kontos1

1 Laboratoire Pierre Aigrain, Ecole Normale Suprieure, CNRS UMR 8551, Laboratoire associ aux universits Pierre et Marie Curie et Denis Diderot, 24, rue Lhomond, 75231 Paris Cedex 05, France. Correspondence and requests for materials should be addressed to T.K. (email: mailto:[email protected]

Web End [email protected] ).

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Cavity QED allows one to study the interaction between light and matter at the most elementary level, for instance, by using Rydberg atoms coupled to cavity photons1.

Recently, it has become possible to perform similar experiments on-chip, by using articial two-level systems made from superconducting circuits instead of atoms2. This circuitQED offers unexplored potentialities, as other degrees of freedom than those of superconducting circuits could be used310, and in particular, those of quantum dots1114. Such a hybrid circuit QED would allow one to study a large variety of situations not accessible with standard cavity QED, owing to the versatility of nanofabricated circuits7,15.

A highly desired functionality required for the use of quantum dots for quantum information processing is the controlled coupling between distant quantum dots. This would allow one to scale up the existing spin quantum bit architectures1618. Among the possible methods for coupling quantum dots16,19,20, the use of photons is particularly attractive because they can mediate a coherent interaction over macroscopic distances1,21,22. Recently, the coupling of single quantum dot circuits to microwave photons has been demonstrated. So far, this architecture has been used to directly measure the electron photon coupling strength11,12,14 or to read-out capacitively a spin quantum bit13.

Here, we demonstrate a distant interaction by embedding two quantum dot circuits in a microwave cavity. Our quantum dots

are separated by 200 times their own size, with no direct tunnel and/or electrostatic coupling. We use one of the two dots as a detector or perform a joint read-out of the two quantum dots with the microwave cavity in order to probe the distant interaction. The spectroscopy of the detector dot is shown to display a dispersion as well as crossings of charge states as a function of its own gate and that of the other distant dot. The joint read-out allows us to probe in a complementary fashion the distant coupling scheme through the phase of the microwave signal.

ResultsSample geometry. The device presented here is shown in Fig. 1a,c (see Methods). The two quantum dots, made using different and well separated single-wall carbon nanotubes, are rst studied using the traditional transport spectroscopy. The colourscale plots of the differential conductance of dot 1 (QD1) and dot 2 (QD2) as a function of their gate and source-drain bias are represented in Fig. 1d,e, respectively. For QD1, the conductance is close to 2e2/h and displays modulations with the characteristic checkerboard pattern of an electronic FabryPerot interferometer23. The vertical distance between the bright spots allows us to determine a level spacing of about 7 meV. This corresponds to a length of about 200 nm, which is slightly smaller than the lithographically dened length. The quantum dot 1 can be considered as an open

a b

0.1

(rad)

0.05

Amplitude (V)

dl/dV QD1 (2e2 / h )

dl/dV QD2 (2e2 / h )

5 5.5 6 6.5

10 11 12 13

Vg1 (V)

f (GHz)

c d

QD1

V sd1(mV)

V sd2(mV)

12.2 12 11.8 11.6

1.2

0.8

0.4

0.030.020.01 0

QD1

QD2

Vg2 (V)

Figure 1 | The coupling architecture. (a) Optical micrograph of the Al microwave resonator in coplanar waveguide geometry. Four sets of three DC lines allow to contact up to four QDs inside the cavity. The bar is 1 mm. (b) Transmitted amplitude (red curve) and phase (black curve) of the microwave eld versus frequency, giving f0 5.75 GHz and QE40. (c) False colours optical micrograph showing the two quantum dots structures inside the microwave

cavity. Aluminium elements of the resonator are represented in blue (transmission line and ground plane). The Pd DC lines contacting the carbon nanotubes (not visible here), are represented in yellow. The two QDs are separated by 80 mm and are designed by the standard source-drain and side gate geometry. The bar is 50 mm. (d) Colour scale plot of QD1 differential conductance in units of 2e2/h in the gate voltage Vg1-bias voltage Vsd1 plane. The spectroscopy shows a FabryPerot behaviour with a level spacing of 7 meV, corresponding to a 200 nm long quantum dot. (e) Colour scale plot of QD2 differential conductance in units of 2e2/h in the gate voltage Vg2-bias voltage Vsd2 plane. The spectroscopy shows a Coulomb diamond with excited states.

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quantum dot. The differential conductance of QD2 is much smaller and peaks only up to 0.03 2e2/h. The colourscale plot of

its differential conductance displays Coulomb diamonds and excited states characteristic of the Coulomb blockade regime24. In particular, one can read off a charging energy of about 12 meV. Another important parameter for both dots is the width G1(2) of

their energy levels. From the colourscale plots, one nds, G1E7 meV and G2E1 meV. This shows that the two electronic systems still relax much faster than the characteristic period of the photonic mode, which corresponds to 5.75 GHzD30 meV as shown in Fig. 1b.

Calibration of electronphoton coupling for each dot. Before going to the main result of the present work, which is the distant interaction between the above described quantum dots, it is essential to calibrate the electronphoton coupling for each dot. In order to do this, we apply a strong, classical, microwave drive to the cavity at the resonance frequency. We rst focus on the closed dot that allows us to obtain the highest spectral resolution as it has the smallest coupling rate to its reservoirs. The colour-scale plots of the differential conductance of QD2 as a function of Vg2 and Vsd2 for no microwave power, 45, and 35 dBm at

the input of the cavity are presented in Fig. 2ac, respectively. The microwave eld splits dynamically the charge degeneracy line crossing at about Vg2 11.7 V. This splitting is about 3 meV at 45 dBm and increases up to 10 meV at 35 dBm. One can

understand classically the observed phenomenon by using the fact that the dots electrons have a relaxation rate to the electrodes,

which is about 45 times higher than the microwave frequency of the cavity. The AC eld of the resonator modulates the dots energy levels with an amplitude lVAC and frequency f0. The averaging of this modulation leads to split conductance peaks (dashed lines) at E0lVAC, as depicted schematically in Fig. 2d.

The differential conductance Gon (Vg, Vsd) at nite microwave

drive can be obtained by assuming that the microwave eld only couples to the electronic levels, leading to: GonVg; Vsd 1/2p

2p0 dyGoffVg lVAC cos y; Vsd, l char

acterizing the electronphoton coupling strength, VAC being the amplitude of the AC voltage at the resonator frequency f0 and Goff(Vg, Vsd) being the differential conductance of QD2 with no microwave power. The results of this procedure is shown in Fig. 2e,f, respectively, for l 15.5, using the measured data of

Fig. 2a. The conductance peak splitting and in particular the peculiar diamond-shaped region emerging at the charge degeneracy point around Vg2 11.7 V are very well accounted for.

Note that not all the features in Fig. 2b,c are reproduced with our procedure, indicating that some of the excited states might have a different coupling to the photons than the ground state25.

The specic origin of the conductance peak splitting can be conrmed by studying its dependence as a function of the power of the microwave signal at the input of the cavity, or equivalently the average number of photons stored in the cavity. Although we have used a classical eld so far, such a study allows us to determine accurately the electronphoton coupling terms entering into the AndersonHolstein-like Hamiltonian of the system11, g12

bN12a aw,

bN12 being the total number of electrons in QD1(2) and a(aw) being the annihilation (creation)

0.01 0.02 0.03

dl/dV QD2 (2e2/h)

dl/dV NT2 (2e2/h)

b c

0 0.005 0.015

0.01

0 0.005 0.01 0.015

10 POFF 45 dBm

10 45 dBm theory 35 dBm theory

35 dBm

V sd2(mV)

12.2 12 11.8 11.6

12 11.8 11.6 12 11.8 11.6

Vg2 (V) Vg2 (V)

Vg2 (V)

d e f

0.005 0.01 0.015

0 0.005 0.01

VAC

CAC

E0+ VAC

V sd2(mV)

E0 VAC

Vg2

12.2 12 11.8 11.6 12 11.8 11.6

Figure 2 | Driving quantum dot 2 through the cavity. Inuence of the microwave power driving the cavity on QD2 spectroscopy. Differential conductance of QD2 as a function of the gate voltage Vg2 and the bias voltage Vsd2 for different powers at the input port of the cavity: (a) no power, (b) Pin 45 dBm

and (c) Pin 35 dBm. The conductance peaks of the quantum dot split, as emphasized by the formation of a new diamond-shaped region around

Vg2 11.7 V. (d) Scheme showing the capacitive coupling of the quantum dot electronic levels to the gate electrode (green) and to the resonator (red).

The DC gate electrode drives the level energy E0 in the quantum dot. The AC eld of the resonator modulates the energy levels with an amplitude lVAC and frequency f0. The averaging of this modulation leads to two split conductance peaks (dashed lines) at E0lVAC. Simulations of the splitting based on the scheme depicted in (d) and applied to the spectroscopy of (a) with l 15.5, respectively, for Pin 45 dBm (e) and Pin 35 dBm (f).

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V sd1(mV)

V sd2(mV)

1.1

0.9

dl/dV QD1 (2e2 /h)

dl/dV QD2 (2e2 /h)

10 0 g1 = 97 +/ 22 MHz

g2 = 98 +/ 22 MHz

E(mV)

0 30

0.8

0.03

0.02

0.01

103 n

10 00 20

E(mV)

60 50 40 30

Pin (dBm)

Figure 3 | Measurement of the electronphoton coupling strength for each dot. Microwave power dependence of conductance peak splitting. (a,b) Colour scale plots of the differential conductance of QD1 (respectively QD2) in the input power Pin-bias voltage Vsd1 (respectively Vsd2) plane. Insets: the dependence of the conductance peak separation DE versus

for each QD (red dots). The slope of the linear t (white line) gives the coupling g1(2) between the quantum dot 1(2) and the microwave eld as DE/h 2g12

p .

operators for the photons in the cavity at the fundamental frequency. This will allow us below to predict the expected form and magnitude of the interaction between the two dots, which are dominated by single-photon processes. Such a study is shown in Fig. 3a,b for QD1 and QD2, respectively. The colourscale plot of the differential conductance of the dots as a function the source-drain bias and the input power displays a characteristic funnel shape showing the conductance peak splitting in QD1 and QD2 as the power is increased. According to the picture developed previously, the splitting should scale linearly with the amplitude of the eld, or equivalently with the square root of the average number of photons. The latter viewpoint is particularly convenient as it allows us to determine directly the coupling constants of QD1 and QD2 to a single photon, g1 and g2, respectively. Indeed, the splitting D1(2) reads: D12 2g12 n

p ,

where

n is the average number of photons in the cavity. The dependence of D1(2) as a function of

p is shown in the inset of Fig. 3a,b for QD1 and 2, respectively. The linear behaviour observed fully conrms our level driving picture and allows us to determine g1 9722 MHz and g2 9822 MHz. We

emphasize again that we obtain them by re-expressing the driving amplitude for each dot using l12VAC g12

p . The

uncertainty in the determination of g1(2) is mainly related to

the systematic errors made in the determination of the exact power at the input of our cavity.

Gate tunable electronphoton coupling. We now investigate with greater detail the coupling mechanism between the photons and the electrons on the carbon nanotubes. This section will allow us rst to demonstrate interesting toggles for the electric control of the electronphoton coupling strength. Importantly, it will allow us to rule out all the spurious DC electrostatic effects, which could hinder the study of the photon-mediated distant interaction between the dots. We focus on QD1 for which electronic interactions can be neglected. The AC response of QD1 can be conveniently predicted with a self-consistent theory of AC transport similar to that of ref. 26 (see Methods). We measure g1 for various gate voltages Vg1 using the previously described method. The result is presented in Fig. 4b in inset by the lled dots. The value of g1 displays modulations as Vg1 is swept.

Strikingly, the change in g1 is proportional to the linear conductance G1 of QD1 as shown in Fig. 4b main panel. The value of g1 increases by almost 100 MHz when the conductance G1 is increased from 0.3 2e2/h to 1.1 2e2/h. Such an increase

is accounted for by the increase of the quantum capacitance of QD1, which is directly proportional to the conductance G1 (see Methods). As shown in Fig. 4b, the sign of the slope of g1 versus G1 allows to discriminate between direct coupling between the cavity photons to the dots energy level and indirect coupling to the dots energy level via the source-drain electrodes. As expected from the efcient screening of the source-drain electrodes, the coupling mechanism is mainly via the source-drain leads, as shown by the positive slope of g1 versus G1 (blue continuous line). One can further specify this by comparing the intercept of the blue line with the vertical axis. It corresponds to the geometrical part of the coupling strength of dot 1 as it corresponds to zero conductance. We nd 48 MHz. An upper bound of the direct coupling can be obtained by reading off from Fig. 1d the DC side gate lever arm, of about 0.01. This corresponds to only 2 MHz and supports further that the coupling mechanism involves the source and drain leads, and that Fig. 4a best displays the corresponding circuit diagram. Note that this mechanism is only possible at the frequency of the cavity as the source and drain electrodes are DC shunted in order to measure simultaneously the low frequency conductance (below 1 kHz). Therefore, DC potentials on the central conductor of the cavity do not affect the energy levels of QD1 and any change in QD1s potential does not affect the potential of the central conductor.

Distant coupling between the dots. We now probe the coupling between the states of QD1 and those of QD2, which is the central result of this article. We rst use QD2 as a detector and measure the evolution of its energy levels via transport spectroscopy as its own gate and the distant gate of QD1 are swept. We present in Fig. 5a the differential conductance of QD2 as a function of Vg1

and Vg2 for one specic region. Strikingly, the gate of QD1 acts on QD2 even though the direct capacitive coupling is vanishingly smallthe wide central conductor has a capacitance to the ground, which is of about 0.7 pF, more than 5 orders of magnitude bigger than the gate capacitances of about 1 aF. Furthermore, as discussed above, the two dots, separated by the central conductor of the cavity, are only AC coupled to the cavity, via the source-drain leads. This strongly points to a distant interaction between the two dots without any direct electrostatic or tunnel coupling. In particular, it is not an electrostatic interaction mediated by a oating gate27.

Both quantum dots are coupled to the fundamental mode of the cavity. They can a priori be coupled via the virtual exchange of photons in a similar fashion as in atomic physics1,21,28. The equivalent interaction mechanism, found in condensed matter

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a b

Cavity

CACL CAC CACR

CL CR

150

2 / h )

g 1(MHz)

G 1 (2e

150

100

0 0 0.5 1

dl/dV QD1 (2e2/h)

8 12

Vg1 (V)

0.5

g 1 (MHz)

Vs VG VD

200

12.2 12 11.8 11.6

0.03

0.02

0.01

G1= 0.5 (2e2/h)

G1= 1.1 (2e2/h)

V sd2 (mV)

0.9

0.8

0.7

Vg2 (V)

g 2

00 20

103n

E(mV)

V sd2 (mV)

108 7.8 7.6 7.4 7.2

0.04

dl/dV (QD2) (2e2 / h ) dl/dV (QD2) (2 e2 / h )

0.02

0.3 0.6 0.9 1.2

dl/dV (QD1) (2e2/h)

Figure 4 | Coupling mechanism and tunable coupling strength. (a) Schematics of the coupling mechanism between the quantum dots and the cavity. (b) Variation of the electronphoton coupling strength of QD1, in open dots, as a function of its own linear conductance. The error bars correspond to the systematic error of the linear tting for the plots in inset of Fig. 3. The solid line is the theory described in the Methods section. Inset: variations of the electronphoton coupling strength and of the conductance of QD1 as a function of Vg1. (c,d) Transport spectroscopy of QD2 for two different values of the conductance of QD1 (corresponding to two different sets of (Vg2, Vg1) for the same energy level of QD2). (e) Relative variation of the electronphoton coupling strength of QD2 as a function of the linear conductance of QD1. The open circles correspond to a change of the conductance of dot 1 by changing its gate. The open triangles and diamonds correspond to a change of conductance by changing the source-drain bias. The solid black line is the theory.

Inset: the dependence of the conductance peak separation DE versus

p for the two points indicated by the arrows. The error bars correspond to the systematic error of the linear tting for the plots in the inset of Fig. 3.

whenever electronic states are coupled to bosonic modes, leads to a polaronic shift of each of the levels of QD1(2): Dpolaron12 4pg1g2N21/f0; N21 being the number of electrons

in QD2(1), respectively29,30. This shift can be simply found from a LangFirsov transformation of the dots total Hamiltonian (see Methods). As QD1 can contain a large number of electrons, D2polaron can become large, comparable to the energy scales relevant for QD2. This shift therefore survives at our moderately low temperatures even though the essence of this interaction is mainly quantum mechanical. One nds, for example, D2polaronE 1 meV for N1E104. As shown in Fig. 4, g1 and g2

strongly depend on the gate voltage of QD1 and QD2, respectively. This means that even though N2(1) are large

numbers and do not vary substantially as the gates of each dot are swept, the shift Dpolaron12 can still evolve thanks to the gate

tunability of the g0s as Vg1(2) are varied. We now use g12

g012 a12Vg12 with the range of a1(2) extracted from Fig. 4.

Note that the hierarchy of energy scales here is the same as for phonons in solids, which are usually much slower than the electrons, the electrons being delocalized over many sites.

Physically, electrons on QD1 and QD2 shift the electric eld corresponding to the cavity photons. The modication of the circuit energy by this effect is twofold. First, as QD2 is coupled to the cavity, the electrons of QD2 directly feel the photonic electric eld deformation induced by the electrons of QD1. Second, as the energy of the cavity involves quadratically the photonic electric eld, the shift of this electric eld by electrons of both QD1 and QD2 produces crossed terms between QD1 and QD2. These two phenomena lead to a global effective interaction between the two dots electronic states.

In addition to the shift, the coupling of electrons to photons can lead to a splitting of the energy levels in each dot, proportional to the coupling constant of each dot to the photons. This can arise for instance owing to photon-induced KK mixing as predicted recently25. The general form of the energy levels of QD2 is therefore E 2 E0 a0Vg2 4pg1Vg1g2Vg2N1/f0 g2Vg2d, E0 being a

constant, a0 characterizing the bare coupling of QD2 to its gate and d characterizing the magnitude of the splitting. The gate dependence of g2 implies a variation of the splitting as gate 2 and

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V g2 (V)

12 13 14 15

0.5 1 1.5

0.1

0.05

dl/dV NT2 (2e2/h)

V g2 (V)

10.5 11 0

Vg1 (V)

QD1

U 2N+3

2N+2

2N+1

QD2 f=5.75 GHz

U+

V g2 (V)

17 5 6 7 8 9

0.8450.840.835

(rad)

Vg1 (V)

Figure 5 | Distant coupling between the quantum dots. (a) Colour scale plot of the differential conductance of QD2 versus its own gate voltage Vg2 and

the gate voltage Vg1 of QD1. The levels of QD2 evolve with different slopes, leading to level crossings. (b) Modelling of the conductance map of QD2 using the form of E 2 presented in the main text and a simplied equation of motion theory. (c) Diagram of the transport spectroscopy of QD2 versus E2 and E1, the energies of QD2 and QD1, respectively. Two orbitals are represented (red and blue levels), separated by U D, with D the orbital spacing. The two

conductance peaks corresponding to the same orbital are separated by the charging energy U. The interaction between QD1 and QD2 shifts the levels of QD2. They are also split as shown by the dashed lines. The conductance peaks separated by less than U in the same orbital are forbidden (shown by a cross). Crossings between conductance peaks originating from different orbitals are allowed. The resulting spectroscopic pattern exhibits an alternation of conductance peaks moving away and coming closer, as observed in (a). (d) Joint readout of the two QDs in the phase j of the transmitted RF signal. Both levels of QD1 and QD2, indicated by green arrows, can be observed and cross.

gate 1 are swept, which explains the crossings of conductance peaks observed in Fig. 5a. As shown in Fig. 5b, we are able to reproduce with a simple modelling all the features observed in Fig. 5a (see Methods). The values of a1(2) and N1 are the three

tting parameters used to produce the main slope in Fig. 5b (see Methods). The origin of the crossings is sketched schematically in Fig. 5c.

The distant coupling can be used to control in a non-local manner the electronphoton coupling strength, as it shifts the energy levels of each dot. We illustrate this fact by studying the evolution of g2 as a function of G1, as shown in Fig. 4e. The open circles are obtained for zero Vsd and different gate voltages whereas all the other points are obtained by varying the source-drain bias at constant gate voltage. The systematic variation (roughly linear) of g2 as a function of G1 in the two different conditions depicted above shows that it is G1 and therefore the quantum capacitance of dot 1 that matters for the shift of the energy levels rather than the gate voltage Vg1 itself. As a consequence, it further supports our distant coupling mechanism. As shown in Fig. 5a, we can follow the evolution of the energy levels of QD2 in the (Vg2Vg1) plane. We present in Fig. 4c,d the bias spectroscopy of QD2 along one of the tilted lines of Fig. 5a for two different sets of (Vg2Vg1). The size of the Coulomb

diamonds is different for each of the sets of (Vg2Vg1), which are

conveniently expressed in terms of G1 for similar reasons as above for the study of the local control of g1. Measuring g2 along this line gives the open symbols of Fig. 4e. For each of these points, we extract the capacitive lever arm for the dot from the slope of the Coulomb diamonds. This allows us to predict the dependence of g2, according to the coupling mechanism described in the Methods section. The resulting theoretical curve is the black solid line, which accounts very well for our measurements of g2.

DiscussionAs both quantum dots are coupled to the microwave eld, one can perform a joint read-out of their state by measuring the phase of the transmitted microwave signal through the cavity. Such a measurement is presented in Fig. 5d. The levels of QD1 and QD2 are clearly observed as tilted narrow and almost vertical shallow stripes, respectively, in the colourscale plot of the phase at5.75 GHz as a function of Vg1 and Vg2. As expected, the levels of QD1 are wider than those of QD2. Their dependance on the distant gate (of QD2 now) is weaker than for QD2. We explain this fact by the smaller number of electrons N2 in the closed quantum dot QD2. Note that the levels of QD1 and QD2 cross, which means that they are not coherently coupled by the photons as expected because the gammas are too large. The origin of the observed interaction relies, however, on the exchange of virtual photons like for the mechanism leading to two quantum bit operations for Rydberg atoms or superconducting quantum bits1,21,22. As the g1,g2 electronphoton couplings are the same as for the transmon-type quantum bits, we can therefore envision to perform SWAP operations as fast as for superconducting quantum bits. This will require converting the coupling to the charge to a coupling to the the spin5,6 and using more isolated quantum dots.

Methods

Experimental. Two single quantum dot circuits are fabricated using two different and well separated single-wall carbon nanotubes connected to normal (Pd) electrodes, inside an Al microwave cavity shown in Fig. 1a (see ref. 11 for details). Both dots share the same antinode of the electric eld of the fundamental mode of the resonator, but they are located on opposite sides of the central conductor of the resonator, as shown in Fig. 1c. Their lateral size dened by nanolithography is about 400 nm, which is more than two orders of magnitude below the distance between them, about 80 mm. The characteristics of the resonator are shown in

Fig. 1b. The fundamental mode frequency is at 5.75 GHz with a quality factor

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we get:

of about 40, as shown by the peak in the amplitude of the transmitted signal in red line. The phase of the microwave signal, shown in black line, displays the characteristic jump of p at the resonance. The temperature of the experiment is1.5 K throughout the paper.

Modelling of the evolution of the conductance peaks. One can model the inuence of the level structure depicted in the main text on transport in a Coulomb-blockaded quantum dot as QD2. We present in Fig. 5b the result of such a modelling using a simplied equation of motion theory for the Greens functions and the MeirWingreen formula for the linear conductance31. The colourscale plot of Fig. 5b has been obtained using N1 106 and a1(2)E25 MHz V 1 in the

{Vg1, Vg2} region considered here. In particular, the alternation between level crossings and splittings as gate 2 is swept towards negative values is reproduced. This phenomenon can be explained qualitatively by the picture of Fig. 5c. In a quantum dot in the Coulomb blockade regime, a splitting of energy levels in the intrinsic spectrum leads to a separation between conductance peaks corresponding to the same orbital and to crossing of conductance peaks corresponding to different orbital levels.

Electronphoton coupling mechanism. The electrostatics of the dots allows us to determine the level energy at the mean-eld level26:

xd Ed eaACVAC eaL aRACVAC EChNi where /NS is the average

number of charge on the island, Ed is the bare dot energy and EC e2/(2CS) is the

charging energy. The other important terms in the above expression are: aAC the

direct cavitydot capacitive coupling, aL,R the L, R capacitances of the source-drain electrodes andAC is the source(drain) leadcavity capacitive coupling that we assume to be the same for source and drain here (symmetric coupling). The self-consistent equation that governs the number of charges on the dot is

hNi R

eH ho0aya

1 ho0

Xdsgdnds

!2 X

Edsnds Hint 8

The second term contains two quadratic terms and one bilinear term in the number of electrons occupying each dot. While the rst can be absorbed in the interaction term of Hamiltonian of each dot, the bilinear term can be factorized in each orbital part of the Hamiltonian, leading to the expression used in the main text for the polaronic shift. This leads to the distant coupling of the main text. Note that, contrarily to transport where only the states close to Fermi energy matter, it is the total charge that is coupled to the electromagnetic eld (the sum goes over all occupied states). This can lead to a very large shift of the levels of each dot provided the distant dot has a high enough charge density. In addition, contrarily to its atomic physics counterpart, the above expression is non-perturbative and is exact when the dotlead coupling is zero.

Using the above equation, we get:

E0 a0Vg2 4pg2Vg2

Xdsg1;jn1;js/f0 9

This can be further simplied in the continuum limit as:

E0 a0Vg2

4pg2Vg2f0 N1 R

V dEg1EAQD1E

V dEAQD1E

where N1

V dEAQD1E for QD1. By dening hg1i R

dEg EA E

implicit equation for {Vg2,Vg1} reads:

a0Vg2

dEA E

, the

dEf E mecAE xd, with f the Fermi function, mec the electrochemical

potential of the leads and A the spectral density of the dot. The electrochemical

potential of the dot is changed via the AC drive VAC of the cavity as mec m eACVAC. From the above equations, one can derive

@hNi

eaAC eaL aR 1AC EC @hNi@V n o

4pg2Vg2hg1iVg1

f0 N1 const 11

This is the most important part of the equation of the main text and controls the dispersion of the energy levels of dot 2. As /g1S(Vg1) is an integral over all the occupied energy levels, its variations are smooth and roughly linear over a large gate voltage scale. The variations of g2(Vg2) can be more pronounced and can lead in general to a nonlinear dispersion of the energy levels in the {Vg2,Vg1} plane, as

observed in the joint read-out of Fig. 5d.

Using the values given in the maintext, we are able to reproduce the observed general slope of the energy levels of QD2 in the {Vg2,Vg1} plane. Note that the above analysis, carried out for a single mode of the electromagnetic eld, can be straightforwardly extended to the multimode case. As the coupling mechanism demonstrated here is non-resonant, essentially all the modes within the line width of the energy levels can contribute30. This can reduce the number of electrons needed to explain our data by a factor of 100300.

References

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@hNi@E. The electronphoton coupling then is obtained by the derivative of the dot energy level with respect to the uctuating potential of the resonator:

g Vrms

@xd @VAC

eVrms aAC aL aRAC EC

The charge susceptibility q/NS/qEd of the dot is the opposite of the quantum capacitance CQ of the dot in the non-interacting limit ((q/NS/qEd) CQ/e2)

and is proportional to the linear conductance G of the dot26. If the dot is directly coupled to the cavity, the above expression shows a negative slope as a function of G whereas if the dot is indirectly coupled via the leads, the above expression shows a positive slope, which arises from the sum rule aL aRo1. This allows us to plot

the blue line of Fig. 4b and to conclude that our dots are coupled to the cavity photons via the dots source-drain leads.

Distant coupling and LangFirsov transformation. We reproduce here the Lang Firsov transformation, which allows us to derive the interaction between the distant dots. We rst consider the AndersonHolstein Hamiltonian of the main text:

H H0 Hc 2

H0 ho0aya

Xds

Edscydscds Hint 3

( )

@hNi

@Ed

aAC

aL aR 1 1 EC @hNi@E

Xdsgday acydscds 4

d corresponds to the different dots. We assume gd dependent on the dot d as the dots are different and located in different positions. The operator Hint accounts for the electronic interaction on each dot.

We follow ref. 29 and use the unitary transformation for an operator A:

eA eSAe S 5

with

Xdsgd ho0 ay acydscds 6

Using the identity:

eO O S; O

2 ! S; S; O

13 ! S; S; S; O . . . : 7

NATURE COMMUNICATIONS | 4:1400 | DOI: 10.1038/ncomms2407 | http://www.nature.com/naturecommunications

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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms2407

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Acknowledgements

We are indebted to P. Simon, C. Bergenfeldt, P. Samuelsson, C. Mora, K. Le Hur and

G. Zarand for fruitful discussions. The devices have been made within the consortium

Salle Blanche Paris Centre. This work is supported by the ANR contracts DOCFLUC,

HYFONT, SPINLOC and the EU-FP7 project SE2ND[271554].

Authors contributions

M.R.D. and T.K. conceived the experiment, carried out the measurements and analysed

the data with inputs from A.C. and J.J.V., M.R.D. carried out the nanofabrication process

with inputs from L.E.B. and S.D., A.C. provided theoretical material; and M.R.D. and

T.K. cowrote the manuscript with the help of A.C.

Additional information

Competing nancial interests: The authors declare no competing nancial interests.

Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/

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How to cite this article: Delbecq, M.R. et al. Photon-mediated interaction between

distant quantum dot circuits. Nat. Commun. 4:1400 doi: 10.1038/ncomms2407 (2013).

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Engineering the interaction between light and matter is an important goal in the emerging field of quantum opto-electronics. Thanks to the use of cavity quantum electrodynamics architectures, one can envision a fully hybrid multiplexing of quantum conductors. Here we use such an architecture to couple two quantum dot circuits. Our quantum dots are separated by 200 times their own size, with no direct tunnel and electrostatic couplings between them. We demonstrate their interaction, mediated by the cavity photons. This could be used to scale up quantum bit architectures based on quantum dot circuits or simulate on-chip phonon-mediated interactions between strongly correlated electrons.

Details

Title

Photon-mediated interaction between distant quantum dot circuits

Author

Delbecq, Mr; Bruhat, Le; Viennot, Jj; Datta, S; Cottet, A; Kontos, T

Pages

1400

Publication year

2013

Publication date

Jan 2013

Publisher

Nature Publishing Group

e-ISSN

20411723

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1038/ncomms2407

ProQuest document ID

1282557765

Photon-mediated interaction between distant quantum dot circuits

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