florian marquardt- efficient on-chip source of microwave photon pairs in superconducting circuit qed

8/3/2019 Florian Marquardt- Efficient on-chip source of microwave photon pairs in superconducting circuit QED

1/5

arXiv:cond-mat/0605232v1[cond-mat.mes-hall]9May2

006

Efficient on-chip source of microwave photon pairs in superconducting circuit QED

Florian MarquardtPhysics Department, Center for NanoScience, and Arnold Sommerfeld Center for Theoretical Physics,

Ludwig-Maximilians University Munich, Theresienstr. 37, 80333 Munich, Germany

We describe a scheme for the efficient generation of microwave photon pairs by parametric down-conversion in a superconducting transmission line resonator coupled to a Cooper pair box servingas an artificial atom. By properly tuning the first three levels with respect to the cavity modes,the down-conversion probability may become higher than in the most efficient schemes for opticalphotons. We show this by numerically simulating the dissipative quantum dynamics of the coupledcavity-box system and discuss the effects of dephasing and relaxation. The setup analyzed heremight form the basis for a future on-chip source of entangled microwave photons.

Introduction. - The generation of photon pairs by para-metric down conversion (PDC)[1, 2, 3, 4, 5] representsone of the basic ways to create nonclassical states of theelectromagnetic field, which has found numerous appli-cations so far. The conditional detection of one of thephotons enables the production of single photon Fockstates [2, 6]. Furthermore, PDC is the primary method

to generate entangled pairs of particles. Apart from thepossibility of testing Bell inequalities [3, 4, 5, 7], this rep-resents a crucial ingredient for a multitude of applicationsin the field of quantum information science, ranging fromquantum teleportation through quantum dense coding toquantum key distribution [8].

With the advent of superconducting circuit quantumelectrodynamics[9, 10], it will now be possible to takeover many of the concepts that have been successful inthe field of quantum/atom optics and transfer them tothe domain of microwave photons guided along coplanarwaveguides on a chip, interacting with superconductingqubits [11, 12, 13, 14]. Recent experiments have real-ized the strong-coupling limit of the Jaynes-Cummingsmodel known in atom optics, employing a superconduct-ing qubit as an artificial two-level atom, and coupling itresonantly to a harmonic oscillator (i.e. a cavity mode[10]or a SQUID [15]). Dispersive QND measurements of thequbit state, Rabi oscillations and Ramsey fringes havebeen demonstrated [16, 17], leading to a fairly detailedquantitative understanding of the system, which behavesalmost ideally as predicted by theory[9].

In this paper, we will analyze a scheme that imple-ments parametric down-conversion of microwave pho-tons entering a transmission line resonator coupled to

a Cooper pair box (CPB) providing the required non-linearity (Fig. 1). This represents the limit of a singleartificial atom taking the place of the nonlinear crystalusually employed in optical PDC experiments [1, 2, 3, 5],with the cavity enhancing the PDC rate (cf. [18]). Incontrast to other solid state PDC proposals [19, 20, 21],both the basic cavity setup and the possibility of eject-ing the generated photons into single-mode transmissionlines with a high degree of reliability are already an exper-imentally proven reality[10, 16, 17]. Recently, squeezing

L

CPB

cavity

3

2

Figure 1: (color online) Schematic setup for the proposedparametric-down conversion (PDC) experiment in supercon-ducting circuit cavity electrodynamics, with a Cooper-pairbox (CPB) interacting with the three lowest modes of a trans-mission line resonator, whose voltage distributions are shown.

and degenerate parametric down-conversion have beenanalyzed theoretically [22] for a circuit QED setup cou-pling a charge qubit to two cavity modes. However, un-

like the experiments and most of the theoretical investi-gations mentioned above, in this paper we propose to gobeyond the regime where the box may be regarded as atwo-level system (qubit), making use of its first three lev-els. By further employing its advantage over real atoms,namely its tunability via the applied magnetic flux andthe gate voltage, this enables us to bring the transitionsbetween the first three box levels into (near) resonancewith the first three cavity modes (Fig. 2), thereby drasti-cally enhancing the resulting probability of (nondegener-ate) parametric down-conversion |3 ||2. Thisrepresents the major advantage of the present scheme.We treat the full quantum dissipative dynamics of the

box-cavity system, incorporating the radiation of pho-tons from the cavity as well as nonradiative decay pro-cesses and dephasing in the CPB. We will present resultsfor the down-conversion efficiency, discuss the minimiza-tion of unwanted loss processes, and comment on possibleapplications in the end.

The model. - The CPB is a device [14] in which Cooperpairs can tunnel between two superconducting islandsdue to a Josephson-coupling EJ (tunable by an externalmagnetic flux in a split-junction geometry). The number
http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1http://arxiv.org/abs/cond-mat/0605232v1


2/5

2

0 0.1 0.2 0.3 0.4 0.5

0

5

10

15

Cooperpairboxlevels

|0

|2

|1,2

|0, , 2

2

2

2

Gate charge NG PDC process

1

2

3

in 3

Figure 2: (color online) Left: Energy-level diagram of theCPB - transition energies in units ofEC, for EJ/EC = 3, asa function of gate charge NG. At particular gate values (awayfrom the qubit regime near NG = 1/2), the CPB transitionsfrequencies are related in an integer ratio, (2 1)/1 = 1 : 1or 2 : 1, respectively. This enables to match them with thecavity modes, giving rise to particularly efficient degenerateor non-degenerate PDC (2 + or 3 + 2, re-spectively). Right: Simplified transition scheme for the non-degenerate PDC process considered in the text, with a detun-ing of the intermediate state |1, 2: 1 = + .

of transferred Cooper pairs N determines the chargingenergy, whose scale EC = e2/2C is set by the total ca-pacitance of the box island, and which can be controlledby application of an external gate voltage (expressed interms of a gate charge NG):

HCPB = 4EC(N NG)2

EJ2

N

|N + 1c N|c + h.c. (1)

Here |Nc represents a charge state of the CPB. Twogaps in a superconducting coplanar waveguide act asmirrors of a cavity for microwave photons, Hcavity =3

j=1 jaj aj , where we will focus attention on the three

lowest-lying cavity modes with j = j (we set 1).The electric field inside the cavity acts on the CPB viaa capacitive coupling, adding a fluctuating quantum-mechanical component to the gate charge NG. This re-sults [9] in an interaction

Hint =

3j=1

gj(aj + aj)

N . (2)

The coupling constants (vacuum Rabi frequencies) gj =g0

jj(x), with g0 = 2

eCgC

/Lc, are given in terms

of the mode functions 1(x) = sin(x/L), 2(x) =cos(2x/L) and 3(x) = sin(3x/L), which are definedon the interval x = L/2 . . . L /2.

The full Hamiltonian forming the basis of our analysisis thus given by

H = Hcavity + HCPB + Hint + Henv = H0 + Henv , (3)

where Henv includes the coupling to the various reservoirsforming the environment. This involves both the possi-bility for microwave photons to leak out of the cavity, aswell as the various possible nonradiative decay and deco-herence processes acting on the CPB. The details will bespecified below when setting up the master equation.

Basic considerations. - At least three basic featuresdistinguish such a setup from the usual optical pho-

ton PDC experiments employing nonlinear crystals: (i)There is no momentum conservation, as the system isessentially zero-dimensional. (ii) However, energy con-servation is much more restrictive, as the set of possiblefrequencies is limited to the discrete cavity modes. Forappropriate input frequency, this results in a resonant en-hancement of the PDC process. In contrast to a passivefiltering scheme, the bandwidth of the generated photonsis reduced without diminishing the signal intensity. (iii)The microwave polarization is fixed and thus cannot beused for entanglement. At the end of this paper we willpoint out other options that can be explored.

Estimating the PDC rate. - If the CPB is operated asa two-level system (qubit) [22], the decay of a 3 pho-ton into two lower-energy photons comes about throughmulti-step transitions, where at least one of the interme-diate virtual states will have an energy detuning of theorder of, which contributes a small factor (g/)2 to thePDC rate.

However, we can enhance the PDC rate by exploitingmore than the first two levels of the CPB. Indeed, byproperly tuning the Josephson coupling EJ and the gatecharge NG, it is possible to make the transitions betweenthe first three CPB energy levels |0 , |1 , |2 resonantwith the cavity modes. The PDC process we want to

consider is thus |0, 3 |2 |1, 2 |0, , 2,with 2 3 and 1 = + , such that all the tran-sitions are (nearly) resonant. This reduces the largestenergy denominator to the detuning , resulting in anenhancement of the PDC rate by a factor of (/)2.What limits the enhancement? If is made too small,one may end up with less than a complete pair of down-converted photons: Instead of passing virtually throughthe intermediate state |1, 2 containing a 2-photon andthe CPB in its first excited level, that state will acquirea significant real population. As a result, the tempo-ral correlation between photons would be destroyed, andnonradiative decays |1, 2 |0, 2 may occur, withoutemitting the second photon of frequency . Clearly, thereis a tradeoff between the achieved PDC probability andthe fidelity of down-conversion. This will be confirmedby the detailed simulations below.

We will find that it is possible to achieve a PDC proba-bility in the percent range that surpasses that of the mostefficient modern optical PDC schemes[23] (which havea PDC probability of about 104; though the absolutePDC rate in those experiments is about 109 times largerdue to the drastically higher input power). Earlier well-


3/5

3

known optical PDC experiments[5] generated less thanone usable coincidence detection event for every 1013 in-coming photons.

Simulation of the quantum-dissipative dynamics. - Inthe ideal case, one could integrate out the intermedi-ate state, yielding an effective PDC term of the form|0 2| a1a2 + h.c.. However, here we take into accountall loss processes, by solving for the full dynamics of the

CPB/cavity-system under an external microwave driveof frequency in 3, using a Markoff master equationof Lindblad form:

d

dt= (L0 + Ldrive + Ldecaycavity + LrelaxCPB + LdephCPB) (4)

Here L0 = i[H0, ], and the external microwave input,at a frequency in 3 and with an amplitude , isdescribed by Ldrive = i[Hdrive(t), ], with Hdrive(t) =a3e

iint + h.c..The dissipative terms in the Liouvillian are of Lindblad

form

L[A] AA 12

AA 12

AA . (5)

They describe: the decay of each cavity mode at a ratej (j = 1, 2, 3),

Ldecaycavity =j

j L[aj ], (6)

pure dephasing processes in the CPB that do not lead totransitions between levels (at rates ,j),

LdephCPB =j

,jL[|j j|], (7)

and nonradiative relaxation processes leading from a levell to a lower energy level j of the qubit:

LrelaxCPB =j 0). For reference, note that 10 GHz in typical experiments[10]. We have placedthe CPB at x = 0.3

L.

Discussion. - In order to interpret the results, we notethat the production of photon pairs at a rate PDC isbalanced by the decay of photons out of the cavity, ata rate . Thus, in an ideal lossless cavity PDC scheme,the probabilities to find the cavity in the states |, 2,| and |2 all become equal to PDC/(2). Therefore,we define PDC/(2) P|,2. The rate R of incomingphotons is given by R = 2||2/. Thus, the PDC prob-ability (chance of a given photon undergoing PDC) canbe set to PPDC = PDC/R = P|,2

2/||2 (we will usethis as a definition even where the scheme deviates fromideal conditions, discussing the PDC fidelity separately).

In Fig. 3, we have plotted the PDC probability asa function of the input frequency in and the detuning = 1 . In general, PPDC becomes maximal whenin matches the doublet frequencies 2 and 3 (modi-fied by the vacuum Rabi splitting): See the two verticalridges, independent of . There is a third resonanceat in = 3 g21/, with a dispersive shift depending on, for which the input frequency matches the energy ofthe outgoing state (dashed curve). Here we have defined


4/5

4

-0.001 -0.0005 0 0.0005 0.0010

0.005

0.01

0.015

0.02

0.025

-0.001 -0.0005 0 0.0005 0.0010

0.05

0.1

0.15

0.2

Q110

Q2Q2

Input frequency (in-1-2)/

(low deph.)

PPDC

P2

P

P2/50

Figure 4: (color online) Top: The parametric down con-version probability as a function of input frequency alongthe cross-section indicated in Fig. 3, and the probabili-ties P = fP(|) and P2 = fP(|2) for one or twophotons in the cavity, rescaled in the same manner, withf = 2/||2 (ideally PPDC = P = P2). The probabilityof an excited qubit, P2 = fP(|2), is also shown. Bottom:The non-ideality parameters Q1 = P|1,2/(P|,2) and

Q2

= P|/P|,21, reaching a minimum at the middle peak(thin line: for half the dephasing rate of / = 2 104).

g1 = g1

1N 0

as the actual vacuum Rabi frequency.

Although PPDC becomes maximal when any of these res-onances cross, this does not yield ideal photon pairs, aswill become clear shortly. Turning to Fig. 4 (top), we seea cross-section of PPDC (as a function of in). In addi-tion, we have included the probabilities of having one ortwo photons inside the cavity, which (in this normaliza-tion) should be identical to PPDC in an ideal scheme. Theapparent surplus in -photons is due to a nonradiativeprocess, as we will discuss now.

The relevant decay routes are: (i) |2 |1 | |0, (ii) |, 2 | |1 |0, (iii) |1, 2 |1 |0, and (iv) |1, 2 |2 |0. Process (i) leads to asingle photon being emitted, while (ii)-(iv) produce asingle 2 photon with no corresponding partner photon.All of these processes get suppressed with an increasingenergy mismatch || = |1 | between the states |1and |. A larger broadening of the levels (producedby dephasing or decay) may partially overcome this en-ergy mismatch, leading to a higher rate of unwanted lossprocesses. In order to quantify these processes, we have

plotted, in Fig. 4 (bottom), the non-ideality measuresQ1 = P|1,2/(P|,2) and Q2 = P|/P|,2 1 .Here Q1 measures the ratio of nonradiative relaxationfrom the intermediate state |1, 2 to the rate of pairemission, while Q2 indicates the degree to which the pop-ulations of| and |, 2 are identical (which is the caseideally, for j ). Both Q1 and |Q2| should be as smallas possible. The doublet peaks yield a large PDC rate,but also a large population of the CPB excited state |2,leading to the decay process (i) and a resulting surplus of

0.02 0.04 0.06 0.08 0.100.010

0.005

0.000

0.005

0.010

0.02 0.04 0.06 0.08 0.100.010

0.005

0.000

0.005

0.010

2

3)/

(2

3)/

/ = (1 )/ / = (1 )/

(a)

0.170.97

0.070

(b)

PPDC Q2

Figure 5: (color online) (a) Parametric down-conversion prob-ability PPDC, and (b) the non-ideality parameter Q2, as afunction of the detunings between the Cooper-pair box en-ergy levels 1,2 and the cavity modes (controlled by EJ andNG). At each point, the microwave input frequency in hasbeen chosen to minimize the parameter Q1 (cf. Fig. 4). Thedotted line indicates the cross-section displayed in Fig. 3.

-photons (Fig. 4, top). Thus, we observe that the vac-uum Rabi splitting of the doublet is essential: it allows

for the appearance of the third (middle) peak in PPDCthat has a far lower qubit population and correspondingrate of unwanted loss processes (minima in Q1,2). Anyreduction in the broadening of the peaks (set by , , )helps to further increase the quality of PDC.

The PDC quality can also be measured byevaluating the 2-photon correlator Kjl(t) =

al aj(t)aj(t)al

/

al al

ajaj

, which deter-

mines the probability to detect a mode j photon attime t inside the cavity, provided a mode l photon hasbeen detected at t = 0. Using the quantum regressiontheorem applied to our master equations, we have

checked that at small values of Q1,2, the ideal caseis indeed observed, where the correlator decays at arate , both for (j,l) = (1, 2) and (j,l) = (2, 1), whilenonradiative processes change these decay rates andmake them unequal.

The gate charge NG and the Josephson coupling EJchange the relevant energies 1 and 2 of the first twoexcited states of the Cooper pair box, and, to a lesserdegree, the matrix elements for the coupling to the cavityfield. At a given value of EJ, one can tune to NG =NG[EJ], such that the bare resonance condition 2 =3 is fulfilled. In the following, we consider the effectsof a small additional offset gate charge NG, which

mainly changes 2, while EJ itself is used to tune 1.The plots discussed so far have been obtained at fixedNG, while changing EJ. At any given value of EJ andNG, it is possible to select an input frequency in whichminimizes Q1. The resulting PDC rate has been plottedas a function of the detunings of the two CPB energylevels, 1 and 2, in Fig. 5 (a), while the correspondingparameter Q2 is shown in Fig. 5 (b) (Q1 is below 0.01 inthe relevant region where Q2 is small).

We conclude that, while maintaining a good reliability


5/5

5

of the PDC process, the down conversion probability canbecome on the order of a few percent in the present setup,which thus indeed represents a highly efficient source ofphoton pairs. Similar results have been found for otherreasonable parameter sets (e.g. g0/ = 102, j/ =105, jl/ = 10

4 (j < l), ,j/ = 103 (j > 0)).

Generation of entanglement. - The down-converted

and 2-photons can independently leak out of either

side of the cavity. By post-selecting (cf. [3, 24]) onlyevents where a photon is detected both in the left and theright arm each, one ends up with a frequency-entangledstate that is directly equivalent to the entangled tripletstate: |2L |R + |L |2R .We note, however,that a full Bell test requires measurements in a superpo-sition basis, which is hard to realize for states of differentenergies.

Another, simpler, possibility is to test for energy-timeentanglement as first proposed by Franson [4, 7, 25].This requires feeding the generated photons into Mach-Zehnder interferometers, each of them containing a shortand a long arm as well as a variable phase-shifter in oneof the arms. By measuring the photon-detection corre-lation between the altogether four output ports of thetwo interferometers, it is possible to violate the usualkinds of Bell inequalities. The great advantage of sucha scheme (particularly in the context of superconductingcircuit QED) is that it does not require the polarizationas a degree of freedom.

A less demanding, first experimental test of the PDCsource described here might measure the intensity cross-correlations of the microwave output beams (at and 2)or implement homodyning techniques [6, 26] to charac-terize the quantum state. Finally, it is worth noting that

for 3 2 a sufficiently strong vacuum Rabi splittingbetween |3 and |2 in principle enables a scheme wherea Rabi pulse is used to put exactly one excitation intothe system, which then decays in the way described here,thus realizing a source of microwave photon pairs on de-mand.

Conclusions. - In this paper we have described andanalyzed a setup for parametric down conversion in su-perconducting circuit cavity QED, suitable for the gen-eration of pairs of entangled microwave photons. In con-trast to earlier discussions, we have considered employ-ing a transition via the first three levels of the artificialatom (Cooper pair box), which can be tuned to achieve

a drastically enhanced PDC rate. We have analyzed thetrade-off between optimizing the PDC rate and minimiz-ing loss processes, by carrying out extensive numericalsimulations of the quantum dissipative dynamics. Thesetup described here can be realized by moderate mod-ifications of existing experiments, and it can hopefullyform the basis for more detailed investigations into thenonclassical properties of the microwave field in circuitQED experiments.

Acknowledgments. I thank S. Girvin, A. Wallraff, A.Blais, J. Majer, D. Schuster, M. Mariantoni, E. Solanoand R. Schoelkopf for discussions, and especially M. H.Devoret for pointing out the potential use of a three-levelconfiguration. This work was supported by the DFG.

[1] D. C. Burnham and D. L. Weinberg, Phys. Rev. Lett.25, 84 (1970).

[2] C. K. Hong and L. Mandel, Phys. Rev. Lett. 56, 58(1986).

[3] Y. H. Shih and C. O. Alley, Phys. Rev. Lett. 61, 2921(1988).

[4] J. Brendel, E. Mohler, and W. Martienssen, Europhys.Lett. 20, 575 (1992).

[5] P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V.Sergienko, and Y. Shih, Phys. Rev. Lett. 75, 4337 (1995).

[6] A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson,J. Mlynek, and S. Schiller, Phys. Rev. Lett. 87, 050402(2001).

[7] P. R. Tapster, J. G. Rarity, and P. C. M. Owens, Phys.Rev. Lett. 73, 1923 (1994).

[8] C. H. Bennett and D. P. DiVincenzo, Nature 404, 247(2000).

[9] A. Blais, R. S. Huang, A. Wallraff, S. M. Girvin, andR. J. Schoelkopf, Phys. Rev. A 69, 062320 (2004).

[10] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S.Huang, J. Majer, S. Kumar, S. M. Girvin, and R. S.Schoelkopf, Nature 431, 162 (2004).

[11] V. Bouchiat, D. Vion, P. Joyez, D. Esteve, and M. H.Devoret, Physica Scripta T76, 165 (1998).

[12] Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, Nature 398,786 (1999).

[13] J. E. Mooij, T. P. Orlando, L. Levitov, L. Tian, C. H.van der Wal, and S. Lloyd, Science 285, 1036 (1999).

[14] Y. Makhlin, G. Schon, and A. Shnirman, Rev. Mod.Phys. 73, 357 (2001).[15] I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C. J.

P. M. Harmans, and J. E. Mooij, Nature 431, 159 (2004).[16] D. I. Schuster et al., Phys. Rev. Lett. 94, 123602 (2005).[17] A. Wallraff et al., Phys. Rev. Lett. 95, 060501 (2005).[18] M. Oberparleiter and H. Weinfurter, Optics Communi-

cations 183, 133 (2000).[19] O. Benson, C. Santori, M. Pelton, and Y. Yamamoto,

Phys. Rev. Lett. 84, 2513 (2000).[20] O. Gywat, G. Burkard, and D. Loss, Phys. Rev. B 65,

205329 (2002).[21] C. Emary, B. Trauzettel, and C. W. J. Beenakker, Phys.

Rev. Lett. 95, 127401 (2005).[22] K. Moon and S. M. Girvin, Phys. Rev. Lett. 95, 140504

(2005).[23] K. Edamatsu, G. Oohata, R. Shimizu, and T. Itoh, Na-

ture 431, 167 (2004).[24] A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett.

49, 1804 (1982).[25] J. D. Franson, Phys. Rev. Lett. 62, 2205 (1989).[26] M. Mariantoni, M. J. Storcz, F. K. Wilhelm, W. D.

Oliver, A. Emmert, A. Marx, R. Gross, H. Christ, andE. Solano, cond-mat/0509737 (2005).

florian marquardt- efficient on-chip source of microwave photon pairs in superconducting circuit qed

Documents