Two–photon lasing by a superconducting qubit

P. Neilinger, M. Rehak, and M. GrajcarDepartment of Experimental Physics, Comenius University, SK-84248 Bratislava, Slovakia and

Institute of Physics, Slovak Academy of Sciences, 845 11 Bratislava, Slovakia

G. Oelsner and U. HubnerLeibniz Institute of Photonic Technology, P.O. Box 100239, D-07702 Jena, Germany

E. Il’ichevLeibniz Institute of Photonic Technology, P.O. Box 100239, D-07702 Jena, Germany andNovosibirsk State Technical University, 20 K. Marx Ave., 630092 Novosibirsk, Russia

(Dated: May 28, 2021)

We study the response of a magnetic-field-driven superconducting qubit strongly coupled toa superconducting coplanar waveguide resonator. We observed a strong amplification/dampingof a probing signal at different resonance points corresponding to a one and two-photon emis-sion/absorption. The sign of the detuning between the qubit frequency and the probe determineswhether amplification or damping is observed. The larger blue detuned driving leads to two-photonlasing while the larger red detuning cools the resonator. Our experimental results are in goodagreement with the theoretical model of qubit lasing and cooling at the Rabi frequency.

Motivated by the first experiment demonstrating theenergy exchange between a strongly driven superconduct-ing qubit and a resonator at the Rabi frequency ΩR

1,Hauss et al.2 elaborated a theoretical model to quantifythis phenomenon. Their model predicts large resonanteffects for the one- and two-photon resonance conditionsΩR = ωr−g3n and ΩR = 2ωr−g3n, where ωr is the fun-damental frequency of the resonator, g3 is the effectivecoupling energy, and n is the average number of photonsin the resonator at frequency ωr. Depending on the de-tuning between the driving frequency ωd and the qubiteigenfrequency ωq, either a lasing behavior (blue detun-ing ωd - ωq > 0) of the oscillator can be realized or thequbit can cool the oscillator (red detuning ωd - ωq < 0).According to the theory, one-photon lasing/cooling ef-fects vanish at the symmetry (degeneracy) point of thequbit. However, the two-photon processes persist at thesymmetry point where the qubit-oscillator coupling isquadratic and decoherence effects are minimized. There,the system realizes a single-atom-two-photon laser. Notea similar two-photon lasing by a quantum dot in a mi-crocavity, which was investigated theoretically in Ref. 3.

Experimentally, a single qubit one-photon lasing wasdemonstrated by the NEC group4. Here, a single chargequbit was used and a population inversion was providedby single-electron tunneling. Later on, the amplifica-tion/deamplification of a transmitted signal trough acoplanar waveguide resonator was achieved by a stronglydriven single flux qubit.5 However, the two-photon lasinghas not been experimentally demonstrated yet. In thispaper, we demonstrate the two-photon lasing, as well asconsiderable enhancement of one-photon lasing of a su-perconducting qubit by one order of magnitude in com-parison with Ref. 5. This enhancement was achieved bya much stronger coupling of the superconducting qubitto the resonator.

The lasing effect was investigated by making use of a

standard arrangement: a superconducting qubit placedin the middle of a niobium λ/2 coplanar waveguideresonator. The latter was fabricated by conventionalsputtering and dry etching of a 150-nm-thick niobiumfilm. The patterning uses an electron beam lithographyand a CF4 ion etching process. The aluminum qubitswere fabricated by the shadow evaporation technique.The coupling between the qubit and the resonator wasimplemented by a shared Josephson junction (Fig. 1).The dimensions of the qubit’s Josephson junctions are0.2× 0.3 µm2, 0.2× 0.2 µm2 and 0.2× 0.3 µm2, the crit-ical current density is about 200 A/cm2, and the areaof the qubit loop is 5 × 4.5 µm2. The resonance fre-quency and the quality factor of the resonator’s funda-mental mode taken for a weak probing (-141 dBm) areωr = 2π × 2.482 GHz, Q0 = 18 000. The same parame-ters of the third harmonics taken at the same power areωr3 = 2π × 7.446 GHz, Q3 = 3750. These values weredetermined from the transmission spectra of the coplanarwaveguide resonator.

a) b)

top and bottom layerJosephson junction

FIG. 1. a) Scanning electron microscope image of the qubitsincorporated into the coplanar waveguide resonator. b) De-tailed scheme of the qubits. The qubits share a Josephsonjunction with each other as well as with the resonator.

In practice, we measured a two-qubit sample whichrepresents a unit cell of a one-dimensional array of fer-romagnetically coupled qubits exhibiting a large Kerrnonlinearity.6 However, by applying a certain energy bias,one qubit can be set to a localized state, while the second

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is in the vicinity of its degeneracy point. This way, wecan measure the qubits separately to reconstruct theirparameters7, and the dynamics of the system is definedby a single qubit only. Therefore, to describe our find-ings, we will use the one-qubit model elaborated in Ref. 2,in which the corresponding Hamiltonian reads in the fluxbasis of the qubit:

H = −1

2εσz −

1

2∆σx − ~ΩR0 cos(ωdt)σz (1)

+ ~ωra†a+ gσz(a† + a)

where ∆ is the energy level separation of the two levelsystem at zero energy bias ε = 0, ΩR0 is the drivingamplitude of the applied microwave magnetic flux withfrequency ωd, and g is the coupling energy between thequbit and the resonator. The coupling energy scales withthe ratio of the magnitude of the persistent current in thequbit Iq and the critical current of the coupling Joseph-son junction Ic0 as

g ≡ ~ωg =~ωr

2π

IqIc0

√1

G0Zr(2)

where Zr = 50Ω is the wave impedance of the coplanarwaveguide resonator and G0 ≡ 2e2/h is the quantumconductance.

This Hamiltonian can be transformed by the Schrieffer-Wolff transformation U = exp(iS) with the generatorS = (g/~ωq) cos η(a+ a†)σy and a rotating wave approx-imation UR = exp(−iωdσzt/2) to the Hamiltonian2

H = ~ωra†a+

1

2~ΩRσz

+ g sin η[sinβσz − cosβσx](a+ a†)

− g2

~ωqcos2 η[sinβσz − cosβσx](a+ a†)2

Here, ~ωq =√ε2 + ∆2, ΩR =

√Ω2

R0 cos2 η + δω2,tan η = ε/∆, tanβ = δω/(ΩR0 cos η) and δω = ωd − ωq.The transmission of the resonator

t ∝ 〈a〉 (3)

where

〈a〉 = tr(ρa) (4)

was calculated numerically by the quantum toolboxQutip,8 solving the Liouville equation for the density ma-trix of the system in the rotating frame

˙ρ = − i~

[H, ρ

]+ Lqρ+ Lrρ (5)

where Lq and Lr are Lindblad superoperators

Lqρ =Γ↓2

(2σ−ρσ+ − ρσ+σ− − σ+σ−ρ)

+Γ↑2

(2σ+ρσ− − ρσ−σ+ − σ−σ+ρ)

+Γϕ

2(σz ρσz − ρ), (6)

Lrρ =κ

2(Nth + 1)(2aρa† − ρa†a− a†aρ)

+κ

2Nth(2a†ρa− aa†ρ− ρaa†). (7)

Here Nth = 1/ [exp(~ωT /kBT )− 1] is the thermal distri-bution function of photons in the resonator, κ is resonatorloss rate, Γ↓,↑ and Γϕ are the relaxation, excitation anddephasing rates in the rotating frame derived in Ref. 2

Γ↑,↓ =Γ0

4cos2 η(1± sinβ)2 +

Γϕ

2sin2 η cos2 β

Γϕ =Γ0

2cos2 η cos2 β + Γϕ sin2 η sin2 β (8)

0.48 0.50 0.52 0.54 0.56 0.58Φ/Φ0

−30

−20

−10

0

10

20

30

(ωs−ωr3

)/2π

(MHZ

)

a)

−32−28−24−20−16−12−8−40

0.495 0.500 0.505Φ/Φ0

−30

−20

−10

0

10

20

30(ω

s−ωr3

)/2π

(MHZ

)

b)

−32−28−24−20−16−12−8−40

0.495 0.500 0.505Φ/Φ0

−50

−40

−30

−20

−10

0

T[dB

]

c)

FIG. 2. (a) Resonator transmission as a function of themagnetic flux and the detuning of the resonator from the thirdharmonic resonance frequency ωr3/2π. The anticrossings ofqubit A and B are separated by a magnetic flux of about55×10−3Φ0. (b) Close view of the transmission in the vicinityof the left qubit’s (A) degeneracy point and (c) the cut of thetransmission map along the dashed line in (b). The crossesare experimental data and the solid line is a theoretical curvecalculated from Eq. 9.

The qubit parameters used for the numerical calcula-tions were determined independently from the transmis-sion of the resonator t coupled to the undriven qubit. Fora weak microwave signal with frequency ωs, the transmis-sion can be expressed in simple form9

t = −iκext2

δωq + iγ

ω2g cos2 η − (δωr + iκ/2) (δωq + iγ)

(9)

where δωq = ωq − ωs, δωr = ωr − ωs, κext is the ex-ternal loss rate of the resonator and γ is the qubit deco-herence rate. The experimental data was fitted by Eq.9(see Fig. 2) and the qubit parameters obtained from thefitting procedure are given in the table I.

We have investigated the stimulated emission effect ob-served when strongly driving the system at a frequency

3

Qubit Ip ∆/2π g/2π γ/2π(nA) (GHz) (MHz) (MHz)

A 208 6.39 109 15B 138 5.28 77 20

TABLE I. Qubit parameters determined from the fitting pro-cedure.

0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58Φ/Φ0

−2.0

−1.5

−1.0

−0.5

0.0

0.5

(ωs−ωr)/

2π(M

HZ

)

−24−21−18−15−12−9−6−30

FIG. 3. a) Resonator transmission in dB units as a functionof the magnetic flux and the detuning of the resonator fromthe resonance frequency 2.482 GHz.

ωd/2π = 9ωr/2π = 22.338 GHz for qubit A. The res-onator transmission was measured by a network analyzerat resonance ωr for magnetic fluxes marked by the blackrectangular area in Fig. 3 and is shown in Fig. 4a. Ata driving power Pd = −103 dBm, two emission peaks(e.1, e.2) accompanied by two attenuation dips (a.1, a.2)appear in the transmission spectra. The increase of thetransmission is accompanied by a narrowing of the res-onance curve. The one-photon (a.1, e.1) and two pho-ton (a.2, e.2) processes are enhanced at resonance withthe Rabi frequency of the qubit ΩR = ωr − g3n andΩR = 2ωr − g3n, respectively. These results are in goodagreement with the theoretical model2 described abovefor parameters given in Table I. By a strong couplingof the qubit to the resonator, we have achieved a con-siderable enhancement of the lasing, nearly one order ofmagnitude, in comparison with the results presented inRef. 5. Further improvement is possible by increasingthe relaxation rate of the qubit, for instance, by placinga gold resistor close to the qubit loop.

To conclude, we have experimentally demonstratedsingle-qubit one-photon and two-photon lasing. The ex-perimental results are in good agreement with the theo-retical model developed by Hauss et al.2 The considerableenhancement of lasing effect was achieved by strongercoupling of the superconducting qubit to the resonator,and theoretical calculations show that it can be enhancedfurther by increasing the relaxation rate of the qubit.Such improvement could enable to observe even higher-order processes analysed in Ref. 10.

The research leading to these results has received fund-ing from the European Communitys Seventh FrameworkProgramme (FP7/2007-2013) under Grant No. 270843(iQIT) and APVV-DO7RP003211. This work was alsosupported by the Slovak Research and Development

0.472 0.474 0.476 0.478 0.480 0.482Φ/Φ0

0.0

0.5

1.0

1.5

2.0

Transm

ission

a.2 a.1

e.1 e.2

Driving ONDriving OFF

0.472 0.474 0.476 0.478 0.480 0.482Φ/Φ0

1.0

1.2

1.4

1.6

1.8

2.0

2.2

nFIG. 4. Transmission of the resonator at fixed frequencyωs/2π for a driving signal with frequency ωd/2π = 9ωr/2π =22.338 GHz and power pd = −102 dBm for the drivingswitched off and on (a). Regions e.1, e.2 and a.1, a.2 ex-hibit amplification and attenuation of the signal, respectively.Panel (b) shows the simulated average photon number in theresonator obtained for the qubit parameters.

−100 −50 0 50 100(ωs − ωr)/2π (KHz)

2

4

6

8

10

n

Driving ONDriving OFF

FIG. 5. Resonance curve of the resonator at driving switchedoff (solid line) and on (dashed line) with frequency ωd/2π =9ωr/2π = 22.338 GHz and power pd = −101 dBm in regione.1. The amplitude increases by a factor of ∼ 9 and band-width is reduced by factor of 10.

Agency under the contract APVV-0515-10 and APVV-0808-12(former projects No. VVCE-0058-07, APVV-0432-07) and LPP-0159-09. The authors gratefully ac-knowledge the financial support of the EU through theERDF OP R&D, Project CE QUTE & metaQUTE,ITMS: 24240120032 and CE SAS QUTE. EI acknowl-

4

edges partial support of Russian Ministry of Educa- tion and Science, in the framework of state assignment8.337.2014/K.

1 E. Il’ichev, N. Oukhanski, A. Izmalkov, Th.Wagner,M. Grajcar, H.-G. Meyer, A. Smirnov, A. Maassen van denBrink, M. H. S. Amin, and A. Zagoskin, Phys. Rev. Lett.91, 097906 (2003).

2 J. Hauss, A. Fedorov, C. Hutter, A. Shnirman, andG. Schon, Physical Review Letters 100, 037003 (2008).

3 E. del Valle, S. Zippilli, F. P. Laussy, A. Gonzalez-Tudela,G. Morigi, and C. Tejedor, Phys. Rev. B 81, 035302(2010).

4 O. Astafiev, K. Inomata, A. O. Niskanen, T. Yamamoto,Y. A. Pashkin, Y. Nakamura, and J. S. Tsai, Nature 449,588 (2007).

5 G. Oelsner, P. Macha, O. V. Astafiev, E. Il’ichev, M. Gra-jcar, U. Hubner, B. I. Ivanov, P. Neilinger, and H.-G.Meyer, Phys. Rev. Lett. 110, 053602 (2013).

6 M. Rehak, P. Neilinger, M. Grajcar, G. Oelsner,U. Hubner, E. Il’ichev, and H.-G. Meyer, Applied PhysicsLetters 104, 162604 (2014).

7 E. Il’ichev, N. Oukhanski, T. Wagner, H.-G. Meyer,A. Smirnov, M. Grajcar, A. Izmalkov, D. Born, W. Krech,and A. Zagoskin, Low Temp. Phys. 30, 620 (2004).

8 J. Johansson, P. Nation, and F. Nori, Computer PhysicsCommunications 184, 1234 (2013).

9 A. N. Omelyanchouk, S. N. Shevchenko, Y. S. Greenberg,O. Astafiev, and E. Il’ichev, Low Temperature Physics 36,893 (2010).

10 S. N. Shevchenko, G. Oelsner, Y. S. Greenberg, P. Macha,D. S. Karpov, M. Grajcar, U. Hubner, A. N. Omelyan-chouk, and E. Il’ichev, Phys. Rev. B 89, 184504 (2014).