Large collective Lamb shift of two distant superconducting artificial atoms
P. Y. Wen, K.-T. Lin, A. F. Kockum, B. Suri, H. Ian, J. C. Chen, S. Y. Mao, C. C. Chiu, P. Delsing, F. Nori, G.-D. Lin, I.-C. Hoi
LLarge collective Lamb shift of two distant superconducting artificial atoms
P. Y. Wen,
1, 2, ∗ K.-T. Lin, ∗ A. F. Kockum,
4, 5, ∗ B. Suri,
6, 4
H. Ian,
7, 8
J. C. Chen,
1, 2
S. Y. Mao, C. C. Chiu, P. Delsing, F. Nori,
5, 11
G.-D. Lin, and I.-C. Hoi
1, 2, † Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan Center for Quantum Technology, National Tsing Hua University, Hsinchu 30013, Taiwan CQSE, Department of Physics, National Taiwan University, Taipei 10617, Taiwan Department of Microtechnology and Nanoscience,Chalmers University of Technology, 412 96 Gothenburg, Sweden Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan Department of Instrumentation and Applied Physics,Indian Institute of Science, Bengaluru 560012, India Institute of Applied Physics and Materials Engineering, University of Macau, Macau UMacau Zhuhai Research Institute, Zhuhai, Guangdong, China Institute of Electro-Optical Engineering, National Chiao Tung University, Hsinchu 30013, Taiwan Department of Electrical Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA (Dated: April 30, 2019)Virtual photons can mediate interaction between atoms, resulting in an energy shift known as acollective Lamb shift. Observing the collective Lamb shift is challenging, since it can be obscuredby radiative decay and direct atom-atom interactions. Here, we place two superconducting qubits ina transmission line terminated by a mirror, which suppresses decay. We measure a collective Lambshift reaching 0.8% of the qubit transition frequency and exceeding the transition linewidth. Wealso show that the qubits can interact via the transmission line even if one of them does not decayinto it.
Introduction.
In 1947, when attempting to pinpointthe fine structure of the hydrogen atom, Lamb andRetherford [1] discovered a small energy difference be-tween the levels S / and P / , which were thought tobe degenerate according to Dirac’s theory of electrons.This energy difference between the two levels can be un-derstood when vacuum fluctuations are included in thepicture, as was verified later by self-energy calculationsin the framework of quantum field theory [2–4]. Brieflyput, a hydrogen atom will emit photons which are instan-taneously reabsorbed; while these “virtual” photons arenot detectable by themselves, they leave their traces inthe Lamb shift.The hydrogen atoms that Lamb and Retherford usedfor their experiment were obtained from molecular hydro-gen through tungsten catalyzation. Since the conversionrate for this process was very low, the S / level wasonly populated in a few atoms. Hence, the observableeffects of virtual photon processes were limited to self-interaction; exchanges of virtual photons between atomscould not be detected. However, it was later realized thatatom-atom interaction mediated by virtual photons alsogives rise to an energy shift, referred to as a collective, orcooperative, Lamb shift [5–9]. The atom-atom interac-tion also underpins the collective decay known as Dickesuperradiance [10, 11].There are several obstacles impeding the experimentalobservation of the collective Lamb shift. The shift can beenhanced by using many atoms, but, if these atoms aretoo close together, direct atom-atom interactions (not viavirtual photons) can obscure the effect. Furthermore, the interaction giving rise to the collective Lamb shift is rel-atively weak in three dimensions, and the shift can alsobe hidden by the radiative linewidth (e.g., due to the col-lective decay). Despite these obstacles, there have beena few experimental demonstrations of collective Lambshifts: in xenon gas [12], iron nuclei [13], rubidium va-por [14], strontium ions [15], cold rubidium atoms [16],and potassium vapor [17]. Mostly, these experimentsused developments in atomic trapping and cooling [18]that have enabled higher densities of atomic ensembles,leading to a strong coupling between atomic condensatesand cavity fields [19, 20]. An improved theoretical un-derstanding [21–23] of collective Dicke states also aidedsome of the experiments.With the single exception of Ref. [15], these previ-ous experiments all required a large number of atomsto demonstrate a collective Lamb shift. The experimentof Ref. [15] only used two atoms, but the measured shiftwas small, 0.2% of the transition linewidth. In this Let-ter, we demonstrate a large collective Lamb shift for twoartificial atoms that significantly exceeds the linewidthand reaches 0.8% of the atomic transition frequency.Our experimental setup, depicted in Fig. 1, is a super-conducting quantum circuit [24, 25] with two transmonqubits [26] coupled to a one-dimensional (1D) waveguide.In such superconducting circuits, strong [25, 27, 28],and even ultrastrong [29–32], coupling can be engi-neered between the qubits and photons in the waveguide.Compared to three-dimensional setups, the 1D versionstrengthens the interaction between qubits and reducesthe decay into unwanted modes. These features have en- a r X i v : . [ qu a n t - ph ] A p r Figure 1. The experimental setup. (a) A conceptual sketch ofthe setup. Two atoms are placed in front of a mirror and inter-act via virtual photons of different frequencies. (b) A photoof the device. Qubit 1 (shown in the zoom-in on the left;the two long bright parts form the qubit capacitance and thegap in the middle between them is bridged by two Josephsonjunctions forming a SQUID loop) is placed L ’
33 mm fromQubit 2, which sits at the end of the transmission line (i.e.,at the mirror). The characteristic impedance of the transmis-sion line is Z ’
50 Ω . The relatively long distance L makesit easier to tune Qubit 1 between nodes and anti-nodes of theelectromagnetic (EM) field in the transmission line by tun-ing the qubit transition frequency. This tuning can be usedto calibrate the velocity v of the EM field in the transmissionline [68]. (c) A sketch of the signal routing for the experiment.Each qubit frequency can be tuned by local magnetic fieldsvia local voltage biases ( V , V ) and both frequencies can betuned by a global magnetic field from a superconducting coilcontrolled by V . For measurements, a coherent signal at fre-quency ω p is generated by a vector network analyzer (VNA) atroom temperature and fed through attenuators (red squares)to the sample, which sits in a cryostat cooled to
20 mK toavoid thermal fluctuations affecting the experiment. The re-flected signal passes a bandpass filter (BPF) and amplifiers,and is then measured with the VNA. abled many important quantum-optical experiments in1D waveguide QED in superconducting circuits in thepast decade [25, 28, 30, 33–43] and inspired a wealth oftheoretical studies for this platform [25, 33, 44–67].As shown in Fig. 1, the transmission line to whichthe qubits couple is terminated in a capacitive couplingto ground, which is equivalent to placing a mirror in awaveguide. The presence of this mirror separates our ex-periment from that of Ref. [36], where two superconduct-ing qubits were coupled to an open transmission line. Insuch an open waveguide, the connection between collec-tive decay and the collective Lamb shift entails that theshift always will be smaller than the linewidth [55], andthe measurements of elastic scattering in Ref. [36] couldthus not resolve the collective Lamb shift. Although asplitting in the fluorescence spectrum (inelastic scatter-ing) indicated the presence of the collective Lamb shift,it is not straightforward to extract the size of the shift from the size of the splitting [36, 55]. In our setup, thepresence of the mirror introduces interference effects thatsuppresses the collective decay more than the collectiveLamb shift [67, 68], allowing us to clearly resolve the shiftin simple reflection measurements of elastic scattering.Interestingly, it turns out that these interference effectsallow us to couple the two qubits via the transmissionline even when one of the qubits is unable to relax intothe transmission line.
Device and characterization.
In our device, the in-terqubit separation L is fixed. However, we can varythe qubit transition frequencies ω by applying a localmagnetic flux [see Fig. 1(c)] and thus change the effec-tive distance L/λ , where the wavelength λ = 2 πv/ω and v is the propagation velocity of the electromagnetic(EM) field in the waveguide [38]. Since Qubit 2 is placednext to the mirror, it will always be at an antinode ofthe voltage field in the waveguide [see Fig. 2(b)]. Qubit1, on the other hand, can be tuned to a voltage node.In this case, Qubit 1 will not couple to the waveguide atits transition frequency, and thus will not contribute toany decay [38, 67]. However, the collective Lamb shiftarises due to emission and absorption of virtual photonsin all other modes of the continuum in the waveguide,which results in an interaction of strength ∆ between thequbits. This interaction (collective Lamb shift) leads toan avoided level crossing between the two qubits, whichshows up as a frequency splitting of ∆ in reflection mea-surements of the system using a weak coherent probe atfrequency ω p . Our experiment thus clearly demonstrateshow the collective Lamb shift has contributions from vir-tual photons of many frequencies.We first characterize each of the two transmon qubitsthrough spectroscopy. We detune the transition fre-quency of one of the qubits far away and measure the am-plitude reflection coefficient | r | of a weak coherent probetone (i.e., the probe Rabi frequency Ω p is much smallerthan the decoherence rate γ of the qubit) as a function ofthe flux controlling the other qubit’s transition frequencyand of the probe frequency ω p . The results are shown inFig. 2 (Qubit 1 in the left column and Qubit 2 in theright column). For Qubit 1, which is placed at a distance L from the mirror, the spectroscopy data in Fig. 2(c)shows a linewidth narrowing [compare the linecuts A andB from Fig. 2(c), plotted in Fig. 2(e)] and a disappearingresponse around .
75 GHz . At this frequency, the effec-tive distance between Qubit 1 and the mirror is L = 7 λ/ ,which places the qubit at a node for the EM field in thetransmission line, as illustrated in Fig. 2(a), and thus ef-fectively decouples the qubit from the transmission line,reducing its relaxation rate to zero [38]. Qubit 2, on theother hand, is always at an antinode of the EM field inthe transmission line [Fig. 2(b)] and thus has an equallystrong response at all frequencies [Fig. 2(d), (f)].We perform further spectroscopy in the full range − , which is the bandwidth of the cryogenic low- Figure 2. Single-tone spectroscopy of the individual qubits.(a), (b) Electromagnetic mode structure (red curve) in thetransmission line seen by Qubit 1 (Q1) and Qubit 2 (Q2),respectively. (c), (d) Amplitude reflection coefficient | r | fora weak coherent probe as a function of probe frequency ω p and qubit transition frequency (controlled by the voltages V and V for Qubit 1 and Qubit 2, respectively). For Qubit 1,panel (c) shows how the response disappears when the qubitends up at a node for the EM field around .
75 GHz . Dur-ing these measurements, the frequency of the other qubit istuned far from resonance with the probe. (e), (f) Linecutsfrom panels (c) and (d) as indicated. Crosses are experimen-tal data and solid curves are fits following Ref. [37]. Theextracted parameters are given in Table S1 in the supplemen-tary material [68]. The linewidth of the dip, which occurs atthe resonance ω p = ω , is set by the qubit decoherence rate γ = Γ/ γ φ , where Γ is the relaxation rate and γ φ is thepure dephasing rate. Relaxation into other channels than thetransmission line will affect the extracted value of γ φ . Thedepth of the dip is set by the ratio Γ/γ φ ; since Γ decreasesclose to the node of the field, the dip in linecut A is moreshallow than that in B. noise amplifier in our experimental setup. The maximumqubit frequency is outside this bandwidth. This datais presented in the supplementary material [68]. Fromthese measurements, we extract [37] the qubit relaxationrate Γ into the transmission line, the pure dephasing rate γ φ (which also contains contributions from relaxation toother channels), and the speed of light in the transmis-sion line. We further use two-tone spectroscopy, drivingat the qubit frequency ω and probing around the tran-sition frequency ω from the first excited state to thesecond excited state, to determine the anharmonicity ofthe qubits. All extracted and derived parameters aresummarized in Table I. Collective Lamb shift.
We now turn to experimentswhere both qubits are involved and the collective Lambshift is measured. We fix the transition frequency ofQubit 2 to ω / π = 4 .
75 GHz , the frequency at whichQubit 1 is at a node of the EM field [see Fig. 2(c)]. Wethen tune the frequency of Qubit 1 to values around this point and measure the reflection of a weak probe signalon the system for frequencies close to ω . The resultsof these measurements are displayed in Fig. 3(a). Weobserve a clear anti-crossing between the vertical reso-nance, corresponding to Qubit 2, and the diagonal res-onance, corresponding to Qubit 1. The observation ofthis anti-crossing indicates that the two qubits are cou-pled on resonance with strength ∆ through a coherentinteraction, which must be mediated by the transmissionline since the qubits are distant from each other. Theminimum size of the separation, shown in the linecut inFig. 3(c), is ∆ ’ π ×
38 MHz .If the qubits were uncoupled, they would have eigen-states | i , | i , | i , and | i , with energies , ~ ω , ~ ω , and ~ ω , respectively. Here, and denoteground and excited states of a single qubit, respectively;the first number in the kets is for Qubit 1 and the sec-ond number is for Qubit 2. Due to the coupling, theeigenstates | i and | i are replaced by the symmet-ric and anti-symmetric eigenstates | s i = √ ( | i + | i ) and | a i = √ ( | i − | i ) , respectively, with eigenener-gies ~ ( ω ± ∆ ) [55]. When the coupling is due to virtualphotons, as in our experiment, this thus gives a collectiveLamb shift of ~ ∆ , as illustrated in the inset in Fig. 3(c).If the two qubits were placed in an open transmissionline, it would not be possible to observe the collectiveLamb shift in this measurement, since each of the tworesonances would have a linewidth set by a relaxationrate Γ = 2 ∆ [55]. This is not easily circumvented, sinceit is the coupling to the transmission line of the two qubitsthat determines both the relaxation into the transmissionline and the strength of the interaction that is mediatedvia the transmission line. However, the presence of themirror in our setup breaks this close connection betweenthe linewidth and the collective Lamb shift. In our setup,the collective Lamb shift is given by [67, 68] ∆ = Γ n sin h ω v ( x + x ) i + sin h ω v | x − x | io , (1)where x j denotes the distance of Qubit j from the mir-ror and Γ = p Γ ( ω ) Γ ( ω ) , with Γ j ( ω ) the bare re-laxation rate of Qubit j at frequency ω into an opentransmission line. This is calculated using the standardmaster-equation approach with the Born-Markov approx-imation and tracing out the photonic modes of the trans-mission line [68, 69]. When x = 0 and x correspondsto Qubit 1 being at a node of the field in the transmis-sion line, as in Fig. 3, the collective Lamb shift becomes ∆ = 2 Γ . However, since Qubit 1 is at a node, boththe effective relaxation rate of Qubit 1 and the collectivedecay rate of the two qubits becomes zero. The only con-tribution from relaxation to the linewidths for the states | s i and | a i is half of Γ , the effective relaxation rate ofQubit 2. In this experiment, we used Γ ≈ Γ (giving ashift ∆ ≈ Γ and a linewidth γ c ≈ Γ ), but we note Qubit E C /h [GHz] ω / π [GHz] Γ/ π [MHz] γ φ / π [MHz] γ/ π [MHz] β v [ m/s] Q . (Antinode) .
068 27 .
18 2 .
15 15 .
74 0 .
717 0 . Q . (Antinode) .
746 28 .
03 2 .
785 16 . . Table I. Extracted and derived qubit parameters. We extract ω , Γ and γ from fitting the spectroscopic magnitude and phasedata according to Ref. [37]. Note that the effective relaxation rate Γ at an antinode is twice what the relaxation rate wouldbe in an open transmission line. The velocity v is extracted by finding multiple nodes of the field for Qubit 1 [68]. From thetwo-tone spectroscopy, we extract the anharmonicity, which approximately equals the charging energy E C of the transmonqubits. We calculate γ φ from Γ and γ , and the ratio β = C c /C Σ between the coupling capacitance C c to the transmission lineand the qubit capacitance C Σ from Γ and E C .Figure 3. Collective Lamb shift. (a) The amplitude reflection coefficient for a weak probe as a function of the probe frequency ω p and the transition frequency of Qubit 1 (controlled through the voltage V ). The frequency of Qubit 2 is fixed at ω = 4 .
75 GHz and the frequency of Qubit 1 is tuned through resonance with this frequency. (b) Theory simulation [68] of the single-tonespectroscopy data in panel (a). The simulation is done with previously fitted parameters from Table I, with the exceptions ofthe free paramaters β = 0 . , β = 0 . , and γ φ / π = 2 . for Qubit 1, which all are close to the values in Table I. Theagreement between the data in (a) and the simulation in (b) is excellent. (c) A linecut of the data and theory, marked by thedashed line in panels (a) and (b), at the point where the two qubits are on resonance and the collective Lamb shift ∆ is mostclearly visible. From this figure, we extract a collective Lamb shift of ∆ ’ π ×
38 MHz . The inset on the right shows the levelstructure of the eigenstates of the qubits that are coupled through the collective Lamb shift. The two dips in the reflectioncorresponds to the symmetric and anti-symmetric eigenstates | s i and | a i . Each of these two states have the same decay rate,giving a linewidth γ c smaller than the collective Lamb shift due to the presence of the mirror, as explained in the text. that the collective Lamb shift could be made many timeslarger than the linewidths by instead designing the qubitssuch that Γ (cid:29) Γ .The fact that we can measure the collective Lamb shifteven though Qubit 1 ostensibly is decoupled from thetransmission line confirms several predictions about howvirtual photons influence relaxation and qubit-qubit in-teraction. The relaxation from Qubit 1 is stimulated byvirtual photons in the transmission line at the transi-tion frequency ω . The relaxation is suppressed whenQubit 1 is placed at a node for the virtual photons atthis frequency [38]. However, Qubit 1 is clearly coupledvia virtual photons to Qubit 2. Thus, the virtual photonsmediating this coupling, and causing the collective Lambshift, must have frequencies that are not equal to ω .In fact, the coupling is given by a sum over all virtualmodes at frequencies separate from ω [55].Finally, we note that there are several processes, withreal photons, where a strong drive shifts or dresses energylevels of qubits to create an effect that could look similar to what we have observed. To rule out such effects, e.g.,the Mollow triplet [70] and Autler-Townes splitting [71],we measure ∆ as a function of the power P of the coher-ent probe. The results are shown in Fig. 4. Clearly, theenergy shift ∆ is independent of P (before the power ishigh enough to saturate the qubits), indicating that thecollective Lamb shift we measure really is due to virtualphotons. Summary and outlook.
In this Letter, we demon-strated a large collective Lamb shift with two distantsuperconducting qubits in front of an effective mirror ina 1D transmission-line waveguide. Using interference ef-fects due to the mirror, we overcame previous limitationson the size of the shift compared to the linewidth, allow-ing us to observe a shift reaching 0.8% of the qubit tran-sition frequency and exceeding the transition linewidth.We explained how future experiments could increase theshift relative to the linewidth even more. This experi-ment also demonstrated that a qubit can couple to an-other qubit via the transmission line even though the first
Figure 4. Energy shift as a function of input power. (a)Amplitude reflection coefficient | r | as a function of probe fre-quency ω p and probe power P for both qubit frequencies fixedat ω = 4 .
75 GHz . The data agrees very well with theoreticalsimulations [68]. At high probe powers, the qubits are satu-rated and most photons are simply reflected from the mirror,resulting in | r | ≈ . (b) The extracted splitting ∆ from panel(a) as a function of P . The splitting is clearly independent ofthe input power in this wide range. qubit is prevented from decaying into the transmissionline. These results give further insight into how virtualphotons affect both atomic relaxation rates and inter-atomic coupling, and how these effects can be controlledusing interference, which could have applications for de-signing, e.g., devices that process quantum information. Acknowledgements.
I.-C.H. and J.C.C. would like tothank I. A. Yu and C.-Y. Mou for fruitful discussions.This work was financially supported by the Center forQuantum Technology from the Featured Areas ResearchCenter Program within the framework of the Higher Ed-ucation Sprout Project by the Ministry of Education(MOE) in Taiwan. I.-C.H. acknowledges financial sup-port from the MOST of Taiwan under project 107-2112-M-007-008-MY3. B.S., A.F.K., and P.D. acknowledgesupport from the Knut and Alice Wallenberg Founda-tion. G.-D.L acknowledges support from the MOST ofTaiwan under Grant No. 105-2112-M-002-015-MY3 andNational Taiwan University under Grant No. NTUCC-108L893206. H. I. acknowledges the support by FDCTof Macau under grant 065/2016/A2, by University ofMacau under grant MYRG2018-00088-IAPME, and byNNSFC under grant 11404415. J.C.C. acknowledges fi-nancial support from the MOST of Taiwan under project107-2112-M-007-003-MY3. F.N. acknowledges supportfrom the MURI Center for Dynamic Magneto-Opticsvia the Air Force Office of Scientific Research (AFOSR)award No. FA9550-14-1-0040, the Army Research Of-fice (ARO) under grant No. W911NF-18-1-0358, theAsian Office of Aerospace Research and Development(AOARD) grant No. FA2386-18-1-4045, the Japan Sci-ence and Technology Agency (JST) through the Q-LEAPprogram, the ImPACT program, and CREST GrantNo. JPMJCR1676, the Japan Society for the Promotionof Science (JSPS) through the JSPS-RFBR grant No. 17-52-50023 and the JSPS-FWO grant No. VS.059.18N, the RIKEN-AIST Challenge Research Fund, and the JohnTempleton Foundation. ∗ These authors contributed equally † e-mail:[email protected][1] W. E. Lamb and R. C. Retherford, Fine Structure of theHydrogen Atom by a Microwave Method, Phys. Rev. ,241 (1947).[2] H. A. Bethe, The Electromagnetic Shift of Energy Levels,Phys. Rev. , 339 (1947).[3] T. A. Welton, Some Observable Effects of the Quantum-Mechanical Fluctuations of the Electromagnetic Field,Phys. Rev. , 1157 (1948).[4] C. Cohen-Tannoudji, Fluctuations in Radiative Pro-cesses, Phys. Scr. T12 , 19 (1986).[5] V. M. Fain, On the Theory of the Coherent SpontaneousEmission, J. Exp. Theor. Phys. , 798 (1959).[6] R. H. Lehmberg, Radiation from an N-Atom System. I.General Formalism, Phys. Rev. A , 883 (1970).[7] F. T. Arecchi and D. M. Kim, Line shifts in cooperativespontaneous emission, Opt. Commun. , 324 (1970).[8] R. Friedberg, S. R. Hartmann, and J. T. Manassah, Fre-quency shifts in emission and absorption by resonant sys-tems of two-level atoms, Phys. Rep. , 101 (1973).[9] M. O. Scully and A. A. Svidzinsky, The Lamb Shift - Yes-terday, Today, and Tomorrow, Science , 1239 (2010).[10] R. H. Dicke, Coherence in Spontaneous Radiation Pro-cesses, Phys. Rev. , 99 (1954).[11] N. Shammah, S. Ahmed, N. Lambert, S. De Liberato,and F. Nori, Open quantum systems with local and col-lective incoherent processes: Efficient numerical simula-tions using permutational invariance, Phys. Rev. A ,063815 (2018), arXiv:1805.05129.[12] W. R. Garrett, R. C. Hart, J. E. Wray, I. Datskou, andM. G. Payne, Large multiple collective line shifts ob-served in three-photon excitations of Xe, Phys. Rev. Lett. , 1717 (1990).[13] R. R¨ohlsberger, K. Schlage, B. Sahoo, S. Couet, andR. R¨uffer, Collective Lamb Shift in Single-Photon Su-perradiance, Science , 1248 (2010).[14] J. Keaveney, A. Sargsyan, U. Krohn, I. G. Hughes,D. Sarkisyan, and C. S. Adams, Cooperative Lamb Shiftin an Atomic Vapor Layer of Nanometer Thickness, Phys.Rev. Lett. , 173601 (2012).[15] Z. Meir, O. Schwartz, E. Shahmoon, D. Oron, andR. Ozeri, Cooperative Lamb Shift in a MesoscopicAtomic Array, Phys. Rev. Lett. , 193002 (2014),arXiv:1312.5933.[16] S. J. Roof, K. J. Kemp, M. D. Havey, and I. M.Sokolov, Observation of Single-Photon Superradianceand the Cooperative Lamb Shift in an Extended Sam-ple of Cold Atoms, Phys. Rev. Lett. , 073003 (2016),arXiv:1603.07268.[17] T. Peyrot, Y. R. P. Sortais, A. Browaeys, A. Sargsyan,D. Sarkisyan, J. Keaveney, I. G. Hughes, and C. S.Adams, Collective Lamb Shift of a Nanoscale Atomic Va-por Layer within a Sapphire Cavity, Phys. Rev. Lett. ,243401 (2018), arXiv:1801.01773.[18] M. D. Lukin, Colloquium: Trapping and manipulatingphoton states in atomic ensembles, Rev. Mod. Phys. ,
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Rev. , 703 (1955). upplementary Material for “Large collective Lamb shift of two distantsuperconducting artificial atoms”
P. Y. Wen,
1, 2, ∗ K.-T. Lin, ∗ A. F. Kockum,
4, 5, ∗ B. Suri,
6, 4
H. Ian,
7, 8
J. C. Chen,
1, 2
S. Y. Mao, C. C. Chiu, P. Delsing, F. Nori,
5, 11
G.-D. Lin, and I.-C. Hoi
1, 2, † Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan Center for Quantum Technology, National Tsing Hua University, Hsinchu 30013, Taiwan Department of Physics, National Taiwan University, Taipei 10617, Taiwan Department of Microtechnology and Nanoscience,Chalmers University of Technology, 412 96 Gothenburg, Sweden Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan Department of Instrumentation and Applied Physics,Indian Institute of Science, Bengaluru 560012, India Institute of Applied Physics and Materials Engineering, University of Macau, Macau UMacau Zhuhai Research Institute, Zhuhai, Guangdong, China Institute of Electro-Optical Engineering, National Chiao Tung University, Hsinchu 30013, Taiwan Department of Electrical Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA (Dated: April 30, 2019)
CONTENTS
S1. Derivation of master equation and qubit-qubit interaction 1S2. Reflection coefficient 3S3. Full spectroscopy 3S4. Additional information for figures in the main text 5References 5
S1. DERIVATION OF MASTER EQUATION AND QUBIT-QUBIT INTERACTION
In this section, we outline the derivation of qubit-qubit coupling through virtual photons in the continuum ofphotonic modes in a 1D transmission line terminated by a mirror. We consider N transmon qubits, placed atpositions x n in the transmission line. The coordinate x n measures the distance from qubit n to the mirror at x = 0 .The Hamiltonian for this system can be expressed as H = H S + H B + H int with H S = N X n =1 ~ ω n σ + n σ − n , (S1) H B = Z dω ~ ωa † ω a ω , (S2) H int = i N X n =1 Z dω ~ g n ( ω ) cos( k ω x n ) (cid:0) a ω σ + n − σ − n a † ω (cid:1) . (S3)Here, H S is the bare Hamiltonian of the qubits, with σ + n ( σ − n ) the raising (lowering) operator of qubit n and ω n the transition frequency of qubit n . The bare Hamiltonian for the continuum of photonic modes in the transmission ∗ These authors contributed equally † e-mail:[email protected] a r X i v : . [ qu a n t - ph ] A p r line is given by H B , where a † ω ( a ω ) is the creation (annihilation) operator for excitations at mode frequency ω . Theinteraction between the qubits and the photons is described by H int , where the interaction strength is given by [S1] g n ( ω ) = eβ n E ( n )J E ( n )C ! / r Z ω ~ π , (S4)where β n = C n c /C nΣ is the ratio between the coupling capacitance C n c to the transmission line and the qubit capacitance C nΣ for qubit n , E ( n )C and E ( n )J are the charging and Josephson energies, respectively, of qubit n , e is the elementarycharge, and Z is the characteristic impedance for the transmission line. The cosine function in H int reflects thepresence of a mirror giving an open boundary condition at x = 0 .Using the standard procedure of eliminating the photonic degrees of freedom under the Born-Markov approxima-tion [S2], we obtain the interaction-picture master equation dρdt = i N X n =1 δ n (cid:2) σ + n σ − n , ρ (cid:3) + i N X n =1 Ω n p cos( k p x n )[ σ xn , ρ ] − i N X n = m =1 (cid:0) ∆ + nm − iΓ − nm (cid:1)(cid:2) σ + n σ − m , ρ (cid:3) + N X n,m =1 (cid:0) Γ + nm + i∆ − nm (cid:1)(cid:0) σ − m ρσ + n − σ + n σ − m ρ − ρσ + n σ − m (cid:1) + N X n =1 γ φ n (cid:0) σ + n σ − n ρσ + n σ − n − σ + n σ − n ρ − ρσ + n σ − n (cid:1) , (S5)where the qubit-qubit interaction is determined by Γ + nm = γ nm + γ mn , (S6) Γ − nm = γ nm − γ mn , (S7) ∆ + nm = ∆ nm + ∆ mn , (S8) ∆ − nm = ∆ nm − ∆ mn , (S9)with γ nm = πα nm ω m { cos( k m [ x n + x m ]) + cos( k m | x n − x m | ) } , (S10) ∆ nm = πα nm ω m { sin( k m [ x n + x m ]) + sin( k m | x n − x m | ) } , (S11) α nm = 2 β n β m e Z ~ π E ( n )J E ( n )C ! / E ( m )J E ( m )C ! / . (S12)In these expressions, the subscripts n and m refer to qubits n and m ; in general, these indices are not interchangeablein terms where they occur together if the two qubits they refer to are non-identical. The first term in Eq. (S5) is theHamiltonian for the individual qubits. Here, we have absorbed single-qubit Lamb shifts into the detuning δ n betweenthe frequency of qubit n and the frequency ω p of a probe field: δ n = ω p − ω n − ∆ nn . (S13)The second term in Eq. (S5) is the Hamiltonian showing qubit n is driven by the probe field, which is characterizedby the Rabi frequency ~ Ω n p = 2 √ eβ n E ( n )J E ( n )C ! / V , (S14)where V is the input voltage. The third term in Eq. (S5) is the qubit-qubit interaction that gives rise to the collectiveLamb shift. The fourth term in Eq. (S5) describes individual and collective relaxation processes for the qubits. Wenote that the individual bare decay rate for qubit n is given by γ n = πg n ( ω n ) = πα nn ω n . (S15)Finally, the fifth term in Eq. (S5) describes pure dephasing. The pure dephasing rate of qubit n is γ φ n . S2. REFLECTION COEFFICIENT
In this section, we summarize the calculation for obtaining the reflection coefficient r ≡ | V out /V in | (S16)from the qubits in the semi-infinite transmission line for an input voltage V in . The output voltage is given by V out ( x, t ) = V in ( x, t ) + V s ( x, t ) , (S17)where the scattered signal is V s ( x, t ) = − i r ~ Z π Z ∞ dω √ ωa ω ( t ) e ikx . (S18)Here, the photonic operator a ω ( t ) = ˜ a ω ( t ) e − iωt can be expressed in terms of the slowly varying amplitude ˜ a ω ( t ) = − N X n =1 g n ( ω ) Z t ˜ σ − n ( s ) e i ( ω − ω n ) s ds, (S19)with ˜ σ − n ( t ) = σ n ( t ) e iω n t . Substituting ˜ a ω into V s and performing the integration, we obtain V s ( x, t ) = i N X n =1 √ eβ n Z ω n E ( n )J E ( n )C ! / cos( k p x n )˜ σ − n ( t ) . (S20)Since the input signal V in is connected to the Rabi frequency of the pumping field through Eq. (S14) by taking n = N ,we immediately obtain r = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i X m η Nm γ m Ω N p cos( k p x m ) (cid:10) σ − m (cid:11)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (S21)with η Nm = β N β m E ( N )J E ( m )C E ( m )J E ( N )C ! / . (S22)The reflection coefficient can then be computed numerically by evolving the master equation in Eq. (S5). S3. FULL SPECTROSCOPY
In this section, we present the full data from the single-qubit spectroscopy, part of which was shown in Fig. 2 inthe main text. Figure S1 shows the amplitude reflection coefficient | r | as a function of probe frequency ω p and qubitfrequency in the full range − , which is the bandwidth of the cryogenic low-noise amplifier in our experimentalsetup. As explained in the caption, we use this data to extract the speed of light in the transmission line.In each of our transmon qubits, two capacitively shunted Josephson junctions form a SQUID loop. The externalflux Φ through this loop affects the transition energy of the qubit [S3]: ~ ω ( Φ ) ≈ p E J ( Φ ) E C − E C . (S23)The transition energy is determined by the charging energy E C = e / C Σ and the Josephson energy E J ( Φ ) = E J | cos( πΦ/Φ ) | , (S24)where Φ = h/ e is the magnetic flux quantum. The Josephson energy can be tuned from its maximum value E J bythe external flux Φ via a magnetic coil or local flux line. Figure S1. Amplitude reflection coefficient | r | as a function of probe frequency ω p and a magnetic flux tuning the qubitfrequencies for the full bandwidth of our measurement setup. In each measurement, the other qubit is detuned far fromresonance. (a) The data for Qubit 1, which is located away from the mirror. The dashed arrows indicate frequencies wherethe response shows that Qubit 1 sits at a node for the electromagnetic field in the transmission line. The marked frequenciesare f = 4 .
745 GHz , f = 6 .
094 GHz , and f = 7 .
414 GHz ; they correspond to L = 7 λ / , L = 9 λ / , and L = 11 λ / ,respectively. Knowing that L = 33 mm , this lets us calculate the speed of light in the transmission line. We find v = f λ =0 . × m / s ≈ f λ ≈ f λ . (b) The data for Qubit 2, which is located right by the mirror.Qubit (Bias) ω / π [GHz] Γ / π [MHz] γ φ / π [MHz] γ/ π [MHz]Q1 (A) .
697 0 . . . Q1 (B) .
01 8 1 . . Q2 (C) .
692 21 2 .
15 12 . Q2 (D) .
014 21 2 12 . Table S1. Extracted parameters from the linecuts A - D in Fig. 2(e) and (f). The fit to theory is performed following Ref. [S4].Figure S2. Theoretical simulation of the results in Fig. 4(a) in the main text. The simulation uses parameter values extractedin earlier measurements given in the main text. S4. ADDITIONAL INFORMATION FOR FIGURES IN THE MAIN TEXT
For completeness, we here present the parameters extracted from fitting the linecuts in the single-tone spectroscopyshown in Fig. 2(e) and (f) in the main text. These parameters are given in Table S1.Finally, we also provide the theoretical simulation of the experimental results presented in Fig. 4(a) in the maintext. These simulations are shown in Fig. S2. [S1] I.-C. Hoi, A. F. Kockum, L. Tornberg, A. Pourkabirian, G. Johansson, P. Delsing, and C. M. Wilson, Probing the quantumvacuum with an artificial atom in front of a mirror, Nat. Phys. , 1045 (2015), arXiv:1410.8840.[S2] H. J. Carmichael, Statistical Methods in Quantum Optics 1 (Springer, 1999).[S3] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, andR. J. Schoelkopf, Charge-insensitive qubit design derived from the Cooper pair box, Phys. Rev. A , 042319 (2007),arXiv:0703002 [cond-mat].[S4] I.-C. Hoi, A. F. Kockum, T. Palomaki, T. M. Stace, B. Fan, L. Tornberg, S. R. Sathyamoorthy, G. Johansson, P. Delsing,and C. M. Wilson, Giant Cross-Kerr Effect for Propagating Microwaves Induced by an Artificial Atom, Phys. Rev. Lett.111