Fast microwave beam splitters from superconducting resonators
M. Haeberlein, D. Zueco, P. Assum, T. Weißl, E. Hoffmann, B. Peropadre, J.J. García-Ripoll, E. Solano, F. Deppe, A. Marx, R. Gross
FFast microwave beam splitters from superconducting resonators
M. Haeberlein,
1, 2, ∗ D. Zueco,
3, 4
P. Assum, T. Weißl, E. Hoffmann,
1, 2
B.Peropadre, J.J. Garc´ıa-Ripoll, E. Solano,
7, 8
F. Deppe,
1, 2
A. Marx, and R. Gross
1, 2 Walther-Meißner-Institut, Bayrische Akademie der Wissenschaften, D-85748 Garching, Germany Physik-Department, Technische Universit¨at M¨unchen, D-85748 Garching, Germany Instituto de Ciencia de Materiales de Arag´on y Departamento de F´ısica de la Materia Condensada,CSIC-Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain Fundaci´on ARAID, Paseo Mar´ıa Agust´ın 36, 50004 Zaragoza, Spain Institut N´eel, CNRS, F-38042 Grenoble cedex 9 Instituto de F´ısica Fundamental, IFF-CSIC, Serrano 113-B, E-28006 Madrid, Spain Departamento de Qu´ımica F´ısica, Universidad del Pa´ıs Vasco UPV/EHU, Apartado 644, 48080 Bilbao, Spain IKERBASQUE, Basque Foundation for Science, Alameda Urquijo 36, 48011 Bilbao, Spain (Dated: November 4, 2018)Coupled superconducting transmission line resonators have applications in quantum informationprocessing and fundamental quantum mechanics. A particular example is the realization of fast beamsplitters, which however is hampered by two-mode squeezer terms. Here, we experimentally studysuperconducting microstrip resonators which are coupled over one third of their length. By varyingthe position of this coupling region we can tune the strength of the two-mode squeezer coupling from2 . . .
44 GHz. Nevertheless, the beam splitter couplingrate for maximally suppressed two-mode squeezing is 810 MHz, enabling the construction of a fastand pure beam splitter.
Recent advances in quantum electrodynamics with su-perconducting circuits (circuit QED) allowed for the ex-perimental implementation of basic quantum computa-tion algorithms [1]. Based on important results such assingle photon generation [2] and multi-qubit gates [3, 4],advanced schemes for quantum error correction [5] andquantum feedback control [6] were proposed. In such dig-ital approaches, superconducting quantum circuits sub-stitute classical bits and bus systems, allowing one toconstruct a general purpose quantum computation de-vice. However, digital quantum simulations typically re-quire a large number of qubits and sophisticated errorcorrection schemes [7], which is still a significant techno-logical challenge to date. Therefore, in the short termit is more promising to focus on what is called analogquantum computation or simulation. In this approach,a model quantum system is used to set up a quantummechanical evolution similar to the physical system ofinterest. However, contrary to the physical system, theinput and output channels of the model system are easilyaccessible. Superconducting quantum circuits interactingwith quantum microwave fields represent a particularlyattractive model system [8]. If the microwave fields areconfined inside cavities, proposals and early experimentstowards the simulation of manybody Hamiltonians ex-ist [9, 10]. Beyond that, recent work on systems involv-ing propagating quantum microwaves [11, 12] suggeststhat it is possible implement all-optical quantum simu-lation schemes [13] in the microwave regime. This routeseems particularly attractive, because superconductingcircuits offer extraordinarily large nonlinearities [14] andtherefore promise deterministic gates. A qubit can, forexample, be encoded in an entangled state of two spa- tially separated superconducting waveguides. In such asituation, linear microwave beam splitters play an im-portant role for the realization of single qubit rotationsand two qubit Knill-Laflamme-Milburn gates [15, 16]. Atthis point, it is important to consider decoherence ef-fects. In order to minimize them, a beam splitter shouldbe fast in the sense that its coupling rate is a signif-icant fraction of the frequency of the propagating mi-crowaves. In such an ultrastrong coupling scenario, it iswell-known [14, 17] that nonlinear effects arise for dipolarcoupling. Hence, these nonlinear effects must be takencare of in order to ensure a pure beam splitter function-ality. In this work, we first develop a theoretical modelfor fast and pure microwave beam splitters based on twofrequency-degenerate coupled superconducting transmis-sion line resonators with low external quality factors. Weconfirm this model with proof-of-principle experimentsusing microstrip resonators with a resonance frequencyof ω / π = 5 .
44 GHz and medium quality factors rangingbetween 150 and 600. Notably, we reach a beam splittercoupling strength of above 800 MHz while suppressingthe nonlinear coupling by a factor of six by exploitingthe 90 ◦ phase shift between the inductive and the capac-itive coupling channel. This allows for many operationswithin decoherence times of superconducting tramsissionline circuits [18]. We first introduce our model, whichis based on Ref. 19. As we aim at the realization of apure beam splitter, the Hamiltonian describing our ex-perimental system ideally should read as H = ¯ hω (cid:0) a † a + b † b (cid:1) + ¯ hg BS (cid:0) a † b + ab † (cid:1) . (1)Here, a † , b † , a , and b are the bosonic creation and an-nihilation operators of the two resonators and g BS is the a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b beam splitter coupling rate. The beam splitter interac-tion term g BS ( a † b + ab † ) splits the single resonance sym-metrically, resulting in two new normal modes at the an-gular frequencies ω ± = ω ± g BS . We can apply Eq. (1) tothe case of two transmission line resonators coupled ina small region around a position where either the mag-netic field (current) or the electric field (voltage) has anantinode. While this scenario allows one to neglect eitherthe capacitive or the inductive coupling channel, it limitspractical devices to coupling rates smaller than approx-imately g / ω ≈ g TMS , we find H = ¯ h (cid:101) ω (cid:0) a † a + b † b (cid:1) +¯ hg BS (cid:0) a † b + ab † (cid:1) +¯ hg TMS (cid:0) a † b † + ab (cid:1) . (2)This Hamiltonian describes two coupled harmonic oscil-lators of renormalized frequency (cid:101) ω , which is split – in gen-eral asymmetrically with respect to ω – into two normalmodes of frequencies ω ± . The detailed definition of (cid:101) ω isnot relevant for this work and can be found in Ref. 19.The total coupling rate results from a superposition ofa capacitive ( g c ) and an inductive ( g i ) coupling channel.The corresponding two channels couple via 90 ◦ -shiftedsingle mode fields. Therefore, we find g BS ≡ ( g i + g c ) and g TMS ≡ ( g i − g c ). The coupling rates g c and g i dependsolely on the resonator geometry. For a suitable res-onator design, we can achieve g c = g i and thus g TMS = 0.In other words, our distributed coupling approach allowsfor the realization of a pure beam splitter described bythe Hamiltonian of Eq. (1) with large coupling rates g BS .In the remainder of this work, we experimentally vali-date the distributed coupling model of Eq. (2) by varying g c and g i in a controlled way. To this end, we fabricatesamples containing two coupled microstrip resonators.Our design is shown in Fig. 1. For the fabrication of thechip shown in Fig. 1(a), we first sputter 100 nm Niobiumon both sides of a 250 µ m thick SiO (50 nm) coated sil-icon wafer. One side is then patterned by optical lithog-raphy and reactive ion etching, the other one serves asground plane. Our microstrip waveguides have a widthof 204 µ m to match an impedance of 50 Ω. As shown inFig. 1(a), the two resonators have the same shape. In or-der to avoid geometry effects, we shift the position of thecoupling capacitors defining both ends of the resonatorsrather than redesigning the coupling region. In thisway, we investigate seven different configurations wherethe resonators are coupled over a length of (cid:96) c = 3 mm atdifferent physical coupling positions (cid:96) left [see Fig. 1(b)].For each two-resonator sample, we fabricate a single res-onator sample with the same parameters for comparison. (c) (d)(a) (b) ℓ c ℓ left ℓ left (e) ℓ ℓ FIG. 1. Sample layout. (a) Reworked photograph of twocoupled resonators on a 10 mm × (cid:96) c . The coupling region starts at the elec-trical length (cid:96) left . (c, d) Enlarged view of the region markedwith the blue (green) box in (a). (e) Sketch of the regionmarked with the purple box in (a). - H GHz L T r an s m i ss i on H d B L FIG. 2. Transmission spectra of three resonators coupledover a length of 3 mm, with δ(cid:96) = 0 µ m (bottom), 300 µ m (mid-dle), and 600 µ m (top). The top and middle curves are shiftedby 160 dB and 80 dB, respectively. The dashed lines show therespective single resonator transmission spectra. In our experiments, we measure transmission spectra ofthe fundamental mode of single and coupled microstripresonators at 4 . ω and ω ± , respectively. Typical examples are shown inFig. 2. It can be seen that the two transmission peaks ofthe coupled resonators split asymmetrically with respectto the peak of the single resonator. This already indicatesthat the coupling is described by Eq. 2. Furthermore, wefind that ω = 5 .
44 GHz is independent of the position ofthe coupling capacitors as expected. Hence, also the to-tal electrical length (cid:96) tot = πc / ( ω √ (cid:15) eff ) is the same for allconfigurations. With c = 2 . × m/s and the effectivedielectric constant (cid:15) eff = 7 .
59, we find (cid:96) tot = 9 .
963 mm.In order to extract the coupling parameters g c and g i ,we make use of the microscopic model [19] leading toEq. (2). The input parameters to this model are theratio of the self-inductance (capacitance) per length tothe mutual inductance (capacitance) per unit length L rat ( C rat ) and the electrical position of the coupling region.The latter cannot be determined directly from the samplegeometry because the physical length of the transmissionline differs from the electrical one whenever there is abend in the resonator. Furthermore, the ratios L rat and C rat depend implicitly on the electrical coupling position.Hence, the first step in our analysis is the determinationof the electrical position of the coupling (cid:96) left = (cid:96) + δ(cid:96). (3)Here, as shown in Fig. 1(e), (cid:96) is the minimum distancebetween the coupling capacitor and the border of thecoupling region and δ(cid:96) accounts for the varying positionof the coupling capacitor. We obtain δ(cid:96) directly from theresonator geometry because in good approximation theelectrical length of a straight segment of the resonatorequals its physical length. With the definition of Eq. (3)and the model in Ref. 19, we can write L rat = ν L ω + ω − ν L ω + C rat = ν C ( ω − ω - ) + ν C ω ω + / ( ω − ν L ω + )2 ω - . Here, ν L,C = (cid:96) tot / ∆ L,C are geometry factors. Theexpressions ∆
L,C represent the overlap integralsof the magnetic (electric) field modes. For ourscenario of homogeneous resonators and funda-mental mode coupling, we can set 2 π ∆ L,C = (cid:96) c ∓ (cid:96) tot [sin(2 π ( (cid:96) left + (cid:96) c ) /(cid:96) tot ) − sin(2 π(cid:96) left /(cid:96) tot )]. Inorder to extract (cid:96) from the measured peak positions ω and ω ± , we first assume that the field in the resonatorsis a TEM-mode and, consequently, L rat and C rat areindependent of (cid:96) left . Subsequent minimization of thenormalized variance of L rat + C rat for all seven capacitorconfigurations yields (cid:96) = 1 .
271 mm. Figure 3(a) showsthat indeed for this value of (cid:96) , the parameters L rat and C rat do not deviate more than 3 % from theiraverage value. This gives evidence that our model isself-consistent.In the next step, we use L rat , C rat , and (cid:96) + δ(cid:96) to cal-culate g BS , g TMS , and ˜ ω . In Fig. 3(b), we show g BS /ω and g BS /g TMS as a function of (cid:96) rel ≡ (cid:96) left / ( (cid:96) tot − (cid:96) c ). Weobserve a maximum suppression of the TMS couplingrate to g TMS /g BS = 16% and a minimum suppression of g TMS /g BS = 43% while the beam splitter coupling ratestays nearly constant at g BS = (816 ±
7) MHz. An extrap-olation of the model prediction suggests that the TMScoupling should vanish at the relative coupling position (cid:96) rel = 14 %. Nevertheless, the beam splitter coupling rate { tot C Rat L rat { rel g TMS (cid:144) g BS g BS (cid:144) w FIG. 3. Top: The parameters L rat , C rat , and (cid:96) tot divided bytheir respective average value displayed against the relativecoupling position. Bottom: Coupling ratios as a function ofthe relative coupling position. The solid line is obtained usingthe average values of L rat , C rat , and (cid:96) tot . at this position still exceeds 780 MHz. This configurationis ideally suited for the realization of a fast beam split-ter and can in principle be reached with our geometry.Finally, we analyze the potential of our devices for theinvestigation of ultrastrong coupling. In this context, wenote that the coupling between the two resonators can beultrastrong in the same way as the qubit-resonator cou-pling discussed in Ref. 14. For our samples, we achieve amaximum TMS coupling rate of 351 MHz for (cid:96) rel = 27%.When moving the coupling region to the center of the res-onators, the maximum rate would become 702 MHz and g TMS /ω = 12 . g TMS /g BS between 16 % and 43 %.An extrapolation of our result shows that an ultrastrongcoupling scenario as well as a pure beam splitter Hamilto-nian can be reached with our sample design. This pavesthe way for studying ultrastrong coupling dynamics and,by design of a suitable capacitor configuration [16], build-ing fast beam splitter circuits for analog quantum com-putation and simulation with both standing-wave andpropagating quantum microwaves.We acknowledge support from the DeutscheForschungsgemeinschaft via SFB 631, the Germanexcellence initiative via the Nanosystems Initia-tive Munich (NIM), from the EU projects SOLID,CCQED and PROMISCE, EPSRC EP/H050434/1,Basque Government IT472-10, and Spanish MINECOFIS2009-12773-C02-01, FIS2011-25167, and FIS2012-36673-C03-02. ∗ [email protected][1] E. Lucero, R. Barends, Y. Chen, J. Kelly, M. Mariantoni,A. Megrant, P. O’Malley, D. Sank, A. Vainsencher,J. Wenner, T. White, Y. Yin, A. N. Cleland, and J. M.Martinis, Nat. Phys. , 719 (2012).[2] M. Hofheinz, E. M. Weig, M. Ansmann, R. C. Bialczak,E. Lucero, M. Neeley, A. D. OConnell, H. Wang, J. M.Martinis, and A. N. Cleland, Nature , 310 (2008).[3] L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop,B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frun-zio, S. M. Girvin, and R. J. Schoelkopf, Nature , 240(2009).[4] A. Fedorov, L. Steffen, M. Baur, M. P. d. Silva, andA. Wallraff, Nature , 170 (2012).[5] D. P. DiVincenzo and P. W. Shor, Phys. Rev. Lett. ,3260 (1996).[6] R. Vijay, C. Macklin, D. H. Slichter, S. J. Weber, K. W.Murch, R. Naik, A. N. Korotkov, and I. Siddiqi, Nature , 77 (2012).[7] I. Buluta and F. Nori, Science , 108 (2009).[8] R. J. Schoelkopf and S. M. Girvin, Nature , 664(2008).[9] M. Leib and M. J. Hartmann, New J. Phys. , 093031 (2010).[10] A. A. Houck, H. E. Tureci, and J. Koch, Nat. Phys. ,292 (2012).[11] E. P. Menzel, R. Di Candia, F. Deppe, P. Eder, L. Zhong,M. Ihmig, M. Haeberlein, A. Baust, E. Hoffmann,D. Ballester, K. Inomata, T. Yamamoto, Y. Nakamura,E. Solano, A. Marx, and R. Gross, Phys. Rev. Lett. ,250502 (2012).[12] I.-C. Hoi, C. M. Wilson, G. Johansson, T. Palomaki,T. M. Stace, B. Fan, and P. Delsing, ArXiv e-prints(2012), arXiv:1207.1203 [quant-ph].[13] J. L. O’Brien, Science , 1567 (2007).[14] T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel,F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco,T. Hmmer, E. Solano, A. Marx, and R. Gross, Nat. Phys. , 772 (2010).[15] E. Knill, R. Laflamme, and G. J. Milburn, Nature ,46 (2001).[16] L. Chirolli, G. Burkard, S. Kumar, and D. P. DiVin-cenzo, Phys. Rev. Lett. , 230502 (2010).[17] J. Casanova, G. Romero, I. Lizuain, J. J. Garcia-Ripoll,and E. Solano, Phys. Rev. Lett. , 263603 (2010).[18] R. Barends, J. Wenner, M. Lenander, Y. Chen,R. C. Bialczak, J. Kelly, E. Lucero, P. O’Malley,M. Mariantoni, D. Sank, H. Wang, T. C. White, Y. Yin,J. Zhao, A. N. Cleland, J. M. Martinis, and J. J. A.Baselmans, Appl. Phys. Lett.99