Geometric Landau-Zener interferometry in a superconducting charge pump
GGeometric Landau-Zener interferometry
S. Gasparinetti, ∗ P. Solinas,
1, 2 and J. P. Pekola Low Temperature Laboratory, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland Department of Applied Physics/COMP, Aalto University, P.O. Box 14100, FI-00076 AALTO, Finland (Dated: April 7, 2019)We propose new type of interferometry, based on geometric phases accumulated by a periodicallydriven two-level system undergoing multiple Landau-Zener transitions. As a specific example, westudy its implementation in a superconducting charge pump. We find that interference patternsappear as a function of the pumping frequency and the phase bias, and clearly manifest themselvesin the pumped charge. We also show that the effects described should persist in the presence ofrealistic decoherence.
A driven quantum two-level system traversing anavoided energy-level crossing can undergo nonadiabatictransitions, known as the Landau-Zener effect. If morethan one crossing is involved and the dynamics is overallcoherent, then transition paths can interfere accordingto the different phase accumulated by the ground andexcited-state wavefunctions between subsequent cross-ings. This phenomenon, sometimes referred to asLandau-Zener-Stückelberg (LZS) interferometry [1], wasfirst observed in atomic and optical systems, and recentlyproposed [2] and measured also in superconducting qubitsystems [3–7]. In all these realizations, the system isdriven in such a way that the interference effects havea purely dynamical nature. In general, though, a quan-tum state subject to steered evolution acquires both adynamic and a geometric phase. While the study of ge-ometric phases in solid-state systems is an active field ofresearch [8–10], their relevance to LZS interferometry hasso far been unexplored [11].In this Letter, we elucidate the link between LZS inter-ference and geometric phases, opening new possibilitiesfor the geometric control of quantum systems. Our re-sults apply to a broad range of devices, namely those forwhich the parametric driving possesses a nontrivial geo-metric structure and the induced energy-gap modulationpresents multiple avoided crossings. As a pertinent ex-ample, we consider a superconducting charge pump, theCooper-pair sluice [12]. The connection between Cooper-pair pumping and geometric phases was highlighted inprevious theoretical works both for the abelian [13, 14]and nonabelian [15] case, yet always in the adiabaticlimit, where the system stays in the instantaneous groundstate and excitations are treated as small corrections [16].We instead consider higher frequency regimes and predictthe appearence of interference patterns depending on thepumping frequency and the superconducting phase bias,the latter embodying the geometric contribution to in-terference. We then show that LZS resonances directlymanifest themselves in the pumped charge, which is anadvantage of using a charge pump rather than a conven-tional qubit as an interferometer. Finally, we introducedecoherence in our model and show that interference ef-fects are still detectable. This should make our proposal (b) E ne r g i e s ( G H z ) Time (periods)1/6 1/3 (c)(a) 1 2 3
Bx ByBz t=0 t=T/6t=T/3t=2T/3t=5T/6 t = T / t=T/4t=3T/4 n g J m a x ϕ / FIG. 1. (Color online) (a) Schematic drawing of the “sluice”.(b) Effective magnetic field corresponding to the pumping cy-cle considered in this Letter. (c) Adiabatic (instantaneous)energy versus time. Avoided level crossings occur at times t = T / t = 3 T /
4. Green and red arrows outline twopossibly interfering paths. feasible for experimental observation.The Cooper-pair sluice, schematically shown inFig. 1(a), consists of a superconducting island coupledto the leads by two superconducting quantum interfer-ence devices (SQUIDs), whose Josephson energies J L,R ,can be tuned by changing the magnetic fluxes Φ
L,R . Agate electrode capacitively coupled to the island is usedto induce a polarization charge n g on the latter, therebyproviding a third control parameter. During a pumpingcycle, the parameters are steered so as to couple the is-land to the left lead, attract a Cooper-pair, switch thecoupling to the right lead, and release the Cooper-pair[17]. We will assume that the superconducting phase dif-ference φ across the device is kept constant. This canbe achieved by shunting the sluice with a large Joseph-son junction. In this case, the switching statistics of theadditional junction also provides a way to measure thepumped charge [18].The device is operated in the Coulomb-blockaderegime E C (cid:29) J max , where J max = max { J L , J R } and a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec E C is the charging energy of the island. The system dy-namics is then best described in the basis of eigenstatesof charge on the island. Also, as long as n g stays closeto the degeneracy point 1 /
2, only two such states arerelevant, namely those with zero and one excess Cooperpair on the island. This allows us to use a pseudo-spinformalism and write the sluice Hamiltonian as H = ~σ · ~B ,where { σ i } are the Pauli matrices and the effective mag-netic field ~B has components B x ( t ) = J + ( t ) cos φ , (1) B y ( t ) = J − ( t ) sin φ , (2) B z ( t ) = E C [1 / − n g ( t )] , (3)where we put J ± ( t ) = J L ( t ) ± J R ( t ). As ~B is steered alongthe path shown in Fig. 1(b), it spans a solid angle whichis responsible for the geometric effects under discussion.This situation is clearly different from that considered ine.g. Refs. [3, 4], where ~B moves on a definite plane (say, x - z ), leaving no room for non-trivial geometric effects totake place.In Fig. 1(c) we plot the energies of the adiabatic states | g i , | e i as a function of time for a pumping cycle, ob-tained by instantaneous diagonalization of the Hamilto-nian. The avoided level crossings at t = T / , T / T is the pumping period) correspond to the gate chargecrossing the degeneracy point. The probability of a nona-diabatic (Landau-Zener) transition at such a crossing isgiven by P LZ = e − πδ , where the adiabatic parameter δ depends on the velocity at which the crossing is traversedand on the energy gap at the crossing [1]. For our case, δ = πJ max / (48 E C ¯ n g hν ), where ¯ n g = max { n g }− / ν = 1 /T .In the limits E C (cid:29) J max and hν (cid:46) E C ¯ n g , nonadia-batic transitions are strongly localized at level crossings.The system dynamics can thus be seen as a sequence ofadiabatic evolutions and localized transitions. For thisreason, the calculation is most conveniently performed inthe adiabatic basis {| g i , | e i} . In the so-called adiabatic-impulse model [1, 19], Landau-Zener tunneling at anti-crossings is treated as instantaneous and described in theadiabatic basis by a transfer matrix of the form: N LZ = (cid:18) √ − P LZ e i ˜ ϕ S −√ P LZ √ P LZ √ − P LZ e − i ˜ ϕ S (cid:19) , (4)where ˜ ϕ S = δ (log δ − − iδ ) − π/ j = 1 , , t j − to t j is described by a diagonalmatrix of the form: U j = exp [ iϕ j σ z ] , where ϕ j is thetotal phase difference acquired by the adiabatic states.The latter can be written as ϕ j = ξ j + γ j , where wehave distinguished a dynamic ( ξ j ) and a geometric ( γ j ) contribution. The dynamic phase difference ξ j is givenby: ξ j = 12 (cid:126) Z t j t j − dt p | H − H | + 4 | H | . (5)By contrast, the gauge-invariant, noncyclic geometricphase difference γ j can be calculated as [21]: γ j = i Z t j t j − dt (cid:20)(cid:28) g (cid:12)(cid:12)(cid:12)(cid:12) ddt (cid:12)(cid:12)(cid:12)(cid:12) g (cid:29) − (cid:28) e (cid:12)(cid:12)(cid:12)(cid:12) ddt (cid:12)(cid:12)(cid:12)(cid:12) e (cid:29)(cid:21) . (6)This phase is uniquely determined by the path drawn bythe system in parameter space, and reduces to the Berryphase for cyclic ground-state evolution [22].Putting things together, the evolution operator over aperiod can be calculated as U = U N LZ U N LZ U = U (cid:18) α − β ∗ β α ∗ (cid:19) U , (7)where α = (cid:2) (1 − P LZ ) e i ˜ ϕ S + iξ + iγ − P LZ e − iξ − iγ (cid:3) , (8) β = 2 p P LZ (1 − P LZ ) cos ( ˜ ϕ S + ξ + γ ) . (9) U and U play no role in the upcoming resonance con-dition and will not be considered further. From (7), wecan calculate the excitation probability after one periodstarting from the ground state. This is given by: P = 4 P LZ (1 − P LZ ) cos ( ˜ ϕ S + ξ + γ ) . (10)This probability oscillates between 0 and 4 P LZ (1 − P LZ )as a function of the accumulated phase. In the fast-passage limit, δ (cid:28)
1, we can approximate ˜ ϕ S ≈ − π/ δ → ∞ and ˜ ϕ S → − π/ J max / ( E C ¯ n g )] , we find: ξ = 5 π E C ¯ n g hν , (11) γ = φ/ . (12)As expected, the dynamic phase ξ is inversely pro-portional to the pumping frequency ν . By contrast, thegeometric contribution γ does not depend on ν , andin this particular case equals half the superconductingphase bias φ . We have thus derived a resonance condi-tion involving the superconducting phase bias φ and thepumping frequency ν . In particular, in the region where˜ ϕ S attains a constant value, the resonances drift in the φ − ν plane as branches of hyperbolae.This analysis predicts the position of resonances andexplains their origin. Its regime of validity lies in be-tween the strictly adiabatic and the fully nonadiabaticone. As a matter of fact, a lower bound for the pumping P u m p i ng f r equen cy ( G H z ) Q P /2e (a)(b) 0 1/6 1/3 1/2 2/3 5/6 100.51 FIG. 2. (Color online) (a) Pumped charge (in units of e ) afterone cycle versus phase bias and frequency. The parametersare: E C /k B = 2 . J max = 0 . E C , 0 . ≤ n g ≤ .
7. Dashedlines enclose the regions where the ground state populationat the end of the cycle is at least 0.9. Dotted lines have thesame meaning but they are calculated according to (10) andin the fast-passage approximation ˜ ϕ S ≈ − π/
4. (b) Ground-state population versus time for a case of destructive (dashedline) and constructive (solid line) interference. The pumpingfrequency is 1.56 GHz for both cases, the phase biases are0.22 and 3.36, respectively. frequency is set by the requirement for time evolution tobe coherent over one pumping cycle. On the other hand,at frequencies comparable to the adiabatic level spacing( hν ≈ E C ¯ n g ) transitions are no longer restricted to thedegeneracy points and the adiabatic-impulse model is ex-pected to break down.The superconducting phase bias φ enters the resonancecondition through the geometric phase accumulated be-tween subsequent transitions. This relationship is trivialfor the case considered, as the geometric phase is simplyproportional to φ . Yet, this example clearly illustratesthe role of geometric phases in Landau-Zener interfer-ence. In particular, by choosing the pumping frequencyso that ˜ ϕ S + ξ is an integer multiple of π , the dynamiccontribution in (10) is washed out, resulting in a purelygeometric Landau-Zener interference effect.We now proceed to show that the predicted resonancesmanifest themselves in the charge pumped by the device,thus providing the most straightforward way of observingthem. To do so, we first obtain the full system dynam-ics from numerical solution of the Schrödinger equation. We then calculate the pumped charge by integrating theinstantaneous current operator [24]. In Fig. 2 (a), weplot the pumped charge over a period versus the phasebias φ and the pumping frequency ν . The parametersare chosen so as to be consistent with our model. Inparticular, microscopic excitations in the superconduct-ing circuit can be neglected provided hf eff (cid:28) ∆, where f eff is the effective frequency of the driving fields and∆ the superconducting gap. Furthermore, for small val-ues of ¯ n g the system is sufficiently anharmonic for thetwo-level approximation to hold in the given frequencyrange. The lines drawn on top of the image plot cor-respond to 90% probability of the system being in theground state at the end of the cycle. Dashed lines are cal-culated numerically, dotted lines according to (10). Thestrong correlation between the ground state populationand the pumped charge demonstrates the possibility toaccess interference patterns simply by measuring the lat-ter. Moreover, the accuracy of the approximations madein deriving (10) is confirmed by the good agreement be-tween analytical and numerical calculations.In Fig. 2 (b), we show the time evolution of ground-state populations for one case of constructive and one ofdestructive interference. In both cases there is a popula-tion transfer to the excited state after the first crossing.Yet, while constructive interference (solid line) enhancesthe excitation after the second crossing, destructive in-terference (dashed line) brings the system back to theground state. In particular, this implies that for a givenpumping frequency, the phase bias can be chosen so asto pump a significant fraction of Cooper pairs (about 0.5in this case) even in the nonadiabatic regime.A complementary and instructive way to understandthese features is provided by Floquet analysis [25]. Infact, we can explicitly calculate the quasienergy spectrumby diagonalizing the evolution operator U in (7). We findthat destructive resonances occur at exact quasienergycrossings, where time evolution over a period is trivialand tunneling between adiabatic states is dynamicallyfrozen. This phenomenon is known as coherent destruc-tion of tunneling [26]. At the opposite end, constructiveinterference enhances such transitions, resulting in Flo-quet states being the maximal mix of the adiabatic ones.This is revealed in the quasienergies as the opening of agap, similarly to a time-independent system with a cou-pling interaction switched on.The LZS interferometry discussed above is a unitaryand coherent process. However, in any experimental im-plementation the undesired and unavoidable coupling toexternal degrees of freedom leads to decoherence and thusit affects the observed pumped charge. This fact pre-cludes a measurement of coherent LZS pumping over agreat number of cycles and must be taken into accountin any experimental proposal. To this end, we will nowdiscuss the case in which the system is affected by chargenoise. The dissipative dynamics of the system is numer-ically obtained from the master equation including thedriving field and the environment. The latter is describedby a bath of harmonic oscillators with ohmic spectrumat zero temperature [27, 28]. To be able to detect theeffect of coherence loss, the pumping period must besmaller than the expected decoherence time. By choos-ing a superconducting island with high charging energy E C /k B = 5 K, this condition is fulfilled at frequencies aslow as 0 . ∗ [email protected].fi[1] S. Shevchenko, S. Ashhab, and F. Nori, Phys. Rep., ,1 (2010). (b)(a) cyclescycles Q P / Q P / FIG. 3. (Color online) Pumped charge per cycle versus timefor a case of constructive (squares) and destructive interfer-ence (circles), without decoherence (top) and with decoher-ence induced by zero-temperature gate-charge noise (bottom).Dashed lines are guides for the eye. The pumping frequencyis 500 MHz for both cases, the phase biases are φ c = 2 . φ d = − .
45, respectively. We have also assumed a residualJosephson coupling J min = min { J L , J R } = 0 . J max .[2] A. V. Shytov, D. A. Ivanov, and M. V. Feigel’man, Eur.Phys. J. B, , 263 (2003).[3] W. D. Oliver et al. , Science, , 1653 (2005).[4] M. Sillanpää, T. Lehtinen, A. Paila, Y. Makhlin, andP. Hakonen, Phys. Rev. Lett., , 187002 (2006).[5] C. M. Wilson, T. Duty, F. Persson, M. Sandberg, G. Jo-hansson, and P. Delsing, Phys. Rev. Lett., , 257003(2007).[6] A. Izmalkov et al. , Phys. Rev. Lett., , 017003 (2008).[7] G. Sun, X. Wen, B. Mao, Y. Yu, J. Chen, W. Xu,L. Kang, P. Wu, and S. Han, Phys. Rev. B, , 180507(2011).[8] G. Falci, R. Fazio, G. Palma, J. Siewert, and V. Vedral,Nature, , 355 (2000).[9] P. J. Leek et al. , Science, , 1889 (2007).[10] M. Möttönen, J. J. Vartiainen, and J. P. Pekola, Phys.Rev. Lett., , 177201 (2008).[11] An optical implementation of Landau-Zener interferencewith geometric phases was realized in D. Bouwmeester,G. P. Karman, C. A. Schrama, and J. P. Woerdman,Phys. Rev. A, , 985 (1996).[12] A. O. Niskanen, J. P. Pekola, and H. Seppä, Phys. Rev.Lett., , 177003 (2003).[13] M. Aunola and J. J. Toppari, Phys. Rev. B, ,020502(R) (2003).[14] M. Möttönen, J. P. Pekola, J. J. Vartiainen, V. Brosco,and F. W. J. Hekking, Phys. Rev. B, , 214523 (2006).[15] V. Brosco, R. Fazio, F. W. J. Hekking, and A. Joye,Phys. Rev. Lett., , 027002 (2008).[16] Recent attempts to go beyond the adiabatic limit can befound in A. Russomanno, S. Pugnetti, V. Brosco, and R.Fazio, Phys. Rev. B 83 (2011), and J. Salmilehto, and M.Mottonen, arXiv:1106.2689v1 (2011). [17] We consider the same pumping cycle as in Refs. [14, 28].[18] This technique was experimentally demonstrated in D.Vion et al. , Science , 886 (2002), and for the sluice inRef. [10].[19] B. Damski and W. H. Zurek, Phys. Rev. A, , 063405(2006).[20] Y. Kayanuma, Phys. Rev. A, , R2495 (1997).[21] G. García de Polavieja and E. Sjöqvist, Am. J. of Phys., , 431 (1998).[22] A relationship connecting the Berry phase to the pumpedcharge in the adiabatic limit was first derived in Ref. [13]and experimentally investigated in Ref. [10].[23] Explicit expressions for | g i , | e i can be found in Ref. [28].[24] For our case, the pumped charge can be safely identifed with the total charge transferred over a period.[25] M. Grifoni and P. Hänggi, Phys. Rep., , 229 (1998),and references therein.[26] F. Grossmann, T. Dittrich, P. Jung, and P. Hanggi,Phys. Rev. Lett., , 516 (1991).[27] J. P. Pekola, V. Brosco, M. Möttönen, P. Solinas, andA. Shnirman, Phys. Rev. Lett., , 030401 (2010).[28] P. Solinas, M. Möttönen, J. Salmilehto, and J. P. Pekola,Phys. Rev. B, , 134517 (2010).[29] The effective system-environment coupling constant usedis g = 0 .
05. This gives for the decoherence rate the valueΓ ≈ . et al. , Phys. Rev. B72