expectation value of the Higgs field in the broken-symmetry state. The parameters
λ,g2control the strength of the fluctuations of the matter and gauge fields. Our HamiltonianH is obtained from that of the Higgs modelHH by using the approximation −π2∂2
x/2 ≈ cos(π∂x)−1 for x = φm,θm and truncating the an-
gular variables to the valuesφm,θm ∈ {0,π}. Within the truncated Hilbert space,
cos(π∂φm) =σx
mand cos(π∂θm) =τ
x
m. Hence,HHreduces (up to an irrelevant additive
constant) to the spin chain Hamiltonian
HZ2= N X m=1 λ π2σ x m+ N−1 X m=1 g2 π2τ x m+v 2σz mτ z mσ z m+1 . (11.5)
The Hamiltonian HZ2 is precisely that of theZ2 Higgs model[313]. Finally, the
Jordan-Wigner transformationam=σmx Qm−1 j=1 σ z j,bm=σmy Qm−1 j=1 σ z
jshows that the
Z2Higgs model is equivalent to our HamiltonianH, provided we identifyh=−λ/π2,
κ=−g2/π2, andEM=−v2. (Note that the phase diagram does not depend on the
signs ofλ,κ,EM.)
As our effective HamiltonianH is related to the Higgs model, we might expect the Higgs mechanism to be present. As a result, gapless excitations should become gapped for arbitrarily small values ofκ, that is, for arbitrarily weak fluctuations of the superconducting order parameter. In other words, the small but finite charging energy ECof each island breaks the ground state degeneracy and splits the otherwise unpaired Majorana modes. In this way, the Higgs-mechanism offers a way to locally break the topological degeneracy of the Majorana chain. It is known that this expectation is indeed correct in the thermodynamic limit, as atκ6=0 the Hamiltonian (11.5) has no phase transitions and describes a gapped phase with a single ground state
[293, 313, 326]. However, in a finite chain signatures of the topologically non-trivial phase, which is present atκ=0 andh<EM, should survive up to a finite value ofκ.
If this is true, then the Hamiltonian of a finite chain should be gapless along a line in the(h,κ)plane.
11.3
Numerical study of the linear response of the chain
In the following, we will show that in the linear response regime, the topological transition reflects itself in the thermal conductanceK through the system also at finite κ, whereas upon increasing κ, the local probe of Andreev conductance G quickly becomes blind to it. To this end, we couple the left and right end of the chain with Hamiltonian H to normal leads through tunneling Hamiltonians[36] HL =γLc†La1e−iφ1/2+H.c.,HR =γRc †
RbNe−iφN/2+H.c.; here,γL andγR denote the
amplitudes for tunneling events into the left (L) and right (R) leads, andcL†,c†Rare the creation operators for electrons in the non-interacting leads. We fix the gauge by choosingφ1=0, so thatφN=
PN−1
m=1ϕm. In the low-energy limit, e−iφN/2=
QN−1
m=1τ
z m,
so we get HL=γL(cL†+cL)a1, HR=γR(cR†+cR)bN N−1 Y m=1 τz m. (11.6)
The tunneling Hamiltonians must break one of the gauge symmetries, since a tunneling event changes the total fermion parity. Due to our gauge choice, we obtain{HL,R,C1}= 0 while[HL,R,Cm] =0 form=2, . . . ,N.
The Andreev conductanceGis determined by the charge transport across a normal metal-superconductor interface. To computeG, we setγR=0,γL=γ, apply a bias
voltage V to the left lead, and ground the rightmost superconducting island. In contrast, the thermal conductanceKis determined by the heat transport between two normal leads. To computeK we setγL=γR=γand establish a small temperature
difference between the right lead at temperatureT and the left lead at temperature T+δT. In the limitT,V →0, we obtainGasG=G0ΓIm[G11(0)][327], andKas K=4K0Γ2|G1N(0)|2, in terms of the tunnel couplingΓ =2π|γ|2ρ0to a wide-band
lead with density of statesρ0and the retarded Green’s functions1
G11(ω) =−i Z ∞ 0 dt eiωt¬{a1(t),a1(0)}¶, (11.7a) G1N(ω) =−i Z ∞ 0 dt eiωt¬{bN(t)QN−1 m=1τ z m(t),a1(0)} ¶ . (11.7b) The averages in Eqs. (11.7a) are taken over the ground state wave function|0〉of our effective HamiltonianH. For anyκ,h,EM >0, the ground state ofHis unique
and belongs to the gauge-invariant sector withCm|0〉=|0〉for allm. Note that at
κ=0, the ground manifold is 2N−1-times degenerate, but we continue to choose|0〉
to be the unique gauge-invariant state in the ground manifold. The time-evolution in Eqs. (11.7a) is determined by the total HamiltonianHtot=H+HR+HL. The retarded Green’s functionG11(t) =R(dω/2π)e−iωtG11(ω)is the amplitude for a reflection process whereby an electron enters the chain from the left lead at timeti=0 and
exit again from the left lead after a timetf =t. Similarly,G1N(t)is the amplitude for
a transmission process whereby the electron enters atti=0 from the left lead and
exits from the right lead after a timetf =t.
We highlight that the thermal transport probesnon-localquasiparticle transfer processes through the chain characterized by the string correlator
G1N(t) =−i〈{bN(t)
QN−1
m=1τ
z
m(t),a1(0)}〉, (11.8)
which is a generalization of the conventional correlator−i〈{bN(t),a1(0)}〉studied in the context of the Majorana chain without the gauge degrees of freedom[303]. Due to the presence of the gauge stringQNm−=11τzm, the Green’s functionG1N is similar to 1In this formula forKwe neglect inelastic processes mediated by phonons and assume that the occupation
11.3 Numerical study of the linear response of the chain 179
Figure 11.2: Numerical results: Andreev conductance (top left, in units ofG0=2e2/h),
Fredenhagen-Marcu order parameter MMF (top right), and thermal conductance (bottom, in units ofK0=π2kB2T/6h) plotted as a function ofhandκ. The results are obtained for a chain ofN =11 islands with coupling constantΓ =0.01EM to
the leads, in the limit of vanishing temperatureTand small applied voltageV. The Andreev conductanceGis averaged over a small voltage interval 10−4E
M to account
for the finite-size energy splitting of the Majorana modes as for a finite size wire we trivially haveG=0[88, 303]. The Andreev conductance only shows a signal along the axis. On the other hand, the string-order parameter allows to distinguish a confined regime - corresponding to the topological regime with Majorana end modes - from the trivial Higgs-regime. The separation between these two regimes can be clearly identified by the peak in the thermal conductance. The inset in the right panel shows line cuts of the thermal conductance atκ/EM=0, 0.01, 0.02, going from right
to left as shown by the arrow. Due to finite size effects, the transition atκ=0 is shifted fromh/EM =1 toh/EM '(Γ/EM)1/11'0.7, as expected. Increasingκ, the
the Fredenhagen-Marcu string-order parameter[314], MFM=−i〈bN QN−1 m=1τ z ma1〉, (11.9)
which measures the presence of the topological phase in the model with fluctuation gauge degrees of freedom. In the following we probe this relation numerically.
To calculate the Green’s functionGmn(ω), we follow the approach[328]of de- coupling the gauged Majorana chain from the leads to first order in the lead coupling
Γ and neglecting higher-order (co-)tunneling processes. The bare Green’s functions without the leads are calculated by exact diagonalization of the Hamiltonian (11.2), using a Lehmann spectral representation in terms of the exact eigenstates. The pres- ence of the symmetries (11.3) greatly simplifies the task of computingGmn. In fact,
we only need to know the energy and the wave function of the ground state|0〉and of all the states|ψ〉such thatC1|ψ〉=−|ψ〉whileCm6=1|ψ〉=|ψ〉. Indeed, sinceC1
is the only symmetry ofHwhich does not commute with the tunneling Hamiltonian HL,R, but anti-commutes instead, these are the only excited states to which transitions from the ground states are possible upon tunneling of an electron from the leads. For a chain ofNislands, there are 2N−1of these states—against a dimension of 22N−1of
the total Hilbert space2.
The numerical results for a chain ofN =11 islands are shown in Fig. 11.2. At
κ=0, coherently with known results, we observe an Andreev reflection plateau at G0in the non-trivial regime and a thermal conductance peak at the transition, which appears shifted toh'EM(Γ/EM)1/Ndue to finite chain size and coupling to the leads. At finiteκ, the Andreev plateau is quickly suppressed, except close toh=0, a limit where two isolated Majorana modes are always present. However, the quantized peak in thermal conductance persists in the interacting part of the parameter space, indicating the presence of a gapless transmitting mode and hence a strong signature of the existence of a topological regime. In fact, the position of the thermal conductance peak qualitatively follows the line of maximum change in the order parameter. We have checked that the agreement persists when varying the system sizeN.
11.4
Conclusions
To conclude, we have shown that QPS in a Majorana chain implement theZ2Higgs
model where the fluctuations of the gauge field are determined by the rateκ/ħhfor
QPS. QPSlocallydestroy the topological phase of the Kitaev model at fixed fermion parity via aZ2version of the Higgs mechanism. However, for finite system size and
smallκ, signatures of the topological phase remain visible in the thermal conductance through the system. The reason is that it is linked to the Fredenhagen-Marcu order parameter for theZ2 Higgs theory, which indicates the topological regime with
2Note that when the total Hamiltonian is projected onto a single gauge sector, it takes the form of a
transverse-field Ising model in a longitudinal fieldκ. This projected Hamiltonian is the one we numerically diagonalize. Within this simpler model in a fixed gauge sector, it is well known that any finiteκ >0 introduces a relevant perturbation[326].
11.4 Conclusions 181
gauge fluctuations present. The thermal conductance provides a clear transport signature of the transition from the topological to the trivial regimes in the presence of the interactions with the gauge field, whereas no signature of the transition is present in the Andreev conductance at a finite rate of QPS. Our results suggest that in topological quantum matter, bulk transport measurements offer access to non-local order parameters, just like susceptibility measurements do for local order parameters in broken-symmetry phases. It remains an interesting question for further studies how this scenario can be generalized to higher dimensions and non-Abelian gauge fields.
Chapter 12
Outlook
This thesis contains the theoretical design of a topological quantum computer based on Majorana modes in superconducting circuits. Will the theoretical ideas contained in this thesis be realized in practice? What is the status of the field, and what are its prospects? To guide the discussion of these crucial questions, I have assembled the schematic table in Fig. 12.1, which identifies some intermediate stages of development, and which might be useful to make a rough measurement of future experimental progress.
Where are we now? Chapter 7 of this thesis reaches the first milestone along this road, the successful realization of microwave quantum circuits with semiconducting nanowire Josephson elements embedded in a cQED chip [329]1. These circuits contained no Majorana modes, so current efforts are focusing on achieving the second stage in the figure. Mainly three different platforms are under scrutiny in this race: semiconducting nanowires[38, 74, 77], quantum spin Hall systems[116, 154, 330], and chains of magnetic adatoms[117], all covered by a large body of theoretical work.
While realizing Majorana modes is obviously a prerequisite for all stages in the figure beyond the second, there will certainly be many other practical challenges. A commonly mentioned one is the ubiquitous presence of non-equilibrium quasiparticles in superconducting circuits, which breaks the topological protection of Majorana qubits. Recently, there has been a lot of progress in the suppression of quasiparticle poisoning[32–34], even though the origins and dynamics of these stray quasiparticles remain largely unknown. Quasiparticle poisoning can prevent a clear observation of the 4π-periodic Josephson effect, which represents the third step - see for instance the discussion in Chapter 2. It certainly poses an upper bound on the coherence time of a Majorana qubit, which must always be operated in a parity-protected fashion.
Chapters 3 to 6 describe in great detail stages four to seven in Fig. 12.1, from the characterization of a Majorana qubit to the full execution of a quantum algorithm
Realization of hybrid superconducting circuits with semiconducting Josephson elements Stabilizing Majorana modes in hybrid superconducting circuits
Observation of 4π Josephson effect in two- and multi-terminal junctions Non-topological operations on parity-protected Majorana qubit
Braiding, demonstration of non-Abelian statistics
Topological operations on parity-protected Majorana qubit Algorithms
Time
Com
pexity
Figure 12.1: Development stages for the realization of a topological quantum com- puter with Majorana modes, following the ideas presented in this thesis. The drawing is inspired by a similar one by Devoret and Schoelkopf appearing in Ref.[31], out- lining the realization of a universal quantum computer with conventional qubits. It should in fact be interpreted by quoting their words: “Each advancement requires mastery of the preceding stages, but each also represents a continuing task that must be perfected in parallel with the others”[31].
in a topological fashion. The major milestone is certainly the demonstration of non-Abelian statistics, which stands out in this research program as a fundamental discovery. Whether or not a topological quantum computer is eventually realized, non- Abelian statistics are a fascinating feature of quantum mechanics. Their observation would once again demonstrate the seemingly inextinguishable ability of quantum mechanics to predict new phenomena, and to surprise us.
As usual, the next generations of experiments will yield unexpected results, and most probably offer opportunities to deviate from the path outlined here and explore territories which are at the moment unforeseen. Perhaps, these unknown unknowns will be what will keep us busy and curious in the years to come.
References
[1] M. A. Nielsen and I. L. Chuang,Quantum computation and quantum information (Cambridge university press, 2010).
[2] S. Aaronson, Quantum computing since Democritus (Cambridge University Press, 2013).
[3] D. Gottesman, arXiv:quant-ph/9705052 (1997).
[4] J. Preskill, Proc. R. Soc. London, Ser. A454, 385 (1998).
[5] A. Y. Kitaev, Ann. Phys.303, 2 (2003).
[6] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys.80, 1083 (2008).
[7] G. Moore and N. Seiberg, Phys. Lett. B212, 451 (1988).
[8] G. Moore and N. Seiberg, Comm. Math. Phys123, 177 (1989).
[9] E. Witten, Comm. Math. Phys121, 351 (1989).
[10] J. Fröhlich and F. Gabbiani, Rev. Math. Phys.2, 251 (1990).
[11] G. Moore and N. Read, Nucl. Phys. B360, 362 (1991).
[12] C. Nayak and F. Wilczek, Nucl. Phys. B479, 529 (1996).
[13] A. Stern, Ann. Phys.323, 204 (2008).
[14] N. Read and D. Green, Phys. Rev. B61, 10267 (2000).
[15] L. E. Ballentine,Quantum mechanics, vol. 280 (Prentice Hall Englewood Cliffs, 1990).
[16] J. Preskill,Lecture Notes for Physics 219: Quantum Computation., chap. Topo- logical Quantum Computation (2004).
[18] M. V. Berry, Proc. R. Soc. London, Ser. A392, 45 (1984).
[19] B. Simon, Phys. Rev. Lett.51, 2167 (1983).
[20] F. Wilczek and A. Zee, Phys. Rev. Lett.52, 2111 (1984).
[21] J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, Nat. Phys.7, 412 (2011).
[22] J. Alicea, Rep. Progr. Phys.75, 076501 (2012).
[23] C. Beenakker, Annu. Rev. Condens. Matter Phys.4, 113 (2013).
[24] M. Leijnse and K. Flensberg, Semi. Sci. Tech.27, 124003 (2012).
[25] G. Volovik, JETP Lett.70, 609 (1999).
[26] A. Y. Kitaev, Phys.-Usp.44, 131 (2001).
[27] D. A. Ivanov, Phys. Rev. Lett.86, 268 (2001).
[28] M. H. Devoret, A. Wallraff, and J. M. Martinis, arXiv:cond-mat/0411174 (2004).
[29] R. Schoelkopf and S. Girvin, Nature451, 664 (2008).
[30] J. Clarke and F. K. Wilhelm, Nature453, 1031 (2008).
[31] M. H. Devoret and R. J. Schoelkopf, Science339, 1169 (2013).
[32] I. M. Pop, K. Geerlings, G. Catelani, R. J. Schoelkopf, L. I. Glazman, and M. H. Devoret, Nature508, 369 (2014).
[33] U. Vool, I. M. Pop, K. Sliwa, B. Abdo, C. Wang, T. Brecht, Y. Y. Gao, S. Shankar, M. Hatridge, G. Catelani,et al., Physical review letters113, 247001 (2014).
[34] D. J. van Woerkom, A. Geresdi, and L. P. Kouwenhoven, arXiv preprint arXiv:1501.03855 (2015).
[35] F. Hassler, A. R. Akhmerov, C.-Y. Hou, and C. W. J. Beenakker, New. J. Phys.12, 125002 (2010).
[36] L. Fu, Phys. Rev. Lett.104, 056402 (2010).
[37] J. Koch, T. M. Yu, J. M. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A76, 042319 (2007).
[38] S. R. Plissard, I. van Weperen, D. Car, M. A. Verheijen, G. W. G. Immink, J. Kammhuber, L. J. Cornelissen, D. B. Szombati, A. Geresdi, S. M. Frolov, L. P. Kouwenhoven, and E. P. A. M. Bakkers, Nature Nano.8, 859 (2013).
REFERENCES 187
[39] B. D. Josephson, Phys. Lett.1, 251 (1962).
[40] M. Tinkham, Introduction to superconductivity(Courier Dover Publications, 2012).
[41] H.-J. Kwon, K. Sengupta, and V. M. Yakovenko, Eur. Phys. J. B37, 349 (2003).
[42] L. Fu and C. L. Kane, Phys. Rev. B79, 161408 (2009).
[43] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys.82, 3045 (2010).
[44] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys.83, 1057 (2011).
[45] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett.105, 077001 (2010).
[46] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett.105, 177002 (2010).
[47] P. A. Ioselevich and M. V. Feigel’man, Phys. Rev. Lett.106, 077003 (2011).
[48] F. S. Nogueira and I. Eremin, J. Phys. Cond. Mat.24, 325701 (2012).
[49] K. T. Law and P. A. Lee, Phys. Rev. B84, 081304 (2011).
[50] L. Jiang, D. Pekker, J. Alicea, G. Refael, Y. Oreg, and F. von Oppen, Phys. Rev. Lett.107, 236401 (2011).
[51] R. F. Service, Science332, 193 (2011).
[52] C. Xu and L. Fu, Phys. Rev. B81, 134435 (2010).
[53] J. R. Friedman and D. V. Averin, Phys. Rev. Lett.88, 050403 (2002).
[54] R. P. Tiwari and D. Stroud, Phys. Rev. B76, 220505 (2007).
[55] E. Grosfeld and A. Stern, Proc. Nat. Acad. Sci.108, 11810 (2011).
[56] E. Grosfeld, B. Seradjeh, and S. Vishveshwara, Phys. Rev. B83, 104513 (2011).
[57] P. J. de Visser, J. J. A. Baselmans, P. Diener, S. J. C. Yates, A. Endo, and T. M. Klapwijk, Phys. Rev. Lett.106, 167004 (2011).
[58] S. Ryu, J. E. Moore, and A. W. W. Ludwig, Phys. Rev. B85, 045104 (2012).
[59] D. V. Averin and Y. V. Nazarov, Phys. Rev. Lett.69, 1993 (1992).
[60] J. A. Schreier, A. A. Houck, J. Koch, D. I. Schuster, B. R. Johnson, J. M. Chow, J. M. Gambetta, J. Majer, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. B77, 180502 (2008).
[61] F. Hassler, A. R. Akhmerov, and C. W. J. Beenakker, New. J. Phys.13, 095004 (2011).
[62] S. Bravyi and A. Y. Kitaev, Phys. Rev. A71, 022316 (2005).
[63] J. D. Sau, S. Tewari, and S. Das Sarma, Phys. Rev. A82, 052322 (2010).
[64] K. Flensberg, Phys. Rev. Lett.106, 090503 (2011).
[65] L. Jiang, C. L. Kane, and J. Preskill, Phys. Rev. Lett.106, 130504 (2011).
[66] P. Bonderson and R. M. Lutchyn, Phys. Rev. Lett.106, 130505 (2011).
[67] J. D. Sau, D. J. Clarke, and S. Tewari, Phys. Rev. B84, 094505 (2011).
[68] A. Romito, J. Alicea, G. Refael, and F. von Oppen, Phys. Rev. B85, 020502 (2012).
[69] B. van Heck, F. Hassler, A. R. Akhmerov, and C. W. J. Beenakker, Phys. Rev. B
84, 180502 (2011).
[70] Y. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys.73, 357 (2001).
[71] P. Bonderson, M. Freedman, and C. Nayak, Phys. Rev. Lett.101, 010501 (2008).
[72] M. Cheng, V. Galitski, and S. Das Sarma, Phys. Rev. B84, 104529 (2011).
[73] Y. V. Nazarov and Y. M. Blanter,Quantum transport: introduction to nanoscience (Cambridge University Press, 2009).
[74] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Science336, 1003 (2012).
[75] M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P. Caroff, and H. Q. Xu, Nano Lett.12, 6414 (2012).
[76] L. P. Rokhinson, X. Liu, and J. K. Furdyna, Nat. Phys.8, 795 (2012).
[77] A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. Shtrikman, Nat. Phys.
8, 887 (2012).
[78] A. Y. Kitaev, Ann. Phys.321, 2 (2006).
[79] S. Das Sarma, M. Freedman, and C. Nayak, Phys. Rev. Lett.94, 166802 (2005).
[80] A. Stern and B. I. Halperin, Phys. Rev. Lett.96, 016802 (2006).
[81] P. Bonderson, A. Kitaev, and K. Shtengel, Phys. Rev. Let..96, 016803 (2006).
[82] L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Nature460, 240 (2009).
[83] A. A. Houck, J. Koch, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf,