Majorana Fermions in Equilibrium and Driven Cold Atom Quantum Wires

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Citation Jiang, Liang, Takuya Kitagawa, Jason Alicea, A. Akhmerov, David Pekker, Gil Refael, J. Cirac, Eugene Demler, Mikhail Lukin, and Peter Zoller. 2011. Majorana fermions in equilibrium and driven cold atom quantum wires. Physical Review Letters 106(22): 220402.

Published Version doi:10.1103/PhysRevLett.106.220402

Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:8152137

Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#OAP Majorana Fermions in Equilibrium and Driven Cold Atom Quantum Wires

Liang Jiang,1,2, ∗ Takuya Kitagawa,3,∗ Jason Alicea,4 A. R. Akhmerov,5 David Pekker,2 Gil Refael,2 J. Ignacio Cirac,6 Eugene Demler,3 Mikhail D. Lukin,3 Peter Zoller,7 1 Institute for Quantum Information, California Institute of Technology, Pasadena, CA 91125, USA 2 Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA 3 Department of Physics, Harvard University, Cambridge, MA 02138, USA 4 Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA 5 Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands 6 Max-Planck-Institut fürQuantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany and 7 Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria (Dated: March 3, 2011) We introduce a new approach to create and detect Majorana fermions using optically trapped 1D fermionic atoms. In our proposed setup, two internal states of the atoms couple via an optical Raman transition—simultaneously inducing an effective spin-orbit interaction and magnetic field—while a background molecular BEC cloud generates s-wave pairing for the atoms. The resulting cold atom quantum wire supports Majorana fermions at phase boundaries between topologically trivial and nontrivial regions, as well as ‘Floquet Majorana fermions’ when the system is periodically driven. We analyze experimental parameters, detection schemes, and various imperfections.

Majorana fermions (MFs), which unlike ordinary 2kxˆ (a) (b) (c)    fermions are their own antiparticles, are widely sought for aa†† BEC k2 k1 p,, p their exotic exchange statistics and potential for topolog-  e  RF Fermions ical quantum information processing. Various promising   1 2 g proposals exist for creating MFs as quasiparticles in 2D   0 k1 k2 systems, such as quantum Hall states with filling factor a† a† b† 5/2 [1], p-wave superconductors [2], topological insula- pk, pk, 0 tor/superconductor interfaces [3, 4], and semiconductor heterostructures [5–8]. In addition, MFs can even emerge FIG. 1: (Color online.) (a) Optically trapped fermionic atoms in 1D quantum wires, such as the spinless p-wave super- form a 1D quantum wire inside a 3D molecular BEC. Two Ra- conducting chain [9] which is effectively realized in semi- man beams propagate along ~k1 and ~k2 directions, respectively. conductor wire/bulk superconductor hybrid structures The recoil momentum ~k1 − ~k2 = 2kxˆ is parallel to the quan- with spin-orbit interaction and strong magnetic field [10– tum wire. (b) Raman coupling between two fermionic states 13]. Although there are many theoretical and experimen- a↑ and a↓ induces a 2k momentum change from photon recoil. tal efforts to search for MFs, their unambiguous detection (c) RF-induced atom-molecular conversion. remains an outstanding challenge. Significant advances in cold atom experiments have fective magnetic field. Combined with s-wave pairing in- opened up a new era of studying many-body quantum duced by the surrounding BEC of Feshbach molecules, systems. Cold atoms not only sidestep the issue of dis- the cold atom quantum wire supports MFs at the bound- order which often plagues solid-state systems, but also aries between topologically trivial and non-trivial super- benefit from tunable microwave and optical control of conducting regions [10, 11]. In contrast to the earlier 2D the Hamiltonian. In particular, recent experiments have cold-atom MF proposals that require sophisticated op- demonstrated synthetic magnetic fields by introducing tical control like tilted optical lattices [21] or multiple a spatially dependent optical coupling between different laser beams [18, 20], our scheme simply uses the Raman internal states of the atom [14, 15], which can be gen- transition with photon recoil to obtain spin-orbit interac- eralized to create non-Abelian gauge fields with careful tion. Moreover, compared with the solid-state proposals design of optical couplings [16, 17]. For example, Rashba [3, 10, 11], the cold atom quantum wire offers various ad- spin-orbit interaction can be generated in an optically vantages such as tunability of parameters and, crucially, arXiv:1102.5367v2 [cond-mat.quant-gas] 2 Mar 2011 coupled tripod-level system [18], which can be used for much better control over disorder. generating MFs in 2D [19, 20]. Theoretical Model. We consider a system of optically In this Letter, we propose to create and detect MFs trapped 1D fermionic atoms inside a 3D molecular BEC using optically trapped 1D fermionic atoms. We show (Fig. 1). The Hamiltonian for the system reads that optical Raman transition with photon recoil can in- X † duce both an effective spin-orbit interaction and an ef- H = ap (εp + V + δRF ) ap (1) p X † † † + Bap+k,↑ap−k,↓ + ∆ap,↑a−p,↓ + h.c. . ∗These authors contributed equally to this work. p 2

(a) (b) The fermionic atoms with momentum p have two relevant E(p) Δ/B T internal states, represented by spinor ap = (ap,↑, ap,↓) . p2 1 Trivial The kinetic energy is εp = and optical trapping po- 2m (C>0) tential is V . As shown in Figs. 1(a) and (b), two laser beams Raman couple the states ap−k,↓ and ap+k,↑ with ∗ 0 Topological Ω1Ω2 p coupling strength B = , where δe is the optical de- (C<0) δe ~ ~ tuning, Ω are Rabi frequencies, and k1 − k2 = 2kxˆ 1(2) 0 1 is the photon recoil momentum parallel to the quan- (c) μ/B tum wire. The bulk BEC consists of Feshbach molecules C

(b a↑ + a↓) [22–24] with macroscopic occupation in 0 x the ground state hb0i = Ξ. The interaction between the fermionic atoms and Feshbach molecules can be in- (d) duced by an RF field with Rabi frequency g and detuning MF MF MF MF δRF . The effective pairing energy is ∆ = gΞ for fermionic atoms [25]. FIG. 2: (Color online.) (a) Energy dispersion for spin-orbit- We can recast the Hamiltonian into a more trans- coupled fermions in a magnetic field. There is an avoided parent form by applying a unitary operation that in- crossing at p = 0 with energy splitting 2B (dark solid line). duces a spin-dependent Galilean transformation, U = p 2 2 R † † The horizontal dotted line represents ∆ + µ , which has ik x(a ax,↑−a ax,↓)dx e x,↑ x,↓ , where x is the coordinate along two crossing points when p∆2 + µ2 < B (blue dotted line) the quantum wire. Depending on the spin, the transfor- and four crossing points when p∆2 + µ2 > B (orange dotted † mation changes the momentum by ±k, Uap+k,↑U = ap,↑ line). (b) Phase diagram for topological and trivial phases † and Uap−k,↓U = ap,↓. The transformed Hamiltonian with respect to parameters of ∆ and µ. (c,d) C (x) can take closely resembles the semiconducting wire model studied positive or negative values, which divides the quantum wire in [10, 11] and reads into alternating regions of topological and trivial phases. X H = a† (ε − µ + upσ + Bσ ) a + ∆a† a† + h.c. , p p z x p p,↑ −p,↓ values, which divides the quantum wire into alternat- p (2) ing regions of topological and trivial phases [Figs. 2(c) and (d)]. Exactly one MF mode localizes at each phase where µ ≡ − (δRF + V + εk) is the local chemical potential and the velocity u = k/m determines the strength of boundary. The position of the MFs can be changed the effective spin-orbit interaction. by adiabatically moving a blue-detuned laser beam that Topological and Trivial Phases. The physics of the changes µ (x). Similarly, we can also use focused Raman quantum wire is determined by four parameters: the s- beams to induce spatially dependent B (x) to control the wave pairing energy ∆, the effective magnetic field B, locations of topological and trivial phases. the chemical potential µ, and the spin-orbit interaction Floquet MFs. It has been recently proposed that pe- 2 riodically driven systems can host non-trivial topological energy Eso = mu /2. For p 6= 0, the determinant of 0 orders [26, 27], which may even have unique behaviors Hp is positive definite, so the quantum wire system has an energy gap at non-zero momenta. For p = 0, how- with no analogue in static systems [28]. Our setup in- 0 p 2 2 deed allows one to turn a trivial phase topological by ever, Hp yields an energy E0 = B − ∆ + µ which 2 2 2 introducing time dependence, generating ‘Floquet MFs’. vanishes when the quantity C ≡ ∆ + µ − B equals For concreteness we consider the time-dependent chemi- zero, signaling a phase transition [10, 11] (see Fig. 2b). cal potential When C > 0 the quantum wire realizes a trivial superconducting phase. For example, when B ∆, µ all en- µ for t ∈ [nT, (n + 1/2) T ) µ (t) = 1 , (3) ergy gaps are dominated by the pairing term, yielding µ2 for t ∈ [(n + 1/2) T, (n + 1) T ) an ordinary spinful 1D superconductor. When C < 0 a topological superconducting state emerges. For instance, which can be implemented by varying the optical trap po- when B ∆, µ, Eso the physics is dominated by a single tential V or the RF frequency detuning δRF . In addition, spin component with an effective p-wave pairing energy we assume the presence of a 1D optical lattice which mod- up ∆p ≈ ∆ B ; this is essentially Kitaev’s spinless p-wave ifies the kinetic energy εp → −2J cos (ka) cos(pa) and superconducting chain, which is topologically non-trivial the spin-orbit interaction upσz → 2J sin (ka) sin (pa) σz and supports MFs [9]. in Eq.(2), where J is the tunnel matrix element and a is With spatially dependent parameters (µ, B or ∆), we the lattice spacing. can create boundaries between topological and trivial Let Hj be the Hamiltonian with µ = µj. The time- phases. MFs will emerge at these boundaries [10, 11]. evolution operator after one period is then given by −iH2T/2 −iH1T/2 Spatial dependence of µ (x) can be generated by addi- UT = e e . We define an effective Hamil- −iH T tional laser beams with non-uniform optical trapping po- tonian from the relation UT ≡ e eff , and study the tential V (x). Then C (x) can take positive or negative emergence of MFs in Heff. Eigenstates of Heff are called 3

-1 MF – associated with a different, non-trivial topological Q� Q0 charge Q0. Interestingly, the two flavors of Floquet MFs 1 at E = 0 and E = π/T are separated in quasi-energies, π and therefore, they are stable Floquet MFs as long as the periodicity of the drive is preserved. The presence of �/2 two particle-hole symmetric gaps changes the topological classification of the system from Z2 to Z2 × Z2. Two topological charges Q0 and Qπ are defined as 0 follows. For translationally invariant quantum wire, the evolution operator has momentum decomposition Q UT (τ) = p UT,p(τ) for intermediate time τ ∈ [0,T ]. -�/2 After one evolution period, we have UT ≡ UT (T ) and U ≡ U (T ). The topological charge Q (or Q ) is

Quasi-energy (ET) T,p T,p 0 π the parity of the total number of times that the eigenval- -π ues of U (τ) and U (τ) cross 1 (or −1). The topolog- 0.2 0.4 0.6 0.8 1.0 1.2 1.5 1.6 T,0 T,π ical charges have the closed form Driving period (JT) Q0Qπ = Pf [M0] Pf [Mπ] Q = Pf [N ] Pf [N ] , (4) FIG. 3: (Color online.) Floquet MFs with two distinct flavors. 0 0 π Quasi-energy spectrum of Heff and topological charges (Q0 p and Qπ) are plotted for varying period T of the drive. Since where Mp = log [UT,p] and Np = log UT,p are skew the quasi-energy is defined up to an integer multiple of 2π/T , symmetric matrices associated with the evolution, and p it can support Floquet MFs at E = π/T (thick red line) as Pf[X] is the Pfaffian of matrix X. Here UT,k is de- well as E = 0 (thick blue line). The appearance of the two MF termined by the analytic continuation from the history flavors is not necessarily correlated, and a single Floquet MF of UT,k(τ). Note that the product of topological charges is present in much of the parameter space. The parameters Q0Qπ is analogous to the Z2 invariant suggested for static are µ1 = −J, µ2 = −3J, B = J, ∆ = 2J, and 2ka = π/4. MFs [9]. In Fig. 3, we plot the topological charges Q0 and Qπ for various driving period T . Indeed, Floquet states at E = 0 and E = π/T appear in the range of T Floquet states and represent stationary states of one pe- at which Q0 and Qπ equal to −1, respectively. riod of evolution. The eigenvalues of Heff are called quasi- Probing MFs. RF spectroscopy can be used to probe energies because they are only defined up to an integer MFs in cold atom quantum wires [29–32]. In particu- multiple of 2π/T . This feature, combined with the built- lar, we consider spatially resolved RF spectroscopy [33] in particle-hole symmetry enjoyed by the Bogoliubov-de as an analog of the STM. The idea is to use another Gennes Hamiltonian, allows for Floquet MFs carrying probe RF field to induce a single particle excitation from non-zero quasi-energy. That is, since states with quasi- the fermionic state (say aσ) to an unoccupied fluores- energy E and −E are related by particle-hole symmetry, cent probe state f. Contrary to conventional RF spec- states with E = 0 or E = π/T ≡ −π/T can be their own troscopy, a tightly confined optical lattice strongly local- particle-hole conjugates. izes the atomic state f, yielding a flat energy band for The existence of Floquet MFs is most easily revealed this state. By imaging the population in state f, we gain by plotting the quasi-energy spectrum of Heff in a finite new spatial information about the local density of states. system, which in practice can be created by introducing For example, by applying a weak probe RF field 0 a confinement along the quantum wire. In Fig. 3, we with detuning δRF from the aσ-f transition, the pop- plot the spectrum for a 100-site system with µ1 = −J, ulation change in state f can be computed from the µ = −3J, B = J, ∆ = 2J, 2ka = π/4 for varying drive d † 2 linear response theory I (x, ν) ≡ dt f (x) f (x) ∝ period T . Note that both H and H correspond to the 0 0 1 2 ρaσ (x, −µ˜ (x) − δRF + ε) Θ (˜µ (x) + δRF − ε). Since the trivial phase with C1,C2 > 0. For small T , states with MFs have zero energy in the band gap and are spatially quasi-energy E = 0 or E = π/T are clearly absent from localized at the end of the quantum wire, there will be an the spectrum—i.e., there are no Floquet MFs here. enhanced population transfer to state f with frequency 0 ∗ ∗ As one increases T , the gap at π/T closes, and for δRF = ε − µ (x ) at the phase boundary x . If the aσ- larger T a single Floquet state with E = π/T remains. f transition has good coherence, we can use a resonant We have numerically checked that the amplitude for this RF π-pulse to transfer the zero-energy population from Floquet state peaks near the ends of the 1D system. Thus aσ to f, and then use ionization or in situ imaging tech- it arises from two localized Floquet MFs and this state niques [34, 35] to reliably readout the population in f is associated with non-trivial topological charge Qπ as with single particle resolution. Floquet MFs can also be we will see below. As one further increases T , another detected in a similar fashion. Since a Floquet state at state at quasi-energy E = 0 appears whose wavefunction quasi-energy E is the superposition of energy states with again peaks near the two ends – a second type of Floquet energies E + 2nπ/T for integer n, we should find the 4

Floquet MFs at energies 0 (or π) + 2nπ/T for 0 (or π) the effects can be examined by considering a spatially iφ(~r) quasi-energy Floquet MFs, respectively. dependent order parameter Ξ0e . A large phase gra- Parameters and Imperfections. We now estimate the 2|∆| dient φx ≡ dφ/dx > u will close the energy gap, and experimental parameters for cold atom quantum wires. the MFs will merge into~ the continuum. To sustain the 2 (1) The spin-orbit interaction energy is Eso = mu /2 ≤ energy gap, the fluctuations in the phase gradient should E , with recoil energy E ≈ 30 (2π) kHz for 6Li p 2 p 2 ∗ rec,0 rec be small, i.e., hφx (T )ithermal < hφx (T )ithermal = atoms. If we use n sequential Λ transitions, the spin-orbit 2|∆| ∗ (n) u , with critical temperature T . Thus, the BEC tem- interaction strength can be increased to u = nk/m and ~ ∗ perature should be below min {T ,Tc} ∼ 50nK. (3) Since (n) 2 Eso = n Eso. (2) The s-wave pairing energy ∆ = gΞ the quantum wire has a finite transverse confinement, can be comparable to the BEC transition temperature other transverse modes might be occupied and coupled 2 2/3 kTc ∼ ~ n0 /m before the BEC is locally depleted. For non-resonantly. Nevertheless, recent numerical and ana- 14 −3 molecule density n0 = 10 cm [23, 24], we have |∆| ∼ lytical studies [12, 13, 36, 37] show that MFs can be ro- Ω Ω∗ 10 (2π)kHz. (3) The effective magnetic field B = 1 2 bust even in the presence of multiple transverse modes, as δe 2 long as an odd number of transverse quantization chan- and the depth of the optical trap V ∼ Ω can be MHz, 0 δ nels are occupied. These results may potentially relax the by choosing large detuning δ ∼ 100 (2π)THz and Rabi requirement of tight confinement of the quantum wire. frequencies Ω ∼ 50 (2π)GHz, while still maintaining a 2 In conclusion, we have proposed a scheme to create and low optical scattering rate Γ ≈ Ω γ ∼ 1 (2π)Hz. (4) The δ2 probe MFs in cold atom quantum wires, and suggested transverse oscillation frequency of the 1D optical trap can q the creation of two non-degenerate flavors of Floquet MF 4V0 be ν ≈ mw2 ∼ 150 (2π)kHz for a laser beam with waist at a single edge. We estimated the experimental param- w = 15µm. Since ν is much larger than the energy scales eters to realize such implementation, considered schemes of Eso and |∆|, it is a good approximation to consider a to probe for MFs, and analyzed imperfections from re- single transverse mode of the quantum wire. alistic considerations. Recently, it has been discovered In practice, there are various imperfections such as par- that braiding of non-Abelian anyons can be achieved in ticle losses due to collision and photon scattering, finite networks of 1D quantum wires [38], which would be very temperature of BEC, and multiple transverse modes of interesting to explore in the cold atoms context. the quantum wire. (1) The lifetime associated with pho- We would like to thank Ian Spielman for enlighten- ton scattering induced loss can be improved to seconds ing discussions. This work was supported by the Sher- using large detuning and strong laser intensity, and the man Fairchild Foundation, DARPA OLE program, CUA, collision-induced loss can be suppressed by adding a 1D NSF, AFOSR Quantum Simulation MURI, AFOSR optical lattice to the quantum wire. (2) At finite tem- MURI on Ultracold Molecules, ARO-MURI on Atom- perature the BEC order parameter will fluctuate, and tronics, and Dutch Science Foundation NWO/FOM.

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