Nanoplasmonic planar traps - a tool for engineering p-wave interactions
B. Juliá-Díaz, T. Graß, O. Dutta, D. E. Chang, M. Lewenstein
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Nanoplasmonic planar traps - a tool for engineering p -wave interactions B. Juli´a-D´ıaz , , T. Graß , O. Dutta , D. E. Chang , and M. Lewenstein , Departament d’Estructura i Constituents de la Mat`eria,Universitat de Barcelona, 08028 Barcelona, Spain ICFO-Institut de Ci`encies Fot`oniques, Parc Mediterrani de la Tecnologia, 08860 Barcelona, Spain and ICREA-Instituci´o Catalana de Recerca i Estudis Avan¸cats, 08010 Barcelona, Spain
Engineering strong p -wave interactions betweenfermions is one of the challenges in modern quan-tum physics. Such interactions are responsiblefor a plethora of fascinating quantum phenom-ena such as topological quantum liquids and ex-otic superconductors. In this letter we proposeto combine recent developments of nanoplasmon-ics with the progress in realizing laser-inducedgauge fields. Nanoplasmonics allows for strongconfinement leading to a geometric resonance inthe atom-atom scattering. In combination withthe laser-coupling of the atomic states, this isshown to result in the desired interaction. Weillustrate how this scheme can be used for the sta-bilization of strongly correlated fractional quan-tum Hall states in ultracold fermionic gases. Recently there has been growing interest in plasmonicnanostructures that can be used for various applicationsin quantum optics and atomic physics [1–7]. Particularlyinteresting is the possibility of confining atomic motionover regions in space of order of nanometers, comparableor smaller than typical values of the atom-atom scat-tering length. In such a regime, atomic scattering un-dergoes strong modifications due to confinement inducedresonances [8, 9]. Here we propose to use this effect toengineer strong and robust p -wave interactions betweenfermionic atoms in planar geometries, which overcomesthe challenges associated with creating such interactionsin previously proposed techniques. This opens a newpath towards the realization of exotic fractional quan-tum Hall states [10, 11] and superfluid phases [12, 13].A strong motivation for realizing such states are theirintriguing topological properties which find direct appli-cation in topological quantum computation, protectedquantum qubits, and protected quantum memories [14].Similarly, p -wave repulsion can stabilize low filling frac-tional quantum Hall states [10, 15], including the Moore-Read state [11]. This state has been proposed in the con-text of a pronounced fractional quantum Hall plateau atfilling 5/2 [13], but formally it also resembles the spinlesschiral p -wave superfluid state. In solid-state physics, onlyin Strontium Ruthenate, chiral ( p x + ip y )-wave Cooperpairs are believed to be responsible for the observed su-perfluidity of electrons [16]. In the field of quantum gases,strong p -wave interaction can in principle be achievedby using Feshbach resonances. Due to the inelastic lossprocesses, however, a strong p -wave interaction is hard to achieve experimentally [17]. Also, in Bose-Fermi mix-tures, density fluctuations of bosons can induce attractive p -wave interactions or even higher partial waves betweenthe fermions [18–20]. However, such proposals also runinto difficulties due to the phase separation instability ofBose-Fermi mixtures and stringent constraints on tem-perature.In this letter we propose to combine two importantconcepts: strongly confined two-dimensional (2D) trapsvia nanoplasmonic fields, and strong laser induced syn-thetic gauge fields, as illustrated in Fig. 1. For simplicity,the synthetic gauge field considered is produced througha minimal scheme described in Ref. [21]. It consists ofa laser field coupling two internal levels of the fermionicatoms, and an external electric or magnetic field whichproduces a linear variation of the energy of the internalstates throughout the sample. Preparing the system inthe lower dressed state, the sample is effectively subjectedto a strong synthetic magnetic field [22]. Being polarizedin one dressed state, interactions between the fermionicatoms are prohibited by the Pauli principle. However,the external degrees of freedom provide a small couplingto the higher dressed state. Remarkably, this will beshown to result in a residual p -wave contact interactionbetween the fermions. This contribution can be enhancedthanks to the resonant behavior of the atom-atom scat-tering length in strongly confined 2D settings [9]. Thisnot only allows to strengthen the interaction but also toexplore both attractive and repulsive p -wave interactions,i.e. going from the physics of p -wave pairing to fractionalquantum Hall physics.Experimental difficulties to provide a sufficient trans-verse confinement, that is on the order of the atom-atomscattering length, are surmountable thanks to the newdevelopments in plasmonics. The interaction of coldatoms with nanoplasmonic systems has attracted signif-icant interest recently. A notable feature of surface plas-mon excitations, which exist along a metal-dielectric in-terface, is the lack of a diffraction limit. In the context ofatom trapping, this enables the generation of fields withdramatically reduced effective wavelengths compared tofree space, and a corresponding reduction of parameterssuch as trap confinement. The properties of the plasmonscan also be greatly engineered through the underlyingdevice geometry. The interaction between Bose-Einsteincondensates and tailored plasmonic micro-potentials hasrecently been observed [3], and plasmon-based trappingtechniques for ultracold atoms with applications in quan-tum simulation have been proposed [1, 2, 4]. Figure 1. The ultracold atomic sample is tightly confined above a metallic surface by a nanoplasmonic field produced by anexternal laser (in red) pointing perpendicular to the metal surface. A second laser (in blue), shining in from the side, is usedto generate the artificial magnetic field felt by the atoms. The mechanism for fermion-fermion contact interaction involves thevirtual excitation of one of the atoms into the excited dressed state, as illustrated in the inset.
As a straightforward application of the scheme, we willconsider the case of repulsive interactions and show byexact diagonalization how the induced p -wave interactioncan be used to explore different quantum Hall phases, no-tably going from filled Landau level (LL) physics, to thefractional quantum Hall regime with a ν = 1 / Model
We consider a trapped ultracold gas of fermionic atomswith two internal states | g i , | e i . The single particleHamiltonian H sp = H ext + H AL consists of an exter-nal part H ext = p / (2 M ) + V ( r ) with the anisotropictrapping potential V ( r ), and an atom-laser coupling H AL including also the internal energies. This coupling is re-sponsible for a synthetic gauge field which emerges dueto the accumulation of Berry’s geometrical phase whenan atom moves within the laser field [23]. The key toachieving non-vanishing phases on closed contours is tomake the internal energies, and thus H AL , spatially de-pendent via a Stark or Zeeman shift, such that H AL and H ext do not commute (see Methods).The laser light mixes the ground and excited state,giving rise to position-dependent dressed states , | Ψ i and | Ψ i , which are the eigenstates of the atom-laser interac-tion. As detailed in the Methods section, increasing thelaser strength, and thus the Rabi coupling between thetwo bare states, one can energetically favor one dressedmanifold, say | Ψ i . By adjusting the external trapping,the single particle Hamiltonian projected in this lowestdressed manifold can be written as the usual quantumHall one, H = ( p + A ) M + M ω ⊥ − η )( x + y ) (1)where ω ⊥ is the effective xy trapping frequency, A = ~ η ( y, − x ) /λ ⊥ , and η is the strength of the syntheticgauge field which depends on the laser wavenumber k and the spatial extent of the Stark or Zeeman shift w (see Methods). This Hamiltonian has the well-knownLL structure, and its eigenfunctions are the Fock-Darwinstates. Restricting ourselves to the lowest LL, the corre-sponding wave functions read ϕ FD l ( z ) ∝ z l exp( −| z | /λ ⊥ )where z = x − iy describes the atom position in the ( x, y )plane, and λ ⊥ = p ~ / ( M ω ⊥ ). The adiabatic approxi-mation requires large Rabi frequencies ~ Ω ≫ E R [22],where the recoil energy is E R = ( k λ ⊥ / ~ ω ⊥ ). Withinthis limit, the off-diagonal Hamiltonian elements, H and H , connecting the dressed states are neglected.Then, transitions to the higher dressed manifold are fullysuppressed. A general atomic state, χ ( r ) = ˜ ϕ ( r ) ⊗| Ψ i + ˜ ϕ ( r ) ⊗ | Ψ i , becomes a low-lying solution for˜ ϕ = 0 and ˜ ϕ = ϕ FD l . In our approach, however, someamount of non-adiabaticity is crucial, as it will yield afinite value for ˜ ϕ resulting in non-zero contact interac-tions. p -wave fermion-fermion interaction Now we turn to the atom-atom interactions, whichwe take as contact interactions. In terms of the barefermionic states, it reads V ij = g c ~ M δ ( z i − z j )( | e i | g i h e | h g | + | g i | e i h g | h e | ) . (2)Here, g c is a number quantifying the interaction strength.A more precise definition will be given later. Of course,in the dressed basis the interaction term maintains itsform, such that interactions remain restricted to pairsof one atom in | Ψ i and the other in | Ψ i . Thus, bypolarizing the system in the lower dressed state | Ψ i , nointeractions are present in the adiabatic limit Ω → ∞ .Still, by making the ratio of the Rabi frequency to recoilenergy much bigger than 1, R E ≡ ~ Ω /E R ≫
1, we canwork in a quasi-polarized regime, in which the | Ψ i levelserves only as a virtual manifold.In this limit, the unperturbed many-body Hamiltonianis given by H (0) = N X i =1 H i P i . (3)where the operator P i = | Ψ i i h Ψ | i projects the i th par-ticle onto the low-lying Hilbert space. The off-diagonalterms, H | Ψ i h Ψ | and H | Ψ i h Ψ | , and the atom-atom interaction of Eq. (2) are taken as perturbations.They give second-order corrections. The effective many-body Hamiltonian can then be written as H eff = H (0) + H (1) + H (2) with H (1) = − X i H i H i ~ Ω P i H (2) = X ij P i H i V ij H j ( ~ Ω ) P j . (4)Note that the denominator in H (1) has been set to aconstant equaling the energy difference between dressedstates | Ψ i and | Ψ i . As this is taken to be large, it isthe dominant contribution to the energy gap.In a previous study of a bosonic system [22, 24], wehave analyzed the influence of H (1) , but the many-body contribution H (2) has been negligible due to the bosonicnature of the atoms. We will in the following show thatin the fermionic case, where H (2) is the only many-bodycontribution, it becomes crucial. As illustrated in Fig. 1, H (2) describes a process where one atom is excited from | Ψ i to the virtual | Ψ i manifold, where it interacts withan atom in | Ψ i , to then get de-excited to | Ψ i again.Importantly, acting among fermions, the many-body in-teraction term gives solely non-zero p -wave contributions.This is seen by using Eq. (13) to cast H (2) into p -waveform (cf. Ref. [15]), H (2) ∝ X i,j ˆ p ij δ (2) ( z ij )ˆ p ij P i P j (5)with the relative variables, ˆ p ij = − i ~ ( ∂ z i − ∂ z j ) and z ij = z i − z j . The important feature of Eq. (4) is that theeffective interaction is linear in the bare one V ij , whichallows one to change the interaction from attractive torepulsive. This is in contrast to second order mechanimslike the Kohn-Luttinger [25].The main question which arises at this point is whetherthe residual interaction term, Eq. (5), is strong enoughto significantly modify the physics of the system. Thisbecomes possible by tuning the interaction strength g c .It is well known that this parameter crucially depends onthe geometry of the system. In particular, for transversalconfinements on the order of the scattering length, andconsidering the case of attractive interaction the effective2D coupling is known to behave as [9] g c = 4 π ~ M √ πλ z /a D + log(0 . ~ ω z /πǫ ) , (6)where ǫ is the energy of the motion in the x − y plane and a D the 3D scattering length. For a value of ~ ω z /ǫ = 10 ,it produces a resonant behavior for values of the trans-verse confinement λ z ∼ . | a D | . This confinement-induced resonance behaviour is not present in usual ex-periments with optical traps. There, the trapping on the z direction has at most been of the order of hundreds ofnanometers, far from the resonance region. The trans-verse confinement lengths of λ z ∼ λ eff ∼ r ,even for system sizes far below the free-space wave-length r ≪ λ . This yields a corresponding reductionof ∼ p r/λ in the trap spatial confinement compared tofree-space techniques.This resonant behavior can in principle be used to pro-duce arbitrarily large values of g c and, importantly, al-lows to achieve not only large values of the coupling, but c < L > F i d e lit y Pfaffian ν =1/3 Laughlin QP ν =1/3 Laughlin (b)(a) ν =1 Laughlin Figure 2. Evolution of the ground state as a function of the in-teraction parameter ˜ g c = g c / ( kλ ⊥ ηR E ) . Panel (a) presentsthe overlap of the ground state with the filled LL state (solid-black), the fermionic Pfaffian state (dotted-red), the quasi-particle state over the ν = 1 / ν = 1 / also provides a way of producing both attractive and re-pulsive p -wave interactions between the fermions. Example: stabilization of the ν = 1 / Laughlin state
As an example, we discuss the case of repulsive p -waveinteraction. In the context of quantum Hall physics wehave to ask whether the obtained p -wave interaction iscapable of bringing the system from the integer quantumHall regime of a non-interacting system to the fractionalquantum Hall regime. In that case, a Laughlin-like stateshould show up as the ground state of the system . Notethat in the quantum Hall regime, η →
1, the contributionof H reduces to a constant, as all Fock-Darwin statesbecome (quasi)degenerate. Thus, to bring the systeminto the fractional quantum Hall regime, the interactionterm must be comparable to the contribution of H (1) term, which breaks the rotational symmetry [24].To give definite numerical predictions, we perform anexact diagonalization (see Methods) with a few number Let us recall the Laughlin wave function [10] at filling ν ,Ψ Laughlin = N Y i 98. We discussthe different phases appearing as we vary the interactionstrength, g c . For weak interactions, the ground state ofthe system has a large overlap with the analytical form ofthe filled LL state, ν = 1, as depicted in Fig. 2. The angu-lar momentum of the ground state is found to be slightlylarger than the analytical value, L = 6. As explained inRefs. [22, 24], this is due to the derivation from rotationalsymmetry. When the interaction is increased, the systemundergoes a transition into a phase, where the groundstate has large overlaps with the fermionic Moore-Readstate [11]. Even stronger interactions bring the systeminto a state which resembles the quasiparticle excitationof the ν = 1 / g c , one finally reaches the ν = 1 / Summary We have presented a novel mechanism to realize sizable p -wave interactions between fermionic atoms. The key isthe combination of a strongly confining plasmonic field,which allows to explore confinement-induced resonances,with a simple scheme to generate a strong synthetic gaugefield. To exemplify the potential of our approach, we haveconsidered the case of repulsive p -wave interaction. Wehave shown that our proposal allows to stabilize a num-ber of interesting quantum Hall states, like the Pfaffian,and the ν = 1 / ACKNOWLEDGMENTS BJD and TG are grateful for stimulating discus-sions with the “Ytterbium team” at ILP (Hamburg), C.Becker, S. D¨orscher, B. Hundt, and A. Thobe. This workhas been supported by EU (NAMEQUAM, AQUTE),ERC (QUAGATUA), Spanish MINCIN (FIS2008-00784TOQATA), Generalitat de Catalunya (2009-SGR1289),Alexander von Humboldt Stiftung, and AAII-Hubbard.BJD is supported by the Ram´on y Cajal program. MLacknowledges support from the Joachim Herz Founda-tion and Hamburg University. DEC acknowledges sup-port from Fundaci´o Privada Cellex Barcelona. METHODSSingle particle Hamiltonian The single particle Hamiltonian reads, H sp = H ext + H AL (8)where H ext = p / (2 M ) + V ( r ) with the anisotropic trap-ping potential V ( r ). H AL is the atom-laser coupling,which includes the internal energies. To make H AL spa-tially dependent, such that H AL and H ext do not com-mute, we perform a Stark or Zeeman shift of the internalenergies. The strength of this shift can be characterizedby a length scale w , which is chosen such that the energiesof the bare internal states read E g = − ~ Ω x/ (2 w ) and E e = ~ ω A + ~ Ω x/ (2 w ). Here, ω A is the energy differ-ence of the bare states. In this way, preparing the systemin the ground-state of H AL , the external part will stim-ulate transitions into the excited manifold of H AL . Theprobability of such transitions is controlled by the Rabifrequency Ω of the coupling. The laser frequency is setto resonance with the atomic transition. Furthermore, wechoose the laser to be a running wave in y -direction withwavenumber k . Then, within the rotating-wave approxi-mation, the atom-laser Hamiltonian H AL can be writtenin terms of bare states | e i and | g i as [29]ˆ H AL = ~ Ω2 [cos θ ( | e i h e | − | g i h g | )+ sin θ (cid:0) e iφ | e i h g | + h . c . (cid:1)(cid:3) , (9)where Ω = Ω p x /w , tan θ = w/x , and φ = ky .Note that spontaneous emission processes are not con-sidered in the Hamiltonian of Eq. (9). This is justifiedif the two atomic states are sufficiently long-lived, as isthe case for the S → P clock transition in Ytterbium.In contrast to the bosonic Ytterbium isotopes, the finitespin of the fermionic isotopes yields a small magneticmoment, which allows for a strong coupling of the clockstates at reasonable laser power. Thus, achieving largeRabi frequencies, as required by our proposal, poses noproblem for , Yb [30].Diagonalizing Eq. (9) yields the dressed states, | Ψ i = e − iG (cid:0) C e iφ/ | g i + S e − iφ/ | e i (cid:1) , | Ψ i =e iG (cid:0) − S e iφ/ | g i + C e − iφ/ | e i (cid:1) , where C = cos θ/ S = sin θ/ G = kxy w . The single-particle Hamiltonian H sp can be expressed as a 2 × H ij . In the dressedstate basis, its diagonal terms can be written as [22], H jj = ( p − ǫ j A ) M + U + V + ǫ j ~ Ω2 , (10)with ǫ = 1 and ǫ = − 1. Full expressions for the vec-tor potential A and the scalar potential U are given in Ref. [22]. Note that for w ≪ x, y , we recover the symmet-ric gauge expression A ( r ) = ~ k w ( y, − x ). With a conve-nient choice of the trapping potential, H can be madesymmetric. Then, the Hamiltonian element H reads H = ( p + A ) M + M ω ⊥ − η ) r (11)where ω ⊥ is the effective xy trapping frequency, η =( kλ ⊥ ) / (4 w ), and λ ⊥ = p ~ / ( M ω ⊥ ). The recoil energyof the atoms is defined as E R = ( k λ ⊥ / ~ ω ⊥ ).Retaining up to quadratic terms, the off-diagonalHamiltonian elements, H = H † , explicitly read H ≃ − ~ M (cid:20) − ik∂ y Ψ + (cid:18) k x w + iky w (cid:19) Ψ + 1 w ∂ x Ψ (cid:21) = − ~ M h ˆ ac + ˆ a † c + ˆ bc + ˆ b † c i , (12)with ˆ a † ≡ − λ ⊥ ∂ ¯ z + λ − ⊥ z , ˆ a ≡ λ ⊥ ∂ z + λ − ⊥ ¯ z , ˆ b † ≡− λ ⊥ ∂ z + λ − ⊥ ¯ z , and ˆ b ≡ λ ⊥ ∂ ¯ z + λ − ⊥ z . Acting ona Fock-Darwin state the operators ˆ a (ˆ a † ) decrease (in-crease) the l quantum number by one, while the opera-tors ˆ b and ˆ b † change the Landau level.As we will be interested in the fractional quantumHall regime of large synthetic magnetic field, η ≃ b and ˆ b † contribu-tions. In this limit, we have c = c ≃ w/λ ⊥ , and c = − c ≃ / ( wλ ⊥ ). In our numerics, we will further-more choose w ≃ . λ ⊥ and k = 10 /λ ⊥ , implying η ≃ c ≪ c . We can then write H = − w ~ M λ ⊥ (ˆ a + ˆ a † ) + O [( w/λ ⊥ ) − ] . (13) Exact diagonalization To solve the effective Hamiltonian, we perform exactdiagonalization. Therefore, we build many-body statesusing as single particle states the Fock-Darwin states, | l i . Then the second quantized form of H (2) is H (2) = 12 X ij,kl ˆ f † i ˆ f † j ˆ f k ˆ f l V ij,kl , (14)where ˆ f i anihilates an atom in ϕ FD i ( z ). The matrix el-ement reads, V ij,kl = ( ~ Ω ) − h i |h j | H V H | l i| k i . 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