Simulation of a quantum phase transition of polaritons with trapped ions
P. A. Ivanov, S. S. Ivanov, N. V. Vitanov, A. Mering, M. Fleischhauer, K. Singer
aa r X i v : . [ qu a n t - ph ] M a y Simulation of a quantum phase transition of polaritons with trapped ions
P. A. Ivanov,
1, 2, ∗ S. S. Ivanov, N. V. Vitanov,
2, 3
A. Mering, M. Fleischhauer, and K. Singer Institut f¨ur Quanteninformationsverarbeitung, Universit¨at Ulm, Albert-Einstein-Allee 11, 89081 Ulm, Germany Department of Physics, Sofia University, James Bourchier 5 blvd, 1164 Sofia, Bulgaria Institute of Solid State Physics, Bulgarian Academy of Sciences, Tsarigradsko chauss´ee 72, 1784 Sofia, Bulgaria Fachbereich Physik, Technische Universit¨at Kaiserslautern, D-67663 Kaiserslautern, Germany (Dated: October 23, 2018)We present a novel system for the simulation of quantum phase transitions of collective internalqubit and phononic states with a linear crystal of trapped ions. The laser-ion interaction creates anenergy gap in the excitation spectrum, which induces an effective phonon-phonon repulsion and aJaynes-Cummings-Hubbard interaction. This system shows features equivalent to phase transitionsof polaritons in coupled cavity arrays. Trapped ions allow for easy tunabilty of the hopping frequencyby adjusting the axial trapping frequency, and the phonon-phonon repulsion via the laser detuningand intensity. We propose an experimental protocol to access all observables of the system, whichallows one to obtain signatures of the quantum phase transitions even with a small number of ions.
PACS numbers: 03.67.Ac, 03.67.Lx, 03.67.Bg, 42.50.Dv
Trapped ions are among the most promising physicalsystems for implementing quantum computation [1] andquantum simulation [2]. Long coherence times and indi-vidual addressing allow for the experimental implemen-tation of quantum gates and quantum computing proto-cols such as the Deutsch-Josza algorithm, teleportation,quantum error correction, quantum Fourier transforma-tion and Grover search [3]. Quantum simulation couldbe performed in the future by large-scale quantum com-putation [4]. With the currently available technology,tailored Hamiltonians can be modeled with trapped ionsto simulate mesoscopic Bose-Hubbard systems [5], spin-boson systems [6] and spin systems [7]. Recently, a quan-tum magnet consisting of two spins has been successfullysimulated experimentally with two trapped ions [8].In this Letter, we propose a physical implementationof the Jaynes-Cummings-Hubbard (JCH) model usingtrapped ions. The JCH model was proposed in the con-text of an array of coupled cavities, each containing asingle two-state atom and a photon [9]. Such a system isdescribed by the combination of two well-known physicalmodels: the Hubbard model [10, 11], which describes theinteraction and hopping of bosons in an optical lattice,and the Jaynes-Cummings model, which describes the in-teraction of an atom with a quantum field [12]. The JCHmodel predicts a quantum phase transition of polaritons,which are collective photonic and atomic excitations. Weshall show that the laser-driven ion chain in a linear Paultrap is described by a JCH Hamiltonian, wherein the ionsand the phonons correspond, respectively, to the atomsand the photons in a coupled cavity array, Fig. 1. Asin [5], the position-dependent energy and the non-localhopping frequency of the phonons is controlled by thetrapping frequencies, while the effective on-site repulsionis provided by the interaction of the phonons with the ∗ Electronic address: [email protected]fia.bg | e,0 > | e,2 > | e,1 > | g,2 > | g,1 > | g,0 > local phonon excitation | e,0 > | e,2 > | e,1 > | g,2 > | g,1 > | g,0 > | e,0 > | e,2 > | e,1 > | g,2 > | g,1 > | g,0 > photons in cavity 1 | e,0 > | e,2 > | e,1 > |g,2> | g,1 > | g,0 > photons in cavity 2 a) b) Laser beam Laser beam
Coulomb mediated phonon hopping local phonon excitation photon hopping
FIG. 1: (Color online). a) Coupled cavities each contain-ing photons and single two-state atoms. Intercavity hoppingis provided by an optical fibre. The strong coupling betweenthe atoms and the photons leads to an effective photon-photonrepulsion. b) All ions are simultaneously interacting with atraveling wave in the radial direction. The laser-ion interac-tion creates an effective on-site interaction between the localphonons. The phonon hopping appears due to the Coulombinteraction and can be adjusted by the mutual distance of theions. internal states of the ions and can be adjusted by theparameters of an external laser field, namely the Rabifrequency and the detuning. This on-site interaction isanalogous to the photon blockade (photon-photon repul-sion), where the strong atom-cavity coupling prevents theentering of additional photons into the optical cavity [13].We shall show that many-body effects appear as a quan-tum phase transition between a localized Mott insulator(MI) and delocalized superfluid state (SF) of the com-posite phononic and internal (qubit) states of the ions.Due to the collective nature of the excitations we distin-guish between collective qubit and phononic SF and MIphases, and the pure phononic SF phase, similar to theeffects predicted in [14] for coupled cavity arrays.Consider a chain of N ions confined in a linearPaul trap along the z axis with trap frequencies ω q ( q = x, y, z ), where the radial trap frequencies are muchlarger than the axial trap frequency ( ω x,y ≫ ω z ), so thatthe ions are arranged in a linear configuration and oc-cupy equilibrium positions z i along the z axis. Makinga Taylor expansion around the equilibrium position, andneglecting x , y , zx , zy and higher order terms, theHamiltonian in the radial direction x reads [15]ˆ H x = 12 M N X k =1 ˆ p k + M ω x N X k =1 ˆ x k − M ω z N X k,m =1 k>m (ˆ x k − ˆ x m ) | u k − u m | . (1)Here ˆ p k is the momentum operator, M is the ion mass, ˆ x k is the position operator of the k th ion about its equilib-rium position z k , u k = z k /l , where l = ( e / πε M ω z ) / is the length scale, e is the charge of the ion, and ε isthe permittivity in free space. In the Hamiltonian (1) themotion in the radial direction is decoupled from the axialmotion. In terms of the normal modes ω p , the Hamilto-nian (1) reads ˆ H x = ~ P Np =1 ω p (ˆ e a † p ˆ e a p + ). Here ˆ e a † p and ˆ e a p are the phonon creation and annihilation operators of the p th collective phonon mode. However, if ˆ x k and ˆ p k arewritten in terms of local creation ˆ a † k and annihilation ˆ a k phonon operators, so that ˆ x k = p ~ / M ω x (ˆ a † k + ˆ a k ) andˆ p k = i p ~ M ω x / a † k − ˆ a k ), the Hamiltonian (1) readsˆ H x = ~ N X k =1 ( ω x + ω k )ˆ a † k ˆ a k + ~ N X k,m =1 k>m t km (ˆ a † k ˆ a m + ˆ a k ˆ a † m ) , (2)where we have neglected higher-order (energy non-conserving) terms. The phonons are trapped with aposition-dependent frequency ω k = − αω z N X s =1 s = k | u k − u s | , (3)where α = ω z /ω x , and they may hop between differentions, with non-local hopping strengths t km = αω z | u k − u m | (4)derived from the long-range Coulomb interaction [5].The collective and local creation and annihilation op-erators are connected by the Bogoliubov transformation,ˆ e a † p = N X k =1 b ( p ) k (ˆ a † k cosh θ p − ˆ a k sinh θ p ) , (5)which preserves the commutation relation, [ˆ a k , ˆ a † m ] = δ km . Here θ p = − ln γ p , with γ p = 1 + α (1 − λ p ) / λ p are the eigenvalues, with eigenvectors b ( p ) , of the ma-trix A km = δ km + 2 P s =1 ,s = k ( δ km − δ sm ) / | u k − u s | [15].Using Eq. (5) one finds that the p th collective phonon state with zero phonons | e p i is a product of N localsqueezed states, | e p i = | ζ ( p )1 i . . . | ζ ( p ) N i , where | ζ ( p ) k i = ∞ X n k =0 s (2 n k − n k )!! (tanh θ p ) n k p cosh θ p | n k i . (6)Here | n k i ( k = 1 , . . . N ) is the local Fock state with2 n k phonons. For the center-of-mass phonon mode wehave p = 1 and cosh θ p = 1; hence the collectiveground state | e i is a product of local ground states, | e i = | i . . . | N i . For a sufficiently small number ofions, we have cosh θ p ≈ | ζ ( p ) k i ≈ | k i . Since thelowest-energy collective vibrational mode in the radialdirection is the highest mode p = N , we find that thesuperfluid ground state of the Hamiltonian (2) is | Ψ SF i = 1 √ N ! N X k =1 b ( N ) k ˆ a † k ! N | i| i . . . | N i . (7)Here we have assumed the commensurate case where thenumber of ions is equal to the number of phonons. Wefind that the ratio between the average number of localphonons in the ground state is given by the square of theoscillation amplitudes: h ˆ n k i / h ˆ n m i = ( b ( N ) k /b ( N ) m ) , whereˆ n k = ˆ a † k ˆ a k ( k = 1 , , . . . , N ) is the local phonon numberoperator.We shall show that the laser-ion interaction inducesan effective repulsion between the local phonons. Thisinteraction provides the phase transition from phononicSF state to composite SF and MI phases of the jointphononic and qubit excitations. Consider ion qubits witha transition frequency ω , which interact along the ra-dial direction with a common traveling-wave laser lightaddressing the whole ion chain with frequency ω L . TheHamiltonian of the system after the optical rotating-waveapproximation is given by [16]ˆ H = ˆ H x + ~ Ω " N X k =1 ˆ σ + k e i η (ˆ a † k +ˆ a k ) − i δt + h.c. . (8)Here ˆ σ + k = | e k ih g k | and ˆ σ − k = | g k ih e k | are the spin flip op-erators, | e k i and | g k i are the qubit states of the k th ion,Ω is the real-valued Rabi frequency, δ = ω L − ω is thelaser detuning, and η = | k | x is the Lamb-Dicke param-eter, with k the laser wave vector, and x = p ~ / M ω x is the spread of the ground-state wave function. TheHamiltonian, after transforming into the interaction pic-ture by the unitary transformation ˆ U = e i ˆ H t/ ~ , withˆ H = − ~ ω x P Nk =1 ˆ a † k ˆ a k + ~ ∆ P Nk =1 | e k ih e k | , in the Lamb-Dicke limit and after the vibrational rotating-wave ap- -2-10-10 -5 0 5 10 µ (3) µ (2) µ (1) µ (0)Detuning ∆ (units of g) C h e m i c a l P o t e n t i a l µ ( u n i t s o f g ) FIG. 2: (Color online) Chemical potential µ as a functionof the laser detuning ∆. Each curve corresponds to µ ( n ) = E g ( n + 1) − E g ( n ), where E g is the ground energy of thesingle ion Hamiltonian (9) for zero hopping. Here n is thenumber of excitations per ion. The Mott lobes are defined as δµ k = µ ( k + 1) − µ ( k ) , k = 0 , , . . . proximation, readsˆ H I = ~ N X k =1 ω k ˆ a † k ˆ a k + ~ ∆ N X k =1 | e k ih e k | + ~ g N X k =1 (ˆ σ + k ˆ a k + ˆ σ − k ˆ a † k )+ ~ N X k,m =1 k>m t km (ˆ a † k ˆ a m + ˆ a k ˆ a † m ) − µ ˆ N , (9)where ˆ H I = ˆ U † ˆ H ˆ U − i ~ ˆ U † ∂ t ˆ U . We assume that the laseris tuned near the red motional sideband δ = − ω x − ∆,with a small detuning ∆ (∆ ≪ ω x ). The couplingbetween the internal qubit and local phonon states is g = η Ω. The Hamiltonian (9) is valid when t km , g ≪ ω x ,which ensures that higher terms, which violate the con-servation of the total number of excitations, can be ne-glected. The first three terms in Eq. (9) describe theJaynes-Cummings model. The first two terms correspondto the energies of the local phonons and the ions, whilethe third term describes the laser-ion interaction. Thefourth term in Eq. (9) describes the non-local hopping ofphonons between different ions and allows the compari-son to Hubbard systems. The last term − µ ˆ N is the usualchemical potential.The Hamiltonian (9) commutes with the total excita-tion operator ˆ N = P Nk =1 ˆ N k , hence the total number ofexcitations is conserved. Here ˆ N k = ˆ a † k ˆ a k + | e k ih e k | is thenumber operator of the total qubit and phononic excita-tions at the k th site. The effective on-site interaction isprovided by the interaction of phonons and qubit statesat each site. The strength of the on-site interaction de-pends on the external parameters, such as the Rabi fre-quency Ω and the laser detuning ∆. This interaction rd ion2 nd ion1 st ionDetuning ∆ (units of g) Q u b i t V a r i a n c e rd ion2 nd ion1 st ion C o ll e c t i v e V a r i a n c e FIG. 3: (Color online) Total (qubit+phonon) variance D N k (top) and the qubit variances D N a,k ( k = 1 , ,
3) (bottom)for a chain of five ions with five excitations as a function ofthe laser detuning ∆ for fixed hopping t = 0 . g . Negativevalues of ∆ correspond to blue detuning with respect to thered-sideband transition. creates an energy gap, which prevents the absorption ofadditional phonons by each ion. In Fig. 2 we plot thechemical potential µ , which counts the energy to add aextra excitation into the system, as a function of the laserdetuning ∆ for zero hopping. The Mott lobes δµ k > D N k = q h ˆ N k i − h ˆ N k i of the number operator ˆ N k with respect to the groundstate of the Hamiltonian (9) for fixed number of excita-tions [14]. If the on-site interaction between the phononsdominates the hopping, the ground state wave functionis a product of local qubit and phononic states for eachsite with a fixed number of excitations. Hence in theMI state, the variance D N k for any k vanishes. Whenthe hopping term dominates the on-site interaction, thenthe ground state consists of a superposition of qubit andphononic states with delocalized excitations over the en-tire chain. In this state the variance D N k at each site(i.e. each ion) is non-zero.Figure 3 (top) shows the variance D N k ( k = 1 , , t = αω z / D N a,k with ˆ N a,k = | e k ih e k | , ( k = 1 , ,
3) [14].This allows us to distinguish the following phases: in theregion of large negative detuning ∆ the collective andthe qubit variances are small, indicating that the sys-tem is in the qubit MI phase. Increasing the detuning,the collective variance stays small but the qubit varianceincreases, which shows that the system is indeed in a col-lective MI phase. Approaching ∆ = 0 the system makesa phase transitions into the collective qubit and phononicSF phase as now both collective and qubit variance arelarge. Finally, for sufficiently large positive detuning thequbit variance decreases but the collective variance stayslarge, which shows that the system is in the phononicsuperfluid phase.The experiment is started by initializing the ion chainwith N phonons in the lowest energy radial mode. Toavoid off resonant excitation of unwanted radial modes, α could be increased temporarily. Then the coupling laseris switched on. The experimental proof for the phasetransition can be carried out by local measurements re-peated for each ion which should be performed fasterthan the hopping time. In the case of Ca + ions thequbit states could be represented by the ground state S / and the metastable state D / . The laser, which createsthe phonon-phonon repulsion, would be detuned to thered sideband of the quadrupole qubit transition betweenthe two states. Then the readout could be performed byscattering photons on the dipole transition S / → P / .This would lead to momentum recoil and changes of thephononic excitation, but to circumvent this we have toperform a measurement of the qubit states, which doesnot affect the phononic state as in the following steps.1) Make a random guess for the qubit excitation. 2)If the guess was the S / ground state, then swap thepopulation S / ⇔ D / by a carrier π pulse leaving thephononic excitation unchanged. 3) Now expose the ionto laser light on the dipole transition. 4) If the ion scat-ters light the guess was wrong and we have to discardthe measurement and restart, otherwise the initial guesswas right and we transfer the qubit excitation back to theS / ground state by another carrier π pulse, then driveRabi oscillations on the red sideband by perpendicularRaman light beams with the difference momentum vec-tor in the radial direction. 6) The phononic populationcan now be extracted by a Fourier analysis of the Rabioscillations.In conclusion, we have proposed a novel implementa-tion of the JCH model by trapped ions simulating po-laritonic phase transitions in coupled cavity arrays. Thesystem shows a Mott insulator to superfluid phase transi-tion of the collective qubit and phononic excitation evenwith a small number of ions. The features can be easilymeasured by local laser adressing. Compared to atomsin optical cavities, our implementation is easier to ma-nipulate, as all parameters can be tuned by changing thetrap frequency, laser detuning and intensity. Addition-ally, the system can be extended by adding impurities ofions with different mass to the ion crystals, which allowsfor simpler addressing of the radial phonon modes and aseparation of coexistent phases.This work has been supported by the European Com-mission projects EMALI and FASTQUAST, the Bulgar-ian NSF grants VU-F-205/06, VU-I-301/07, D002-90/08,and the excellence programme of the LandesstiftungBaden-W¨urttemberg. The authors thank F. Schmidt-Kaler for useful discussions. [1] J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995).[2] R. P. Feynman, Int. J. Theoret. Phys. 21, 467–468 (1982).[3] C. Monroe et al ., Phys. Rev. Lett. 75, 4714 (1995); D.Leibfried, et al. , Nature 422, 412 (2003); F. Schmidt-Kaler, et al. Nature 422, 408 (2003); S. Gulde, et al. ,Nature 421, 48–50 (2003); M. D. Barrett et al. , Nature429, 737–739 (2004); M. Riebe, et al.
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