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Trapped-ion quantum simulation of tunable-range Heisenberg chains
© Graßand Lewenstein; licensee Springer on behalf of EPJ. 2014
Received: 26 February 2014
Accepted: 12 May 2014
Published: 3 June 2014
Quantum-optical techniques allow for generating controllable spin-spin interactions between ions, making trapped ions an ideal quantum simulator of Heisenberg chains. A single parameter, the detuning of the Raman coupling, allows to switch between ferromagnetic and antiferromagnetic chains, and to modify the range of the interactions. On the antiferromagnetic side, the system can be tuned from an extreme long-range limit, in which any pair of ions interacts with almost equal strength, to interactions with a decay. By exact diagonalization, we study how a system of up to 20 ions behaves upon tuning the interactions. We find that it undergoes a transition from a dimerized state with extremely short-ranged correlations towards a state with quasi long-range order, that is, algebraically decaying correlations. The dynamical evolution of the system after a local quench is shown to strongly vary in the two regimes: While in the dimerized limit, the excitation remains localized for long times, propagating spinons characterize the dynamics of the quasi-long-range ordered system. Taking a look onto the ferromagnetic side of the system, we demonstrate the feasibility of witnessing non-locality of quantum correlations by measuring two-particle correlators.
A paradigm system of quantum mechanics which may exhibit intriguing quantum properties like entanglement and non-locality are two spins. By increasing the number of spins, more complex behavior may emerge. In fact, a large variety of condensed matter phenomena, ranging from metal-insulator transition to superfluidity or superconductivity, are successfully described by mapping the relevant low-energy Hilbert space onto a spin model . Moreover, spin models may describe spin-liquid phases which exhibit topological order. In recent years, technological progress in manipulating atoms on the quantum level has allowed to explicitly engineer spin models . This has opened the opportunity for testing the foundations of quantum mechanics, for simulating complex many-body behavior, and for studying the dynamics of interacting spin chains [3–5].
A very promising quantum simulator are trapped ions, with widely spread applications ranging from relativistic quantum mechanics [6, 7], topological systems , fermionic lattice models , to neural networks . Trapped ions can be prepared in such a way that their dynamics is mainly restricted to some internal states of the ions, while the external motion is cooled down to only a few phonons. The internal states then represent a (pseudo)spin, and phonon-mediated spin-spin interactions can be implemented [11, 12]. This has already led to the experimental realization of SU(2) Ising models in one and two spatial dimensions [13–16], and, very recently, to the experimental study of entanglement dynamics in Ising and XY chains [17, 18]. The implementation of more complicated spin models has been suggested, e.g. of Heisenberg and XY models [12, 19], or models with higher spin . Furthermore, the tunability of the phonon-mediated interaction allows to study models with long-range interactions. Roughly, interactions with decay have been engineered for .
In this paper, we first provide an overview of the different models in Section 2. We then discuss the ionic setup in Section 3. In Section 4, we show that by increasing the range of interactions, a transition from a Haldane-Shastry-like quasi-long-range order to a Majumdar-Ghosh-like dimerized order can be observed in the ionic system. In Section 5, we apply different quenches to the system, and study the subsequenct dynamical evolution of the system. We consider local quenches, e.g. by a sudden application of a magnetic field to a single ion, and show that the dynamics in the dimerized regime differs significantly from the one in a system with quasi-long-range order. While in the former case the excitations remain localized for a long time, in the latter they spread in good agreement with the velocity obtained analytically for the Haldane-Shastry model. For global quenches, realized by apruptly changing the range of interactions and thereby sweeping through the dimerization transition, we study the correlation spreading, which is found at an increased velocity. In Section 6, we consider the Lipkin-Meshkov-Glick limit on the ferromagnetic side, which has been suggested for witnessing non-locality of quantum correlations . Finally, we provide a summary in Section 7. Moreover, our paper contains two appendices: In Appendix 1, we provide details to the ionic setup, and in Appendix 2, we discuss the relation between the ionic model and a spin model.
2 The models
where , and denotes a Pauli matrix for the spin at position i. In this paper we focus on the antiferromagnetic side of this model, that is, the model with interaction strengths .
The functional behavior of as may crucially influence the physics of the model. Let us in the following discuss the different cases which are important for our application.
2.1 Haldane-Shastry model
With this criterion, the Haldane-Shastry model, despite the long-range interactions, supports the same quantum phase as the Heisenberg chain with nearest-neighbor interactions, also characterized by quasi long-range spin order. In the next subsection, we will introduce a model which is in contrast to this behavior.
2.2 Majumdar-Ghosh model/ model
It is obvious that in such dimer state, any spin is fully correlated to one nearest neighbor, but fully uncorrelated with the other spins. In other words, no long-range spin correlations exist. Note that for a system with open boundary conditions, the spin at position 1 and the spin at position N do not interact, and therefore the ground state is uniquely given by .
2.3 Lipkin-Meshkov-Glick model
However, this degeneracy is lifted by any small spatial dependence of the interactions.
with the unique ground state for and .
3 The ionic system
Tunable-range Ising models have already been implemnted in trapped ions [13, 14]. In these experiments, two-levels ions are confined to a line by a Paul trap, and their motional state is cooled to only a few phonons. Strong spin-spin interactions are then achieved by applying state-dependent forces on the ions. In Ref. , Ising-type interactions of the form are generated, whereas Ref.  describes the production of interactions of the form or . As reviewed in Ref. , both approaches have basically the same footing, and it is possible to combine them. Instead of an Ising coupling, one then obtains, in the first place, an -model. By making all interactions equal, one gets the Heisenberg model.
An important difference between the ionic system and the ideal models discussed above are the boundary conditions: For the most feasible experimental implementation, they are open, while periodic boundary conditions are convenient for a theoretical description. In general, the effect of boundary conditions is minimized by scaling up the system. For systems of up to ions, we will in the following discuss how close the connection to the Haldane-Shastry model and the Majumdar-Ghosh model can be made by tuning the range of the interaction.
Here, is the energy difference between the two internal levels, is the annihilation operator for phonons denoted by m and with frequency . The Rabi frequencies of the couplings are denoted by , and are the frequencies of the fields. Furthermore, the strength of each coupling depends on the Lamb-Dicke paremeters , which are explicitly defined in the Appendix 1. For all couplings α, the wave vector difference of the photons, , is assumed to be transverse to the ion chain, so the sum over m reduces to a sum over N transverse modes.
with all given by Eq. (16). These interactions can be tuned by varying the frequency of the Raman laser. Choosing it close to the frequency of the center-of-mass mode, the strength of the induced interactions will barely depend on the particles’ position, . The sign of J can be made positive (negative) by tuning above (below) the center-of-mass frequency. Close to resonance, the system is similar to the Lipkin-Meshkov-Glick model. Due to the strong nearest-neighbor and next-nearest-neighbor interactions this limit is somewhat similar to the dimerized -model.
By increasing the detuning from the center-of-mass mode, is made spatially dependent, and in the limit of large detuning, a decay can be achieved. For intermediate values of the detuning, the interactions may approximate a quadratic decay for sufficiently small r, as shown in Figure 1(c), (d). The most dominant interactions then agree quantitatively well with the interactions of the Haldane-Shastry model. Further techniques could be applied in order to achieve a better agreement of the ion setup with a particular model, e.g. by individually addressing of the ions . However, here we restrict ourselves to the simplest implementation which already exhibits rich physics.
The feasibility of the experimental implementation depends on different factors which potentially reduce the fidelity. In the first place, one should note that the effective Hamiltonian (17) has been obtained by neglecting a residual spin-phonon coupling. This implies an error , with the mean phonon number . This error is small in the Lamb-Dicke regime, , typically of the order of few percents . Another source of error, which becomes particularly important for small detunings, are Raman-induced phonon excitations, with a probability given by . Keeping this error small at a given detuning therefore sets an upper limit to the coupling parameter J. On the other hand, J should be sufficiently large compared to imperfections like field imhomogenities or spontanous emission rates. The recent experiments in Maryland  and in Innsbruck  with up to 15 trapped ions operate at a , much larger than the estimated inhomogenities <20Hz . We will find in the next section that dimerization occurs at a detuning of the order of 10kHz, that is, slightly below the lowest detuning realized in Ref. . In this regime, photon errors are estimated to be .
An important part of trapped-ion quantum simulations are the well developped read-out techniques. In particular, it is possible with single-site precision to excite ions from one spin level to some excited state, and detect the subsequent fluorescence. This allows local measurements. Repeated measurements at different i will allow to evaluate correlation functions, e.g. , as has been demonstrated, even with temporal resolution, in Refs [17, 18].
A key issue is the scalability of spin model quantum simulations. This topic has been discussed in Refs [15, 35], finding only a slow growth of the errors due to spontaneous emission. Nevertheless, clearly outperforming classical computing resources with trapped ions may require new architectures, as the quantum simulation of a two-dimensional system with hundreds of ions in a Penning trap .
4 Dimerization transition
We have studied by means of exact diagonalization chains of up to 20 ions. The only tunable parameter in our study is the detuning , where refers to the frequency of the center-of-mass mode, that is, the transverse mode of largest energy. For convenience, we have chosen ions, with equilibirum distance of 10μ m at a radial trap frequency . As shown in Figure 1(c), (d), for very small detuning, , the system is close to the Lipkin-Meshkov-Glick limit: Any pair of ions interacts with almost the same strength. For large detuning, , the interactions decay with . For intermediate values, the interactions between near neighbors becomes similar to the Haldane-Shastry interactions.
A sharp criterion for the transition from a dimerized state to a long-range ordered state can be inferred from Figure 2(b), where the energy of the lowest singlet and the lowest triplet excitation is plotted. For the model, it is known that the crossing of these energies mark with high precision, even in small systems, the dimerization transition . In our case, the triplet becomes the low-lying excitation for . In this context, we also note that, at , in total 118 singlet states have lower energy than the triplet state. Still, this is far from , the number of singlet states providing the ground state manifold of the antiferromagnetic Lipkin-Meshkov-Glick model. But in contrast, at , the lowest triplet state already reaches the 7th position in the energy spectrum.
4.1 Entanglement entropy
where is the density matrix of the subsystem A.
In Figure 3, we plot the entanglement entropy of different systems with spins: the ionic system with detuning and , an ideal Heisenberg chain with nearest neighbor interactions, and the Majumdar-Ghosh chain. In all cases, we have applied open boundary conditions, and accordingly we find alternating behavior of the entanglement entropy. As expected, these alternations are strongest for the Majumdar-Ghosh chain, and weakest for the nearest-neighbor Heisenberg model. The entanglement entropy of ionic systems with behaves very similar to the entanglement entropy of the Majumdar-Ghosh model, proving the dimerized nature of the phase. At , the curve alternates less and comes closer to the entanglement entropy of the nearest-neighbor Heisenberg model. This shows that the tendency of nearest neighbors to form singlets is still present, but also more remote spins become entangled.
4.2 Spin-spin correlations
To some extent this restores periodic boundary conditions. Furthermore, we perform a finite-size scaling (taking into account all even system sizes from to ).
5 Quench dynamics
So far we have considered ground state properties of the trapped ion quantum simulation of the Heisenberg model. In two very recent experiments [17, 18], trapped ions have been demonstrated to be particularly useful for studying also the dynamics of interacting spins. This can be done by exciting the system through either a local or a global quench. In the following, we will investigate the system’s response to local quenches in both the dimerized and the quasi-long-range ordered regime. Afterwards, we study a global quench through the dimerization transition.
5.1 Local quenches
In the dimerized system, Figure 6(a), the magnetic field induces a spin oscillation of the first spin. Simultaneously, also the second spin performs an oscillation in the opposite direction. This behavior is an apparent consequence of the strong anticorrelations between the two spins. Interestingly, all other spins remain almost unaffected by the magnetic field for a long time.
An alternative but similar way of performing a local quench is by a projective measurement of one spin. The subsequent dynamics is then described purely by the Hamiltonian H, and is found to be qualitatively similar to the dynamics described above. The main difference is that, in the system with quasi long-range correlations, the projective measurement will immediately affect all spins to a certain extent, obscuring the signal of the spinons.
While in practice it might be difficult to maintain coherence during the whole evolution shown in Figure 6(a), up to times of the order , the striking differences between Figure 6(a) and (b) become apparent already on a much smaller time scale, say . For typical values of J, this is of the order of 10ms, which is smaller than the duration of measurements in Ref. .
5.2 Global quenches
We have also studied the systems’ dynamics following a global quench. Here an interesting possibility is to abruptly change the range of interactions by changing the detuning. Thereby one can quench through the dimerization transition. Starting from the dimerized ground state, where each spin is correlated only to one neighbor, one can observe the creation of entanglement between more distant spins.
6 Nonlocality witness
Here, are the correlations of two measurements taken at sites k and l. On each site, one may take and . If a θ for which inequality (21) is violated exists, the system must have non-local correlations.
While inequality (21) turns out to be unable to detect nonlocality on the antiferromagnetic side of the ionic setup, a chance for violation exists on its ferromagnetic side. For sufficiently small detuning, the sytem is in the Lipkin-Meshkov-Glick limit of spatially independent interactions, and for , states with maximal spin are ground state of the Hamiltonian . The ground state manifold is thus given by all Dicke states. To lift the degeneracy, one could apply a magnetic field: A field in z-direction yields either or as the unique ground state. The state becomes unique ground state if a staggered magnetic field along z-direction is applied. Alternatively, even without applying a symmetry-breaking field, the degeneracy is lifted in favor of by switching off the interactions . The system is then govered by given in Eq. (12).
With this, inequality (21) is violated. To assert the feasibility of such non-locality witness in the ionic system, we have calculated the ground state of a system of 14 ions, with , while and given by Eq. (16). Indeed, the system is found in the Dicke state with fidelity >0.99 for . By increasing the absolute value of the negative detuning, we find that, while and remain constant, decreases. Thus, for below a critical value , inequality (21) will not be violated anymore. For , we numerically found at , while at . In the same parameter range, the overlap of the ground state with the Dicke state drops suddenly: While it remains above 0.95 up to , it reaches zero for .
In this regime of relatively large detuning, other phonons than the center-of-mass mode strongly affect the spin-spin interactions. This gives rise t very peculiar interaction patterns consisting of attractively and repulsively interacting pairs. It would, however, be striking if non-locality of correlations could also be witnessed in a system where interactions are short-ranged. Interestingly, we find that the short-range ferromagnetic Heisenberg chain behaves exactly the same way as the long-range chain does: Dicke states form a ground state manifold in which degeneracies can be lifted by magnetic fields, in particular, in favor of by a staggered magnetic field. Certainly, the ionic setup discussed here does not immediately allow for implementation of the ferromagnetic short-ranged Heisenberg model due to the mentioned mixing of different phonon modes, but it could be achieved by switching from transverse to longitudinal phonons as transmitter of interactions. In that case, the center-of-mass mode is lowest in energy, and the negative detuning will not interfere with other modes. Moreover, additional control could be implemented through additional Raman couplings . Alternatively, also atoms in optical waveguides or phononic crystals allow for implementing spin models with controllable interactions, and could be tuned into a ferromagnetic short-range regime . We also note that by using adiabatic passage techniques the Dicke state has recently been produced in trapped ion systems .
We have simulated a quantum simulation of the Heisenberg model with trapped ions. In particular, we have investigated the influence of a single control parameter, the detuning, on the system. This parameter allows to tune the range of the interactions. We have shown that this can trigger a dimerization transition. To demonstrate dimerization, we have calculated the entanglement entropy and two-spin correlation functions in systems of up to ions. Studying the response to local quenches, we were able to distinguish between the two regimes also by the system’s dynamics: For long times, local excitations do not spread in the dimerized system. In contrast to this, they are carried through the whole chain in form of spinons in the regime with quasi-long-range order. We also have calculated the spreading of correlations following a global quench, and found some signature of non-locality. Apart from this, we have explored the possibility of witnessing non-locality by measurement of two-spin correlation functions on the ferromagnetic side of the model. In summary, the trapped-ion quantum simulation is able to test basic foundations of quantum mechanics, and to study both ground state properties and dynamics of complicated, long-ranged models.
Appendix 1: Photon-phonon interaction
Here, is the energy difference between the two internal levels, is a Pauli matrix. The phonons in the system are counted by , for each mode denoted by m. The different Raman couplings, denoted by the index α, give rise to Rabi frequencies . The fields have wave vector , frequency , and a phase . The position of the ions is denoted by , and the action of the field on the internal state of each ion is expressed by . We choose the polarization of the couplings such that , whereas .
Several assumptions on the Hamiltonian parameters can be made to simplify the expression. First of all, in the Lamb-Dick regime we have , and we therefore approximate . In the latter step, we have replaced the ions’ coordinates by the normal modes around their equilibrium position. The scalar product is then rewritten in terms of Lamb-Dicke parameters . These parameters measure the strength of the couplings between the photons with wave vector and the phononic mode m.
Here, we label the phononic modes by and an additional label specifying the direction of the modes. Having solved Eq. (27), the Lamb-Dicke parameters are expressed by , with M the ion’s mass.
where , , e the unit of charge, and the electric constant. For a simulation of the Haldane-Shastry model, it turns out to be convenient to generate all interactions using transverse mode. Assuming isotropy in the transverse directions, we will be able to suppress the index α in the following.
Here, we have set the phase of the -laser field, , set to zero. For the coupling in y direction, we choose this phase to be , which effectively replaces by the matrix. Note that in the rotating-wave approximation, we have also neglected terms oscillating with , which is justified if we tune towards the upper edge of the phonon spectrum.
Appendix 2: Time evolution
The time evolution is thus identical to the one of a spin model with spin-spin coupling .
The authors thank Remigiusz Augusiak and Ravindra Chhajlany for discussions. This work has been supported by EU (SIQS, FP7/2007-2013 Grant No. 323714 EQUAM), ERC (QUAGATUA, OSYRIS), Spanish MINCIN (FIS2008-00784 TOQATA), Generalitat de Catalunya (2009-SGR1289), Alexander von Humboldt Stiftung, and Fundació Cellex.
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