Simulating the Quantum Magnet
Axel Friedenauer, Hector Schmitz, Jan Tibor Glückert, Diego Porras, Tobias Schätz
aa r X i v : . [ qu a n t - ph ] F e b Simulating the Quantum Magnet
Axel Friedenauer, Hector Schmitz, Jan Gl¨uckert, Diego Porras, and Tobias Sch¨atz ∗ Max-Planck-Institut f¨ur Quantenoptik,Hans-Kopfermann-Straße 1, 85748 Garching, Germany (Dated: November 2, 2018)To gain deeper insight into the dynamics of complex quantum systems we need a quantum leapin computer simulations. We can not translate quantum behaviour arising with superposition statesor entanglement efficiently into the classical language of conventional computers. The final solutionto this problem is a universal quantum computer [1], suggested in 1982 and envisioned to becomefunctional within the next decade(s); a shortcut was proposed via simulating the quantum behaviourof interest in a different quantum system, where all parameters and interactions can be controlledand the outcome detected sufficiently well.Here we study the feasibility of a quantum simulator based on trapped ions [2]. We experimentallysimulate the adiabatic evolution of the smallest non-trivial spin system from the paramagnetic intothe (anti-)ferromagnetic order with a quantum magnetisation for two spins of 98%, controllingand manipulating all relevant parameters of the Hamiltonian independently via electromagneticfields. We prove that the observed transition is not driven by thermal fluctuations, but of quantummechanical origin, the source of quantum fluctuations in quantum phase transitions [3]. We observea final superposition state of the two degenerate spin configurations for the ferromagnetic ( |↑↑i + |↓↓i )and the anti-ferromagnetic ( |↑↓i + |↓↑i ) order, respectively. These correspond to deterministicallyentangled states achieved with a fidelity up to 88%.Our work demonstrates a building block for simulating quantum spin-Hamiltonians with trappedions. The method has potential for scaling to a higher number of coupled spins [2]. PACS numbers:
I. INTRODUCTION
It is not possible to efficiently describe the time evo-lution of quantum systems on a classical device, like aconventional computer, since their memory requirementsgrow exponentially with their size. For example, a clas-sical memory needs to hold 2 numbers to store arbi-trary quantum states of 50 spin-1/2 particles. To beable to calculate its evolution demands to derive a ma-trix of (2 ) = 2 elements, already exceeding by farthe capacity of state of the art computers. Each dou-bling of computational power permits only one additionalspin-1/2 particle to be simulated. To allow for deeper in-sight into quantum dynamics we need a “quantum leap”in simulation efficiency.As proposed by Richard Feynman [1], a universal quan-tum computer would accomplish this step. A huge va-riety of possible systems are under investigation, a verypromising one being trapped ions [4, 5] acting as quan-tum bits (qubits). After addressing the established crite-ria summarized by DiVincenzo [6] on up to 8 ions [7, 8]with operational fidelities exceeding 99% [7, 8, 9], thereseems to be no fundamental reason why such a devicewould not be realisable.An analogue quantum computer, much closer to theoriginal proposal by Feynman, might allow for a short-cut towards quantum simulations. We want to simulate ∗ Electronic address: [email protected] a system by a different one being described by a Hamilto-nian that contains all important features of the originalsystem. The simulator needs to be controlled, manipu-lated and measured in a sufficiently precise manner andhas to be rich enough to address interesting questionsabout the original system. For large coupled spin systemsoptical lattices might be advantageous [10], while smallerspin systems and degenerate quantum gases might besimulated by trapped ions [2, 11]. Instead of implement-ing a Hamiltonian with a universal set of gates, directsimulation of the Hamiltonian typically consists of one(adiabatic) evolution of the initial state into the corre-sponding final state of interest.Here, in a proof-of-principle experiment, we simulatethe adiabatic transition from a quantum para- to a quan-tum (anti-)ferromagnet and illustrate the advantages ofthe adiabatic quantum simulation (see FIG. 1). Wedemonstrate the individual access, via rf- and laser fields,to all relevant parameters in the underlying Hamiltonian,representing one out of a large spectrum of quantum spin-Hamiltonians.
II. ADIABATIC QUANTUM SIMULATION
The adiabatic quantum simulation of generic spin-Hamiltonians proposed by Porras and Cirac [2] can beillustrated considering a string of charged spin-1/2 par-ticles confined in a common harmonic potential. Twoelectronic states of each ion simulate the two-level sys-tem of a spin-1/2 magnetic moment, |↑i and |↓i . Note
FIG. 1: Phase transition of a quantum magnet: Eachion can simulate a magnetic spin, analogue to an elemen-tary magnet. We initialise the spins in the paramagneticstate |→→ . . . →i , the ground state of the Hamiltonian H B = B x ( σ x + σ x + . . . + σ xN ). This is equivalent to align-ing the spins parallel to the simulated magnetic field. Addingan effective spin-spin interaction J ( t ) (at constant B x ) andincreasing it adiabatically to | J max | ≫ B x , we expect the sys-tem to undergo a quantum phase transition into a ferromag-net, the new ground state of the system ( J is symbolized as lit-tle chains, trying to keep neighbouring spins aligned). Ideally,the two possible ferromagnetic orders |↑↑ . . . ↑i and |↓↓ . . . ↓i are degenerate ground states. The spin system should evolveinto the superposition state |↑↑ . . . ↑i + |↓↓ . . . ↓i , a maximallyentangled “Schr¨odinger Cat” state/magnet. that the inter-ion distance of several µ m renders any di-rect spin-spin coupling negligible. The quantum IsingHamiltonian, H Ising = H B + H J = B x X i σ xi + X i
0, the ground state of the spin-system has allspins aligned with B x along the x -axis. This correspondsto the paramagnetically ordered state |→→ . . . →i , theeigenstate of the Hamiltonian B x P i σ xi with the lowestenergy.For the opposite case of B x = 0 and J <
J >
FIG. 2: Simulating the quantum magnet: a) Two perpendicu-lar polarized laser beams of frequency ω and ω induce a statedependent optical dipole force F ↓ = − / F ↑ along the trapaxis a by the AC-Stark shift (here, only F ↑ is depicted). b)For a standing wave ω = ω , the force conditionally changesthe distance between neighbouring spins simulating a spin-spin interaction [2] ( F ↑ ( F ↓ ) symbolised by the arrow to theright (left)). Only if all spins are aligned (top), the totalCoulomb energy of the spin system is not increased, definingferromagnetic order, the quantum magnet, to be the groundstate. For ω = ω , the sinusoidal force pattern can be seenas a wave moving along the trap axis a pushing or pulling theions repeatedly at a frequency ω − ω . We chose ω − ω close to the resonance frequency of the ions oscillating out ofphase (stretch mode). The energies of different spin statesnow depend on the coupling of the spin state to the stretchmode. Energy can be coupled efficiently into the state withdifferent spin orientations (e.g. bottom), defining the not af-fected upper case as a ground state [14]. The interpretation interms of an effective spin-spin interaction is further describedin methods. states, defined by any superposition of the lowestenergy eigenstates of P i
We implemented the experiment as described in thefollowing. We confine two Mg + ions in a linear Paultrap [16] and laser-cool them to the Coulomb-crystallinephase where the ions align along the trap axis a . Themotion of the ions along a can be described in the basis ofnormal modes: the oscillation-in-phase-mode (com) andthe oscillation-out-of-phase-mode (stretch). The relatedoscillation frequencies amount to ω com = 2 π × . ω stretch = 2 π × . |↓i ≡ | F = 3 , m f = 3 i and |↑i ≡ | F = 2 , m f = 2 i in the 2 S / levels separated by ω ∼ = 2 π × . . Anexternal magnetic field B of 5 . ~ m f of each ion’s angularmomentum F . In this field adjacent Zeeman sublevels ofthe F = 3 and F = 2 manifolds split by 2 . |↓i and the |↑i with aresonant radio-frequency field at ω to implement singlespin rotations [14, 17], R (Θ , φ ) = cos(Θ / I − i sin(Θ /
2) cos( φ ) σ x − i sin(Θ /
2) sin( φ ) σ y , (2)where I is the identity operator, σ x and σ y denote thePauli spin matrices acting on |↓i and |↑i , Θ / B x t isproportional to the duration t of the rotation and φ is thephase of the rf-oscillation, defining the axis of rotation inthe x - y -plane of the Bloch sphere.We provide the effective spin-spin interaction by a statedependent optical dipole force [12, 14, 18]. The relativeamplitudes F ↓ = − / F ↑ are due to AC-Stark shifts in-duced by two laser beams at wavelength λ of 280 nm,depicted in FIG. 2a, perpendicular in direction and polar-isation with their effective wave-vector difference point-ing along the trap axis a . They are detuned 80 GHzblue of the P / excited state, with intensities allow-ing J/ ~ above 2 π × . π × .
45 MHz = ω stretch + δ with δ = − π ×
250 kHz.This choice avoids several technical problems of the orig-inal proposal [2] (see methods), while at the same time,resonantly enhancing the effective spin-spin interactionby a factor of | ω stretch /δ | = 14 . |↓i|↓i | n ∼ = 0 i .We rotate both spins in a superposition state via a R ( π/ , − π/ i = |→i|→i| n ∼ = 0 i . Note that thisparamagnetic state |→i|→i ≡ ( |↑i + |↓i )( |↑i + |↓i ) = |↑↑i + |↑↓i + |↓↑i + |↓↓i has a 25% probability to be pro-jected into either |↑↑i or |↓↓i (normalisation factors aresuppressed throughout).After the adiabatic evolution described below, weproject the final spin state into our σ z -measurementbasis by a laser beam tuned resonantly to the |↓i ↔ P / | F = 4 , m f = 4 i cycling transition [14]. Anion in state |↓i fluoresces brightly, leading to the detec-tion of on average 40 photons during a 160 µ s detectionperiod with our photo multiplier tube. In contrast, an ionin state |↑i remains close to dark (on average 6 photons).We repeat each experiment for the same set of parameters10 times and derive the probabilities P ↓↓ , P ↑↑ and P ↓↑ for the final state being projected into state |↓↓i , |↑↑i and |↓↑i or |↑↓i , respectively (and further described inmethods).We simulate the effective magnetic field by continu-ously applying a radio-frequency field with phase φ = 0and an amplitude such that it corresponds to a single FIG. 3: Quantum magnetisation of the spin system: Weinitialise the spins in the paramagnetic state |→i |→i =( |↑i + |↓i )( |↑i + |↓i ) = |↑↑i + |↑↓i + |↓↑i + |↓↓i , the groundstate of the Hamiltonian H B = B x ( σ x + σ x ). A measurementof this superposition state would already project into each |↑↑i and |↓↓i with a probability of 0.25. After applying B x we adiabatically increase the effective spin-spin interaction J ( t = 0) = 0 to J ( T ). State sensitive fluorescence detec-tion allows to distinguish the final states |↑↑i , |↓↓i , |↑↓i or |↓↑i . Averaging over 10 experiments provides us with itsprobability distribution P ↓↓ (two ions fluoresce), P ↑↑ (no ionsfluoresces), and P ↑↓ or P ↓↑ (one ion fluoresces). We repeatthe measurement for increasing ratios J ( T ) /B x . The exper-imental results for the ferromagnetic contributions P ↑↑ and P ↓↓ are depicted as squares, the solid lines representing thetheoretical prediction. For J ( T ) /B x ≪
1, the paramagneticorder is preserved. For J ( T ) /B x ≫
1, the spins undergo atransition into the ferromagnetic order, the ground state ofthe Hamiltonian H J = J max σ z σ z , with a related quantummagnetisation M = P ↓↓ + P ↑↑ of ≥ (98 ± qubit rotation R (Θ ,
0) with full rotation period Θ = 2 π in 118 µ s and deduce B x = 2 π × .
24 kHz. Precise controlof the phase φ of the rf-oscillator relative to the initiali-sation pulse allows to align B x parallel to the spins alongthe x -axis in the equatorial plane of the Bloch sphere, en-suring that | Ψ i i is an eigenstate of this effective magneticfield.At the same time, we switch on the effective spin-spininteraction J ( t ) ( t ∈ [0; T ]) and increase its amplitudeadiabatically up to J ( T ). At time T , we switch off theinteractions and analyse the final state of the two spinsvia the state sensitive detection described above. In asequence of experiments at constant B x we increase T and therefore J ( t ) /B x . After 50 steps of 2.5 µ s eachwe reach the maximal amplitude J ( t = 125 µ s) /B x = J max /B x = 5 . M = P ↓↓ + P ↑↑ , the probability of beingin a state with ferromagnetic order, of M = (98 ± FIG. 4: Entanglement of the quantum magnet: Measurementof the parity P = P ↓↓ + P ↑↑ − ( P ↓↑ + P ↑↓ ) of the final ferromag-netic state after the simulation reached J max /B x = 5 .
2. As wevary the phase φ of a subsequent analysis pulse, the parity ofthe two spins oscillates as C cos(2 φ ). Together with the finalstate populations P ↓↓ and P ↑↑ depicted in FIG. 3, we can de-duce a lower bound for the fidelity F = 1 / P ↓↓ + P ↑↑ ) + C/ ± |↑↑i + |↓↓i , amaximally entangled state, highlighting the quantum natureof this transition. We find qualitatively comparable resultsfor the antiferromagnetic case |↑↓i + |↓↑i . Each data pointaverages 10 experiments. IV. ENTANGLEMENT
In our experiment we can detect both ferromagneticcontributions, P ↑↑ and P ↓↓ , separately. Any imperfec-tion in the simulation acting as a bias field B z along the z -axis, would energetically prefer one of the ferromag-netic states over the other and therefore unbalance theircontribution to the final state. We carefully cancel allbias fields (see methods) to balance the populations P ↑↑ and P ↓↓ , as can be seen in FIG. 3. The results are ingood agreement with theoretical predictions for our ex-periment, shown as solid lines. We expect the final stateto be a coherent superposition of the two ferromagneticstates |↑↑i + |↓↓i , close to a maximally entangled Bellstate. To quantify the experimentally reached coherencewe measure the parity [21] P = P ↓↓ + P ↑↑ − ( P ↓↑ + P ↑↓ )after applying an additional R ( π/ , φ )-pulse to both ionsafter J max is reached, with a variable rf-phase φ relativeto the rf-field simulating B x . The measured data shownin FIG. 4 have a component that oscillates as C cos(2 φ ),where | C | / |↑↑i and |↓↓i components in the state produced. Deducing acontrast C of (78 ± F = 1 / P ↓↓ + P ↑↑ ) + C/ ± |←←i = ( |↓i − |↑i )( |↑i − |↓i ), with the spinsaligned anti-parallel with respect to the simulated mag-netic field via a R ( π/ , π/
2) rf-initialisation pulse. Theadiabatic evolution should preserve the spin system inits excited state leading now into the anti-ferromagneticorder |↑↓i + |↓↑i . After evolution to J = J max wefind P ↓↑ + P ↑↓ ≥ (95 ± |↓↑i and the |↑↓i components, we ro-tate the state via an additional R ( π/ , |↑↓i + |↓↑i −→ |↑↑i + |↓↓i , beforewe continue to measure the parity as explained above.We deduce a lower bound for the fidelity of the anti-ferromagnetic entangled state F = (cid:12)(cid:12)(cid:10) Ψ final (cid:12)(cid:12) ↓↑ + ↑↓ (cid:11)(cid:12)(cid:12) =1 / P ↑↓ + P ↓↑ ) + C/ ± |←←i as the ground state of the Hamiltonian − H Ising . Because the sign of all spin-spin interactionsis also reversed in − H Ising it is equivalent to a change ofsign in the spin-spin interaction J .The entanglement of the final states additionally con-firms that the transition from paramagnetic to (anti-)ferromagnetic order is not caused by thermal fluctua-tions driving thermal phase transitions. The evolutionis coherent and quantum mechanical, the coherent coun-terpart to the so-called quantum fluctuations [3, 15] driv-ing quantum phase transitions in the thermodynamiclimit. In this picture tunnelling processes [15] inducedby B x coherently couple the degenerate (in the rotatingframe) states |↑i and |↓i with an amplitude proportionalto ( B x / | J | ). In a simplified picture for N spins the ampli-tude for the tunnelling process between Ψ N ↑ = |↑↑ . . . ↑i and Ψ N ↓ = |↓↓ . . . ↓i is proportional to ( B x / | J | ) N , sinceall N spins must be flipped. In the thermodynamic limit( N → ∞ ) the system is predicted to undergo a quan-tum phase transition at | J | = B x . At values J > B x the tunnelling between Ψ ∞↑ and Ψ ∞↓ is completely sup-pressed. In our case of a finite system Ψ ↑ and Ψ ↓ re-main coupled and the sharp quantum phase transitionis smoothed into a gradual change from paramagnetic to(anti-)ferromagnetic order. V. CONCLUSION AND OUTLOOK
We demonstrated the feasibility of simple quantumsimulations in an ion trap by implementing the Hamil-tonian of a quantum magnet undergoing a robust transi-tion from a paramagnetic to an entangled ferromagneticor anti-ferromagnetic order. While our system is cur-rently too small to solve classically intractable problems,it uses an approach that is complementary to a universalquantum computer in a way that can become advanta-geous as the approach is scaled to larger systems. Sinceour scheme only requires inducing the same overall spin-dependent optical force on all the ions [2], it does not relyon the use of sequences of quantum gates, thus its scalingto a higher number of ions can be simpler. Furthermorethe desired outcome might not be affected by decoherenceas drastically as typical quantum algorithms, because acontinuous loss of quantum fidelity might not spoil com-pletely the outcome of the experiment (for example the(anti-)ferromagnetic ordering transitions are hardly af-fected by phase decoherence), while universal quantumcomputation will almost certainly require involved sub-agorithms for error correction [4]. Decoherence in thesimulator might even mimic the influence of the naturalenvironment [22] of the studied system, if we judiciouslyconstruct our simulation (for example the decoherencewe mainly observe in our demonstration implements adephasing environment).Despite technical challenges, we expect that this workis the start to extensive experimental research of com-plex many-body phases with trapped ion systems. Lin-ear trapping setups may be used for the quantum simula-tion of quantum dynamics beyond the ground state wherechains of 30 spins would already allow to outperform cur-rent simulations with classical computers. We may alsoadapt our scheme to new ion trapping technologies [13].For example, a modest scaling to systems of 20 ×
20 spinsin 2D would yield insight into open problems in solid statephysics, e.g. related to spin-frustration. This could pavethe way to address a broad range of fundamental issuesin condensed matter physics which are intractable withexact numerical methods, like, for example, spin liquidsin triangular lattices, suspected to be closely related tophases of high- T c superconductors [23]. VI. METHODS
State dependent optical dipole force: An effective(Ising) spin-spin interaction was proposed to be im-plemented via magnetic field gradients [24]. Por-ras and Cirac suggested to use state dependent opti-cal dipole forces [11, 18] displacing the spin state | s i ( s either ↑ or ↓ ) in phase space by an amount that de-pends on | s i . The area swept in phase space changesthe state to e iφ ( s ) | s i . The phase φ ( s ) can be brokendown into single spin terms proportional to σ zi and ap-parent spin-spin interactions proportional to σ ki σ kj andthus gives rise to the desired simulation of spin-spin in-teractions [25]. It can also lead to single spin phasesthat simulate the unwanted contribution of a commonbias fields B z σ z in the Hamiltonian that will unbalancethe probabilities P ↓↓ and P ↑↑ . To achieve a balancedprobability distribution as depicted in FIG. 3, we haveto carefully compensate these single spin phases. To thisend we (1) compensate the residual AC-Stark shifts ofthe individual laser beams by carefully choosing direc-tion and polarisation of the beams [10] and (2) compen-sate for the imbalance caused by single spin phases via adetuning of the order of several kHz of the rf-transitionrelative to ω . (3) The ions have to be separated by aninteger multiple of the effective wavelength λ eff = λ/ √ × λ eff , requiring the control of the axial trapping frequency to better than 100 Hz. Incontrast to phase gate implementations [18] we have todetune the two laser beams far enough to keep the mo-tional excitation and the related errors due to residualspin-motion coupling [2] insignificant. Adjusting the de-tuning to δ = − ( ω stretch − ( ω − ω )) = − π ×
250 kHz redof the stretch mode frequency and terminating the evo-lution after the system returned back into the motionalground state [18] ideally completely cancels the simula-tion errors discussed in [2].State sensitive detection: For two spins, the integratedfluorescence signal does not allow to distinguish betweentwo ( |↑↓i and |↓↑i ) of the four possible spin configura-tions. In addition, the amount of detected photons foreach of the three distinguishable configurations fluctuatesfrom experiment to experiment according to Poissonianstatistics and therefore can only be determined with lim-ited accuracy. For the data reported, we repeated eachexperiment 10 times and fitted the resulting photon-number distribution to the weighted sum of three refer-ence distributions to derive P ↓↓ , P ↑↑ and P ↑↓ + P ↓↑ .Adiabatic evolution: We achieve the best fidelities forthe reported transitions at a duration of the simulationof T = 125 µ s at a B x = 2 π × .
24 kHz. Even though weare not strictly in the adiabatic limit, the robustness ofthe transition allows to minimise decoherence effects re-ducing the duration of the simulation. In addition, tech-nical reasons led to the evolution of J ( t ) linear in timeto up to J ( t = 50 µ s) = 5 × − J max , continued by J ( t ) ≈ (e αt − β ) best fitted by α = 0 .
026 and β = 4.Up to now we did not improve the fidelities by evolvingor terminating J ( t ) or B x in a more adiabatic way. Acknowledgments
This work was supported by the Emmy-Noether Pro-gramme of the German Research Foundation (DFG,Grant No. SCHA 973/1-2), the MPQ Garching, the DFGCluster of Excellence Munich-Centre for Advanced Pho-tonics, and the European project SCALA. We thank Di-etrich Leibfried for his invaluable input and Ignacio Ciracand Gerhard Rempe for most interesting comments andgenerous support. [1] R. P. Feynman, Int. J. Theo. Phys. , 467 (1982).[2] D. Porras and J. I. Cirac, Phys. Rev. Lett. , 207901(2004).[3] S. Sachdev, Quantum Phase Transitions (CambridgeUniversity Press, 1999).[4] M. Nielsen and I. Chuang,
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