Abstract
To gain deeper insight into the dynamics of complex quantum systems we need a quantum leap in computer simulations. We cannot translate quantum behaviour arising from superposition states or entanglement efficiently into the classical language of conventional computers. The solution to this problem, proposed in 1982 (ref. 1), is simulating the quantum behaviour of interest in a different quantum system where the interactions can be controlled and the outcome detected sufficiently well. Here we study the building blocks for simulating quantum spin Hamiltonians with trapped ions^{2}. We experimentally simulate the adiabatic evolution of the smallest nontrivial spin system from paramagnetic into ferromagnetic order with a quantum magnetization for two spins of 98%. We prove that the transition is not driven by thermal fluctuations but is of quantummechanical origin (analogous to quantum fluctuations in quantum phase transitions^{3}). We observe a final superposition state of the two degenerate spin configurations for the ferromagnetic order (↑↑〉+↓↓〉), corresponding to deterministic entanglement achieved with 88% fidelity. This method should allow for scaling to a higher number of coupled spins^{2}, enabling implementation of simulations that are intractable on conventional computers.
Main
It is not possible to efficiently describe the time evolution of quantum systems on a classical device, such as a conventional computer, because their memory requirements grow exponentially with their size. For example, a classical memory would need to hold 2^{50} numbers to store arbitrary quantum states of 50 spin1/2 particles. The ability to calculate their evolution requires the derivation of a matrix of (2^{50})^{2}=2^{100} elements, already exceeding by far the capacity of stateoftheart computers. Further, each doubling of computational power permits only one additional spin1/2 particle to be simulated. To allow for deeper insight into quantum dynamics, we need a ‘quantum leap’ in simulation efficiency.
As proposed in ref. 1, a universal quantum computer would accomplish this step. A huge variety of possible systems are under investigation, with individual trapped ions^{4,5} as quantum bits (qubits) being a very promising architecture. After addressing the established criteria summarized in ref. 6 on up to eight ions^{7,8} with operational fidelities exceeding 99% (refs 7, 8, 9), there seems to be no fundamental reason why such a device would not be realizable.
Alternatively, an analogue quantum computer, much closer to the original proposal in ref. 1, might allow for a shortcut towards quantum simulations. We would like to simulate a given system by a different one described by a Hamiltonian containing all the important features of the original system. The simulator system needs to be controlled, manipulated and measured in a sufficiently precise manner and has to be rich enough to address interesting questions about the original system. For large coupled spin systems, optical lattices might be advantageous^{10}, whereas smaller spin systems and degenerate quantum gases might be simulated by trapped ions^{2,11}. Instead of implementing a Hamiltonian with a universal set of gates, direct simulation of the Hamiltonian can be implemented by one (adiabatic) evolution of the initial state into the corresponding final state of interest.
Here, in a proofofprinciple experiment, we simulate the adiabatic transition from a quantum para to a quantum (anti) ferromagnet and illustrate the advantages of the adiabatic quantum simulation (Fig. 1). We demonstrate the individual access, via radiofrequency (r.f.) and laser fields, to all relevant parameters in the underlying Hamiltonian, representing one out of a large spectrum of quantum spin Hamiltonians.
The adiabatic quantum simulation of generic spin Hamiltonians proposed in ref. 2 can be illustrated considering a string of charged spin1/2 particles confined in a common harmonic potential. Two electronic states of each ion simulate the twolevel system of a spin 1/2 magnetic moment, ↑〉 and ↓〉. The interion distance of several micrometres renders any direct spin–spin coupling negligible. The quantum Ising Hamiltonian,
consists of two terms. The first denotes the interaction of each individual spin, represented by the Pauli operator σ_{i}^{k} (k can be x,y or z), with a uniform magnetic field of amplitude proportional to B_{x} pointing in direction x. The second term represents the spin–spin interaction, which tries to align the spins (σ_{i}^{z}) parallel or antiparallel along the z axis, depending on the sign of the interaction amplitude J_{i j}. To simulate the first term in equation (1), the eigenstates of σ_{i}^{z}, ↑〉_{i} and ↓〉_{i}, can be coupled with an electromagnetic field. The latter is simulated by a statedependent force^{12}, further explained in Fig. 2.
To understand the experiment discussed below, we consider interactions between nearest neighbours only, J_{i,i+1}=J, and two extreme scenarios. For the case of J=0 and B_{x}>0, the ground state of the spin system has all spins aligned with B_{x} along the x axis. This corresponds to the paramagnetically ordered state →→…→〉, the eigenstate of the Hamiltonian with the lowest energy.
For the opposite case of B_{x}=0 and J<(>)0, the system has an infinite number of degenerate ground states, defined by any superposition of the lowestenergy eigenstates of , namely ↑↑…↑〉 and ↓↓…↓〉, which represent ferromagnetic order (or ↑↓↑…↑↓〉 and ↓↑↓…↓↑〉, the antiferromagnetic order, respectively). Initializing the spin system in an eigenstate in the σ^{x} basis, starting with J(t=0)=0, B_{x}>0 and adiabatically increasing J(t) to J_{max}≫B_{x} should evolve the system from the paramagnetic order arbitrarily close to the (anti) ferromagnetic order, as depicted in Fig. 1. A quantum phase transition is supposed to occur at B_{x}=J in the thermodynamic limit of an infinite number of spins^{3,13}.
We experimentally demonstrate the above features on two spins as follows. We confine two ^{25}Mg^{+} ions in a linear Paul trap^{14} and lasercool them to the Coulombcrystalline phase, where the ions align along the trap axis a. The motion of the ions along this axis a can be described in the basis of normal modes. The inphase mode oscillates at a frequency ω_{com}/2π=2.1 MHz, whereas the outofphase mode has .
In our implementations we define the ^{2}S_{1/2} hyperfine ground states as ↓〉≡F=3;m_{f}=3〉 and ↑〉≡F=2;m_{f}=2〉, which are separated by ω_{0}/2π≃1.7 GHz. An external magnetic field B (different from the simulated magnetic field in equation (1)) of 5.5 G orients the magnetization axis for the projection ℏm_{f} of each ion’s angular momentum F. In this field adjacent Zeeman sublevels of the F=3 and 2 manifolds are split by 2.7 MHz per level.
We coherently couple the states ↓〉 and ↑〉 with a resonant r.f. field at ω_{0} to implement single spin rotations^{14,15,16}
where I is the identity operator, σ^{x} and σ^{y} denote the Pauli spin matrices acting on ↑〉_{i} and ↓〉_{i}, Θ/2=B_{x}t/ℏ is proportional to the duration t of the rotation and φ is the phase of the r.f. oscillation defining the axis of rotation in the x–y plane of the Bloch sphere^{15}.
We provide the effective spin–spin interaction by a statedependent opticaldipole force^{12,16,17}. The relative amplitudes F_{↓}=−(3/2)F_{↑} are due to a.c. Stark shifts induced by two laser beams at a wavelength λ of 280 nm, depicted in Fig. 2a, perpendicular in direction and polarization, with their wavevector difference pointing along the trap axis a. They are detuned 80 GHz blue of the ^{2}P_{3/2} excited state, with intensities allowing for J/ℏ as large as 2π×22.1 kHz. We use a walkingwave forcepattern by detuning the two laser frequencies by 2π×3.45 MHz=ω_{stretch}+δ, with δ=−2π×250 kHz. This choice avoids several technical problems of the original proposal^{2} (see the Methods section), while at the same time resonantly enhancing the effective spin–spin interaction by a factor of ω_{stretch}/δ=14.8 compared with the standingwave case^{18}. The enhancement is induced by the fact that the walking wave is closer to resonance (detuned only by δ) with the vibrational mode compared with the standing wave (detuned by ω_{stretch}).
After laser cooling we initialize the quantum simulator by optical pumping^{19} to the state ↓〉↓〉n≃0〉. We rotate both spins in a superposition state via an R(π/2,−π/2) pulse (see equation (2)) on the r.f. transition to initialize the state Ψ_{i}〉=→〉→〉n≃0〉. Note that the paramagnetic state →〉→〉≡(↑〉+↓〉)(↑〉+↓〉)=↑↑〉+↓↑〉+↑↓〉+↓↓〉 has a 25% probability to be projected into either ↑↑〉 or ↓↓〉 (normalization factors are suppressed throughout).
We simulate the effective magnetic field by continuously applying an r.f. field with phase φ=0 and an amplitude such that it corresponds to a single qubit rotation R(Θ,0) with full rotation period Θ=2π in 118 μs and deduce B_{x}/ℏ=2π×4.24 kHz. Precise control of the phase φ of the r.f. oscillator relative to the initialization pulse enables alignment of B_{x} parallel to the spins along the x axis in the equatorial plane of the Bloch sphere, ensuring that Ψ_{i}〉 is an eigenstate of this effective magnetic field.
At the same time, we switch on the effective spin–spin interaction J(t) (t∈[0;T]) and increase its amplitude adiabatically up to J(T) within 50 steps of 2.5 μs each. At time T, we switch off the interactions and analyse the final state of the two spins via the statesensitive detection described below. In a sequence of experiments at constant B_{x} we increase J_{max} and therefore J(T)/B_{x}. Finally we reach the maximal amplitude J(t=125 μs)/B_{x}=J_{max}/B_{x}=5.2 (see the Methods section) and achieve a quantum magnetization M, the probability of being in a state with ferromagnetic order, of M=P_{↓↓}+P_{↑↑}=98%±2%.
After the adiabatic evolution described above, we project the final spin state into the σ^{z} measurement basis by a laser beam tuned resonantly to the ; m_{f}=4〉 cycling transition^{16}. An ion in state ↓〉 fluoresces brightly, leading to the detection of an average of 40 photons during a 160 μs detection period with our photomultiplier tube. In contrast, an ion in state ↑〉 remains close to dark (on average six photons). We repeat each experiment for the same set of parameters 10^{4} times and derive the probabilities P_{↓↓}, P_{↑↑} and P_{↓↑} for the final state being projected into state ↓↓〉, ↑↑〉 and (↓↑〉 or ↑↓〉), respectively (further described in the Methods section).
In our experiment we can detect both ferromagnetic contributions, P_{↑↑} and P_{↓↓}, separately. Any imperfection in the simulation acting as a bias field B_{z} along the z axis would energetically prefer one of the ferromagnetic states over the other and therefore unbalance their contributions to the final state. We carefully cancel all bias fields (see the Methods section), to balance the populations P_{↑↑} and P_{↓↓} as can be seen in Fig. 3. The results are in good agreement with theoretical predictions for our experiment, shown as solid lines. We expect the final state to be a coherent superposition of the two ferromagnetic states ↑↑〉+↓↓〉—a maximally entangled Bell state. To quantify the experimentally reached coherence we measure the parity^{20} P=P_{↓↓}+P_{↑↑}−(P_{↓↑}+P_{↑↓}) after applying an additional R(π/2,φ) pulse to both ions after J_{max} is reached, with a variable r.f. phase φ relative to the r.f. field simulating B_{x}. The measured data shown in Fig. 4 have a component that oscillates as Ccos(2φ), where C/2 characterizes the coherences between the ↑↑〉 and the ↓↓〉 components in the final state Ψ_{final}〉 produced. Deducing a contrast C of 78±2% from the best fit we derive the Bellstate fidelity^{20} F=〈Ψ_{final}↓↓+↑↑〉^{2}=1/2(P_{↓↓}+P_{↑↑})+C/2 of 88±3%.
We also simulate the adiabatic evolution of a system not initialized in the ground state of the initial Hamiltonian. In particular, we prepare the paramagnetic eigenstate , with the spins aligned antiparallel with respect to the simulated magnetic field via an R(π/2,π/2) r.f. initialization pulse. The adiabatic evolution should preserve the spin system in its excited state, leading now into the antiferromagnetic order ↑↓〉+↓↑〉. After evolution to J=J_{max} we find P_{↓↑}+P_{↑↓}≥95±2%. To investigate the coherence between the ↑↓〉 and the ↓↑〉 components, we first rotate the state via an additional R(π/2,0) pulse, which would ideally take ↑↓〉+↓↑〉→↑↑〉+↓↓〉, followed by the measurement of the parity as explained above. We deduce the fidelity of the antiferromagnetic entangled state F=〈Ψ_{final}↓↑+↑↓〉^{2}=1/2(P_{↓↑}+P_{↑↓})+C/2=80±4%.
An equally valid viewpoint of this experiment interprets as the ground state of the Hamiltonian–H_{Ising}. Because the sign of all spin–spin interactions is also reversed in −H_{Ising}, it is equivalent to a change of sign in the spin–spin interaction J.
The entanglement of the final states confirms that the transition from paramagnetic to (anti) ferromagnetic order is not caused by thermal fluctuations driving thermal phase transitions. The evolution is coherent and quantum mechanical, the coherent equivalent to the socalled quantum fluctuations^{3,13} driving quantum phase transitions in the thermodynamic limit. In this picture, tunnelling processes^{13} induced by B_{x} couple the degenerate (in the rotating frame) states ↑〉_{i} and ↓〉_{i} with an amplitude proportional to (B_{x}/J). In a simplified picture for N spins, the amplitude for the tunnelling process between Ψ_{N↑}=↑↑…↑〉 and Ψ_{N↓}=↓↓…↓〉 is proportional to (B_{x}/J)^{N}, because all N spins must be flipped. In the thermodynamic limit (), the system is predicted to undergo a quantum phase transition at J=B_{x}. At values J>B_{x}, the tunnelling between and is completely suppressed. In our case of a finite system, Ψ_{2↑} and Ψ_{2↓} remain coupled and the sharp quantum phase transition is smoothed into a gradual change from paramagnetic to (anti) ferromagnetic order.
In conclusion, we have demonstrated the feasibility of simple quantum simulations in an ion trap by implementing the Hamiltonian of a quantum magnet undergoing a robust transition from a paramagnetic to an entangled ferromagnetic or antiferromagnetic order. Although our system is currently too small to solve classically intractable problems, it uses an approach that is complementary to a universal quantum computer in a way that can become advantageous as the approach is scaled to larger systems. Our scheme does not rely on the use of sequences of quantum gates, thus scaling to a higher number of ions can be simpler, because it only requires inducing the same overall spindependent optical force on all the ions^{2}. Furthermore, the desired outcome might not be affected by decoherence as drastically as typical quantum algorithms, because a continuous loss of quantum fidelity might not completely spoil the outcome of the experiment.
For example, the coherence of a Greenberger–Horne–Zeilingerlike state ↑↑…↑〉+↓↓…↓〉 might be completely lost, but still the output of the quantum simulation describes correctly the ground state in a solidstate system, where spontaneous symmetry breaking implies ferromagnetic ordering randomly along one of two antiparallel orientations. In contrast, universal quantum computation will almost certainly require involved subalgorithms for error correction^{5}. Decoherence in the simulator might even mimic the influence of the natural environment^{21} of the studied system if we judiciously construct our simulation (for example, the decoherence we mainly observe in our demonstration implements a dephasing environment).
Despite technical challenges, we expect that this work is the start to extensive experimental research of complex manybody phases with trappedion systems. Linear trapping setups may be used for the quantum simulation of quantum dynamics beyond the ground state, where chains of 30 spins would already enable us to outperform current simulations with classical computers. We may also adapt our scheme to new iontrapping technologies^{22}. For example, a modest scaling to systems of 20×20 spins in two dimensions would yield insight into open problems in solidstate physics, for example related to spin frustration. This could pave the way to address a broad range of fundamental issues in condensedmatter physics that are intractable with exact numerical methods, such as spin liquids in triangular lattices, suspected to be closely related to phases of hightemperature superconductors^{23}.
Methods
Statedependent opticaldipole force
An effective (Ising) spin–spin interaction was proposed to be implemented via magnetic field gradients^{24}. It was suggested to use statedependent opticaldipole forces^{2,11,17} displacing the spin state S〉 (S either ↓ or ↑) in phase space by an amount that depends on S〉. The area swept in phase space changes the state to e^{iφ(S)}S〉. The phase φ(S) can be broken down into singlespin terms proportional to σ_{i}^{z} and apparent spin–spin interactions proportional to σ_{i}^{k}σ_{j}^{k} and thus gives rise to the desired simulation of spin–spin interactions^{25}. It can also lead to singlespin phases that simulate the unwanted contribution of a common bias field B_{z}σ^{z} in the Hamiltonian that will result in an imbalance of the probabilities P_{↓↓} and P_{↑↑}. To achieve a balanced probability distribution as depicted in Fig. 3, we have to carefully compensate these singlespin phases. To this end we compensate the residual a.c. Stark shifts of the individual laser beams by carefully choosing the direction and polarization of the beams^{12} and also compensate for the imbalance caused by singlespin phases via a detuning of the order of several kilohertz of the r.f. transition relative to ω_{0}. Furthermore, the ions have to be separated by an integer multiple of the effective wavelength ; in our implementation 18λ_{eff}≈3.6 μm, requiring the control of the axial trapping frequency to better than 100 Hz.
To minimize the errors of our simulation, we have to keep the motional excitation small enough during the evolution to enable the system to be described within the Lamb–Dicke regime^{16} (ideally ). The Lamb–Dicke factor η can be interpreted as the ratio between the width of the groundstate wavefunction of the ion and λ_{eff}/2π (for our parameters η≈0.25).
In addition, we have to return the system close to its motional ground state at the end of the simulation to minimize the errors due to residual spin–motion coupling^{2} causing entanglement between the spin states and the motional degrees of freedom. To fulfil the first condition, we detune the two laser beams far enough from resonantly exciting the motional modes. Adjusting the detuning δ=−(ω_{stretch}−(ω_{1}−ω_{2}))=−2π×250 kHz red of the stretchmode frequency reduces the motional excitation during one step of the evolution and on average to . Finally terminating the evolution after the system returned into its motional ground state^{17}, it ideally completely cancels the simulation errors discussed in ref. 2, allowing for the measured contrast of the parity oscillations depicted in Fig. 4.
Statesensitive detection
For two spins, the integrated fluorescence signal does not enable us to distinguish between two (↓↑〉 and ↑↓〉) of the four possible spin configurations. In addition, the number of detected photons for each of the three distinguishable configurations fluctuates from experiment to experiment according to Poissonian statistics and therefore can be determined only with limited accuracy. For the data reported, we repeated each experiment 10^{4} times and fitted the resulting photonnumber distribution to the weighted sum of three reference distributions to derive P_{↓↓},P_{↑↑} and P_{↓↑}+P_{↑↓}.
Adiabatic evolution
We achieve the best fidelities for the reported transitions at a duration of the simulation of T=125 μs at a B_{x} of 2π×4.24 kHz. We are not strictly in the adiabatic limit, where for the final ratio J_{max}/B_{x}=5.2 allows for a maximum quantum magnetization of 93.4%. The robustness of the coupling scheme used enables us to minimize decoherence effects and even to enhance the final quantum magnetization to 98% by reducing the duration of the simulation.
In addition, technical factors related to the specific nonlinear performance of the r.f. attenuator used to control J(t) led to its evolution being linear in time to up to J(t=50 μs)=5×10^{−4}J_{max}, continued by J(t)∼(e^{αt}−β)^{2} best fitted by α=26×10^{3} s^{−1} and β=4. So far we have not improved the fidelities by evolving or terminating J(t) or B_{x} in a more adiabatic way.
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Acknowledgements
This work was supported by the EmmyNoether Programme of the German Research Foundation (DFG, grant No SCHA 973/12) the MPQ Garching, the DFG Cluster of Excellence Munich—Centre for Advanced Photonics and the European project SCALA. We thank D. Leibfried for his input and I. Cirac and G. Rempe for comments and support. We thank D. Moehring for reading and improving the manuscript.
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Friedenauer, A., Schmitz, H., Glueckert, J. et al. Simulating a quantum magnet with trapped ions. Nature Phys 4, 757–761 (2008). https://doi.org/10.1038/nphys1032
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DOI: https://doi.org/10.1038/nphys1032
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