Experimental quantum simulations of many-body physics with trapped ions
TL;DR: An overview of different trapping techniques of ions as well as implementations for coherent manipulation of their quantum states and current approaches for scaling up to more ions and more-dimensional systems are given.
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Abstract: Direct experimental access to some of the most intriguing quantum phenomena is not granted due to the lack of precise control of the relevant parameters in their naturally intricate environment. Their simulation on conventional computers is impossible, since quantum behaviour arising with superposition states or entanglement is not efficiently translatable into the classical language. However, one could gain deeper insight into complex quantum dynamics by experimentally simulating the quantum behaviour of interest in another quantum system, where the relevant parameters and interactions can be controlled and robust effects detected sufficiently well. Systems of trapped ions provide unique control of both the internal (electronic) and external (motional) degrees of freedom. The mutual Coulomb interaction between the ions allows for large interaction strengths at comparatively large mutual ion distances enabling individual control and readout. Systems of trapped ions therefore exhibit a prominent system in several physical disciplines, for example, quantum information processing or metrology. Here, we will give an overview of different trapping techniques of ions as well as implementations for coherent manipulation of their quantum states and discuss the related theoretical basics. We then report on the experimental and theoretical progress in simulating quantum many-body physics with trapped ions and present current approaches for scaling up to more ions and more-dimensional systems.
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Figure 4. Excerpt of the level scheme of 25Mg+ as an example of a hyperfine qubit (not to scale). 25Mg+ has a nuclear spin of I = 5/2 and thus a hyperfine-split ground state (S1/2, F = 3 and S1/2, F = 2). By applying a static magnetic field of a few Gauss, the degeneracy of the Zeeman sublevels is lifted. The Doppler cooling laser (labelled ‘BD’) is σ + polarized and detuned red by ![Figure 6. Comparison between parameters of geometric phase gates [110] with two ions using the axial motional modes and radial motional modes. (a) The parameters correspond to the gate from [123]. The axial centre-of-mass (COM) and stretch (STR) mode have a large frequency difference (2π × 1.6 MHz). The detuning of the Raman beams from the STR mode amounts to δSTR = −2π × 266 MHz. That is why the main contribution to the differential geometric phase between |↓↓〉/|↑↑〉 and |↓↑〉/|↑↓〉 is due to a (single) loop in the phase space of the STR mode. However, as already suggested in [110], the detuning from the COM mode is chosen to be an integer multiple of the detuning from the STR mode (δCOM = −5 × δSTR). Hence, there is no entanglement left between the electronic and motional modes at the gate duration Tg = |2π/δSTR| = 3.75µs. (Note that the spin-echo sequence is not included in Tg.) (b) The parameters correspond to a phase gate on two of the radial motional modes. The radial centre-of-mass (COM) and rocking (ROC) mode have a comparatively small frequency difference of only 2π × 130 kHz. The detunings from both modes are chosen to have the same absolute values resulting in (approximately) equal contributions to the acquired geometric phase from both modes. The gate duration according to the original implementation would amount to Tg = |2π/δSTR| = 15.4µs. As the displacement pulse is repeated in the second gap of the spin-echo sequence (compare figure 7) to cancel dynamic phases (compare [118] and see text), the duration increases by an additional factor of two.](/figures/figure-6-comparison-between-parameters-of-geometric-phase-1o7b7c9n.png)
Figure 6. Comparison between parameters of geometric phase gates [110] with two ions using the axial motional modes and radial motional modes. (a) The parameters correspond to the gate from [123]. The axial centre-of-mass (COM) and stretch (STR) mode have a large frequency difference (2π × 1.6 MHz). The detuning of the Raman beams from the STR mode amounts to δSTR = −2π × 266 MHz. That is why the main contribution to the differential geometric phase between |↓↓〉/|↑↑〉 and |↓↑〉/|↑↓〉 is due to a (single) loop in the phase space of the STR mode. However, as already suggested in [110], the detuning from the COM mode is chosen to be an integer multiple of the detuning from the STR mode (δCOM = −5 × δSTR). Hence, there is no entanglement left between the electronic and motional modes at the gate duration Tg = |2π/δSTR| = 3.75µs. (Note that the spin-echo sequence is not included in Tg.) (b) The parameters correspond to a phase gate on two of the radial motional modes. The radial centre-of-mass (COM) and rocking (ROC) mode have a comparatively small frequency difference of only 2π × 130 kHz. The detunings from both modes are chosen to have the same absolute values resulting in (approximately) equal contributions to the acquired geometric phase from both modes. The gate duration according to the original implementation would amount to Tg = |2π/δSTR| = 15.4µs. As the displacement pulse is repeated in the second gap of the spin-echo sequence (compare figure 7) to cancel dynamic phases (compare [118] and see text), the duration increases by an additional factor of two. 
Figure 10. Example for normal modes in a surface-electrode trap geometry similar to the one discussed in section 6. It consists of three trapping zones (red spheres) arranged in a triangle at mutual distances of 40µm. Each trapping zone has a (bare) frequency of 2π × 2 MHz corresponding to the vibration towards the centre of the triangle, 2π × 1.1 MHz for the out-of-plane vibration and 2π × 0.9 MHz for the third, perpendicular direction. The eigenvectors (green arrows) of the nine normal modes for an ion of mass m = 25 amu in each potential minimum are shown (first and last row: top view for in-plane modes; middle row: side view for out-of-plane modes; see appendix A for details of the calculation). The labels denote the frequencies corresponding to the modes. The (coupled) frequencies make up three triplets close to each bare frequency; two frequencies in each triplet are degenerate. For ![Figure 14. Schematic to illustrate the projection of the electrodes of the RF (yellow) and dc (shaded) electrodes on a surface as a way to scale towards two-dimensional arrays of ions. The black crosses indicate the positions of the minima of the pseudopotentials. (a) Cross section of the electrodes of a conventional linear RF trap with three-dimensional geometry and the electrode structures projected onto a surface. The dashed arrows point at the new location of the electrodes, the white areas represent isolating gaps. (b) Cross section (upper part) and top view (lower part) of the stripe electrodes. It has been proposed to concatenate several linear RF surface-electrode traps as depicted in (a) as a basic unit to span a two-dimensional array of ions [164] (red and blue disks representing ions in opposite spin states). For sufficiently small mutual ion distances and decoherence rates of the ions, this is an approach to scale analogue QS.](/figures/figure-14-schematic-to-illustrate-the-projection-of-the-i9ntj5w1.png)
Figure 14. Schematic to illustrate the projection of the electrodes of the RF (yellow) and dc (shaded) electrodes on a surface as a way to scale towards two-dimensional arrays of ions. The black crosses indicate the positions of the minima of the pseudopotentials. (a) Cross section of the electrodes of a conventional linear RF trap with three-dimensional geometry and the electrode structures projected onto a surface. The dashed arrows point at the new location of the electrodes, the white areas represent isolating gaps. (b) Cross section (upper part) and top view (lower part) of the stripe electrodes. It has been proposed to concatenate several linear RF surface-electrode traps as depicted in (a) as a basic unit to span a two-dimensional array of ions [164] (red and blue disks representing ions in opposite spin states). For sufficiently small mutual ion distances and decoherence rates of the ions, this is an approach to scale analogue QS. 
Figure 15. Illustration of the optimization results for the electrode structure for a basic triangular lattice with respect to the height of the ions above the traps at constant mutual ion distance. The white gap isolates RF and dc patches. The three red disks symbolize three ions at a constant distance of d = 40µm, hovering above the surface at a height of (a) h = 30µm, (b) h = 40µm and (c) h = 50µm. (Courtesy of Roman Schmied.) ![Figure 5. Implementations of different interaction types for hyperfine/Zeeman qubits. (a) An operation of type (a) can be implemented, for example, by two-photon stimulated-Raman transitions driven by a pair of laser beams (shown without motional dependence) or directly by a microwave field. These types of interactions can be used for single-qubit gates in QC and to simulate the effective magnetic field in the simulation of quantum spin Hamiltonians. (b) State-dependent forces (see type (c) in the text) can be created by two beams detuned by approximately the frequency of a motional mode. This interaction is used in the geometric phase gate for the displacement pulse [110, 123] or in the simulation of the quantum Ising Hamiltonian to create the effective spin–spin interaction [21, 47].](/figures/figure-5-implementations-of-different-interaction-types-for-3mxtf5gx.png)
Figure 5. Implementations of different interaction types for hyperfine/Zeeman qubits. (a) An operation of type (a) can be implemented, for example, by two-photon stimulated-Raman transitions driven by a pair of laser beams (shown without motional dependence) or directly by a microwave field. These types of interactions can be used for single-qubit gates in QC and to simulate the effective magnetic field in the simulation of quantum spin Hamiltonians. (b) State-dependent forces (see type (c) in the text) can be created by two beams detuned by approximately the frequency of a motional mode. This interaction is used in the geometric phase gate for the displacement pulse [110, 123] or in the simulation of the quantum Ising Hamiltonian to create the effective spin–spin interaction [21, 47].
![Figure 6. Comparison between parameters of geometric phase gates [110] with two ions using the axial motional modes and radial motional modes. (a) The parameters correspond to the gate from [123]. The axial centre-of-mass (COM) and stretch (STR) mode have a large frequency difference (2π × 1.6 MHz). The detuning of the Raman beams from the STR mode amounts to δSTR = −2π × 266 MHz. That is why the main contribution to the differential geometric phase between |↓↓〉/|↑↑〉 and |↓↑〉/|↑↓〉 is due to a (single) loop in the phase space of the STR mode. However, as already suggested in [110], the detuning from the COM mode is chosen to be an integer multiple of the detuning from the STR mode (δCOM = −5 × δSTR). Hence, there is no entanglement left between the electronic and motional modes at the gate duration Tg = |2π/δSTR| = 3.75µs. (Note that the spin-echo sequence is not included in Tg.) (b) The parameters correspond to a phase gate on two of the radial motional modes. The radial centre-of-mass (COM) and rocking (ROC) mode have a comparatively small frequency difference of only 2π × 130 kHz. The detunings from both modes are chosen to have the same absolute values resulting in (approximately) equal contributions to the acquired geometric phase from both modes. The gate duration according to the original implementation would amount to Tg = |2π/δSTR| = 15.4µs. As the displacement pulse is repeated in the second gap of the spin-echo sequence (compare figure 7) to cancel dynamic phases (compare [118] and see text), the duration increases by an additional factor of two.](/figures/figure-6-comparison-between-parameters-of-geometric-phase-1o7b7c9n.webp)
![Figure 14. Schematic to illustrate the projection of the electrodes of the RF (yellow) and dc (shaded) electrodes on a surface as a way to scale towards two-dimensional arrays of ions. The black crosses indicate the positions of the minima of the pseudopotentials. (a) Cross section of the electrodes of a conventional linear RF trap with three-dimensional geometry and the electrode structures projected onto a surface. The dashed arrows point at the new location of the electrodes, the white areas represent isolating gaps. (b) Cross section (upper part) and top view (lower part) of the stripe electrodes. It has been proposed to concatenate several linear RF surface-electrode traps as depicted in (a) as a basic unit to span a two-dimensional array of ions [164] (red and blue disks representing ions in opposite spin states). For sufficiently small mutual ion distances and decoherence rates of the ions, this is an approach to scale analogue QS.](/figures/figure-14-schematic-to-illustrate-the-projection-of-the-i9ntj5w1.webp)
![Figure 5. Implementations of different interaction types for hyperfine/Zeeman qubits. (a) An operation of type (a) can be implemented, for example, by two-photon stimulated-Raman transitions driven by a pair of laser beams (shown without motional dependence) or directly by a microwave field. These types of interactions can be used for single-qubit gates in QC and to simulate the effective magnetic field in the simulation of quantum spin Hamiltonians. (b) State-dependent forces (see type (c) in the text) can be created by two beams detuned by approximately the frequency of a motional mode. This interaction is used in the geometric phase gate for the displacement pulse [110, 123] or in the simulation of the quantum Ising Hamiltonian to create the effective spin–spin interaction [21, 47].](/figures/figure-5-implementations-of-different-interaction-types-for-3mxtf5gx.webp)
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References
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