Long-range interacting quantum systems

In this review recent investigations are summarized of many-body quantum systems with long-range interactions, which are currently realized in Rydberg atom arrays, dipolar systems, trapped-ion setups, and cold atoms in cavities. In these experimental platforms parameters can be easily changed, and control of the range of the interaction has been achieved. The main aim of the review is to present and identify the common and mostly universal features induced by long-range interactions in the behavior of quantum many-body systems. Discussed are the case of strong nonlocal couplings, i.e., the nonadditive regime, and the one in which energy is extensive, but low-energy, long-wavelength properties are altered with respect to the short-range case. When possible, comparisons with the corresponding results for classical systems are presented. Finally, cases of competition with local effects are also reviewed.

Figure 1

Illustration of long-range interactions in quantum many-body systems. Atoms (red spheres) trapped in a potential landscape (blue) are coupled to the light field inside an optical cavity. Far distant atoms can interact with each other via the exchange of photons (yellow arc) confined in the cavity mode. Using diverse tools of AMO systems (such as those provided by trapped ions, Rydberg atoms, or dipolar atoms), other types of long-range interactions (colored arcs) can also be induced.

Figure 2

Trapped-ion systems. (a) A 77 linear chain of ${}^{171}{\mathrm{Yb}}^{+}$ ions. The harmonic confinement and Coulomb interactions cause the spacing between ions to be inhomogeneous, thereby breaking translational invariance. (b) A laser drive at frequency $\mu$ is detuned from the radial center-of-mass mode frequency ${\omega }_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}}$ to create phonon-mediated spin-spin interactions. (c) Calculated spin-spin interaction (blue circles) for a 1D chain of 20 ions vs the distance from the edge ion. In this case $\delta =\mu -{\omega }_{\mathrm{c}\mathrm{.}\mathrm{m}\mathrm{.}}=2\pi \times 100\mathrm{kHz}$ and $J_{1,1+r}\sim 1/r^{1.3}$ (red solid line). (d) Calculated Ising couplings in a 2D crystal of 217 ions vs a sampling of the distance $d_{ij}$ between ion pairs (empty circles). The solid lines are best-fit power-law exponents ($\alpha =0.01$, 0.12, 0.75, 1.73, and 2.72 from top to bottom) for various detunings from the center-of-mass (c.m.) mode of 795 kHz. Adapted from [104].

Figure 3

Experimental scheme for realizing cavity-mediated interactions and mode softening at the superradiant phase transition. (a) A BEC (shaded cloud) inside an optical cavity is transversally illuminated by a far-red-detuned standing-wave laser field. In a quantized picture, atoms off-resonantly scatter photons from the pump field into a close-detuned cavity mode and back, creating and annihilating pairs of atoms in the superposition of momenta $(p_x,p_y)=(\pm \hslash k,\pm \hslash k)$ (one of four possible processes is shown schematically). This results in global interactions between all atoms. The interaction strength $\mathcal{V}$ is controlled via the power of the transverse laser field and the detuning ${\mathrm{\Delta }}_c$. (b) The cavity-mediated atom-atom interaction causes a softening of a collective excitation mode with energy $\hslash {\omega }_s$ at the momenta $(\pm \hslash k,\pm \hslash k)$ and a diverging susceptibility (shaded area) at a critical interaction strength (dashed line). Adapted from [569].

Figure 4

Tunable-range cavity-mediated interaction in a multimode cavity. The dimensionless interaction strength $\mathcal{D}(x_1,x_1)$ as a function of BEC position in a mode with waist $w_0$ for five different cavities is indicated by the saturation of the color. The darkest data correspond to a confocal cavity at a high degeneracy of modes, while the brighter shades correspond to fewer interacting modes. Inset: enlargement near the cavity center illustrating how a larger number of interacting modes allows a more localized effective atomic interaction to be engineered. From [748].

Figure 5

Dicke model phase diagram. (a) The power of the transverse pump is increased over 10 ms for different values of the pump-cavity detuning ${\mathrm{\Delta }}_c$. The recorded mean intracavity photon number is displayed (intensity scale) as a function of pump power (and the corresponding pump lattice depth) and pump-cavity detuning ${\mathrm{\Delta }}_c$. A sharp phase boundary is observed over a wide range of values; this boundary is in good agreement with a theoretical mean-field model (dashed curve). The dispersively shifted cavity resonance for the nonorganized atom cloud is marked by the arrow on the vertical axis. (b),(c) Typical traces showing the intracavity photon number for different pump-cavity detunings, as indicated by the symbols. From [53].

Figure 6

Lattice models with cavity-mediated long-range interactions. (a) Left image: experimental scheme. A stack of 2D systems along the $y$ axis is exposed to a 2D optical lattice in the $x\text{−}z$ plane (red arrows). Right image: illustration of the three competing energy scales, tunneling $t$, short-range collisional interactions $U_s$, and global-range, cavity-mediated interactions $U_l$. (b) Measured phase diagram as a function of detuning ${\mathrm{\Delta }}_c$ between pump field and cavity and 2D lattice depth $V_{2\mathrm{D}}$, featuring superfluid (SF), lattice supersolid (SS), charge-density wave (CDW), and Mott insulating (MI) phases. From [442].

Figure 7

Long-range interactions in systems of Rydberg atoms for many-body dynamics. (a) Illustration of the interaction between pairs of atoms excited to Rydberg states. Shown are the relevant dipole-coupled pair states labeled by quantum numbers $\alpha ,\beta ,\dots$ with the Förster defect ${\mathrm{\Delta }}_F$ relative to the pair states $|\alpha \beta ⟩$ and $|\gamma \delta ⟩$. (b) Principle of the Rydberg blockade. For two nearby atoms, the van der Waals interaction $\propto C_6/R^6$ ($R$ is the interatomic distance) shifts the doubly excited state $|rr⟩$ preventing the double excitation of the atomic pair when $R<R_b={(C_6/\hslash \mathrm{\Omega })}^{1/6}$. (c) Illustration of a superatom from the collective blockaded lattice of $N$ atoms. Shown is a Bloch sphere with its basis states (labeled by excitation numbers $n_e$) and coupled states highlighted [south pole ($n_e=0$) and singly excited state ($n_e=1$), represented by the shaded (red) plane]. The small pictograms above and below the sphere depict the lattice system with atoms in the ground (below, in red) and Rydberg states (above, in blue). The dashed (red) line indicates an enlargement into the subspace spanned by the lowest two states. The Husimi distribution of these states and their enhanced coupling ${\mathrm{\Omega }}_N$ is shown in the center. Adapted from [799]. (d) The first row displays the experimental image of the initial state of a Rydberg atom array. The following rows represent the atom array after a slow sweep across the phase transition, showing larger average sizes of correlated domains for the slower sweep. The green spots (open circles) represent atoms in the ground (Rydberg) state. The blue rectangles mark the position of domain walls. Courtesy of A. Omran.

Figure 8

Finite-range interactions in spin systems: dynamics and applications. (a) An experimental setup and Rydberg dressing scheme for a cloud of Cs atoms is held in an optical dipole trap and locally illuminated with 319 nm light to generate Ising interactions of a characteristic range $r_c$ and strength $J_0$. The quantization axis is set by a 1 G magnetic field $B$. (b) Energy level diagrams for a pair of atoms. (c) Transverse-field Ising dynamics. Trajectories for initial states $|\theta ,\varphi ⟩$ (square data points) and up to four Floquet cycles were obtained with dressing parameters $(\mathrm{\Omega },\mathrm{\Delta })=2\pi \times (2.8,25)\mathrm{MHz}$. (i)–(iv) ${\mathrm{\Lambda }}_{\mathrm{eff}}=0,1.2(2),1.8(3),2.7(4)$, respectively. The blue flow lines show mean-field theory for best fit $\mathrm{\Lambda }=0,1.1,1.5,2.2$; see the main text. (a)–(c) Adapted from [90]. (d) Loschmidt echo protocol applied to the LMG one-axis twisting model, with a snapshot of the Husimi distribution. Top image: a spin-polarized state is prepared at the north pole of the Bloch sphere. Center image: the interaction is switched on for a time $t_1$ (transformation $U_1$). The state is then rotated of an angle $\theta$ [$R_y(\theta )$]. Bottom image: the interaction is switched on again for a time $t_2$ (transformation $U_2$) such that $U_1{U}_2=1$. In these plots $\theta /\pi =0.01$ and $\tau /\pi =0.05$. (e) Probability $P_0(\theta )$ (solid line) as a function of phase shift and as a function of $\theta$ for $\tau /\pi =0.05$. The dashed line is the second-order expansion involving the quantum Fisher information $F_Q$. Here $N=101$. (d),(e) Adapted from [519].

Figure 9

Inverse correlation length exponent of long-range interactions. The results obtained from MC simulations for 1D long-range percolation are compared to the results of the effective dimension approach. The MC data of [304] (empty blue circles) are compared to the results obtained using an effective dimension and the $ϵ$-expansion result (black dashed line) for the short-range model ([306]). The low accuracy of the analytical result in the $\sigma \to 1$ limit is due to the appearance of BKT scaling ([133]), which cannot be captured by the $ϵ$ expansion. Inset: MC simulations for the long-range Ising model in $d=2$ given by [511] and [20] (blue diamonds and red squares, respectively). The black circles were obtained by mapping the conformal bootstrap results for the short-range critical exponents ([236]) via the effective dimension approach described by [186]. The axis labels of the inset coincide with main axis labels.

Figure 10

Classical correlation length exponent. The correlation length exponent $1/{\nu }_{\mathrm{lr}}$ is given as a function of $\sigma$ in $d=2$ for several values of $\mathcal{N}$ (from top, $\mathcal{N}=1,2,3,4,5,10,100$). The discrepancy between the $\mathcal{N}=1$ and the $\mathcal{N}\ge 2$ cases is in agreement with the expectations of the Mermin-Wagner theorem. The black dashed line is the analytical result obtained for the spherical model $\mathcal{N}=\infty$ ([382]). Adapted from [186], where the FRG calculation of the critical exponent ${\nu }_{\text{lr}}$ was discussed.

Figure 11

Phase diagram of long-range quantum rotors models in the plane $d$, $\sigma$. The universal behavior features the mean-field critical exponents in Eqs. (38), (39), and (40) in the cyan shaded region (upper left corner), while the universal properties are associated with an interacting Wilson-Fisher point in the white region. The red, blue, and green lines represent the boundary between long-range and short-range universality ($\mathcal{N}=1,2,3$ from left to right, respectively). Finally, the gray shaded region (lower right corner) displays no phase transition at all.

Figure 12

Universal properties of long-range quantum rotor models. Upper panel: estimates for the dynamical critical exponent $z=\sigma /(2-{\eta }_{\tau })$ obtained using the fixed point solution of the evolution equations (43) and (44) in the cases $\mathcal{N}=1,2,3$ in red, blue, and green (from top to bottom), respectively. Lower panel: inverse correlation length exponents $\mathcal{N}=1,2,3$, progressing from top to bottom. The MC simulations of [718] are shown as empty circles in the $\mathcal{N}=1,2$ cases top (red) and bottom (blue) lines, respectively. In both panels the upper gray dashed lines represent the mean-field results, while the dashed lower black lines represent the spherical model results ($\mathcal{N}=\infty$).

Figure 13

Comparison of Kac-on and Kac-off finite-size scalings for hard-core bosons. The scaling of the boson single-particle energy, as defined in Eq. (85), is displayed as a function of the system size. The results have been obtained via a DRMG computation at half filling $⟨n_i⟩=0.5$ for $\alpha =1$ and different interaction strengths $V$ (in units of the hopping energy $t$). The difference between the Kac scaled or unscaled scenario is evident, as in the first case the single-particle energy always vanishes in the thermodynamic limit, while in the second case the system remains gapped up to the thermodynamic limit. Adapted from [97].

Figure 14

Luttinger parameter. Thermodynamic limit extrapolation of the Luttinger parameter $K$ given as a function of the long-range interaction strength $V$ at half filling in the case $\alpha =0.5$ (the same scenario has been obtained for several $\alpha$ values in the range $0<\alpha <1$). The three different definitions for the Luttinger parameter have been compared in both the (a) Kac unscaled and (b) Kac scaled cases ([97]). As a function of $\alpha$, given a fixed value of the interaction [$V=1.5$ in the inset of (b)], there is no discrepancy between the Luttinger parameter obtained using a correlation function $K_{1p}=K_{2p}$, confirming the metallic character of the system. Still the conventional Luttinger-liquid theory is not obeyed since the Luttinger parameter does not fit the gap scaling $K_{\mathrm{\Delta }}$. As seen in the inset of (b), the traditional Luttinger-liquid picture is recovered for $\alpha >1$.

Figure 15

Zero-temperature phase diagram of one-dimensional soft-core bosons in the continuum and their excitation spectrum. (a) Phase diagram of one-dimensional soft-core bosons (log-log scale). A star marks the critical point between the Luttinger-liquid and CLL phases for densities commensurate with two-particle clusters. The long-dashed line corresponds to the softening of the Bogoliubov roton. (b) Spectra at $\rho =1.37$ with decreasing $U$, compared to Feynman (${ϵ}_{\mathrm{FA}}$, thin solid lines) and Bogoliubov approximations (${ϵ}_B$, dotted lines), and the harmonic chain acoustic mode ${\omega }_{\mathrm{acou}}$ (dashed lines). At $q\approx q_c$, the secondary mode is fitted by the transverse Ising spectrum ${ϵ}_{\mathrm{TI}}$ (thick solid). Adapted from [661].

Figure 16

Two-dimensional soft-core bosons in the continuum and their excitation spectrum. (a) The phase diagram displays the emergence of SF and normal solid (NS) and SS phases for the varying interaction strength $U$ and density $\rho$. The density on the left $y$ axis was scaled by the soft-core radius $R_c$. The right axis gives the density in units of the inverse area $A=\sqrt{3}(1.6R_c{)}^2/2$ of the unit cell of the high-density solid phase, corresponding to the lattice site occupation $N/N_s$ for a given number of particles and lattice sites $N$ and $N_{\mathrm{s}}$, respectively. For $A\rho \gtrsim 1.5$, the gray region labeled as NS corresponds to a cluster crystal with $N/N_s>1$, as indicated by the gray scale. The horizontal dashed lines refer to the integer filling. Supersolid phases with different occupation numbers are found between two hyperbolas defined by $R_c^2\rho U=\text{const}$ (dotted lines). At high densities ($A\rho \gtrsim 3.5$) they can be understood in terms of density-modulated superfluids. In contrast, superfluidity within the low-density supersolid lobes emerges from delocalized zero-point defects according to the ALC scenario. Adapted from [159]. (b) Left panel: PIMC snapshot illustrating the particle density profile in the SF phase. Right panel: excitation spectrum in the superfluid phase at $\gamma =11.86$ compared to the PIMC data (circles) of [665]. (c) Left panel: PIMC snapshot illustrating the particle density profile in the SS and NS phases. Right panel: mean-field spectra (solid, dashed, and dash-dotted lines) at $\gamma =16.93$ (top) and $\gamma =30.62$ (bottom) numerically computed along the three symmetry directions of the Brillouin zone (see the inset of the bottom panel). The symbols represent the PIMC data of [665] for longitudinal excitations computed along the direction $\mathrm{\Gamma }$–$M$–$\mathrm{\Gamma }$ in the first two Brillouin zones. (b),(c) Adapted from [516].

Figure 17

Signatures of atomic self-organization in an optical cavity. (a) The transverse pump power (dashed line) is gradually increased while the mean intracavity photon number (solid line) is monitored. After the sudden release of the atomic cloud and its subsequent ballistic expansion, absorption images are made for pump powers corresponding to transverse pump lattice depths of (b) $2.6E_{\mathrm{r}}$, (c) $7.0E_{\mathrm{r}}$, and (d) $8.8E_{\mathrm{r}}$. (d) Self-organization is manifested by an abrupt buildup of the cavity field accompanied by the formation of momentum components at $(p_x,p_y)=(\pm \hslash k,\pm \hslash k)$. The weak momentum components at $(0,\pm 2\hslash k)$ result from loading the atoms into the one-dimensional standing-wave potential of the transverse pump laser. From [53].

Figure 18

Excitation spectrum across the self-organization phase transition. The measured resonance frequencies $E_s=\hslash {\omega }_s$, which were obtained from atomic and photonic signals, are shown in blue and red, respectively, for positive (open circles) and negative interaction strengths (solid circles) $\mathcal{V}$. Gray shading indicates the theoretical prediction, including experimental uncertainties. From [569].

Figure 19

Critical behavior of the self-organizing phase transition. (a) Power spectral density (PSD) of the light field leaking out of the cavity shown as a function of frequency shift $\omega$ with respect to the pump laser frequency and the relative transverse pump power $P/P_{\mathrm{cr}}$. Two sidebands are visible, corresponding to the incoherent creation ($\omega <0$) and annihilation ($\omega >0$) of quasiparticles. The energy of these quasiparticles vanishes toward the critical point. At the phase transition, a strong coherent field at the pump frequency appears ($\omega =0$). (b)–(d) Normalized dynamic structure factor for three different values of $P/P_{\mathrm{cr}}$ [see the dashed lines in (a)]. While the positions and widths of the sidebands give direct access to the energy and lifetime of the quasiparticles, the sideband asymmetry can be used to determine the occupation of the quasiparticle mode. (e) Sketch of the measurement setup. The atoms coupled to the cavity mode are illuminated by the transverse pump field at frequency ${\omega }_i$, while the frequency emitted from the cavity is ${\omega }_f$. A heterodyne detection system gives access to the PSD shown as a function of $\omega ={\omega }_i-{\omega }_f$ in (a). (f) The data can be used to extract the divergent density fluctuations and the emerging density modulation. Inset: Density fluctuations shown on a double logarithmic scale, which allows the critical exponents of $0.7(1)$ and $1.1(1)$ to be determined on the normal and self-organized sides, respectively. From [441].

Figure 20

Evidence of QSSs in the long-range Ising chain, the Kitaev chain, and the one-dimensional spherical model, from left to right, respectively. In each panel the top subpanel displays the case of weak long-range interactions $\alpha >1$, where roughly the same equilibration properties of the nearest-neighbor case are found. Conversely, the bottom subpanels show the case of strong long-range interactions $\alpha <d$, where dynamical fluctuations survive in the $t\to \infty$ limit. (a) Transverse magnetization of the long-range Ising model [see the Hamiltonian (61)] after a quench from the fully paramagnetic state at $h\to \infty$ deep into the ordered phase at $h\to 0$. While the observable expectation equilibrates at long times for $\alpha =2.0$ (top subpanel), it persists in its initial value for increasingly long times as the system size increases for $\alpha =0.5$ (bottom subpanel); see the discussion given by [391]. A similar signature is noticed in the case of the Hamiltonian (67), i.e., the Kitaev chain representation of the Ising model, where the dynamics can be exactly solved for any global quench across the phase boundary. (b) Evolution of the spatial and quantum average of the ${\sigma }_z$ in Eq. (62) for a long-range Kitaev chain with $\alpha \gtrsim 12$ (top subpanel) and $\alpha =0.4$ (bottom subpanel for $h_i\gg 1$ to $h_f=0.4$). (c) Lack of equilibration also appears for noncritical quenches, as shown for the potential energy $U$ of a quantum spherical model with long-range interactions. Dynamical fluctuations reduce as size increases for decay rates $\alpha >1$; see the upper subpanel, where the $\alpha \gtrsim 12$ case is shown for increasing system sizes $N\in [2^9,2^{10},2^{11},2^{12}]$ from bottom to top. Conversely, dynamical fluctuations tend to increase for $\alpha <1$, as shown in the lower subpanel for $\alpha =0.2$, again displayed from bottom to top; see [181].

Figure 21

Different models and physical quantities are shown in the different panels, but the overall picture remains the same. Left panel: detection probability for a signal sent through a quantum channel between two sites at distance $\delta$ shown for the long-range Ising chain ([232]). The green dashed line displays the power law $\delta \propto t^{1.7}$. Center panel: connected correlation functions between two sites at distance $\delta$ in a long-range field theory; see the effective action in Eq. (36) with $d=1$ and $\alpha =4$ ([651]). The short-distance spreading resembles the conventional light cone observed with local interactions, while for larger distances long-range effects appear, and power-law scaling is observed. The green dashed curve is a guide for the eye. Right panel: mutual information between two lattice sites at distance $\delta$ in the Kitaev chain described by the Hamiltonian (49) with vanishing pairing and long-range hopping ($\alpha =8$). The decay rate is large enough that only the light cone is observed. From [721].

Figure 22

Propagation of quantum information in long-range trapped-ion systems. (a) Single-site magnetization $⟨{\sigma }_i^z(t)⟩$ as a function of time, following a quantum quench of the long-range $XY$ Hamiltonian (9), with the central eighth ion initially flipped. The dashed red lines are fits to the observed magnon arrival times [bottom of (b)]; dot-dashed white lines, light cone for averaged nearest-neighbor interactions; orange dots, after renormalization by the algebraic tail. The white lines are in clear disagreement with the red lines. (b),(c) Gaussian fits of magnon arrival time [the red lines in (a)] for ions 6 (dark blue on the left) and 13 (light blue on the right) with $\alpha \approx 1.41$ (top) and $\alpha \approx 0.75$ (bottom). A nearest-neighbor Lieb-Robinson bound captures most of the signal (shaded region) in the $\alpha \approx 1.41$ case and does not for $\alpha \approx 0.75$. Adapted from [384]. (d) Spatial and time-dependent correlations following a global quench of a long-range Ising Hamiltonian (6) with $\alpha =0.63$. (e) Correlation propagation velocity. (f) The curvature of the boundary shows an increasing propagation velocity, quickly exceeding the short-range Lieb-Robinson velocity bound $v$ (red dashed line). The solid lines give a power-law fit to the data that depends slightly on the choice of fixed contour $C_{i,j}$. Adapted from [657].

Figure 23

Kibble-Zurek mechanism in long-range Kitaev chains. (a) Dynamical phase diagram reporting the universal slow-dynamics exponents of the anisotropic Kitaev chain in the $(\alpha ,\beta )$ plane. (b),(c) Numerical analysis of Eq. (120) compared with the analytic formula in Eq. (121) for intermediate and small dynamical rates $\delta =0.5,0.05$. Each line represents a different value of $\beta =1.25,1.50,1.75,\infty$, from bottom to top at small $k$. Different values of $\alpha$ are displayed with different symbols; see the legend in (c). Adapted from [185].

Figure 24

Universal slow dynamics of the LMG model. Heat curves obtained via the effective model in Eq. (124) (upper panel) and via a full numerical solution of the time-dependent LMG model in Eq. (97) (lower panel) with a time-dependent coupling $J=h_c-\delta t$ performed by [4]. Each color represents a value of $\mathrm{\Lambda }=N\delta =15$, 3.75, 0.94, progressing from top to bottom, with $N=2^9$ and $N=2^{11}$ (dashed and solid lines in the upper panel). Both the heat and time variables were rescaled following the notation of [4]. As expected, the curves at different sizes but the same $\mathrm{\Lambda }$ collapse both in theory and in the exact simulations. Moreover, the similarity between the theoretical model and the numerics is noteworthy. Despite the fact that the model in Eq. (124) is simply an effective model that does not account for the mean-field energy shift, it captures quantitative features such as the initial smooth increase and the oscillations of the residual energy. The numerical data displayed in the lower panel are courtesy of [4].

Figure 25

Dynamical critical behavior at the self-organization phase transition. (a) Intracavity intensity while the transverse pump lattice depth ${\epsilon }_p$ ramped up (blue) and down (red) in ramps of 1.5 ms each. Shown beneath the panel are momentum spectra (1)–(5), recorded at increasing times during the ${\epsilon }_p$ ramp, indicated by the numbered arrows in (a). (b) Mean-field calculation according to (a) neglecting collisional interactions and assuming an infinite system. The points ${\epsilon }_{p,1}$ and ${\epsilon }_{p,2}$ indicate the upper and lower critical lattice depths. (c) Mean-field calculations of ${\epsilon }_{p,1}$ and ${\epsilon }_{p,2}$ as a function of quench time, resulting in exponents of $(n_1,n_2)=(-0.57,0.85)$ for power-law fits. (d),(e) Experimentally determined dependence of the upper and lower critical lattice depths on the quench time, together with solid lines reproducing the power-law dependences of (c). From [412].

Figure 26

(a),(b) Type-I and (c),(d) type-II dynamical phase transitions. (a) Measured rate function $\lambda$ for three different system sizes at $h/J_0\approx 2.38$, with $\tau =th$ the dimensionless time. The kinks in the evolution become sharper for larger $N$. Here the rate function is defined based on the return probability to the ground-state manifold, namely, $\lambda (t)=N^{-1}\log (P_{|{\psi }_0⟩}+P_{|-{\psi }_0⟩})$, where $|-{\psi }_0⟩=|\uparrow \uparrow \uparrow \cdots \uparrow {⟩}_x$. (b) Comparison between the rate function $\lambda (t)$ and the magnetization evolution $m_x(t)$. The inversion of the magnetization sign corresponds to the nonanalyticity of the rate function $\lambda (t)$. The solid lines are exact numerical predictions based on the experimental parameters ($h/J_0=2$). Adapted from [385]. (c) Long-time averaged values of the two-body correlations $C_2$ for different numbers of spins in the chain. The solid lines are exact numerical solutions to the Schrödinger equation, and the shaded regions take into account uncertainties from experimental Stark shift calibration errors. The dashed lines for $N=12,16$ are calculations using a canonical (thermal) ensemble with an effective temperature corresponding to the initial energy density. (d) Domain statistics and reconstructed single-shot images of 53 spins. Top and bottom images: reconstructed images based on binary detection of the spin state. The top image shows a chain of 53 ions in bright spin states. The other three images show 53 ions in combinations of bright and dark spin states. Center panel: statistics of the sizes of domains for three different values of $h/J_0$ plotted on a logarithmic scale. The dashed lines are fits to the exponential functions that are expected for an infinite-temperature thermal state. Long tails of deviations are clearly visible and vary depending on $h/J_0$. Right panel: mean of the largest domain sizes in every experimental shot. The dashed lines represent a piecewise linear fit used to extract the transition point. The green, yellow, and red data points correspond to the transverse fields shown in the domain statistics data in the center panel on the left. Adapted from [802].

Figure 27

Confinement in long-range spin systems. (a) Magnetic domain walls in Ising spin chains can experience an effective confining potential that increases with distance analogously to the strong nuclear force. This potential results in mesonlike domain-wall bound states (labeled $E_1$ to $E_3$) that influence the postquench dynamics. Adapted from [728]. (b) Magnetization oscillation $⟨{\sigma }_z(t)⟩$ (black line) vs time after quenching to $\alpha =2.3$ and $B_z=0.27J_0$ for $N=20$. The dashed green lines show the magnetization for the transverse-field Ising model with nearest-neighbor interactions only. The numerical calculations were adapted from [506]. (c) Left panel: confinement of correlation in long-range systems for $\alpha =2.3$ starting with the polarized state $|\downarrow \downarrow \cdots \downarrow ⟩$. Adapted from [506]. Right panel: confinement of correlation in long-range systems for $\alpha =1.25$ starting with the highly excited state $|\downarrow \downarrow \cdots \downarrow \uparrow \uparrow \cdots \uparrow ⟩$. Adapted from [472]. (d) Magnetization oscillations ($\alpha \sim 1.1$) starting with low-energy product states to probe the first three meson masses. Adapted from [728]. (e) Confinement dynamics at $B_z/J_0\approx 0.75,L=11,\alpha \sim 1.1$. Top row: absolute value of experimental center-connected correlations $|C_{i,6}^x(t)|$ averaged over 2000 experiments. Middle row: $|C_{i,6}^x(t)|$, calculated by solving the Schrödinger equation. The dashed white lines show correlation propagation bounds (light cones) in the limit $\alpha \to \infty$ (nearest-neighbor interactions). Bottom row: measured individual-spin magnetizations along their initialization axes $⟨{\sigma }_i^z(t)⟩$ averaged over 2000 experiments. The symbols represent magnetization data, and the solid colored curves represent theoretical magnetizations calculated by solving the Schrödinger equation. Purple (green) dashed lines represent thermal expectation values calculated from a canonical (microcanonical) ensemble averaged over the three displayed spins. Adapted from [728].