Ferromagnetism in an Extended Coherently Coupled Atomic Superfluid

Ferromagnetism is an iconic example of a first-order phase transition taking place in spatially extended systems and is characterized by hysteresis and the formation of domain walls. We demonstrate that an extended atomic superfluid in the presence of a coherent coupling between two internal states exhibits a quantum phase transition from a paramagnetic to a ferromagnetic state. The nature of the transition is experimentally assessed by looking at the phase diagram as a function of the control parameters, at hysteresis phenomena, and at the magnetic susceptibility and the magnetization fluctuations around the critical point. We show that the observed features are in good agreement with mean-field calculations. Additionally, we develop experimental protocols to deterministically generate domain walls that separate spatial regions of opposite magnetization in the ferromagnetic state. Thanks to the enhanced coherence properties of our atomic superfluid system compared to standard condensed matter systems, our results open the way toward the study of different aspects of the relaxation dynamics in isolated coherent many-body quantum systems.

Popular Summary

Phase transitions—such as freezing liquid water into ice or boiling it into vapor—occur when a material changes its state from one to another as a result of changing some parameter, usually temperature. However, quantum phase transitions occur at zero temperature and require different tuning knobs to drive them, such as an external field or an intrinsic property of the system like lattice spacing or nonlinearity. One open question is whether such a transition occurs in a system with finite size and nonzero temperature. Here, we report on an example of one such transition.

We observe evidence of a paramagnetic-to-ferromagnetic transition in a system composed of an ultracold atomic gas in the presence of radiation that coherently couples two different internal spin states. Our atomic system reproduces textbook magnetic models in a clean and highly controllable way and is a promising platform for studies of magnetism in the absence of dissipation, a requirement difficult to have in conventional condensed matter systems. The flexibility of our ultracold atomic system allows us to observe typical phenomena related to the transition such as hysteresis, enhancement of the susceptibility, and fluctuations of the magnetization at the critical point. Moreover, we demonstrate the possibility of generating ferromagnetic domains on demand by controlling the position of domain walls.

Our results demonstrate a novel approach to study magnetic systems in an ultracold environment.

Figure 1

Phase diagram of the magnetic model. The relative magnetization $Z$ of the system’s stationary states is shown as a function of the nonlinearity and of the axial magnetic field strength, both in units of the transverse field. The system can be paramagnetic ($|\alpha |n<B_1$), ferromagnetic ($|\alpha |n>B_1\gg B_3$), or saturated ferromagnetic ($|\alpha |n>B_1$ and $B_3\gg B_1$). Panels (A)–(H) show the dependence of the energy [Eq. (2)] on the relative magnetization $Z$ in several points of the phase diagram. Three gray side panels show the value of $Z$ at the energy minimum, as a function of $|\alpha |n/B_1$ for $B_3=0$ (bottom) and as a function of $B_3/B_1$ for $|\alpha |n/B_1=0$ (left) or $|\alpha |n/B_1=3$ (right). Numbered dashed yellow lines mark four different single-shot experimental realizations in the atomic system as reported in Fig. 2. See Sec. 2b and Table 1 for mapping from magnetic to atomic system.

Figure 2

(a),(b) Absorption images of the atoms in the states $|\uparrow ⟩$ and $|\downarrow ⟩$ (only the left half of the system is shown) for the parameters marked by the yellow lines in Fig. 1, for forward (a) and backward (b) ramps of ${\delta }_B$. (c),(d) Bare experimental data of the axial magnetization as a function of ${\delta }_B$ and position $x$ for forward (c) and backward (d) ramps at fixed ${\mathrm{\Omega }}_R/2\pi =400\mathrm{Hz}$. The solid arrows on the side of the plot indicate the direction of the ramp on ${\delta }_B$. The yellow dashed lines mark experimental shots shown in (a) and (b), corresponding to number 1–4 as in Fig. 1 (${\delta }_{B,1}/{\mathrm{\Omega }}_R=-0.8$, ${\delta }_{B,2}/{\mathrm{\Omega }}_R=+1.2$, ${\delta }_{B,3}/{\mathrm{\Omega }}_R=+3.2$, ${\delta }_{B,4}/{\mathrm{\Omega }}_R=+4.2$). The vertical black dashed line marks the position where $|\kappa |n={\mathrm{\Omega }}_R$ and the system switches from PM to FM. Dot-dashed black lines in (c) and (d) mark the local resonance condition ${\delta }_B=-n(x)\mathrm{\Delta }$. (e) Magnetization of the central $10\mathrm{\mu }\mathrm{m}$ region during a forward (dot-dashed line) and backward (dashed line) ramp, showing the hysteretic behavior.

Figure 3

Magnetic hysteresis. (a),(b) Experimental magnetization data from Figs. 2 and 2, rescaled according to the $|\uparrow ⟩\text{−}|\downarrow ⟩$ asymmetry and to the density profile; see main text. White regions in the bottom left-hand corner are due to a lack of data that manifests when applying the vertical axis rescaling. (c),(d) 1D mean-field numerical simulations for the experimental parameters of Figs. 2 and 2. The dotted black and white lines in (c) and (d) mark the border of the hysteresis region calculated from theory. Yellow dashed lines mark experimental shots shown in (a), corresponding to number 1–4, as in Fig. 1. (e) The width of the hysteresis ${\delta }_{\mathrm{hys}}$ is calculated as explained in Appendix pp2. Green points are experimental data with their uncertainties resulting from the binning procedure and systematic errors. The dotted line stands for theory, while the purple points are results from numerical simulations.

Figure 4

(a) Magnetic susceptibility. Green points are experimental data with their uncertainties resulting from the binning procedure and systematic errors, red line is the theory prediction, purple points connected by dashed line are simulation results. (b) Magnetic fluctuations. The variance of $Z$ is extracted in a central region of the cloud and shows a maximum at $|\kappa |n/{\mathrm{\Omega }}_R\approx 1$. Error bars are standard variations resulting from averaging different experimental realization and from different binning procedures.

Figure 5

Deterministic creation of FM domain walls. (a) Experimental protocol used to create DW through a ramp on ${\delta }_B$. (b) Schematics of the spatial variation of the phase diagram as a result of the protocol shown in (a). Different regimes are labeled with the same color as in (c) and (d). (c) Absorption images of the two components (left half of the system only) at the initial point (I) and after a wait time of 25 ms (IV), when ${\delta }_B/2\pi ={\delta }_{\mathrm{ref}}/2\pi =1\mathrm{kHz}$ [dashed line in (a)]. (d) Absorption images of the two components corresponding to the solid line ramp in (a), where ${\delta }_B$ reaches ${\delta }_{\mathrm{DW}}/2\pi =1.13\mathrm{kHz}$ (II) and is then ramped back down to ${\delta }_{\mathrm{ref}}$ (III). PM, SFM, and FM regions are illustrated in the line between the absorption images. The third (III) image in (c) shows the presence of a DW between two FM domains with opposite magnetization. In both (c) and (d), dashed lines mark the position at which $Z=0$. (e) Continuous dependence of the position $x_{\mathrm{DW}}$ of the DW with respect to the initial interface position $x_{\mathrm{ref}}$ (in units of $R_x$) as a function of $({\delta }_{\mathrm{DW}}-{\delta }_{\mathrm{ref}})/{\mathrm{\Omega }}_R$. The red line is extracted from numerical simulations. Error bars show the experimental uncertainties (horizontal axes) and the shot-to-shot standard deviation (vertical axes).

Figure 6

Magnetic hysteresis. As in Fig. 3, but now the comparison is with numerical simulations including transversal direction. (a),(b) Experimental magnetization data from Figs. 2 and 2, rescaled according to the $|\uparrow ⟩\text{−}|\downarrow ⟩$ asymmetry and to the density profile; see main text. White regions in the bottom left-hand corner are due to a lack of data that manifests when applying the vertical axis rescaling. (c),(d) 1D mean-field numerical simulations for the experimental parameters of Figs. 2 and 2. The dotted black and white lines in (c) and (d) mark the border of the hysteresis region calculated from theory. Yellow dashed lines mark experimental shots shown in (a), corresponding to number 1–4, as in Fig. 1. (e) The width of the hysteresis ${\delta }_{\mathrm{hys}}$ is calculated as explained in Appendix pp2. Green points are experimental data with their uncertainties resulting from the binning procedure and systematic errors. The dotted line stands for theory, while the purple points are results from numerical simulations.

Figure 7

Thermal and condensate atomic distribution. (a) Schematic three-dimensional view of the condensate (dark green) and thermal (light green) distribution. A, B, and C cylinders highlight the lines of sight of the central atoms and of the thermal atoms just outside $R_x$. (b) Line density profiles along the imaging direction for $x=-R_x,0,+R_x$, calculated using Hartree-Fock theory for our partially condensed (30%) gas. (c) The integrated density for the thermal component $\int n(x,0,z)dz$ has an almost flat distribution in region occupied by the condensate.

Figure 8

Effects of removing the thermal component. Panel (a) shows the hysteresis width obtained through the four different methods of thermal subtraction, as explained in the text. Correspondingly, panel (b) presents the same comparison performed on the susceptibility. In both panels blue empty symbols correspond to paraboloid, orange to flattop, green to linear, and red to Gaussian. Both panels show good agreement between the four methods. Black dotted lines are the theory predictions. Error bars are standard variations resulting from averaging different experimental realization and from systematic errors.

Figure 9

Unprocessed experimental data for forward (a) and backward (b) ramp as in Figs. 2 and 2 before applying the thermal removal procedure. Dot-dashed black lines mark the local resonance condition ${\delta }_B=-n(x)\mathrm{\Delta }$.