Heralded Magnetism in Non-Hermitian Atomic Systems

Quantum phase transitions are usually studied in terms of Hermitian Hamiltonians. However, cold-atom experiments are intrinsically non-Hermitian because of spontaneous decay. Here, we show that non-Hermitian systems exhibit quantum phase transitions that are beyond the paradigm of Hermitian physics. We consider the non-Hermitian $XY$ model, which can be implemented using three-level atoms with spontaneous decay. We exactly solve the model in one dimension and show that there is a quantum phase transition from short-range order to quasi-long-range order despite the absence of a continuous symmetry in the Hamiltonian. The ordered phase has a frustrated spin pattern. The critical exponent $\nu$ can be 1 or $1/2$. Our results can be seen experimentally with trapped ions, cavity QED, and atoms in optical lattices.

Popular Summary

A quantum magnet is composed of spins that self-organize into patterns. For example, the spins can point in the same direction or in opposite directions. Quantum magnets are typically assumed to be Hermitian, meaning that their energy is constant in time. We theoretically determine what happens when energy leaks out of the system because of spin decay; such a quantum magnet is then non-Hermitian. We investigate a non-Hermitian magnet whose spins are arranged on a one-dimensional chain and show that it exhibits drastically different behavior than Hermitian magnets.

In the non-Hermitian magnet, we find that two spins already exhibit singularities, in contrast to the usual requirement of an infinite number of spins. Furthermore, a long chain of spins forms a frustrated pattern: The spins cannot decide whether to point in the same direction or in opposite directions. This frustration is surprising given the one-dimensional nature of the chain and the fact that the atoms only experience nearest-neighbor interactions. (Hermitian systems, on the other hand, become frustrated when placed on a more complicated lattice or when the interaction is long range.) Additionally, the correlations between spins decrease with distance according to a power law, instead of converging to a nonzero value as in the Hermitian case.

Non-Hermitian magnets can be observed in experiments with cold atoms that spontaneously decay: for example, trapped ions and atoms in optical lattices. We find that ten atoms already resemble an infinite system, which makes observing our predictions easier. Our work paves the way for discovering new types of magnetic behavior.

Figure 1

(a) Experimental setup with a chain of atoms. The population in the auxiliary state $|a⟩$ is measured by scattering photons off of it. (b) The non-Hermitian model is heralded by the absence of population in $|a⟩$, i.e., the absence of fluorescence. (c) $|\uparrow ⟩$ decays into the auxiliary state $|a⟩$. (d) The population of $|a⟩$ is measured by exciting it with a laser and detecting the fluorescence (red arrows). (e) Level scheme for a ${{}^{171}\mathrm{Yb}}^{+}$ ion, showing optical pumping (black arrows) and detection (red arrows).

Figure 2

Population in the six slowest-decaying eigenstates of $H_{\text{eff}}$ during the non-Hermitian evolution (a) without normalization and (b) with normalization. This is from exact diagonalization of $N=10$ atoms with open boundary conditions, the initial condition $|\downarrow \downarrow \cdots \downarrow ⟩$, $J^{\prime }=0.12\gamma$, and $J=0$. The thick red line denotes the steady state.

Figure 3

$⟨{\sigma }_n^z⟩$ for (a) two atoms and (b) an infinite chain with $J=0$ (blue solid line) and $J=0.1\gamma$ (red dashed line). The critical point is $J^{\prime }=\gamma /8$ for all cases. Panel (a) is independent of $J$.

Figure 4

Imaginary part of $\epsilon (k)$ for $J^{\prime }=0.1\gamma$ (black solid line), $J^{\prime }=0.125\gamma$ (red dashed line), and $J^{\prime }=0.15\gamma$ (blue dash-dotted line). (a) $J=0.1\gamma$. (b) $J=0$.

Figure 5

Real part of $\epsilon (k)$ for the same parameters as in Fig. 4.

Figure 6

Ordered-phase correlation functions: $⟨{\sigma }_m^x{\sigma }_n^x⟩$ (blue solid line) and $⟨{\sigma }_m^y{\sigma }_n^y⟩$ (red dashed line) for $J^{\prime }=0.13\gamma$ with (a) $J=0.1\gamma$ and (b) $J=0$.

Figure 7

Correlation functions for $J^{\prime }=0.13\gamma$ (solid line) and $J^{\prime }=0.12\gamma$ (dashed line) for (a) $J=0.1\gamma$ and (b) $J=0$. Panel (b) shows only even distances for $J^{\prime }=0.12\gamma$.

Figure 8

Correlation length $\xi (J^{\prime })$ for $J=0.1\gamma$, found by fitting the exponential decay of $⟨{\sigma }_m^x{\sigma }_n^x⟩$. At the phase transition, $\xi$ diverges with critical exponent $\nu =1$.

Figure 9

Exact diagonalization of $N=10$ atoms with open boundary conditions and $J=0$. (a) Gap (red dashed line) and second gap (red dotted line). The gap for $N=\infty$ (solid black line) is from Eq. (10). (b) Correlations between the first atom and other atoms for $J^{\prime }=0.12\gamma$ (blue solid line, circles) and $J^{\prime }=0.14\gamma$ (green dashed line, triangles).

Figure 10

Contours for the evaluation of pair contraction functions for (a) $J=0$ and (b) an example of $J\ne 0$. Solid lines are the original contours. Dashed lines are the deformed contours. Jagged lines are the branch cuts.