Quantum Entangled States of a Classically Radiating Macroscopic Spin

Entanglement constitutes a main feature that distinguishes quantum from classical physics. Here, however, we show that entanglement may also serve as the essential ingredient for the emergence of classical behavior in a radiating spin system. We consider the relation between the state of a macroscopic spin, such as an atomic ensemble, and the radiation it emits. We introduce a new class of macroscopic spin states, the coherently radiating spin states (CRSSs), defined as the asymptotic eigenstates of the SU(2) lowering operator. We find that a spin emitter in a CRSS radiates classical coherent light, although the CRSS itself is a quantum entangled state exhibiting spin squeezing. We further show that the CRSS is naturally realized in Dicke superradiance and underlies the dissipative Dicke phase transition, hence predicting the optimal scaling of spin squeezing in superradiance. More generally, the CRSS emerges as the ground state of a collective spin Hamiltonian. The CRSS thus provides a promising concept for studying many-body spin systems in various platforms, with applications ranging from quantum metrology and lasing to phase transitions.

Popular Summary

Various recently investigated platforms in quantum science involve systems comprised of collections of atoms, spins, or artificial quantum emitters characterized by a collective macroscopic spin. A common goal is to understand and exploit the generation of entangled states within the macroscopic spin system or in the light it scatters. In this context, collective radiation effects constitute a key element. Our current work reveals unfamiliar quantum physics of collective radiation: It’s shown that, surprisingly, quantum entanglement in a composite system may sometimes be essential for the emergence of its classical response to light. More specifically, for a macroscopic spin to scatter light classically, its constituent microscopic degrees of freedom (atoms and spins) must be in certain collective entangled states, introduced in this work as “coherently radiating spin states” (CRSS).

We develop a theory of CRSS, characterizing their entanglement properties via spin squeezing. Importantly, we find that CRSS are naturally produced by shining classical coherent light on the macroscopic spin system. This allows us to establish a link between CRSS and the iconic Dicke superradiance problem of collective radiation, leading to new predictions on entanglement and radiation in superradiance. Furthermore, CRSS emerge as ground states of a many-body spin Hamiltonian, suggesting the usefulness of a theory of CRSS for collective spin models.

CRSS thus provide a new concept and tool for the analysis of timely quantum platforms comprised of collections of spins or quantum emitters, both in the context of radiation and beyond, and with potential applications in quantum metrology, many-body physics, and lasers.

Figure 1

Spin states represented on the Bloch sphere (southern hemisphere). (a) CSS: the variance contour perpendicular to the mean spin vector (green arrow) exhibits no spin squeezing (circular contour). However, its projection onto the $x$-$y$ “dipole plane” is squeezed (elliptical), leading to nonclassical radiation (see dipole-projected squeezing in Sec. 6a). (b) CRSS exhibits spin squeezing (elliptical contour perpendicular to the mean spin), but no dipole-projected squeezing (circular contour in the $x$-$y$ plane); hence, it radiates classical light.

Figure 2

The CRSS as an approximate eigenstate of the spin-$j$ lowering operator ${\hat{J}}_{-}$ with an eigenvalue magnitude $|\alpha |=jr$. For given $j$ and $r$, the state coefficients $|a_m|$ of the ansatz state, Eqs. (4) and (5), peak at $m=m_{-}$ and are set to zero for $m>m_{+}$ ($m_{-}<m_{+}<j$), where $|a_{m_{+}}|\to 0$ as $j\to \mathrm{\infty }$. The resulting Gaussian shape agrees well with the analytical result in Eq. (10). Here $j=100$ and $r=0.4,0.9$ are taken (black stars represent state coefficients of two respective CSSs with matching “magnetization” values $⟨{\hat{J}}_z⟩$).

Figure 3

Proximity error of the CRSS ansatz state given in Eq. (4) [evaluated numerically using Eq. (5)]. (a) Error ${ϵ}_j(r)$ versus $j$ for $r=0.4,0.7,0.9$ (dashed, dash-dot, solid lines). Exponential decay is observed for large $j$, in agreement with Eq. (6). (b) Error ${ϵ}_j(r)$ as a function of $j$ and $r$. The dashed curve marks $r_j$, defined in Eq. (7a) and calculated using Eq. (6). For given $j$ and $0<r<r_j$, the state (4) well approximates a CRSS ($ϵ\ll 1$), as predicted analytically.

Figure 4

Spin squeezing ${\xi }^2$ in superradiance. (a) Plot of ${\xi }^2$ as a function of $r=|\mathrm{\Omega }|/{\mathrm{\Omega }}_c$ for the steady state of Eq. (8) with $j=N/2=25,100$ (circles, squares). The optimal (minimum) value ${\xi }_{min}^2$ is seen around $r=r_j$ from Eq. (7a) [calculated using Eq. (6)]. Agreement with CRSS theory (solid line) is observed within the validity region $r<r_j$. (b) Plot of ${\xi }_{min}^2$ defined in (a) as a function of $j$ on a log-log scale (circles). Agreement with CRSS theory (11a) (solid line) is observed. Inset: increasing $j$, Eq. (11a) scales as $j^{-0.32}$, in close agreement with the asymptotics $j^{-1/3}$ from Eq. (11b) (solid and dashed lines, respectively).

Figure 5

State coefficients $|a_m/a_{-j}|$ from Eq. (5) with $j=25$ ($m=-j,\dots ,j$). For $r=0.75$ (dashed line), $a_j$ approximately vanishes, whereas for $r=0.84$ (solid line), it does not. For the latter, the maximum and minimum points $m_{\mp }$ are marked. In the CRSS ansatz (4), all coefficients $m>m_{+}$ are set to zero, and condition (3) is satisfied for all $r<1$.

Figure 6

(a) The infidelity between the state $|j,\alpha {⟩}_{min}$ that minimizes the proximity error and the CRSS ansatz $|j,\alpha {⟩}_{\text{ans}}$ decays exponentially with $j$, so that the states coalesce for $j\to \mathrm{\infty }$ ($r=0.4,0.7,0.9$; dashed, dash-dot, solid lines). (b) The infidelity between the exact steady state ${\hat{\rho }}_s$ of Eq. (8) and the CRSS ansatz behaves similar to the proximity error in Fig. 3, as expected. Therefore, ${\hat{\rho }}_s\approx |j,\alpha {⟩}_{\text{ans}}⟨j,\alpha |$ forms an excellent approximation at any finite $j\gg 1$ for $0<r<r_j$ (dashed curve marks $r_j$ and the white parts represent values too small for our numerical evaluation).

Figure 7

Minimal proximity error, ${ϵ}_{\text{min}}$, of the minimizing state $|j,\alpha {⟩}_{min}$. (a) For $r<1$, this error decays exponentially with $j$, as in Fig. 3 ($r=0.7,0.9$; dash-dot, solid lines), whereas for $r>1$, it grows with $j$ ($r=1.05,1.2$; dashed line, crosses). (b) Plot of ${ϵ}_{\text{min}}$ as a function of both $j$ and $r$ exhibits low values within the validity region $r<r_j$ (dashed curve). Plots (a) and (b) thus directly validate the generality of the error and existence condition $r<r_j<1$ derived using the CRSS ansatz state $|j,\alpha {⟩}_{\text{ans}}$.

Figure 8

Deviation of the variance of the radiated field from that of a coherent state, $Q=Q_{\varphi =0}$ from Eq. (E3), calculated numerically as a function of $j$ using the CRSS ansatz state from Eq. (4) ($r=0.4,0.7,0.9$; dashed, dash-dot, solid lines). Exponential decay with $j$ is observed, and agreement with the analytical result ${ϵ}_j^2(r)$ of Eq. (6) is verified.