Proposed realization of the Dicke-model quantum phase transition in an optical cavity QED system

The Dicke model describing an ensemble of two-state atoms interacting with a single quantized mode of the electromagnetic field (with omission of the ${\widehat{A}}^2$ term) exhibits a zero-temperature phase transition at a critical value of the dipole coupling strength. We propose a scheme based on multilevel atoms and cavity-mediated Raman transitions to realize an effective Dicke model operating in the phase transition regime. Optical light from the cavity carries signatures of the critical behavior, which is analyzed for the thermodynamic limit where the number of atoms is very large.

Figure 1

Atomic level scheme. Excited states have energies $\hslash {\omega }_j$ $(j=r,s)$. Such a scheme might be realized, e.g., by alkali atoms, with ∣0⟩ and ∣1⟩ as different ground-state sublevels. Note that $\mid r⟩$ and $\mid s⟩$ may be the same level, provided that the Raman channels remain distinct (which requires ${\omega }_1\ne 0$).

Figure 2

(a) Ring cavity configuration for implementing the proposed realization of the Dicke model. Quantized input and output fields are denoted by ${\widehat{a}}_{\mathrm{in}}$ and ${\widehat{a}}_{\mathrm{out}}$, respectively. (b) Possible atomic excitation scheme based on an $F=1\leftrightarrow F^{\prime }=1$ atomic transition and a linearly polarized cavity field $\widehat{a}$. Note that the magnetic field splittings of the Zeeman sublevels are not drawn to scale; the detunings of the optical fields from the excited atomic states are much larger than the ground-state splittings (i.e., $\mid {\Delta }_{r,s}\mid ⪢\mid \mu B\mid$).

Figure 3

Steady-state field amplitude ${\alpha }_{\mathrm{ss}}$, polarization amplitude ${\beta }_{\mathrm{ss}}$, and atomic inversion $w_{\mathrm{ss}}$, plotted as a function of the coupling strength $\lambda$, for $\omega ={\omega }_0=1$ and $\kappa =0.2$. Only stable steady states are shown.

Figure 4

Imaginary parts (upper row) and real parts (lower row) of the eigenvalues in the linearized Holstein-Primakoff representation as a function of the coupling strength $\lambda$; for $\omega ={\omega }_0=1$ and $\kappa =0.2$. Solid (dashed) lines correspond to the photonic (atomic) branch. The right-hand column magnifies the view around the transition at $\lambda ={\lambda }_c=0.5099$; note the splitting (convergence) at ${\lambda }^{\prime }\simeq 0.5050$ and ${\lambda }^{″}\simeq 0.5124$.

Figure 5

Output photon flux as a function of coupling strength; for $\omega ={\omega }_0=1$ and $\kappa =0.1$ (solid), 0.2 (dashed), and 0.5 (dotted-dashed).

Figure 6

Sum of EPR operator variances as a function of coupling strength; for $\omega ={\omega }_0=1$, $\kappa =0.2$, $\theta ={\tan }^{-1}(\kappa ∕\omega )$, and $\varphi =0$. The choice of $\theta$ minimizes the sum in the vicinity of the critical coupling strength, $\lambda ={\lambda }_{\mathrm{c}}\simeq 0.51$.

Figure 7

Incoherent part of the cavity fluorescence spectrum $S\left(\nu \right)$ for various values of coupling strength $\lambda$; for $\omega ={\omega }_0=1$ and $\kappa =0.2$ $({\lambda }_{\mathrm{c}}\simeq 0.51)$.

Figure 8

Probe transmission spectrum $T\left({\nu }_{\mathrm{p}}\right)$ for various values of the coupling strength $\lambda$; for $\omega ={\omega }_0=1$ and $\kappa =0.2$ $({\lambda }_{\mathrm{c}}\simeq 0.51)$.

Figure 9

Quadrature noise spectra $S_{\mathrm{out},\theta }\left(\nu \right)$ with $\theta =0$ (solid) and $\theta =\pi ∕2$ (dashed); for $\omega ={\omega }_0=1$, $\kappa =0.2$ $({\lambda }_{\mathrm{c}}\simeq 0.51)$, and $\lambda =0.4,0.49,0.505,0.52,0.6$ (top to bottom).

Figure 10

Optimal squeezing at $\nu =0$ as a function of $\lambda$ (top) and the quadrature phase angle at which the optimum occurs (bottom); for $\omega ={\omega }_0=1$ and $\kappa =0.2$ $({\lambda }_{\mathrm{c}}\simeq 0.51)$.

Figure 11

Estimate of the EPR variance, $V_{\mathrm{est}}$, as a function of coupling strength; for $\omega ={\omega }_0=1$, $\kappa =0.2$, and quadrature phase angle $\theta ={\tan }^{-1}(\kappa ∕\omega )\simeq 0.2$. The choice of $\theta$ minimizes $V_{\mathrm{est}}$ close to the threshold at ${\lambda }_{\mathrm{c}}\simeq 0.51$.

Figure 12

Entanglement measures $V_1$ (dashed) and $V_2$ (solid) versus coupling strength; for $\omega ={\omega }_0=1$, $\kappa =0.2$, and $\theta =0$.