Figure 1

(Color online) Geometry of the proposed device. $N$ qubits are placed at the antinode(s) of a “cavity” with resonant frequency $\omega$ (depending on specific realization, they might alternatively be placed at one antinode, or at subsequent antinodes, to minimize qubit-qubit interactions). Their energy splittings are on-resonance with the oscillator ${\omega }_0=\omega$. An additional “open” TQ (shown in the left) is also coupled to the cavity off-resonance $\left(ϵ\ne \omega \right)$ and used to passively readout the state of the cavity mode. In the figure, the solid lines represent tunneling while the dashed lines represent energy gaps. The properties of the transport qubit are defined by an energy splitting $ϵ$, coherent tunneling rate $\Delta$ and transport rates ${\Gamma }_L$ and ${\Gamma }_R$. In addition, the cavity mode has a cavity decay rate ${\gamma }_b$, not shown in the figure. Superconducting artificial atoms coupled to an on-chip cavity (e.g., a quantized LC oscillator or microwave transmission line) are a feasible realization using current technology. The transport qubit can be realized using a charge qubit in the transport regime, i.e., a superconducting single electron transistor. Alternatively, a large number of qubits in the form of two-level atoms in an atom chip, coupled to a transmission line, has recently been proposed as a way to realize the large-$N$ Dicke model (Ref. 22).Reuse & Permissions

Figure 2

(Color online) (a) The current $I/e$ versus multiqubit-oscillator coupling $\lambda$ through the transport qubit for $\Delta =0.1$, ${\Gamma }_L={\Gamma }_R=0.1$, $ϵ=0$, $\omega ={\omega }_0=1$, and $g=0.1$ for $N=4,8,16,20,24,\infty$. (b) The derivative of the current through the transport qubit for the same parameter set versus $\lambda$. (c) and (d) show one particular data curve $\left(N=4,{\gamma }_b=0.1\right)$ for the bosonic occupancy $⟨a^{†}a⟩$ and the current $I/e$ for the three different approximations; zero back action (ground state of the pure Dicke model), master equation with cavity damping, and master equation with cavity damping and transport qubit feedback.Reuse & Permissions

Figure 3

(a) shows the scaling with $N$ and scaling exponent of the position $\left({\lambda }_m\right)$ of the minimum of the current derivative: $\left({\lambda }_m-{\lambda }_c\right)\propto N^{-0.68\pm 0.05}$. (b) shows the scaling of the value of the current at this minimum point to be ${\left(\frac{d\left(I/e\right)}{d\lambda }\right)}_m\propto \left(0.81\pm 0.05\right){\text{log}}_{2}N$. The parameters used here are $\Delta =0.1$, ${\Gamma }_L={\Gamma }_R=0.1$, $ϵ=0$, $\omega ={\omega }_0=1$, and $g=0.1$ with data taken at $N=4,8,16,20,24,40,60$.Reuse & Permissions

Figure 4

(Color online) (a) The current noise $F\left(0\right)=S\left(0\right)/2eI$ versus multiqubit-oscillator coupling $\lambda$ for $\Delta =0.1$, ${\Gamma }_L={\Gamma }_R=0.01$, $ϵ=0$, $\omega ={\omega }_0=1$, $g=0.1$, and for $N=4,8,16,20,24,\infty$. (b) The derivative $\frac{dF\left(0\right)}{d\lambda }$ versus $\lambda$. The peak scales as a power law of $N$, similar to the minimum of the current derivative.Reuse & Permissions

Figure 5

(Color online) The inset shows an example of the positive imaginary components of the eigenvalues $\left(E_i^L\right)$ of the Liouvillian for the damped Dicke model and the main figure shows their probability distribution $P\left(E_i^L\right)$, normalized to the maximum energy gap $S_{\text{max}}$, for $N=6$, $\lambda ={\lambda }_c$, and ${\gamma }_b=0.1$. While this contains every possible eigenvalue separation of the Hamiltonian (up to numerical bosonic cutoff), and has not been unfolded to remove secular variations, level repulsion is still visible.Reuse & Permissions