Selective amplification of a quantum state

We predict a unique effect in a quantum two-level system (TLS) coupled to a resonant cavity. By bringing the TLS in and out of resonance with the cavity by a series of $N$ rectangular bias pulses (the length of the $m\text{th}$ pulse scaling as $1∕\sqrt{m}$), we will coherently excite the $N$-photon state, $\mid N⟩$, of the cavity only if the TLS was initially in an appropriate quantum state (“go” state). Otherwise, the number of photons in the cavity will remain small compared to $N$ (selective amplification). If the TLS was in a coherent superposition of the “go” and “no go” states, the cavity will be in a superposition of states, in which the state $\mid N⟩$ will enter with the same weight as the initial “go” component. The effect is due to $\sqrt{N+1}$ dependence of the Rabi oscillation frequency on the number $N$ of photons in the cavity. It is stable with respect to noise, pulse shape, finite temperature, TLS decoherence, and TLS detuning from resonance with the cavity. The effect can be used as a means to read out a quantum state of a qubit coupled to a resonator. Such coherent amplication of a low-energy signal is most relevant for solid-state quantum information processing, where the energy scale is well below the optical range.

Figure 1

Probability distribution of occupation numbers of the resonator, $P_n^r\equiv {\rho }_{nn}^r$, as a function of number of bias cycles, $m$, calculated using RWA. In the units of $\Delta$, the parameters are $\omega =10$ (resonator frequency), ${\lambda }_z=0$, ${\lambda }_x=0.8$ (TLS-resonator couplings), $\delta =0$ (perfect resonance), $\mid \epsilon \mid =9.9499$. The length of the $m\text{th}$ bias pulse is $3T_0∕\sqrt{m}$, where $T_0=5.9202$ is the Rabi half-period for empty resonator. (a) Zero temperature: empty resonator; initial density matrix of the TLS is ${\rho }^{\mathrm{TLS}}\left(0\right)={\mid 1⟩}_{\epsilon }$ ${⟨1\mid }_{\epsilon }$ (“go” state, left-hand side), ${\mid 0⟩}_{\epsilon }$ ${⟨0\mid }_{\epsilon }$ (“no go” state, right-hand side). The result for $T=1$ is visually the same. (b) The same for finite temperature (resonator in equilibrium at $T=20$).

Figure 2

Importance of effective phase randomization: time dependence of $P_n^r$ for the same parameters as in Fig. 1a, but with the bias pulse length $T_m=T_0∕\sqrt{m}$. The “no go” state (right panel) is also amplified and will not be reliably distinguishable from the “go” state (left panel). Increase of the $\mu$ factor beyond 3 will not qualitatively change the outcome (not shown).

Figure 3

Role of detuning $\delta =1$. Other parameters are the same as given in the caption of Fig. 1.

Figure 4

The role of decoherence (the state of TLS is measured after every bias cycle, as described in the text): $\omega =10$, ${\lambda }_z=0$, ${\lambda }_x=0.8$, $\delta =0$. The temperature $T=0$ (top), $T=20$ (bottom).

Figure 5

Selective amplification: exact numerical results for $\omega =5$, ${\lambda }_z=0$, ${\lambda }_x=0.1$, $\delta =0$, $T=0$, for “go” (curve 1) and “no go” (curve 2) states. The average resonator occupation number $⟨N⟩$ is plotted vs dimensionless time $\tau =\hslash ∕\Delta$. The squares show the results obtained in RWA.