Dynamics of Coherent and Incoherent Emission from an Artificial Atom in a 1D Space

We study dynamics of a two-level superconducting quantum system, analogous to a natural atom in an open space, by measuring the evolution of its coherent and incoherent emission. The emitted waves containing full information about the states of the artificial atom are efficiently collected due to strong atom-transmission-line coupling. This allows us to do simultaneous measurements of all the quantum state projections and perform a full characterization of the system. We derive coherence times and extract the two-time correlation function from the dynamics of the coherent emission.

Figure 1

(a) Measurement circuit diagram. (b) General pulse sequence (upper panel) with schematic atomic dynamics (lower panel): The pulses $P$ and $M$ are used to prepare and manipulate the atomic states, and the emission from the atom is detected during the readout pulse $R$.

Figure 2

(a) Pulse sequence for the preparation and readout the pseudospin and the corresponding rotation in $yz$ plane of the Bloch sphere. (b) Power spectrum of the free-induction decay emission peak with a long excitation pulse $\Delta t_P=100\mathrm{ns}$. (c) Evolution of all projections of the quantum state $⟨{\sigma }_{x,y,z}⟩$. The atomic population measured from the emitted power (upper panel), and the normalized dipole moment $⟨{\sigma }^{-}⟩$ derived from the coherent dipole emission (lower panel). The dashed lines mark the points where the emission is completely incoherent ($⟨{\sigma }^{-}⟩=0$), corresponding to the extrema of the emitted power ($\propto 1+⟨{\sigma }_z⟩$).

Figure 3

(a) Evolution of $⟨{\sigma }_y⟩/2$ at different driving amplitudes $\Omega /2\pi =140\mathrm{MHz}$ (red curve), 44 MHz (blue curve), and 14 MHz (black curve). The stationary value of $⟨{\sigma }^{-}⟩$ strongly depends on the driving power. The dashed black curve depicts decay of the fast oscillations with rate $({\Gamma }_1/2+{\Gamma }_2/2)/2\pi =13.5\mathrm{MHz}$. (b) Direct measurements of dephasing. The red dots show $⟨{\sigma }_y⟩$ as a function of the delay $\Delta t_{PR}$ between the manipulation $\pi /2$ and readout pulses. The black curve is an exponential-decay fit with ${\Gamma }_2/2\pi =9.1\mathrm{MHz}$ (c) Measurements of relaxation. The measured decay of $⟨{\sigma }_y⟩$ (projected $⟨{\sigma }_z⟩$) is presented by the red dots. It changes the sign when $⟨{\sigma }_z⟩$ becomes negative. The black curve is an exponential fit with ${\Gamma }_1/2\pi =18.3\mathrm{MHz}$.

Figure 4

(a) and (b) Evolutions of $⟨\Delta {\sigma }_{x,y}(t)⟩$ with initial conditions $\overrightarrow{\sigma }(0)=\{1,0,0\}$ and $\{0,1,0\}$, respectively. (c) Two-time correlation function of fluctuations $⟨\Delta {\sigma }^{+}(0)\Delta {\sigma }^{-}(t){⟩}_{ss}$ calculated based on evolutions in (a) and (b). The red and blue are the real and imaginary parts, and the dashed black curve is a real part of the solution obtained from Eq. (5) using $B$ with experimentally deduced parameters. Inset: Spectrum [$S(f)=2\pi S(\omega )$] of the resonance fluorescence triplet as a function of detuning $\delta f=(\omega -{\omega }_a)/2\pi$ derived from the two-time correlation function according to Eq. (4) (black curve) and measured directly in the frequency domain SA (red open circles).