Decoherence Dynamics of Complex Photon States in a Superconducting Circuit

Quantum states inevitably decay with time into a probabilistic mixture of classical states due to their interaction with the environment and measurement instrumentation. We present the first measurement of the decoherence dynamics of complex photon states in a condensed-matter system. By controllably preparing a number of distinct quantum-superposed photon states in a superconducting microwave resonator, we show that the subsequent decay dynamics can be quantitatively described by taking into account only two distinct decay channels: energy relaxation and pure dephasing. Our ability to prepare specific initial quantum states allows us to measure the evolution of specific elements in the quantum density matrix in a very detailed manner that can be compared with theory.

Figure 1

Decay of the $|0⟩+|n⟩$ states. (a) Measured density matrices at five delay times for the decay of $|0⟩+|3⟩$. The amplitude and phase of ${\rho }_{mn}$ is represented by the length and direction of an arrow in the complex plane (for scale, see the legend on the right). The element ${\rho }_{03}$ ($={\rho }_{30}^{*}$) is the only off-diagonal term that is nonzero during evolution, as expected. (b) Reconstruction of the Wigner distribution $W(\alpha )$ from data in (a). The distribution decays to the ground state $|0⟩$ for long times. (c) Amplitudes of the density matrix elements versus time for the decay of $|0⟩+|3⟩$. Solid lines are fits to the data for ${\rho }_{33}$ and ${\rho }_{03}$, from which decay times $T_{33}$ and $T_{03}$ are obtained. (d) Plot of $T_{nn}$ (solid circles), $T_{0n}$ (solid squares), and $T_{0n,\varphi }$ (solid hexagons) versus $n$ for $n=1–8$ obtained for a range of states $|0⟩+|n⟩$. Dephasing times $T_{0n,\varphi }$ are calculated from Eq. (2). Gray line (blue online) ($\mathrm{\text{slope}}=-1$) and black line ($\mathrm{\text{slope}}=-2$) are weighted least-square fits to obtain the resonator $T_1$ and $T_{\varphi }$, respectively.

Figure 2

Snapshots of the time evolution of $|2⟩+|3⟩$. (a) Measured and simulated density matrices at five delay times. The simulation starts with the measured density matrix at minimum delay and uses the decay parameters $T_1=2.4\mu \mathrm{s}$ and $T_{\varphi }=70\mu \mathrm{s}$. Note that only the off-diagonal elements ${\rho }_{23}\to {\rho }_{12}\to {\rho }_{01}$ are nonzero. (b) Wigner distributions $W(\alpha )$ as a function of time reconstructed from data in (a).

Figure 3

Evolution of the off-diagonal elements for $|m⟩+|3⟩$ with (a) $m=0$, (b) $m=1$, and (c) $m=2$. Right panels illustrate the evolution of the off-diagonal elements. Left panels display amplitude of matrix elements versus time. Solid lines are simulations based on Eq. (1), using $T_1=2.4\mu \mathrm{s}$ and $T_{\varphi }=70\mu \mathrm{s}$.

Figure 4

Evolution of (a) the state $|0⟩+i|2⟩+|4⟩$, (b) the coherent state $|\alpha =\sqrt{5}⟩$, (c) the odd Schrödinger cat state $|\alpha =i\sqrt{2}⟩-|\alpha =-i\sqrt{2}⟩$, and (d) the even Schrödinger cat state $|\alpha =i\sqrt{2}⟩+|\alpha =-i\sqrt{2}⟩$. Animations of the evolution for all states can be viewed online [22].