Probing Decoherence with Electromagnetically Induced Transparency in Superconductive Quantum Circuits

Superconductive quantum circuits comprise quantized energy levels that may be coupled via microwave electromagnetic fields. Described in this way, one may draw a close analogy to atoms with internal (electronic) levels coupled by laser light fields. In this Letter, we present a superconductive analog to electromagnetically induced transparency that utilizes superconductive quantum circuit designs of present day experimental consideration. We discuss how a superconductive analog to electromagnetically induced transparency can be used to establish macroscopic coherence in such systems and, thereby, be utilized as a sensitive probe of decoherence.

Figure 1

(a) Energy level diagram of a three-level $\Lambda$ system. EIT can occur in atoms possessing two long-lived states $|1\rangle ,|2\rangle$, each of which is coupled via resonant laser light fields to a radiatively decaying state $|3\rangle$. (b) Circuit schematic of the persistent-current (PC) qubit and its readout SQUID. (c) One-dimensional double-well potential and energy-level diagram for a three-level SQC. Using PC qubit parameters [16, 18, 22], we calculate ${\omega }_2-{\omega }_1=(2\pi )36\mathrm{G}\mathrm{H}\mathrm{z}$ and ${\omega }_3-{\omega }_2=(2\pi )32\mathrm{G}\mathrm{H}\mathrm{z}$. The simulated matrix elements $\langle p|\sin (2\pi f+2{\varphi }_m)|q\rangle$ for $(p,q)=(1,2)$, $(2,3)$, and $(1,3)$ are, respectively, 0.0704, $-0.125$, and 0.0158.

Figure 2

(a) Population ${\rho }_{33}$ versus time with EIT fields ${\Omega }_{13}={\Omega }_{23}=(2\pi )150\mathrm{M}\mathrm{H}\mathrm{z}$, an initial dark state ${\rho }_{11}={\rho }_{22}=0.5$ and ${\rho }_{12}=-0.5$, and with a dephasing rate ${\gamma }_{12}=(2\pi )5\mathrm{M}\mathrm{H}\mathrm{z}$. The inset shows ${\rho }_{33}$ for early times. ${\rho }_{33}$ exhibits a rapid rise to a plateau after a short time $T_{\mathrm{s}\mathrm{s}}$, followed by a much slower decay. (b) The total population $P$ remaining in the system versus time for the same simulation (solid curve). The dashed curve shows the population for the out-of-phase case ${\rho }_{11}={\rho }_{22}=0.5$ and ${\rho }_{12}=0.5$ discussed in the text.

Figure 3

(a) The maximum plateau value ${\rho }_{33}^{(\mathrm{m}\mathrm{a}\mathrm{x})}$ for different ${\gamma }_{12}$ (circles). The solid curve shows the prediction (11). (b) The remaining population $P={\rho }_{11}+{\rho }_{22}+{\rho }_{33}$ at the time the plateau is reached $T_{\mathrm{s}\mathrm{s}}$ for the cases in (a).