Flying microwave qubits with nearly perfect transfer efficiency

We propose a procedure for transferring the state a microwave qubit via a transmission line from one resonator to another resonator, with a theoretical efficiency arbitrarily close to 100%. The emission and capture of the microwave energy is performed using tunable couplers, whose transmission coefficients vary in time. Using the superconducting phase qubit technology and tunable couplers with maximum coupling of 100 MHz, the procedure with theoretical efficiency $\eta =0.999$ requires a duration of about 400 ns (excluding propagation time) and an on/off ratio of 45. The procedure may also be used for a quantum state transfer with optical photons.

Figure 1

Our goal is to pass a classical microwave energy from one resonator to another one via a transmission line. Conceptually, this is done by varying in time transmission amplitudes of “barriers” (tunable couplers) separating the resonators (half wavelength or quarter wavelength in size) and the long transmission line.

Figure 2

Sketches of the assumed time dependence $|{\mathbf{t}}^{\mathrm{em}}(t)|$ of the transmission amplitude of the emitting coupler (blue), transmission amplitude $|{\mathbf{t}}^{\mathrm{rec}}(t)|$ of the receiving coupler (black), and the wave amplitude $A(t)$ incident to the receiving coupler from the transmission line (green). The voltage amplitude $A(t)$ is constant ($A_0$) during the buildup period $0<t<t_{\mathrm{s}}$, then increases exponentially until time $t_{\mathrm{m}}$ at which $\left|{\mathbf{t}}^{\mathrm{em}}\right|$ reaches the maximum available value $\left|\mathbf{t}\right|{}_{\max }$, and after that $A(t)$ decreases exponentially until the end of the process $t_{\mathrm{e}}$. The receiving coupling $\left|{\mathbf{t}}^{\mathrm{rec}}\right|$ is kept at the maximum value $\left|\mathbf{t}\right|{}_{\max }$ at $t<t_{\mathrm{m}}$, while at $t>t_{\mathrm{m}}$ the emitting coupling $\left|\mathbf{t}\right|{}^{\mathrm{em}}$ is kept at the maximum $\left|\mathbf{t}\right|{}_{\max }$. The ratio of couplings $|{\mathbf{t}}^{\mathrm{em}}/{\mathbf{t}}^{\mathrm{rec}}|$ is chosen to cancel the back reflection into the transmission line at any time after $t_{\mathrm{s}}$.

Figure 3

Notations for the scattering matrix.

Figure 4

The tunable coupler consisting of inductances $L_1$ and $L_2$ coupled by a tunable mutual inductance $M$. Incoming wave amplitude is $A$ (voltage), the voltages across the inductors are $V$ and $x$, and the transmission amplitude is $\mathbf{t}=x/A$.