$T_1$-echo sequence: Protecting the state of a qubit in the presence of coherent interaction

We propose a sequence of pulses intended to preserve the state of a qubit in the presence of strong, coherent coupling to another quantum system. The sequence can be understood as a generalized swap sequence and works in formal analogy to the well-known spin echo. Since the resulting effective decoherence rate of the qubit state is strongly influenced by the additional system, this sequence might serve to protect its quantum state as well as negating the effects of the coherent coupling. A possible area of application is in large-scale quantum computing architectures, where spectral crowding of the resources might necessitate a method to mitigate residual couplings.

Figure 1

Schematic representation of the pulse sequence for (a) standard spin-echo sequence and (b) the proposed $T_1$-echo sequence. In both cases, the state of the qubit is read out after time $T$. Also in both cases, the application of echo and recovery pulses leads to a final state which is independent of $T$. For details, see the text.

Figure 2

Step-by-step illustration of the $T_1$-echo sequence on the effective one-excitation Bloch sphere for the resonance case $\delta \omega =0$, with a starting state of $|1_q{0}_m\rangle$; for details see the text. Red arrows show the path of the state vector during the two time spans of free evolution; the blue dashed arrow represents the path during the echo pulse ${\widehat{U}}_{\perp }$. The recovery pulse is not shown, since for $\delta \omega =0$ we have $|{\Psi }_T\rangle =-i|{\Psi }_0\rangle$. The illustration depicts a free-evolution time of $T=2\pi /\left(3{\omega }_0\right)$. In this illustration we assume infinitely sharp pulses, i.e., no time is needed for the application of $U_{\perp }$.

Figure 3

Illustration of a full $T_1$-echo sequence on the effective Bloch sphere. Red arrows show the path of the state vector during the free-evolution periods, while the dashed blue arrows show the effects of the echo pulse ${\widehat{U}}_{\perp }$ and the recovery pulse ${\widehat{U}}_R$. The black vertical arrow depicts the state of the system at the start and end of the sequence. The depicted situation corresponds to $\delta \omega =-v_{\perp }$, i.e., ${\xi }_0=3\pi /4$, where we find $|{\Psi }_T\rangle =|x+\rangle$ and a total evolution time of $T=2\pi /{\omega }_0$.

Figure 4

Illustration of the $T_1$-echo sequence using only detuned Hamiltonian evolution. Both pulses ${\widehat{U}}_{\perp }$ and ${\widehat{U}}_R$ can be replaced by a period of detuned free evolution at detuning $\delta {\omega }_1=-v_{\perp }^2/\delta \omega$ for the time $t_{\pi }$. The red line depicts the qubit's level splitting at different times. The resonance condition with the memory, ${\epsilon }_q={\epsilon }_m$, is indicated by the dashed line in the middle. The situation depicted corresponds to the case of initially $\delta \omega =v_{\perp }$ where $\delta {\omega }_1=-\delta \omega$.

Figure 5

Time evolution of the qubit's excited-state population $P_{1_q}$ as a function of free-evolution time $T$ with (red solid) and without (blue dashed) application of the $T_1$-echo sequence and for three different detunings between qubit and memory. The dotted green line shows the decay of the qubit itself without any interactions with other quantum systems. The pulses are assumed to be infinitely sharp in time, and there is relaxation acting on only the qubit, not the memory. Parameters are (in units of ${\gamma }_{1,q}$) ${\gamma }_{1,m}={\gamma }_{\varphi ,q}={\gamma }_{\varphi ,m}=0$, $v_{\perp }=5$, $\delta \omega =0\text{(top),}5\text{(middle),}15\text{(bottom)}$.

Figure 6

Time evolution of the qubit's excited-state population $P_{1_q}$ as a function of time with (solid red) and without (dashed blue) application of the $T_1$-echo sequence and for three different detunings between qubit and memory. The pulses are realized in this case using a detuned Hamiltonian evolution; see the text. The green dotted line shows again simple exponential decay with the rate ${\gamma }_{1,q}$. The red line does not start at the origin due to the finite pulse time $2t_{\pi }$ needed for application of the sequence. Parameters are (in units of ${\gamma }_{1,q}$) ${\gamma }_{\varphi ,q}=0.5$, ${\gamma }_{1,m}={\gamma }_{\varphi ,m}=0$, $v_{\perp }=5$, $\delta \omega =0\text{(top),}1\text{(middle),}2\text{(bottom)}$.

Figure 7

Initial decay of the qubit state vector due to the finite pulse length $t_{\pi }$ obtained from numerical calculations. The red solid curve shows the probability of finding the qubit in its excited state as a function of detuning $\delta \omega$, after only the pulses have been applied to the system via the detuned Hamiltonian evolution. The total time that has passed is thus $2t_{\pi }$, while no free evolution has taken place, so the free-evolution time $T$ is zero. The green dotted curve for comparison shows the value of $e^{-{\gamma }_{1,q}2t_{\pi }}$, meaning the value of qubit excitation probability after application of the pulses without any influence of the TLS. The blue dashed line finally shows decay with the composite rate ${\gamma }_{+}$. One can see that for small initial detuning, the pulses lead to decay dominated by the qubit rate, while for stronger initial detuning it crosses over to decay of the coupled systems. Parameter values are (in units of ${\gamma }_{1,q}$) ${\gamma }_{\varphi ,q}=0.5$, ${\gamma }_{1,m}={\gamma }_{\varphi ,m}=0$, $v_{\perp }=5$.