Time-reversal symmetrization of spontaneous emission for quantum state transfer

We demonstrate the ability to control spontaneous emission from a superconducting qubit coupled to a cavity. The time domain profile of the emitted photon is shaped into a symmetric truncated exponential. The experiment is enabled by a qubit coupled to a cavity, with a coupling strength that can be tuned in tens of nanoseconds while maintaining a constant dressed state emission frequency. Symmetrization of the photonic wave packet will enable use of photons as flying qubits for transferring the quantum state between atoms in distant cavities.

Figure 1

(a) A TCQ is shown coupled to a coplanar waveguide cavity. The aluminum qubit islands are colored red, yellow, and white and as the name suggests, they have a coupling strength that is tunable in situ. Embedded in the cavity ground plane are two control lines for the qubit, labeled 1 and 2, which apply a magnetic field that tunes the TCQ's frequency and coupling [11]. Each control line is biased with a low-noise dc voltage source ($V_1^{\text{dc}}$ and $V_2^{\text{dc}}$) and one channel of a Tektronix 5014 arbitrary waveform generator [$V_1^{\text{AWG}}\left(t\right)$ and $V_2^{\text{AWG}}\left(t\right)$]. For the wave-packet shaping experiment, the dc voltage sources are constant and bias the TCQ at a point of medium coupling. Then time-varying voltages from the AWG are used to change the Josephson energies in such a way that the coupling strength increases as the TCQ relaxes. (b) The effect of asymmetry in the pulse shape on the transfer fidelity is studied in simulation. The blue curve shows the highest state transfer fidelity for a symmetric incoming wave packet combined with the appropriate time reversal of the coupling as given in Eq. (3). The inset shows the wave-packet shapes used in the simulation. The red and green curves keep the same shape as the blue curve on the leading edge, but have time constants 2 (red) and 5 (green) times that of the blue curve on the trailing edge.

Figure 2

(a) If $V_1^{\text{AWG}}\left(t\right)=V_2^{\text{AWG}}\left(t\right)=0$, the coupling strength is constant throughout the relaxation process and the qubit emits a photon with an exponentially decaying wave-packet shape. The $V_1^{\text{dc}}$ and $V_2^{\text{dc}}$ are biased to hold the qubit at ${\omega }_{\mathrm{TCQ}}/2\pi =7.445\mathrm{GHz}$ and the bare cavity frequency and linewidth are ${\omega }_c/2\pi =7.596\mathrm{GHz}$ and $\kappa /2\pi =20\mathrm{MHz}$, respectively. The $\pi /2$ qubit excitation pulse is visible in the inset as a large spike at $200\mathrm{ns}$ in the quadrature measurement channel. The cavity output is mixed with a resonant local oscillator, which reveals the envelope. The emission signal appears on the in-phase channel, orthogonal to the qubit excitation. The signal is spread over a long time period and hence the emission amplitude is weak. (b) The TCQ is initialized with a $\pi /2$ pulse (shown in the inset) at the same operating point as the experiment in (a) that generated the exponentially decaying wave packet, but now $V_1^{\text{AWG}}\left(t\right)$ and $V_2^{\text{AWG}}\left(t\right)$ are varied to shape the emission into a symmetric exponential. The falling edge is generated by holding the coupling strength at a constant value and the coupling strength is varied along the rising edge in a manner to match its shape to the falling one. The small amount of emission in the quadrature channel corresponds to drifts in frequency. (c) The state transfer fidelity $F\left(t\right)=\mathrm{Tr}\left[\rho \left(t\right)\widehat{F}\right]$ is plotted for the case where the experimentally produced wave packet in (b) is sent towards a simulated TCQ-cavity system. Here $\widehat{F}$ is the fidelity operator that is the tensor product of an identity on the photons and the projector onto the target superposition state. The TCQ is initially in the ground state and its coupling strength to the cavity is tuned in a manner time reversed to the form described in Eq. (3), which was used to generate the symmetric shape. Since the TCQ in the destination cavity is initially in the ground state, the overlap fidelity starts at $0.5$ before the wave packet drives it into the symmetric superposition. Effects of nonradiative relaxation are considered for values $T_1^{\text{nr}}=1.0\mu \mathrm{s}$ (solid black), the experimentally observed value of $1.6\mu \mathrm{s}$ (dotted blue), $5\mu \mathrm{s}$ (dashed green), and $\infty$ (solid red). With $T_1^{\text{nr}}=\infty$, the maximum fidelity is $94\%$.

Figure 3

(a) Pulse sequence for constant frequency contour. For state transfer, having the symmetric wave packet at a constant frequency is beneficial since absorbing it at the destination requires time reversing all the parameters used at the source to generate the wave packet. Hence, only the coupling strength needs to be varied at the destination and the frequency can be fixed to the source's emission frequency. The pulse sequence used for constant frequency calibration is shown here; the blue Gaussian envelope is mixed with ${\omega }_{\mathrm{TCQ}}/2\pi =7.445\mathrm{GHz}$ for the qubit $\pi /2$ pulse. Then the two voltages $V_1^{\text{AWG}}\left(t\right)$ and $V_2^{\text{AWG}}\left(t\right)$ are ramped. (b) Calibrating constant frequency emission. The ramp height of $V_1^{\text{AWG}}\left(t\right)$ is fixed, the ramp height of $V_2^{\text{AWG}}\left(t\right)$ is varied ($Y$ axis), and emission (mV) from the qubit as a function of time ($X$ axis) is recorded after the $\pi /2$ pulse. The $V_2^{\text{AWG}}\left(t\right)$ ramp height that keeps the qubit frequency locked to ${\omega }_{\mathrm{TCQ}}$ is denoted by the dashed line. The fringes away from the line correspond to changes in qubit frequency. The exponential decay at constant frequency is equal to the relaxation time $T_1$.

Figure 4

Relaxation time along the contour. The pulse sequence shown in Fig. 3 yields a contour in the 2D space spanned by $V_1^{\text{AWG}}$ and $V_2^{\text{AWG}}$, which keeps the qubit frequency fixed at ${\omega }_{\mathrm{TCQ}}/2\pi =7.445\mathrm{GHz}$, but varies the coupling strength. The contour is parametrized by one of the voltages $V_1^{\text{AWG}}\left(t\right)$ in this plot. The $Y$ axis shows the relaxation time as measured by fitting an exponential decay to the constant frequency emission. The $T_1$ varies from $600\mathrm{ns}$ to less than $50\mathrm{ns}$. The emission amplitude gets extremely weak as the coupling strength decreases and the $T_1$ approaches the nonradiative rate $T_1^{\text{nr}}$. At the point of minimum coupling, emission is undetectable and nonradiative processes are the dominant decay mechanism. Here a $\pi$ pulse and delayed measurement were used to measure $T_1^{\text{nr}}\approx T_1=1.6\mu \mathrm{s}$.