Experimental demonstration of efficient and selective population transfer and qubit distillation in a rare-earth-metal-ion-doped crystal

In optically controlled quantum computers it may be favorable to address different qubits using light with different frequencies, since the optical diffraction does not then limit the distance between qubits. Using qubits that are close to each other enables qubit-qubit interactions and gate operations that are strong and fast in comparison to qubit-environment interactions and decoherence rates. However, as qubits are addressed in frequency space, great care has to be taken when designing the laser pulses, so that they perform the desired operation on one qubit, without affecting other qubits. Complex hyperbolic secant pulses have theoretically been shown to be excellent for such frequency-addressed quantum computing [I. Roos and K. Molmer, Phys. Rev. A 69, 022321 (2004)]—e.g., for use in quantum computers based on optical interactions in rare-earth-metal-ion-doped crystals. The optical transition lines of the rare-earth-metal-ions are inhomogeneously broadened and therefore the frequency of the excitation pulses can be used to selectively address qubit ions that are spatially separated by a distance much less than a wavelength. Here, frequency-selective transfer of qubit ions between qubit states using complex hyperbolic secant pulses is experimentally demonstrated. Transfer efficiencies better than 90% were obtained. Using the complex hyperbolic secant pulses it was also possible to create two groups of ions, absorbing at specific frequencies, where 85% of the ions at one of the frequencies was shifted out of resonance with the field when ions in the other frequency group were excited. This procedure of selecting interacting ions, called qubit distillation, was carried out in preparation for two-qubit gate operations in the rare-earth-metal-ion-doped crystals. The techniques for frequency-selective state-to-state transfer developed here may be also useful also for other quantum optics and quantum information experiments in these long-coherence-time solid-state systems.

Figure 1

(Color online) Conceptual illustration of the creation of a peak. (a) When looking at a small part of the inhomogeneous absorption line, the initial absorption appears flat. (b) Scanning the laser frequency back and forth over a region transfers all the ions absorbing within that region to other hyperfine states. (c) A peak of ions absorbing on one hyperfine transition can be transferred back into the pit [27].

Figure 2

(Color online) (a) Two of the ground-state hyperfine levels are used as qubit states. (b) In principle a NOT operation can be performed with a radio frequency $\pi$ pulse. However, this would affect all qubits. (c) Since the optical transition frequency is different for different ions, optical pulses can be used to selectively address one group of ions at a time. The NOT operation between states ∣1⟩ and ∣0⟩ now requires three pulses.

Figure 3

(Color online) Schematic description of a controlled-NOT gate. (a, top) If the control qubit is originally in state ∣0⟩, pulse 1 will transfer it to the excited state ∣e⟩, and since the static dipole moment is different for the ground state and the excited state, the electric field in the vicinity of the ion changes. (b, top) This causes a Stark shift of optical resonance frequency of the neighboring ions. (c, top) Gate pulses on the original resonance frequencies of the target will no longer have any effect. A final pulse (not shown) transfers the control ion back to state ∣e⟩. The lower part of the figure shows the effect of the same pulse sequence when the control qubit is originally in state ∣1⟩. The first pulse will not be resonant with the ions, and the qubit NOT operation will be executed on the target qubit.

Figure 4

(Color online) Theoretical calculations of population transfer using optical pulses. The left column shows the intensity envelope of the pulses, the middle column shows the instantaneous frequency of the pulse, and the right column shows the simulated, solid line (blue online), and analytically calculated, dashed line (dashed red online), transfer for the same pulse. In simulations (a)–(d) a laser coherence time of $10\mu \mathrm{s}$ is used. All pulse lengths are given as the intensity FWHM. (a) Transfer using a Gaussian pulse with $t_{\mathrm{FWHM}}=0.33\mu \mathrm{s}$ and $t_{\text{cutoff}}=2\mu \mathrm{s}$ gives effective transfer for a narrow frequency interval since it is short and therefore less sensitive to decoherence. But only a small change, decrease or increase, in intensity will make the transfer much less effective. (b) Transfer using a sech pulse, $t_{\mathrm{FWHM}}=1\mu \mathrm{s}$, $t_{\text{cutoff}}=3\mu \mathrm{s}$, and ${\nu }_{\text{width}}=1.5\mathrm{MHz}$ gives a transfer efficiency of $\sim 93\%$. These pulse settings were used for the three-state transfer experiments shown in Figs. 9, 10. (c) Transfer using a sech pulse, $t_{\mathrm{FWHM}}=2\mu \mathrm{s}$, $t_{\text{cutoff}}=6\mu \mathrm{s}$, and ${\nu }_{\text{width}}=1.5\mathrm{MHz}$ yields an efficiency of $\sim 85\%$. A longer pulse makes the inversion flatter but smaller due to larger susceptibility to decoherence. This pulse was used for the two-level transfer experiments shown in Fig. 8(g). (d) Sech transfer with $t_{\mathrm{FWHM}}=1\mu \mathrm{s}$, $t_{\text{cutoff}}=2.5\mu \mathrm{s}$, and ${\nu }_{\text{width}}=1.5\mathrm{MHz}$. Even though the change in cutoff from case (b) is barely visible in the intensity plot, the effect is evident in the population transfer. (e) If the coherence time is increased to $100\mu \mathrm{s}$ the transfer efficiency is predicted to be 98%, $t_{\text{cutoff}}=5\mu \mathrm{s}$ and ${\nu }_{\text{width}}=1.5\mathrm{MHz}$.

Figure 5

Summary of relevant data for the ${}^3\mathrm{H}_4\to {}^1\mathrm{D}_2$ transition of site I in ${\mathrm{Pr}}^{3+}:{\mathrm{Y}}_2\mathrm{Si}{\mathrm{O}}_5$ and state labels used throughout this paper. $T_1$, $T_2$, and ${\Gamma }_{\mathrm{hom}}$ are given according to Ref. 42.

Figure 6

(Color online) Shift of the resonance frequency of ions due to changes in local electric fields caused by the excitation of nearby ions, absorbing at a different frequency. (a) The peak before excitation. (b), (c), and (d) Exciting 3-, 6-, and $20\text{−}\mathrm{MHz}$ intervals, respectively, corresponding to an increasing density of excited ions. (e) The control ions have returned to the ground state and the peak regains its initial shape. Single-shot data.

Figure 7

(Color online) Setup used in the experiments. See the text, Sec. 3, for details.

Figure 8

(Color online) Population transfer between the ground and optically excited states. (a) The qubit starts in state ∣1⟩. (b) Applying one sech pulse to the $\mid 1⟩\to \mid e2⟩$ transition transfers the qubit to state $\mid e2⟩$. (c). Another sech pulse transfers the qubit back to state ∣1⟩. (d) A third sech pulse will again transfer the qubit to state $\mid e2⟩$. Single-shot data.

Figure 9

(Color online) Multiple population transfer between two ground state levels via an excited state. The graphs on the left show the absorption during readout scans after completion of the operation, while the energy diagrams on the right schematically show the state of the qubit. Peaks in parentheses are the same ions absorbing to different excited levels. (a) The qubit is originally in state ∣0⟩. (b) The qubit is transferred to state $\mid e2⟩$ with a sech pulse resonant with the $\mid 0⟩\to \mid e2⟩$ transition, which causes strong stimulated emission on the $\mid e2⟩\to \mid 1⟩$ transition at the beginning of the scan. During the readout scan $\left(130\mu \mathrm{s}\right)$ the qubit partly relaxes back to the ground state and therefore absorption on the transition $\mid 0⟩\to \mid e1⟩$ can be seen in the middle of the scan and the $\mid e2⟩\to \mid 0⟩$ inversion at the end of the scan is small. (c) A second sech pulse brings the qubit down to state ∣1⟩. (d) A third sech pulse transfers the qubit to the state $\mid e2⟩$, (e) followed by a fourth sech pulse, transferring the qubit back to ∣0⟩. (f) Ten complete ground state-to-ground state transfers, each caused by two sech pulses. Single-shot data.

Figure 10

(Color online) Repeated population transfer between ground-state sublevels via an excited state, with a $500\text{−}\mu \mathrm{s}$ delay between each transfer. The graphs on the left show the readout scan $500\mu \mathrm{s}$ after completion of the operation. The energy diagrams on the right schematically show the state of the qubit. The peaks in parentheses are the same ions absorbing to different excited levels. (a) The qubit is originally in state ∣0⟩. (b) The qubit has been transferred to state ∣1⟩ using two sech pulses, resonant with the $\mid 0⟩\to \mid e2⟩$ and $\mid e2⟩\to \mid 1⟩$ transitions, respectively. (c) The qubit has been transferred back to state ∣0⟩. (d) Fifteen transfers. (e) Sixty transfers between the two ground-state sublevels. (e) Sixty-one transfers. Single-shot data.

Figure 11

(Color online) Selecting target ions that interact strongly with the control ions. The leftmost column shows data using a $3\text{−}\mathrm{MHz}$ control interval and the next column using a $20\text{−}\mathrm{MHz}$ control interval. The third and fourth columns schematically depict the state of the target and control ions, respectively. (a) The original peak consists of both controllable and noncontrollable ions. (b) Exciting the control ions shifts the resonance frequencies of the strongly interacting ions. (c) The ions that do not shift are transferred to state ∣aux⟩, using two sech pulses. (d) After the control ions have relaxed back to the ground state, the controlled ions return to their original resonance frequencies. (e) If the two sech pulses used in (c) are used to remove the peak of ions without first exciting the control ions, almost all of the peak is removed. The difference between (d) and (e) shows the efficiency of the selection. Single-shot data.