Quantum gates with phase stability over space and time

The performance of a quantum information processor depends on the precise control of phases introduced into the system during quantum gate operations. As the number of operations increases with the complexity of a computation, the phases of the gates at different locations and different times must be controlled, which can be challenging for optically driven operations. We circumvent this issue by demonstrating an entangling gate between two trapped atomic ions that is insensitive to the optical phases of the driving fields while using a common master reference clock for all coherent qubit operations. Such techniques may be crucial for scaling to large quantum information processors in many physical platforms.

Figure 1

(a) The qubit is driven from atomic levels $\left|0\right.〉$ to $\left|1\right.〉$ via a two-photon-stimulated Raman process by absorbing from the ${\nu }_{B,1}$ comb and emitting into the ${\nu }_{B,2}$ comb. The phase written to the qubit in this transition is ${\Phi }_{B,1}-{\Phi }_{B,2}$, where ${\Phi }_{B,1}$ and ${\Phi }_{B,2}$ are the optical phases of the two combs at the ion position. The inverse process from $\left|1\right.〉$ to $\left|0\right.〉$ reverses these phases. This coherent transition can also be driven directly with microwaves at frequency ${\nu }_0$. (b) Simplified experimental diagram. The master microwave oscillator and pulsed laser repetition rate are locked through a feedforward system. AOM B is used for copropagating transitions, and AOM A is used in conjunction with AOM B for multiqubit entangling gates.

Figure 2

(a) $\pi /2$ microwave rotation followed by a $\pi /2$ Raman rotation. The phase of the Raman rotation ${\varphi }_R$ is scanned for two different microwave phases ${\varphi }_{\mu }=0,\pi$. If these operations are phase coherent, the final state of the qubit can be controlled by adjusting the phase of either operation. The probability of being in state $\left|1\right.〉$ is fit to $P\left(\left|1\right.〉\right)=0.5+Asin({\varphi }_0+{\varphi }_R-{\varphi }_{\mu })$, where ${\varphi }_0$ is a static offset phase that stems from the path-length difference between the microwave or optical fields and the ion. (b) A Ramsey experiment with delay $T$ is carried out with an initial Raman rotation and a final microwave rotation. The phases of the rotations are adjusted to give $P\left(\left|1\right.〉\right)=1$ at $T=0$. A Gaussian fit to the data gives a $1/e$ decay time of 1.8 s.

Figure 3

(a) and (b) Representations of the optical combs in the frequency domain and (c) and (d) orientations of the Raman beams with respect to the addressed vibrational mode $\mathbf{X}$ and magnetic field $\mathbf{B}$. Beam ${\mathbf{k}}_A$ is polarized perpendicular to $\mathbf{B}$, whereas beams ${\mathbf{k}}_{B,r}$ and ${\mathbf{k}}_{B,b}$ have ${\sigma }_{+}$ polarization. This orientation allows copropagating Raman transitions to be driven by AOM B and the entangling gates to be driven by AOMs A and B. In order to drive the gate, AOMs A and B shift the reference $0\mathrm{th}$ comb tooth by ${\nu }_A,{\nu }_{B,r}$, and ${\nu }_{B,b}$ from the 0 modulation line (vertical dashed line), and the negative shift for ${\nu }_A$ is obtained by taking negative first-order diffracted beam. The beat notes between the combs, represented by the dashed arrows, have the required frequencies for the gate, and the optical-field gradient (purple shading) addresses the transverse modes. (a) In the optical phase-insensitive geometry, off-resonant blue sideband transition is driven by absorption from the $m\mathrm{th}$ comb tooth of the ${\mathbf{k}}_{B,b}$ beam and emission into the $0\mathrm{th}$ comb tooth of the ${\mathbf{k}}_A$ beam. The absorption and emission directions of the red sideband transition is opposite that of the blue sideband transition such that the gate is driven by absorbing from the $n\mathrm{th}$ comb tooth of the ${\mathbf{k}}_A$ beam and emitting into the $0\mathrm{th}$ comb tooth of the ${\mathbf{k}}_{B,r}$ beam. (b) In the optical phase-sensitive geometry, off-resonant red and blue sideband transitions are driven by absorption from the $m\mathrm{th}$ comb tooth of the ${\mathbf{k}}_{B,r},{\mathbf{k}}_{B,b}$ beams and emission into the $0\mathrm{th}$ comb tooth of the ${\mathbf{k}}_A$ beam. (c) In the Mølmer-Sørensen protocol, the gate phase is ${\varphi }_G=-({\varphi }_{\mathrm{rsb}}+{\varphi }_{\mathrm{bsb}})$, where ${\varphi }_{\mathrm{rsb}},{\varphi }_{\mathrm{bsb}}$ are phases associated with the red and blue sideband transitions. Drifts of the optical path length from the source to the ions $\delta x$, along the ${\mathbf{k}}_{B,r},{\mathbf{k}}_{B,b}$ beam path change the optical phases of these fields at the ion position resulting in a phase shift of ${\varphi }_{\mathrm{rsb}}$ and ${\varphi }_{\mathrm{bsb}}$ by $\delta \varphi =k_{B,r}\delta x\approx k_{B,b}\delta x$ [see Fig. 1 and Eq. (4)]. In the optical phase-insensitive geometry since the directions of the red and blue sideband transitions are opposite, the phase changes nearly cancel out so that ${\varphi }_G^{\prime }=({\varphi }_{\mathrm{rsb}}-\delta \varphi )+({\varphi }_{\mathrm{bsb}}+\delta \varphi )\approx {\varphi }_G$, providing optical path-length independence to the gate. (d) For the optical phase-sensitive case, this change is directly imprinted onto the ions: ${\varphi }_G^{\prime }=({\varphi }_{\mathrm{rsb}}+\delta \varphi )+({\varphi }_{\mathrm{bsb}}+\delta \varphi )\approx {\varphi }_G+2\delta \varphi$. Similar uncorrelated phase sensitivity is also present on path-length drifts of the ${\mathbf{k}}_A$ beam.

Figure 4

(a) If the wave fronts of the optical-field gradient (purple lines) are misaligned with respect to the trapping axis by an angle ${\theta }_{\epsilon }$, the ions experience different state-dependent force phases resulting in gate errors. The wave fronts are separated by ${\lambda }^{\prime }=\frac{2\pi }{\Delta k}\approx \frac{2\pi }{\sqrt{2}k}\approx 250\mathrm{nm}$. As an example, in order to realize a phase variation of $<{10}^{\circ }$ along a $30\text{−}\mu \mathrm{m}$ ion chain, ${\theta }_{\epsilon }$ must be $<0.{02}^{\circ }$. (b) Experimental sequence for the wave-front alignment and the expected signal. A single ion in state $\left|0\right.〉$ is rotated by a resonant noncopropagating Raman $\pi /2$ pulse and is shuttled by $d$ along the trapping axis. In the new position, the ion is rotated again by another noncopropagating Raman $\pi /2$ pulse before fluorescent detection of the final state. The blue (red) curve shows the expected ion brightness corresponding to a $1^{\circ }(0.{05}^{\circ })$ misalignment. The oscillation on the final qubit state is a result of the phase difference between the resonant $\pi /2$ rotations and is given by $P\left(\left|1\right.〉\right)={cos}^2[\pi dsin\left({\theta }_{\epsilon }\right)/{\lambda }^{\prime }]$.

Figure 5

Parity $P\left(\left|00\right.〉\right)+P\left(\left|11\right.〉\right)-P\left(\left|01\right.〉\right)-P\left(\left|10\right.〉\right)=Acos({\varphi }_G+2\varphi +{\varphi }^{\prime })$ of the two-qubit entangled state. Ions are first optically pumped to the $\left|00\right.〉$ state, and following the phase-insensitive gate, a $\pi /2$ analysis rotation with phase $\varphi$ is applied. The blue circles are the result of analysis with a copropagating Raman rotation, and the red squares are analyzed with a microwave rotation. The phase shift between the parity curves is due to different ${\varphi }^{\prime }$ static offsets between the gate and the $\pi /2$ analysis rotations.

Figure 6

(a) and (b) Changes in the phase of the red $\left({\varphi }_{\mathrm{sb}}\equiv {\varphi }_{B,r}\right)$ and blue $\left({\varphi }_{\mathrm{sb}}\equiv {\varphi }_{B,b}\right)$ sideband addressing frequencies cause ${\varphi }_G$ to shift in opposite directions for the phase-insensitive gate, whereas ${\varphi }_G$ shifts in the same direction for the phase-sensitive gate. This behavior is verified by the phase shift in the parity oscillation. (c) and (d) To simulate a change in the relative optical path length, a random phase is added to frequencies provided by the AWG during the gate at each point. The parity curve is not affected for the phase-insensitive gate, whereas the phase-sensitive gate parity curve becomes randomized from point to point as verified by three data sets.

Figure 7

(a) Phase-coherence circuit. An AWG is used to provide the necessary frequencies for quantum operations while at the same time maintaining phase relations between different frequency components. Filters are used throughout the circuit to remove undesired frequency components of the mixer output. The second-harmonic light of a mode-locked Nd:YAG laser at 532 nm is directed to a fast photodiode which generates a frequency comb with tooth separation ${\nu }_r$. The third harmonic at 355 nm is used to drive atomic transitions. Direct digital synthesizer is represented by DDS. (b) The photodiode signal is mixed (No. 1) with the master oscillator (HP 8672A) and is sent to three different PLLs which use this signal to output $\sim 198$ and $\sim 285\mathrm{MHz}$, matching the difference between the oscillator and the $m=154,n=160$ comb teeth. (c) The PLL 1 and 2 signals are first combined and then are mixed (No. 2) with the AWG to address the detuned sideband frequencies of the trapped ions. During the gate, switch $\mathrm{a}\to 3$ and switch $\mathrm{b}\to 1$. (d) Phase-coherent microwave rotations with gates are realized by mixing (No. 3) the AWG signal with the master oscillator to drive carrier transitions. For the microwave rotations, switch $\mathrm{a}\to 1$. The third PLL provides phase-coherent copropagating carrier transitions using the $p=157$ comb tooth and AOM B with switch $\mathrm{a}\to 2$ and switch $\mathrm{b}\to 2$.

Figure 8

Phase-coherent circuit with a single PLL for phase-coherent qubit operations.