Cavity State Manipulation Using Photon-Number Selective Phase Gates

The large available Hilbert space and high coherence of cavity resonators make these systems an interesting resource for storing encoded quantum bits. To perform a quantum gate on this encoded information, however, complex nonlinear operations must be applied to the many levels of the oscillator simultaneously. In this work, we introduce the selective number-dependent arbitrary phase (snap) gate, which imparts a different phase to each Fock-state component using an off-resonantly coupled qubit. We show that the snap gate allows control over the quantum phases by correcting the unwanted phase evolution due to the Kerr effect. Furthermore, by combining the snap gate with oscillator displacements, we create a one-photon Fock state with high fidelity. Using just these two controls, one can construct arbitrary unitary operations, offering a scalable route to performing logical manipulations on oscillator-encoded qubits.

Figure 1

Qubit spectroscopy and geometric phase gate. (a) Qubit spectrum with storage cavity population $\overline{n}\approx 2$, showing that the qubit transition frequency depends on the number of photons in the cavity. (b) Phasor representation of a cavity state. The arrow $c_n$ corresponds to the complex amplitude of state $|n⟩$; the area of the circle is proportional to $|c_n{|}^2=p(n)$. (c) An example of the snap gate. Two $\pi$ pulses on the qubit along different axes result in a trajectory that encloses a geometric phase and allows a rotation on each $c_n$ selectively. (d) Final state after operation in (c), which results in a controllable phase evolution on each $c_n$. (e) Quantum circuit representation of the gates used in this work, where $\mathbf{Q}$ and $\mathbf{C}$ correspond to the qubit and the cavity state, respectively. $R_{y|n}(\varphi )$ should be read as “a rotation by angle $\varphi$ around $y$ conditional on $n$ photons in the cavity”; “%” is the modulo operator.

Figure 2

Phase evolution measurement. (a) Quantum circuit describing the pulse sequence. State preparation consists of an optional $\pi$ pulse on the qubit (not shown), a displacement of the cavity $D(\alpha )$, and a variable waiting time $\mathrm{\Delta }T$. The phase between $c_n$ and $c_{n+1}$ is determined by varying the applied quantum phase using the snap gate $S_{n+1}(\theta )$, displacing by $\epsilon$ and measuring the population $p(n)$. (b) Typical measurement signal $p(n)$ versus applied phase $\theta$ at fixed $\mathrm{\Delta }T$: a sine wave with minimum $p_{\min }$ and maximum $p_{\max }$. The phase between $c_n$ and $c_{n+1}$ corresponds to the case of destructive interference and is determined from the fit. (c) Phase evolution with qubit in the ground state. Each trace is the (unwrapped) measured phase difference between neighboring Fock-state components versus waiting time $\mathrm{\Delta }T$. The rotating frame is chosen such that there the phase difference between $|0⟩$ and $|1⟩$ is close to zero. From the fit we extract (see Supplemental Material [19]) the cavity drive detuning $\mathrm{\Delta }/2\pi =(-1.1\pm 0.7)\mathrm{kHz}$ and Hamiltonian parameters $K/2\pi =(-107.9\pm 0.5)\mathrm{kHz}$ and $K^{\prime }/2\pi =(3.4\pm 0.1)\mathrm{kHz}$. (d) Same as in (c) but with the qubit starting in the excited state. The frame on the cavity generator is adjusted by $-8300.0\mathrm{kHz}$, and from the fit we find additional detuning $\mathrm{\Delta }/2\pi =(17.6\pm 0.9)\mathrm{kHz}$ to give $\chi /2\pi =(-8281.3\pm 1)\mathrm{kHz}$, ${\chi }^{\prime }/2\pi =(48.8\pm 0.8)\mathrm{kHz}$, and ${\chi }^{\prime \prime }/2\pi =(0.5\pm 0.2)\mathrm{kHz}$.

Figure 3

Kerr correction. (a) Wigner function of the cavity in a coherent state directly after a displacement operation $D(2)$. (b) Wigner function after $1\mu \mathrm{s}$ of free evolution dominated by the Kerr effect. (c) Wigner function after one step of the Kerr-correction scheme (taking $1\mu \mathrm{s}$). The fidelity to a coherent state is $F=0.97$ (see Ref. [19]).

Figure 4

Fock state creation. Phasor representation, qubit spectrum, and Wigner function after each of the steps in the Fock-state creation scheme. (a) Displacement ${\beta }_1=1.14$. (b) Applying a $\pi$ phase shift on $|0⟩$ as well as cancelling Kerr-induced phases on other components. (c) Final displacement ${\beta }_2=-0.58$ to complete Fock-state $|1⟩$ creation.