Experimental Implementation of a Raman-Assisted Eight-Wave Mixing Process

S.O. Mundhada, A. Grimm, J. Venkatraman, Z.K. Minev, S. Touzard, N.E. Frattini, V.V. Sivak, K. Sliwa, P. Reinhold, S. Shankar, M. Mirrahimi, and M.H. Devoret

Phys. Rev. Applied 12, 054051 – Published 21 November 2019

Abstract

Nonlinear processes in the quantum regime are essential for many applications, such as quantum-limited amplification, measurement, and control of quantum systems. In particular, the field of quantum error correction relies heavily on high-order nonlinear interactions between various modes of a quantum system. However, the required order of nonlinearity is often not directly available or weak compared to dissipation present in the system. Here, we experimentally demonstrate a route to obtain higher-order nonlinearity by combining more easily available lower-order nonlinear processes, using a generalization of the Raman transition. In particular, we show a transformation of four photons of a high- $Q$ superconducting resonator into two excitations of a superconducting transmon mode and two pump photons, and vice versa. The resulting eight-wave mixing process is obtained by cascading two fourth-order nonlinear processes through a virtual state. We expect this type of process to become a key component of hardware-efficient quantum error correction using continuous-variable error-correction codes.

Received 4 December 2018
Revised 21 August 2019

DOI:https://doi.org/10.1103/PhysRevApplied.12.054051

Physics Subject Headings (PhySH)

Nonlinear optics Photonics Quantum control Quantum error correction Quantum information processing with continuous variables Quantum information with solid state qubits

Superconducting qubits

Raman spectroscopy

Quantum Information, Science & Technology

Authors & Affiliations

S.O. Mundhada ^1,*, A. Grimm¹, J. Venkatraman¹, Z.K. Minev¹, S. Touzard¹, N.E. Frattini¹, V.V. Sivak¹, K. Sliwa^1,‡, P. Reinhold¹, S. Shankar¹, M. Mirrahimi², and M.H. Devoret^1,†

¹Department of Applied Physics, Yale University, New Haven, Connecticut 06511, USA
²QUANTIC team, INRIA de Paris, 2 Rue Simone Iff, Paris 75012, France

^*shantanu.mundhada@yale.edu
^†michel.devoret@yale.edu
^‡Present address: Quantum Circuits Inc., New Haven, Connecticut 06511, USA

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Vol. 12, Iss. 5 — November 2019

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Figure 1
Schematic of Raman-assisted nonlinear processes and their experimental implementation. (a) The target eight-wave mixing process that exchanges four photons of a high- $Q$ resonator (magenta) with two excitations of a transmon mode (green) and two pump photons (blue, brown), and vice versa. (b) Energy-level diagram of a high- $Q$ storage resonator at frequency $ω_{a}$ coupled to a transmon mode at frequency $ω_{b}$ (called conversion mode). The first three transmon eigenstates (denoted by $| g ⟩$ , $| e ⟩$ , and $| f ⟩$ ) and the first five eigenstates of the storage resonator (denoted by $| 0 ⟩$ to $| 4 ⟩$ ) are considered. Starting in $| g 0 ⟩$ , the system is prepared in $| f 0 ⟩$ by applying $| g ⟩ \to | e ⟩$ and $| e ⟩ \to | f ⟩$ Rabi pulses (green arrows). A pump at frequency $ω_{p 1}$ (blue) connects $| f 0 ⟩$ to a virtual (non-energy-conserving) state, represented by the dashed line, detuned from $| e 2 ⟩$ with a detuning $Δ$ . This virtual state acts as an intermediate metastable excitation of the transmon. A second pump at frequency $ω_{p 2}$ (brown) connects the virtual state to $| g 4 ⟩$ , thus converting the two transmon excitations into four resonator excitations. (c) Frequencies of the pumps and the transitions involved in the scheme. (d) Schematic of the implementation. The high- $Q$ storage mode is formed by an aluminum $λ / 4$ -type three-dimensional superconducting resonator (magenta), which is dispersively coupled to the conversion transmon (green) and the tomography transmon (red). The two $λ / 2$ stripline resonators coupled to the transmons are used for performing single-shot readout of the respective transmons.
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Figure 2
Pulse sequence and Rabi oscillations of the cascading process. (a) Pulse sequence used for locating the $| f 0 ⟩ \leftrightarrow | g 4 ⟩$ resonance of the system. The system is initialized in $| f 0 ⟩$ by using $π$ pulses on $| g ⟩ \leftrightarrow | e ⟩$ and $| e ⟩ \leftrightarrow | f ⟩$ transitions. Following this, the two pumps are applied with varying frequency and duration. The frequency difference of the two pumps is maintained constant at $χ_{b b} - 4 χ_{a b} + 2 Δ$ . Finally an indirect measurement of the storage resonator population is performed using a photon-number selective $π$ pulse on the tomography transmon and a measurement pulse on the tomography resonator. Optionally, a measurement of the conversion transmon state can also be performed using a measurement pulse on the conversion resonator. (b) Rabi oscillations in the population of Fock state $| 0 ⟩$ ( $p_{0}$ , colorbar). The $x$ axis shows the detuning of pump 1 from the $| f 0 ⟩ \leftrightarrow | e 2 ⟩$ transition, the $y$ axis shows the duration for which the two pumps are applied. The frequency landscape above the data explains the origin of the two chevronlike features.
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Figure 3
Partial tomography of $| f 0 ⟩ \leftrightarrow | g 4 ⟩$ oscillations as a function of time. The system is prepared in $| f 0 ⟩$ and the two pumps are applied for a variable period of time on resonance with the $| f 0 ⟩ \leftrightarrow | g 4 ⟩$ transition. Following this, a selective pulse with a variable frequency is applied on the tomography transmon enabling an indirect measurement of various Fock-state populations of the storage resonator. (a) Excited-state population of tomography transmon (colorbar) versus pump duration ( $x$ axis) and the detuning of the selective $π$ pulse on the tomography transmon ( $y$ axis). The $y$ axis on the right shows the frequency of the tomography transmon ( $ω_{T n}$ ) conditioned on the number of photons $n$ in the storage mode. (b) From top to bottom, $| 0 ⟩$ , $| 2 ⟩$ , and $| 4 ⟩$ Fock-state populations (magenta), measured along the dashed lines shown in (a). Independently measured populations in $| f ⟩$ , $| e ⟩$ , and $| g ⟩$ states of the conversion mode (green) are also plotted, respectively, from top to bottom. The plots on the left are experimental data and the ones on the right are obtained from numerical simulation (see text).
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Figure 4
Conditional Wigner tomography of the storage resonator after a quarter period of the $| f 0 ⟩ \leftrightarrow | g 4 ⟩$ oscillation. After quarter period of $| f 0 ⟩ \leftrightarrow | g 4 ⟩$ oscillation the system is in the state $(| f 0 ⟩ + | g 4 ⟩) / \sqrt{2}$ . (a),(b) Experimental Wigner function of the storage resonator after postselecting the conversion mode in the $| g ⟩$ and $| e ⟩$ states. This leaves the storage resonator in Fock states $| 4 ⟩$ , $| 0 ⟩$ , respectively. (d),(e) Ideal Wigner functions of Fock states $| 4 ⟩$ , $| 0 ⟩$ for comparison. (c) Wigner function of the resonator after photon-number selective $π$ pulses from $| f 0 ⟩$ to $| e 0 ⟩$ and $| e 0 ⟩$ to $| g 0 ⟩$ (indicated by $U_{sel}$ ) and postselecting the conversion transmon in $| g ⟩$ . Comparing (c) with the ideal Wigner function of $(| 0 ⟩ + | 4 ⟩) / \sqrt{2}$ state in (f), shows that the storage resonator is in a coherent superposition of $| 0 ⟩$ and $| 4 ⟩$ , thus indicating that the $| f 0 ⟩ \leftrightarrow | g 4 ⟩$ oscillations are coherent.
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Figure 5
Wiring diagram.
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