Manipulating Fock states of a harmonic oscillator while preserving its linearity

K. Juliusson, S. Bernon, X. Zhou, V. Schmitt, H. le Sueur, P. Bertet, D. Vion, M. Mirrahimi, P. Rouchon, and D. Esteve

Phys. Rev. A 94, 063861 – Published 29 December 2016

Abstract

We present a scheme for controlling the quantum state of a harmonic oscillator by coupling it to an anharmonic multilevel system (MLS) with first- to second-excited-state transition on resonance with the oscillator. In this scheme, which we call ef-resonant, the spurious oscillator Kerr nonlinearity inherited from the MLS is very small, while its Fock states can still be selectively addressed via an MLS transition at a frequency that depends on the number of photons. We implement this concept in a circuit-QED setup with a microwave three-dimensional cavity (the oscillator, with frequency 6.4 GHz and quality factor $Q_{O} = 2 \times 10^{6}$ ) embedding a frequency tunable transmon qubit (the MLS). We characterize the system spectroscopically and demonstrate selective addressing of Fock states and a Kerr nonlinearity below 350 Hz. At times much longer than the transmon coherence times, a nonlinear cavity response with driving power is also observed and explained.

Received 20 July 2016

DOI:https://doi.org/10.1103/PhysRevA.94.063861

Physics Subject Headings (PhySH)

Cavity quantum electrodynamics Hybrid quantum systems Josephson effect Quantum control Quantum information with solid state qubits Superconducting qubits

Nonlinear DynamicsCondensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

K. Juliusson¹, S. Bernon¹, X. Zhou¹, V. Schmitt¹, H. le Sueur², P. Bertet¹, D. Vion^1,*, M. Mirrahimi³, P. Rouchon⁴, and D. Esteve¹

¹Quantronics Group, SPEC, CEA, CNRS, Université Paris-Saclay, CEA Saclay, 91191 Gif-sur-Yvette, France
²Centre de Sciences Nucléaires et de Sciences de la Matière, 91405 Orsay, France
³INRIA Paris-Rocquencourt, Domaine de Voluceau, Boîte Potale 105, 78153 Le Chesnay Cedex, France
⁴Centre Automatique et Systèmes, Mines-ParisTech, PSL Research University, 60, boulevard Saint-Michel, 75006 Paris, France

^*Corresponding author: denis.vion@cea.fr

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Vol. 94, Iss. 6 — December 2016

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Figure 1
The $ef$ -resonant scheme for manipulating Fock states $|n〉$ of a quantum harmonic oscillator. Top: The oscillator with frequency $ν_{O}$ is coupled with a coupling frequency $g_{O}$ to a multilevel system (MLS) with eigenstates $\{|g〉, |e〉, |f〉, \dots\}$ ; the $|e〉 \leftrightarrow |f〉$ transition is resonant with $ν_{O}$ , whereas the $|g〉 \leftrightarrow |e〉$ one is detuned by $α$ . Bottom: The resulting energy diagram consists of a quasiharmonic ladder $\{|\tilde{g n}〉\}$ when the MLS is left unexcited, and two anharmonic ladders of levels $|\pm n〉$ that correspond approximately to symmetric and antisymmetric superpositions of $|e (n - 1)〉$ and $|f (n - 2)〉$ states for $n \geq 2$ . The $|\tilde{g n}〉 \leftrightarrow |\pm (n + 1)〉$ transitions can be driven at different frequencies $ν_{\pm n} \sim ν_{ge} \pm \sqrt{2 n} g_{O}$ to manipulate selectively any $|\tilde{g n}〉$ .
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Figure 2
Circuit-QED implementation of the $ef$ -resonant scheme. (a) The harmonic oscillator $O$ is mode 120 of a superconducting Al cavity, and the MLS is a tunable transmon qubit with a SQUID. The transmon chip is inserted only partly in the cavity, with the SQUID being exposed to a dc magnetic field $\vec{B}$ . The transmon is weakly coupled to $O$ and strongly to cavity mode 210 used for dispersive readout of the transmon state. The two modes have quality factors $Q_{O} = 2 \times 10^{6}$ and $Q_{R} = 15 \times 10^{3}$ . (b) Picture of one half cavity with chip and Cu cap. (c) Equivalent electric circuit of the system with relevant frequencies $f_{R, O}$ , quality factors $Q_{R, O}$ , and coupling frequencies $g_{R, O}$ . $O$ is driven coherently and resonantly through port 1 (left purple pulse). Fock states $|\tilde{g n}〉$ are manipulated by driving the $|\tilde{g n}〉 \leftrightarrow |+ n〉$ transition (middle red pulse). A projective measurement on the Fock state $|\tilde{g n}〉$ is obtained by a $π$ pulse at $ν_{+ n}$ followed by a readout pulse (right blue pulse). (d) Electrical setup at room temperature (300 K) and inside the dilution refrigerator. Cavity resonances are measured with continuous waves using a vectorial network analyzer (VNA), whereas pulsed experiments use heterodyne modulation and demodulation. Microwave pulses at the cavity (qubit) frequencies $ν_{O, R}$ ( $ν_{+ n}$ ) are obtained by single-sideband mixing of a continuous microwave (LO) with an intermediate frequency-modulated pulse generated by two channels of an arbitrary waveform generator (AWG). All pulses travel along an attenuated and filtered line to cavity port 1. The readout signal transmitted at port 2 is filtered, isolated from backward propagating noise, amplified with a parametric amplifier (JPA) in reflection, a high-electron-mobility transistor (HEMT), and room-temperature amplifiers, and then demodulated to produce two quadratures, which are finally filtered, amplified, and digitized (ADC).
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Figure 3
Spectroscopic characterization of the system. (a) Measured transition frequencies as a function of the magnetic flux $Φ$ applied through the transmon SQUID. Horizontal dashed lines correspond to the readout (top blue line) and oscillator frequencies (bottom purple line), $ν_{R}$ and $ν_{O}$ ; the red line with dots corresponds to the hybridization of $ν_{ge}$ and $ν_{R}$ , and the thick green line at the bottom corresponds to $ν_{ef}$ . The $ef$ -resonant condition is obtained at $ν_{O} = ν_{ef}$ . (b) Spectra determining the qubit-resonator coupling frequencies $g_{R, O}$ , as well as the qubit anharmonicity $α$ . Top: Spectra around $ν_{R}$ with $ν_{ge}$ either far away (central blue peak) or anticrossing $ν_{R}$ (lateral red peaks). Bottom: Spectra around $ν_{O}$ with $ν_{ge}$ either far away (central purple peak) or anticrossing $ν_{O}$ (lateral violet peaks). Middle: Spectrum at the $ef$ -resonance showing the $|g〉 \leftrightarrow |e〉$ transition (top red peak), as well as transitions $|\tilde{e 0}〉 \leftrightarrow |\pm 2〉$ (bottom magenta doublet; see also Fig. 1) at frequencies $ν_{\tilde{O}} + Δ ν_{\pm 0}$ . (c) Readout resonance when the qubit is left in $|g〉$ or excited in $|e〉$ or $|f〉$ . Note that the residual thermal population of level $e$ is well below 1%. The dashed line indicates the frequency at which the readout mode transmission $S_{21}$ is measured in (d). (d) Qubit spectra measured at the $ef$ resonance for three different fillings of $\tilde{O}$ ( $β ≃ 0.54$ , 4.5, and 10; the last two are horizontally shifted for clarity), showing $ν_{\pm n}$ transitions (denoted $\pm n$ here) from $- n = - 1$ to $n = 17$ . (e) Transition frequencies $ν_{\pm n}$ deduced (orange dots) from spectroscopy in (d), calculated in Sec. 2 (+ and $-$ symbols), and numerically computed by diagonalization of the system Hamiltonian (open circles). The dashed line corresponds to the parabolic approximation $ν_{\pm n} ≃ ν_{ge} \pm \sqrt{2 n} g_{O}$ .
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Figure 4
Generation and measurement of coherent states $|β〉$ in the oscillator mode: a state is obtained from a 0.3-μs-long rectangular pulse with frequency $ν_{\tilde{O}}$ and amplitude $A$ [expressed here in volts in the AWG of Fig. 2]; a $π$ pulse on one of the $+ n$ transition then excites conditionally the transmon, and the variation $Δ S_{21} (ν_{R}, A, n)$ of the cavity transmission is measured. (a) Occupation probability $p (n)$ of Fock state $|n〉$ for $n \in [0, 10]$ and for increasing $A$ . $p (n)$ is obtained by dividing $Δ S_{21} (ν_{R}, A, n)$ by a calibration factor $c_{n}$ resulting from a fit of the coherent model $β (A) = k A$ to the whole data set: the fitted coefficients ${c_{0}, \dots, c_{10}} = {0.764, 0.835, 0.847, 0.846, 0.833, 0.854, 0.846, 0.834, 0.847, 0.832, 0.841}$ differ by less than 3%, except for $c_{0}$ , which corresponds to a transition between transmon states not hybridized with mode $O$ . (b) $p (n)$ cuts [also shown in (a)] at $A = 0$ (solid blue squares), 0.45 (solid green dots), 0.8 (open magenta circles), and 1.3 V (open red squares), showing both measured data (dots with $\pm 2 σ$ error bars) and expected Poisson distributions (lines). Residual errors between data and fit are homogeneously distributed over the whole data set (not shown) and Gaussianly distributed with a standard deviation $σ = 0.6 %$ and a shift of 0.1% (left inset). The right inset shows the reconstruction of the Wigner function of a targeted state $|β = - \sqrt{5} + i \sqrt{2}〉$ by tomography and maximum-likelihood analysis (see text).
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Figure 5
Steady-state nonlinearities of oscillator $\tilde{O}$ at the $ef$ resonance. (a) Transmitted power (in linear units) measured with the VNA (large dots joined by dotted lines) for input powers $P_{1}$ varying from $- 135$ to $- 126 dBm$ in steps of 3 dB (bottom to top). Solid peaks are the steady-state average photon numbers $\bar{n}$ in mode $\tilde{O}$ simulated numerically (see text) for the same $P_{1}$ values (according to calibration) and for $P_{1} = - 149.6 dBm$ ( $\bar{n} = 1$ at resonance). The nonlinearity in frequency calculated with four transmon levels (dark blue dashed line with dots) is smaller than the $- 346 Hz / photon$ Kerr constant (dark blue solid line with dots). A large nonlinearity in power is, however, observed (see text). (b) Calibration of $P_{1}$ versus $\bar{n}$ . Downward red peak: Variation of the transmitted readout amplitude $|P_{2} (ν)|$ when keeping mode $\tilde{O}$ in its ground state and applying a $π$ pulse to the qubit at variable frequency $ν$ around the $+ 0$ transition at $ν_{ge}$ (dashed vertical line). Right blue curve: Same readout amplitude after the resonant qubit $π$ pulse when filling $\tilde{O}$ with a continuous tone with frequency $ν_{\tilde{O}}$ and variable input power $P_{1}$ . The average photon number $\bar{n}$ reaches 1 at $P_{1} = - 149.6 dBm$ (short-dashed lines; see text). (c) Simulated photon-number distributions $p_{\infty} (n)$ at the top of each simulated resonance of (a). The curve for $\bar{n} = 1$ has been multiplied by 0.25 for clarity.
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