Preparation of maximally-entangled states with multiple cat-state qutrits in circuit QED

doi:10.1007/s11467-023-1357-4

Front. Phys.

2024, Vol. 19

Issue (3) : 31201 https://doi.org/10.1007/s11467-023-1357-4

RESEARCH ARTICLE

Preparation of maximally-entangled states with multiple cat-state qutrits in circuit QED

Chui-Ping Yang¹, Jia-Heng Ni¹, Liang Bin¹, Yu Zhang², Yang Yu², Qi-Ping Su^1,³(

)

¹. Department of Physics, Hangzhou Normal University, Hangzhou 311121, China
². School of Physics, Nanjing University, Nanjing 210093, China
³. Institute for Quantum Science and Technology, University of Calgary, Alberta T2N 1N4, Canada

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Abstract

In recent years, cat-state encoding and high-dimensional entanglement have attracted much attention. However, previous works are limited to generation of entangled states of cat-state qubits (two-dimensional entanglement with cat-state encoding), while how to prepare entangled states of cat-state qutrits or qudits (high-dimensional entanglement with cat-state encoding) has not been investigated. We here propose to generate a maximally-entangled state of multiple cat-state qutrits (three-dimensional entanglement by cat-state encoding) in circuit QED. The entangled state is prepared with multiple microwave cavities coupled to a superconducting flux ququart (a four-level quantum system). This proposal operates essentially by the cavity-qutrit dispersive interaction. The circuit hardware resource is minimized because only a coupler ququart is employed. The higher intermediate level of the ququart is occupied only for a short time, thereby decoherence from this level is greatly suppressed during the state preparation. Remarkably, the state preparation time does not depend on the number of the qutrits, thus it does not increase with the number of the qutrits. As an example, our numerical simulations demonstrate that, with the present circuit QED technology, the high-fidelity preparation is feasible for a maximally-entangled state of two cat-state qutrits. Furthermore, we numerically analyze the effect of the inter-cavity crosstalk on the scalability of this proposal. This proposal is universal and can be extended to accomplish the same task with multiple microwave or optical cavities coupled to a natural or artificial four-level atom.

Keywords maximally-entangled states cat state qutrit circuit QED

Corresponding Author(s): Qi-Ping Su

Issue Date: 04 December 2023

Cite this article:

Chui-Ping Yang,Jia-Heng Ni,Liang Bin, et al. Preparation of maximally-entangled states with multiple cat-state qutrits in circuit QED[J]. Front. Phys. , 2024, 19(3): 31201.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1357-4
https://academic.hep.com.cn/fop/EN/Y2024/V19/I3/31201

Fig.1 (a) Schematic diagram of

n

microwave cavities coupled to a flux ququart. Each square represents a one-dimensional (1D) or three-dimensional (3D) cavity. The circle

A

in the middle represents the flux ququart, which is capacitively or inductively coupled to each cavity. (b) Cavity

j

(

j = 1, 2, ?, n

) is dispersively coupled to the

| f ? ↔ | h ?

transition with coupling constant

g j

and detuning

Δ j

, while highly detuned (decoupled) from the transitions between any other two levels. It is noted that the transition between the two lowest levels

| g ?

and

| e ?

can be made weak by increasing the barrier between the two potential wells.

Fig.2 Setup for two 1D microwave cavities and a SC flux ququart (the circle

A

in the middle). Each cavity is a transmission line resonator. The flux ququart consists of three Josephson junctions and a superconducting loop, which is linked to each cavity via a capacitor.

Fig.3 Illustration of the unwanted coupling between cavity

j

(

j = 1, 2

) and the

| e ? ↔ | h ?

transition with coupling constant

g j ′

and detuning

Δ j ′

. Illustration of cavity

j

(

j = 1, 2

) and the

| g ? ↔ | f ?

transition with coupling constant

g j ′ ′

and detuning

Δ j ′ ′

. Illustration of cavity

j

(

j = 1, 2

) and the

| e ? ↔ | f ?

transition with coupling constant

g j ′ ′ ′

and detuning

Δ j ′ ′ ′

. Red lines correspond to cavity 1, while green lines correspond to cavity 2.

$ω e g / (2 π) = 3 G H z$	$ω f g / (2 π) = 11 G H z$	$ω h e / (2 π) = 17 G H z$
$ω h f / (2 π) = 9 G H z$	$ω h g / (2 π) = 20 G H z$	$ω c 1 / (2 π) = 7.5 G H z$
$ω c 2 / (2 π) = 6.5 G H z$	$△ 1 / (2 π) = 1.5 G H z$	$△ 1 ′ / (2 π) = 9.5 G H z$
$△ 1 ′ ′ / (2 π) = 3.5 G H z$	$△ 1 ′ ′ ′ / (2 π) = 0.5 G H z$	$△ 2 / (2 π) = 2.5 G H z$
$△ 2 ′ / (2 π) = 10.5 G H z$	$△ 2 ′ ′ / (2 π) = 4.5 G H z$	$△ 2 ′ ′ ′ / (2 π) = 1.5 G H z$
$△ 12 / (2 π) = 1.0 G H z$	$g 1 / (2 π) = 75 M H z$	$g 1 ′ / (2 π) = 75 M H z$
$g 1 ′ ′ / (2 π) = 53.04 M H z$	$g 1 ′ ′ ′ / (2 π) = 7.5 M H z$	$g 2 / (2 π) = 96.82 M H z$
$g 2 ′ / (2 π) = 96.82 M H z$	$g 2 ′ ′ / (2 π) = 68.47 M H z$	$g 2 ′ ′ ′ / (2 π) = 9.682 M H z$

Tab.1 Parameters used in the numerical simulation. For the definitions of the parameters, please refer to the text.

Fig.4 Fidelity versus

κ − 1

for

g 12 / g m a x = 0, 0.01, 0.05

. Here,

κ

is the cavity decay rate, and

g 12

is the inter-cavity crosstalk strength between cavities 1 and 2. The parameters used in the numerical simulation are referred to the text and Tab.1.

Fig.5 Fidelity versus

g c r

for three cavities (

n = 3

), four cavities (

n = 4

), and five cavities (

n = 5

). Here, assume that the crosstalk strength

g c r

between any two cavities is identical. The system dissipation and the unwanted cavity-ququart couplings are not considered in the numerical simulation.

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