Quantum simulation of the Rabi model in a trapped ionsystem

Dingshun Lv1, Shuoming An1,Zhenyu Liu1, Jing-Ning Zhang1

Julen S. Pedernales2, Lucas Lamata2, Enrique Solano2,3, Kihwan Kim1

1CQI, IIIS, Tsinghua University2Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado

644, 48080 Bilbao, Spain3IKERBASQUE, Basque Foundation for Science, Alameda Urquijo 36, 48011 Bilbao, Spain

lds12@mails.tsinghua.edu.cn

September 7, 2017

(CQI, IIIS, Tsinghua University, Beijing) Kihwan Kim’s group September 7, 2017 1 / 15

Outline

1 Background of the Rabi model

2 Experimental setup and proposal to simulate QRM with trapped ionsystem

3 Experimental results

(CQI, IIIS, Tsinghua University, Beijing) Kihwan Kim’s group September 7, 2017 2 / 15

Background of the Rabi model:Why quantum Rabi model?

• Originally, Rabi model was proposed in 1936 to deal with the effect of a varying, weak magnetic field on an oriented atom possessing nuclear.

Rabi I I. On the process of space quantization[J]. Physical Review, 1936, 49(4): 324.

• With full quantized field, we get the quantum version of Rabi model, describe as

𝐻 𝑄𝑅𝑀 =1

2ℏ𝜔 𝜎𝑧 + ℏ𝜔𝑚 𝑎† 𝑎 + ℏ𝑔 𝜎+ + 𝜎− 𝑎† + 𝑎

it can extend to Jaynes-Cummings model via the rotate wave

approximation(RWA)(g ≪ 𝜔𝑚)

𝐻 𝐽𝐶𝑀 =1

2ℏ𝜔 𝜎𝑧 + ℏ𝜔𝑚 𝑎† 𝑎 + ℏg 𝑎 𝜎+ + 𝑎† 𝜎−

• Recently an analytic solution has been proposaled that cover all the coupling regime and experiments has push into the USC/DSC regime, where RWA, JCM is not applicable, and full QRM is required. (USC: 0.1 ≤g/𝜔𝑚, DSC: 1 ≤g/𝜔𝑚)

Braak D. Integrability of the Rabi model[J]. Physical Review Letters, 2011, 107(10):

(CQI, IIIS, Tsinghua University, Beijing) Kihwan Kim’s group September 7, 2017 3 / 15

Recent achievements in USC/DSC regime

Crespi et al. "Photonic realization of the quantum Rabi model." Physical review letters 108.16 (2012):

163601.

Langford, N. K., et al. "Experimentally simulating the dynamics of quantum light and matter at ultrastrongcoupling." arXiv preprint arXiv:1610.10065 (2016).

Yoshihara, Fumiki, et al. "Superconducting qubit-oscillator circuit beyond the ultrastrong-coupling

regime." Nature Physics (2016).

Forn-Díaz, P., et al. "Ultrastrong coupling of a single artificial atom to an electromagnetic continuum in the

nonperturbative regime." Nature Physics 13.1 (2017): 39-43.

(CQI, IIIS, Tsinghua University, Beijing) Kihwan Kim’s group September 7, 2017 4 / 15

Experimental setup and proposal to simulate QRM withtrapped ion system

𝛿𝑏

𝛿𝑟

|↑, n〉

|↑, n +1〉

|↑, n -1〉

|↓, n〉

|↓, n +1〉

|↓, n -1〉

Bichromatic lasers

Raman1

F=1

F=0

ωHF

171Yb+

2S1/2

2P1/2

Pedernales J S, Lizuain I, Felicetti S, et al. Quantum Rabi model with trapped ions[J]. Scientific reports, 2015, 5.

State initialization to: |↓, 0〉

State detection: florensence collection

|↑〉: photons collected

|↓〉: no photons collected

(CQI, IIIS, Tsinghua University, Beijing) Kihwan Kim’s group September 7, 2017 5 / 15

Ion-light interaction

The total Hamiltonian of the ion-light matter interaction include threeparts, written as

H = Hm + He + Hi (1)

Here Hm of the motional model X can be written as

Hm = ~ωX (a†a +1

2) (2)

He describes the internal electronic level structure of spin down|↓〉 andspin up |↑〉, written as

He = ~ωHF

2σZ (3)

the ion-light interaction part

Hi = ~Ω

2(σ+ + σ−)(e i(

~k~x−ωt+φ) + e−i(~k~x−ωt+φ)), (4)

(CQI, IIIS, Tsinghua University, Beijing) Kihwan Kim’s group September 7, 2017 6 / 15

Ion-light interaction

In the interaction picture H0 = Hm + He , ∆ = ω − ωHF

If ∆ = 0, we get the carrier transition Hamiltonian,

Hcar (φ) =~Ω

2(σ+e

−iφ + σ−eiφ). (5)

If ∆ = −ωX , we get the red sideband transition Hamiltonian or theJaynes-Cummings(JC) coupling,

HJC (φ) =i~ηΩ

2(σ−a

†e iφ − σ+ae−iφ), (6)

If ∆ = ωX , we get the blue sideband transition Hamiltonian or theanti-Jaynes-Cummings(aJC)coupling

HaJC (φ) =i~ηΩ

2(σ+a

†e−iφ − σ−ae iφ). (7)

(CQI, IIIS, Tsinghua University, Beijing) Kihwan Kim’s group September 7, 2017 7 / 15

Simulate Quantum Rabi model

Remind that the Quantum Rabi Model(QRM) is written as

HR =1

2~ωσz + ~ωma

†a + ~g σx(a + a†) (8)

In our system, if we employ a new transformation

Hv = V †HV + i~dV †

dt V under V = exp(−i H0v t/~) where

H0v = 12~ωv σz + ~νv a†a is the free energy of the virtual trapped ion

system. The resulting Hamiltonian in the interaction picture reads

Hv =1

2~ωv σz + ~νv a†a

+ ~Ω[e iψe i(ωl−ω+δω)t σ− + e−iψe−i(ωl−ω+δω)t σ+]

− i~Ωη

2[ae−i(ν−δν)t + a†e i(ν−δν)t ]

× [e iψe i(ωl−ω+δω)t σ− + e−iψe−i(ωl−ω+δω)t σ+]

Here, the virtual detuning parameters are detuned as δω = ω − ωv

and δν = ν − νv . For simplicity, we choose ψ = π2 from now on.

(CQI, IIIS, Tsinghua University, Beijing) Kihwan Kim’s group September 7, 2017 8 / 15

For ωl = (ω − δω) + (ν − δνv ), we again get the anti-JCM-typeHamiltonian

H1 =1

2~δωσz + ~δνa†a + ~

Ωη

2(aσ− + a†σ+) (9)

For ωl = (ω − δω)− (ν − δνv ), we again get the JCM-type Hamiltonian

H2 =1

2~δωσz + ~δνa†a + ~

Ωη

2(aσ+ + a†σ−) (10)

Now we come to the time-evolution of a quantum state under the sum

H1 + H2 = ~δωσz + 2~δνa†a + ~Ωη

2σx(a + a†) (11)

The striking similarity between the QRM Hamiltonian 8 and Eq. 11 leadsto the interpretation of s pesudo effective atomic and trap frequency givenby ω = 2δω, ωm = 2δν and g = ηΩ

2

(CQI, IIIS, Tsinghua University, Beijing) Kihwan Kim’s group September 7, 2017 9 / 15

Spin and motion evolution at different coupling regime

0 100 200 300

0.

0.5

1.

0 100 200 3000.

0.5

1.

0 100 200 3000.

0.5

1.

0 100 200 3000.

0.5

1.

0 100 200 300

0.

1.

2.

0 100 200 3000.

2.

4.

6.

8.

0 100 200 3000.

2.

4.

6.

8.

a b c

d e f

g

time(μs)

‹n› ‹N›=p(↑)+‹n›p(↑)

g/ωm=0.04 g/ωm=0.6 g/ωm=1.2

(CQI, IIIS, Tsinghua University, Beijing) Kihwan Kim’s group September 7, 2017 10 / 15

Phonon bounce back and forth in the DSC regime for thedegenerate ω0 = 0 and non-degenerate case ω0 6= 0 case

0 2 4 6 8 100.

0.5

1.

0 2 4 6 8 100.

0.5

1.

0 2 4 6 8 100.

0.5

1.

0 2 4 6 8 100.

0.5

1.

0 2 4 6 8 100.

0.5

1.

0 2 4 6 8 100.

0.5

1.

n

0 100 200 3000.

0.5

1.

0 100 200 3000.

0.5

1.

a

c

p(n)

p(↑)

p(n)

p(n=0)

b

time(μs)

t=0μs t=10μs t=30μs

t=100μst=70μst=50μs

(CQI, IIIS, Tsinghua University, Beijing) Kihwan Kim’s group September 7, 2017 11 / 15

Adiabatically prepare the ground state in the DSC regime

n

a

p(↑)

,pur

ity

b

time(μs)

0 100 200 300 400 500 6000.

0.5

1.Ttarget Trevival

time(μs)

0 2 4 6 8 10 120.4

0.2

0.

0.2

0.4

0 2 4 6 80.6

0.4

0.2

0.

0.2

0.4

0.6

0 2 4 6 80.6

0.4

0.2

0.

0.2

0.4

0.6

0.8

p(↓,n)

p(↑,n)

c d

e

Adiabaticity=‹n|∂H/∂t|m›/ΔEnm

ωm

g

0 100 200 3000.0

0.0125

0.05

0.1

0.0

0.005

0.01

ωm

,g(MHz)

Adi

abat

icity

Fidelity=0.89(0.02) Fidelity=0.87(0.03) Fidelity=0.79(0.03)

g/ωm=1.2 g/ωm=1.5 g/ωm=2.0

(CQI, IIIS, Tsinghua University, Beijing) Kihwan Kim’s group September 7, 2017 12 / 15

Energy spectrum

Add modulate field that break the parity

Hmodulate = HQRM + ggg ∗ ~Ωσx (12)

Hmodulate = HQRM + ggg ∗ ~Ωη(a + a†) (13)

0 20 40 60 80 1000.

0.2

0.4

0.6

0 20 400.

0.2

0.4

0.6

0. 0.5 1.0

1

2

Probe frequency(kHz) g/ωm

0

1

p(↑

)p

(↑)

b

ca

(CQI, IIIS, Tsinghua University, Beijing) Kihwan Kim’s group September 7, 2017 13 / 15

Group photos

(CQI, IIIS, Tsinghua University, Beijing) Kihwan Kim’s group September 7, 2017 14 / 15

Thanks for your attention!

(CQI, IIIS, Tsinghua University, Beijing) Kihwan Kim’s group September 7, 2017 15 / 15