quantum electrodynamics near a photonic bandgap ... · 5e ﬀective master equation in photonic...

Supplementary material

Yanbing Liu and Andrew. A. Houck

Contents1 Single photon bound state calculation 1

2 Band structure and tranfer matrix technique 2

3 Nonlinear response and comments on multiphoton bound states 2

4 Multilevel analysis of dressed-state transitions 4

5 Effective master equation in photonic crystal 4

6 Analysis of reservoir engineering 5

1 Single photon bound state calculationHere we show the derivation of the eigenenergy of the single photon bound state within the band gap.

The Hamiltonian of our system is

H =∑

k

ωka†kak + ωq

2σz +

∑k

gk(a†kσ− + akσ

+) (1)

The number of excitations N = (σz + I)/2 +∫

B.Z. a†kak is a conserved quantity in our model. So in the

single-excitation manifold, the eigenstate is in the form of |φb〉 = cos θ |0〉 |e〉+ sin θ∫

B.Z. dkcka†k |0〉 |g〉 and theeigenstate equation is H |φb〉 = ωb |φb〉. This yields

(ωb − ωa) =∫

B.Z.dk

2g2k

Eb − ωk(2)

tan2(θ) =∫

B.Z.dk

2g2k

(ωb − ωk)2 (3)

ck =gk

tan θ(ωb − ωk)(4)

To get an analytical result, we further make two assumptions: the coupling is independant of wavevectorgk ≈ g and the dispersion is quadratic ωk = ω0 + α(k − k0)2. Then extending the integral limits to infinityand performing the integrals yield the following results:

(ωq − ωb) =πg2

α√(ω0 − ωb)

(5)

tan2(θ) =ωq − ωb

2(ω0 − ωb)(6)

Thus, the photonic part of the wavefunction |φb〉 is∫

B.Z. dk gtan θ(ωb−ωk) a

†k |0〉 |g〉. By performing a fourier trans-

form ak → ax and approximating the Bloch wavefunction ψk(x)eikx with ψk0(x)eikx, we get the photonic part

1

Quantum electrodynamics near a photonic bandgap

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS3834

NATURE PHYSICS | www.nature.com/naturephysics 1

© 2016 Macmillan Publishers Limited. All rights reserved.

of the wavefunction is in the following form:∫

dxe−x/λa†x |0〉 |g〉 (7)

where λ is the penetration depth defined as λ =√α/(ω0 − ωb). The means that photonic part of this

polarition state is exponentially localized around the qubit, hence this is often called single photon boundstate.

2 Band structure and tranfer matrix techniqueWe first analyze the mode structure (Bloch wavefunction) of an infinite microwave photonic crystal (PhC).

It can be obtained by solving the following wave equations for the periodic structure. This calculation leadsto the choice of the qubit’s position, the midde of the unit cell, for maximal coupling with the second band.

ddx

V(x, t) = −l(x)ddt

I(x, t) (8)

ddx

I(x, t) = −c(x)ddt

V(x, t)

Here l(x), c(x) are the inductance and capacitance per unit length, leading to the following wave equation

∂

∂x[

vp

Zc(x)∂V(x, t)∂x

] =1

vpZc(x)∂2

∂t2 V(x, t) (9)

Here vp =1√lc

is the phase velocity and Zc(x) =√

l(x)c(x) is the characteristic impedance. Zc(x) is periodically

modulated by changing the center pin and gap widths of the coplanar waveguide. This 1D wave equationcan be solved using standard fourier transform technique assuming Vk(x) = ΣnCn(k)ei2nπx/d+ikx and 1/Zc(x) =Σmηmei2mπx/d. Then the algebraic equation can be solved numerically. The results are shown in Fig. 1.

To simulate the transmission of our device with a finite number of periods, the transfer matrix (also calledABCD matrix in microwave engineering literature) technique can be employed. The ABCD matrix of asection of coplanar waveguide with characteristic impedance Z can be written as

MZ =

[cos(ω/vp) jZsin(ω/vp)

jsin(ω/vp)/Z cos(ω/vp)

](10)

Then the ABCD matrix for one unit cell is Munit = MZhi MZlo .The ABCD matrix for an atom in the single photon regime[1] is given by:

Ma =

[1 0

− j γ(ωa)ω−ωa

1Z0

1

](11)

Here γ(ωa) is the decay constant in a normal waveguide with characteristic impedance Z0. This constant isset by the coupling strength between the qubit and the waveguide. To maximize this coupling, one capac-itor island of the transmon qubit is fabricated very close (2µm) to the center pin of the waveguide. In thesimulation shown in Fig. 2, we have assumed that γ(ωa) is constant for all ωa.

The ABCD matrix for the whole device can then be written as M = (Munit)N × Ma × (Munit)N and thetransmission coefficient can then derived. The explicit expression is too cumbersome to be presented here, soinstead we give the numerical results.

3 Nonlinear response and comments on multiphoton bound statesNeither of the previous theoretical approaches can be easily generalized to the nonlienar regime. A few

very recent theoretical studies have used variational ansatz and numerical methods to prove the existence ofmulti-photon bound states[7, 8]. We did not observe a direct signature of these multiphoton bound states in

2

−π/d 00

5

10

Wavevector

Fre

qu

en

cy (

GH

z)

6 7 8

−50

−40

−30

−20

−10

Frequency (GHz)

S1

2 (

dB

)

Wavevector

Po

siti

on

in u

nit

ce

ll

-d/2

0

d/2

Wavevector

Simulation

Data

π/d

−π/d 0 π/d −π/d 0 π/d

(b)(a)

(c) (d)

0.5

1

1.5

0

Figure S1: Band structure of the bare photonic crystal. (a) Dispersion of the PhC. (b) Measured (at 10mK)and simulated (with ABCD matrix technique) transmission amplitude in the log scale. The small discrep-ancy comes from fabrication imperfections, additional loss and slight impedance mismatch in the wholemeasurement chain. Note there are no pronounced resonances in the second band. Mode structure (Blochwavefunction) of the first (c) and second band (d). The most strongly-coupled modes, which lie near theband-gap, have wavevector k = π/d. In the middle of the unit cell, clearly the mode in the first band reachesminimum while the mode in the second band reaches maximum.

Flux Bias (V)

Freq

unen

cy (G

Hz)

−1 0 1 2

6.5

7

7.5

8

8.5

0 0.5 1−55

−50

−45

−40

−35

−30

−25

−20

−15

7.5 8 8.5−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

Frequency (GHz)

S12

(dB)

SimulationDataBand edge

Flux Bias (V)

Freq

unen

cy (G

Hz)

6.5

7

7.5

8

8.5Single photon bound state

Qubit

(a) Data (b) Simulation (c)

Figure S2: ABCD matrix simulation. (a) Measured low-power transmisson versus flux bias in log scale. (b)Simulated transmission with ABCD matrix technique in log scale. Here we have assumed γ/2π = 0.5GHz.This calculation gives qualitative agreement. But we only use the Hamiltonian approach to fit the experimen-tal data. (c) S21 at ωa/2π = 8.5GHz.

32 NATURE PHYSICS | www.nature.com/naturephysics

SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHYS3834


of the wavefunction is in the following form:∫

dxe−x/λa†x |0〉 |g〉 (7)

where λ is the penetration depth defined as λ =√α/(ω0 − ωb). The means that photonic part of this

polarition state is exponentially localized around the qubit, hence this is often called single photon boundstate.

2 Band structure and tranfer matrix techniqueWe first analyze the mode structure (Bloch wavefunction) of an infinite microwave photonic crystal (PhC).

It can be obtained by solving the following wave equations for the periodic structure. This calculation leadsto the choice of the qubit’s position, the midde of the unit cell, for maximal coupling with the second band.

ddx

V(x, t) = −l(x)ddt

I(x, t) (8)

ddx

I(x, t) = −c(x)ddt

V(x, t)

Here l(x), c(x) are the inductance and capacitance per unit length, leading to the following wave equation

∂

∂x[

vp

Zc(x)∂V(x, t)∂x

] =1

vpZc(x)∂2

∂t2 V(x, t) (9)

Here vp =1√lc

is the phase velocity and Zc(x) =√

l(x)c(x) is the characteristic impedance. Zc(x) is periodically

modulated by changing the center pin and gap widths of the coplanar waveguide. This 1D wave equationcan be solved using standard fourier transform technique assuming Vk(x) = ΣnCn(k)ei2nπx/d+ikx and 1/Zc(x) =Σmηmei2mπx/d. Then the algebraic equation can be solved numerically. The results are shown in Fig. 1.

To simulate the transmission of our device with a finite number of periods, the transfer matrix (also calledABCD matrix in microwave engineering literature) technique can be employed. The ABCD matrix of asection of coplanar waveguide with characteristic impedance Z can be written as

MZ =

[cos(ω/vp) jZsin(ω/vp)

jsin(ω/vp)/Z cos(ω/vp)

](10)

Then the ABCD matrix for one unit cell is Munit = MZhi MZlo .The ABCD matrix for an atom in the single photon regime[1] is given by:

Ma =

[1 0

− j γ(ωa)ω−ωa

1Z0

1

](11)

Here γ(ωa) is the decay constant in a normal waveguide with characteristic impedance Z0. This constant isset by the coupling strength between the qubit and the waveguide. To maximize this coupling, one capac-itor island of the transmon qubit is fabricated very close (2µm) to the center pin of the waveguide. In thesimulation shown in Fig. 2, we have assumed that γ(ωa) is constant for all ωa.

The ABCD matrix for the whole device can then be written as M = (Munit)N × Ma × (Munit)N and thetransmission coefficient can then derived. The explicit expression is too cumbersome to be presented here, soinstead we give the numerical results.

3 Nonlinear response and comments on multiphoton bound statesNeither of the previous theoretical approaches can be easily generalized to the nonlienar regime. A few

very recent theoretical studies have used variational ansatz and numerical methods to prove the existence ofmulti-photon bound states[7, 8]. We did not observe a direct signature of these multiphoton bound states in

2

−π/d 00

5

10

Wavevector

Fre

qu

en

cy (

GH

z)

6 7 8

−50

−40

−30

−20

−10

Frequency (GHz)

S1

2 (

dB

)

Wavevector

Po

siti

on

in u

nit

ce

ll

-d/2

0

d/2

Wavevector

Simulation

Data

π/d

−π/d 0 π/d −π/d 0 π/d

(b)(a)

(c) (d)

0.5

1

1.5

0

Figure S1: Band structure of the bare photonic crystal. (a) Dispersion of the PhC. (b) Measured (at 10mK)and simulated (with ABCD matrix technique) transmission amplitude in the log scale. The small discrep-ancy comes from fabrication imperfections, additional loss and slight impedance mismatch in the wholemeasurement chain. Note there are no pronounced resonances in the second band. Mode structure (Blochwavefunction) of the first (c) and second band (d). The most strongly-coupled modes, which lie near theband-gap, have wavevector k = π/d. In the middle of the unit cell, clearly the mode in the first band reachesminimum while the mode in the second band reaches maximum.

Flux Bias (V)

Freq

unen

cy (G

Hz)

−1 0 1 2

6.5

7

7.5

8

8.5

0 0.5 1−55

−50

−45

−40

−35

−30

−25

−20

−15

7.5 8 8.5−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

Frequency (GHz)

S12

(dB)

SimulationDataBand edge

Flux Bias (V)

Freq

unen

cy (G

Hz)

6.5

7

7.5

8

8.5Single photon bound state

Qubit

(a) Data (b) Simulation (c)

Figure S2: ABCD matrix simulation. (a) Measured low-power transmisson versus flux bias in log scale. (b)Simulated transmission with ABCD matrix technique in log scale. Here we have assumed γ/2π = 0.5GHz.This calculation gives qualitative agreement. But we only use the Hamiltonian approach to fit the experimen-tal data. (c) S21 at ωa/2π = 8.5GHz.

3NATURE PHYSICS | www.nature.com/naturephysics 3



our experiments. As we increase input power, the bound state assisted transmission coefficient decreases ac-companying with supersplitting. The supersplitting doublet appears due to saturation of the two-level systemand does not correspond to multiphoton transitions. We think that the absence of multiphoton resonancescan be attributed to insufficiently strong coupling and relatively large bandwidth of the single photon boundstate. The Mollow triplet structure and Autler-Townes splitting clearly shows the quantum nature of this statewithin the band-gap. Incorporation of low noise amplifiers in the future may reveal multitphoton effects incorrelation measurements.

4 Multilevel analysis of dressed-state transitionsThe Hamiltonian of the qubit subject to coherent drive has the following form,

H =∑

n=0,1...

[(ωn − nωd) |n〉〈n| + Ωn

2(|n〉〈n + 1| + h.c.)] (12)

The above Hamiltonian is written in the frame of the drive and multiple excited states are taken into ac-count. Where Ωn =

√γnE is the effective Rabi rate for the nth transition and Fermi’s golden rule dictates

that√γn is proportional to the transition matrix of transmon qubit. Note that the energy levels here have

taken into account the effect of the photonic crystal, so ω01 is really just the bound state frequency ωb. Theother transition frequencies ω12, ω23 are found empirically in the experiments, rather than using the transmonanharmonicity Ec. Hence Ωn =

√n + 1Ω0. We note that the full quantum analysis has to use input-output

theory and consider the coherent coupling between the qubit and photonic crystal modes. However, if thequbit is deep within the gap, the effect of the photonic modes can be described perturbatively with the decayrate γn. Diagonizing the matrix involving a few energy levels yields the complete dressed states. We considertwo experimentally relevant cases where ωd = ω1 − ω0 and ωd = ω2 − ω1. These dressed states are fitted tothe experimental data. The only fitting parameter is Ω0. The dressed state transition shown in Figure 3 in themain text thus correspond to |a〉 ↔ |b〉 for ωd = ω01. and for ωd = ω12.

5 Effective master equation in photonic crystalTo understand resonance fluorescenc in the band-gap medium, we first start from the complete Hamilto-

nian including both the coherently-driven qubit and the modes of the photonic crystal.

H =∆a

2σz +

Ω0

2(σ+ + σ−) +

∑k

∆ka†kak +∑

k


+) (13)

Where ∆a = ωa − ωd,∆k = ωk − ωd. We first go to the dressed state of the qubit, defining |0〉 = cos(θ) |0〉 −sin(θ) |1〉 , |1〉 = sin(θ) |0〉 + cos(θ) |1〉, where cos2(θ) = 1

2 +∆a

2√Ω2

0+∆2a.

H =Ω

2σz +

∑k

∆ka†kak +∑

k

gk(cos2(θ)a†kσ− − sin2(θ)a†kσ

+ + sin(θ) cos(θ)a†kσz) (14)

The Rabi rate is given by Ω =√Ω2

0 + ∆2a. We work in the rotating frame, applying a unitary transformation

U = exp[i(Ωσz/2 + i∑

k ∆ka†kak)t]:

H(t) =∑

k

gk(cos2(θ)a†kσ−ei(∆k−Ω)t − sin2(θ)a†kσ

+ei(∆k+Ω)t + sin(θ) cos(θ)ei∆kta†kσz) (15)

From the above Hamiltonian, we can follow the standard procedure involving Born-Markov approximationand rotating wave approximation, deriving the reduced master equation for the qubit. But before we do that,let us analyze it to get some physical intuition. If we assume that the upper sideband is close to the bandedge ω0 ≈ ωd + Ω, then the effective interaction strength will be gk cos2(θ), and the coupling of the sideband

4

will be effectively smaller than the direct coupling between the qubit and the photonic modes. The reduceddensity matrix of the qubit is govened by the following equation,

dρdt= −∫ t

0dτTrR[H(t), [H(τ), ρ(τ) ⊗ ρR(τ)]] (16)

We assume ρ(τ)→ ρ(t) and ρR(τ)→ ρR(0)(Born-Markov approximation), then we can get

2∂ρ

∂t= γo sin2(θ) cos2(θ)(σzρσz − σzσzρ) + γ− sin4(θ)(σ+ρσ− − σ−σ+ρ) (17)

+γ+ cos4(θ)(σ−ρσ+ − σ+σ−ρ) + h.c.

Where γ0 = 2π∑

k g2kδ(ωk − ωL), γ− = 2π

∑k g2

kδ(ωk − ωL + Ω) and γ+ = 2π∑

k g2kδ(ωk − ωL − Ω). It is easy

then to get the steady state of the master equation and get

〈1| ρ |1〉 = γ− sin4(θ)γ− sin4(θ) + γ+ cos4(θ)

(18)

〈0| ρ |0〉 = γ+ cos4(θ)γ− sin4(θ) + γ+ cos4(θ)

(19)

If the dephasing is also included, then the above equations shall be modified to be

〈1| ρ |1〉 =γ− sin4(θ) + γp sin2(2θ)

γ− sin4(θ) + γ+ cos4(θ) + 2γp sin2(2θ)(20)

〈0| ρ |0〉 =γ+ cos4(θ) + γp sin2(2θ)


In the case of resonant driving θ = π/4, the target dressed state is simply 1√2(|0〉 − |1〉). The results can be

generalized to many atoms[6].

6 Analysis of reservoir engineeringSince we could not directly perform qubit state tomography, we employ two methods to simulate the

state purity ρ−− in the reservoir engineering scheme, relying on two different sets of experimental data. Bothresults are presented in the main text and provide consistent estimates of state fidelity.

The first approach (named A in the main text) takes advantage of the linewidth data to estimate γ±. It isapparent that near the band edge, the linewidth within the band reaches a maximum due to strong coupling tothe electric field as shown in Fig. 1(d) and high density of states DOS ∼ 1√

ω−ω0. In the perturbative regime, the

lifetime of the qubit in the gap can be several orders of magnitude smaller than that in the band. In the strongcoupling regime (which is our case here), the difference is smaller due to the formation of the leaky photonbound state. To account for this effect, we added an additional phenomelogical dissipation γ0/2π = 25MHzobserved in our experiment to the perturbative formula 2πRe[Y(ω)]

CΣ[4], where Y(ω) is the input admittance

from the port of the qubit and CΣ is the total capacitance of the qubit. The final simulated linewidth (Fig. S3)agrees with the experimental data well both in the gap and in the band. In essense, γ(ω) in the gap is still anorder of magnitude smaller than γ(ω) in the band. We will use the simulated linewidth to represent γ(ω). Theprevious section shows that the formula ρ−− = 1 − γ−

γ++γ−≈ 1 − γ−

γ+gives the dressed state purity in the ideal

two-level system. In the presence of higher transmon level, leakage to the other dressed state will decreasethe purity of the target state. Nevertheless, all the other dressed state transitions fall within the band gap. Thismeans that their effects will be at most of the order of γ/γ+ where γ ≈ γ− is the decay rate in the gap. Basedon this understanding, we approximate the state purity as

ρ−− ≈ 1 − γ− + γγ+

≈ 1 − 2γ−/γ+ (22)




our experiments. As we increase input power, the bound state assisted transmission coefficient decreases ac-companying with supersplitting. The supersplitting doublet appears due to saturation of the two-level systemand does not correspond to multiphoton transitions. We think that the absence of multiphoton resonancescan be attributed to insufficiently strong coupling and relatively large bandwidth of the single photon boundstate. The Mollow triplet structure and Autler-Townes splitting clearly shows the quantum nature of this statewithin the band-gap. Incorporation of low noise amplifiers in the future may reveal multitphoton effects incorrelation measurements.

4 Multilevel analysis of dressed-state transitionsThe Hamiltonian of the qubit subject to coherent drive has the following form,

H =∑

n=0,1...

[(ωn − nωd) |n〉〈n| + Ωn

2(|n〉〈n + 1| + h.c.)] (12)

The above Hamiltonian is written in the frame of the drive and multiple excited states are taken into ac-count. Where Ωn =

√γnE is the effective Rabi rate for the nth transition and Fermi’s golden rule dictates

that√γn is proportional to the transition matrix of transmon qubit. Note that the energy levels here have

taken into account the effect of the photonic crystal, so ω01 is really just the bound state frequency ωb. Theother transition frequencies ω12, ω23 are found empirically in the experiments, rather than using the transmonanharmonicity Ec. Hence Ωn =

√n + 1Ω0. We note that the full quantum analysis has to use input-output

theory and consider the coherent coupling between the qubit and photonic crystal modes. However, if thequbit is deep within the gap, the effect of the photonic modes can be described perturbatively with the decayrate γn. Diagonizing the matrix involving a few energy levels yields the complete dressed states. We considertwo experimentally relevant cases where ωd = ω1 − ω0 and ωd = ω2 − ω1. These dressed states are fitted tothe experimental data. The only fitting parameter is Ω0. The dressed state transition shown in Figure 3 in themain text thus correspond to |a〉 ↔ |b〉 for ωd = ω01. and for ωd = ω12.

5 Effective master equation in photonic crystalTo understand resonance fluorescenc in the band-gap medium, we first start from the complete Hamilto-

nian including both the coherently-driven qubit and the modes of the photonic crystal.

H =∆a

2σz +

Ω0

2(σ+ + σ−) +

∑k

∆ka†kak +∑

k


+) (13)

Where ∆a = ωa − ωd,∆k = ωk − ωd. We first go to the dressed state of the qubit, defining |0〉 = cos(θ) |0〉 −sin(θ) |1〉 , |1〉 = sin(θ) |0〉 + cos(θ) |1〉, where cos2(θ) = 1

2 +∆a

2√Ω2

0+∆2a.

H =Ω

2σz +

∑k

∆ka†kak +∑

k

gk(cos2(θ)a†kσ− − sin2(θ)a†kσ

+ + sin(θ) cos(θ)a†kσz) (14)

The Rabi rate is given by Ω =√Ω2

0 + ∆2a. We work in the rotating frame, applying a unitary transformation

U = exp[i(Ωσz/2 + i∑

k ∆ka†kak)t]:

H(t) =∑

k

gk(cos2(θ)a†kσ−ei(∆k−Ω)t − sin2(θ)a†kσ

+ei(∆k+Ω)t + sin(θ) cos(θ)ei∆kta†kσz) (15)

From the above Hamiltonian, we can follow the standard procedure involving Born-Markov approximationand rotating wave approximation, deriving the reduced master equation for the qubit. But before we do that,let us analyze it to get some physical intuition. If we assume that the upper sideband is close to the bandedge ω0 ≈ ωd + Ω, then the effective interaction strength will be gk cos2(θ), and the coupling of the sideband

4

will be effectively smaller than the direct coupling between the qubit and the photonic modes. The reduceddensity matrix of the qubit is govened by the following equation,

dρdt= −∫ t

0dτTrR[H(t), [H(τ), ρ(τ) ⊗ ρR(τ)]] (16)

We assume ρ(τ)→ ρ(t) and ρR(τ)→ ρR(0)(Born-Markov approximation), then we can get

2∂ρ

∂t= γo sin2(θ) cos2(θ)(σzρσz − σzσzρ) + γ− sin4(θ)(σ+ρσ− − σ−σ+ρ) (17)

+γ+ cos4(θ)(σ−ρσ+ − σ+σ−ρ) + h.c.

Where γ0 = 2π∑

k g2kδ(ωk − ωL), γ− = 2π

∑k g2

kδ(ωk − ωL + Ω) and γ+ = 2π∑

k g2kδ(ωk − ωL − Ω). It is easy

then to get the steady state of the master equation and get

〈1| ρ |1〉 = γ− sin4(θ)γ− sin4(θ) + γ+ cos4(θ)

(18)

〈0| ρ |0〉 = γ+ cos4(θ)γ− sin4(θ) + γ+ cos4(θ)

(19)

If the dephasing is also included, then the above equations shall be modified to be

〈1| ρ |1〉 =γ− sin4(θ) + γp sin2(2θ)


〈0| ρ |0〉 =γ+ cos4(θ) + γp sin2(2θ)


In the case of resonant driving θ = π/4, the target dressed state is simply 1√2(|0〉 − |1〉). The results can be

generalized to many atoms[6].

6 Analysis of reservoir engineeringSince we could not directly perform qubit state tomography, we employ two methods to simulate the

state purity ρ−− in the reservoir engineering scheme, relying on two different sets of experimental data. Bothresults are presented in the main text and provide consistent estimates of state fidelity.

The first approach (named A in the main text) takes advantage of the linewidth data to estimate γ±. It isapparent that near the band edge, the linewidth within the band reaches a maximum due to strong coupling tothe electric field as shown in Fig. 1(d) and high density of states DOS ∼ 1√

ω−ω0. In the perturbative regime, the

lifetime of the qubit in the gap can be several orders of magnitude smaller than that in the band. In the strongcoupling regime (which is our case here), the difference is smaller due to the formation of the leaky photonbound state. To account for this effect, we added an additional phenomelogical dissipation γ0/2π = 25MHzobserved in our experiment to the perturbative formula 2πRe[Y(ω)]

CΣ[4], where Y(ω) is the input admittance

from the port of the qubit and CΣ is the total capacitance of the qubit. The final simulated linewidth (Fig. S3)agrees with the experimental data well both in the gap and in the band. In essense, γ(ω) in the gap is still anorder of magnitude smaller than γ(ω) in the band. We will use the simulated linewidth to represent γ(ω). Theprevious section shows that the formula ρ−− = 1 − γ−

γ++γ−≈ 1 − γ−

γ+gives the dressed state purity in the ideal

two-level system. In the presence of higher transmon level, leakage to the other dressed state will decreasethe purity of the target state. Nevertheless, all the other dressed state transitions fall within the band gap. Thismeans that their effects will be at most of the order of γ/γ+ where γ ≈ γ− is the decay rate in the gap. Basedon this understanding, we approximate the state purity as

ρ−− ≈ 1 − γ− + γγ+

≈ 1 − 2γ−/γ+ (22)




6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 8.2 8.4

101

102

Bound state linewidthTransmission dip widthBand edgePerturbative calculation

Line

wid

th (M

Hz)

Frequency (GHz)

Photonic Gap Photonic Band

Sidebands

Figure S3: Linewidths of the bound state and the qubit transmission dip. The errorbar represents the standarddeviation of the Lorenzian fit. This perturbative calculation can be seen as the first order approximation tothe real linewidth quantities in the strong coupling regime. The dressed state cooling is most effective in theshaded regions where γ+ is an order of magnitude larger than γ−.

7.8 7.85 7.9 7.95 8

25

30

35

40

45

50

55

60

Tran

smis

sion

line

wid

th (M

Hz)

Frequency (GHz)7.8 7.85 7.9 7.95 8

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Tran

smis

sion

am

plitu

de

Frequency (GHz)

(a) (b)

Figure S4: Further analysis of the pump-probe data. (a) The resonance linewidth of the upper sideband. Theerrorbar represents the standard deviation of the Lorenzian fit. (b) The attenutation of the probe signal. Thedata shows that these two quantities are approximately inversely proportional to each other, which agreeswith the theoretical model discussed in this section.

6

In the reservoir engineering experiment, the upper (lower) sideband is from 7.72GHz (7.16GHz) to 8GHz(6.86GHz) as shown in shaded regions in Fig. 3. The maximum γ+/2π is approximately 250MHz and thecorresponding γ−/2π is about 25MHz. The above equation yields maximum ρ−− ≈ 80%.

The second approach (named B in the main text) uses the pump-probe data of the upper sideband in thephotonic band to infer ρ−−. In Fig. 4, we have extracted the sideband linewidth and amplitude by fittingthe dip to a Lorenzian. We first note that they are inversely proportional to each other. For instance, nearthe band edge ∼ 7.75GHz, γ reaches a maximum and the transmission amplitude reaches minimum. Thisactually means that the dressed state cooling is most effective here. In fact, Eq. (4) in the main text directlyrelates transmission amplitude to the dressed state inversion [5]. The exact expression for η depends onspecific dressed states and local decay rates. In the strong driving limit Ωp γ we regard it as a constant forsimplicity. Ignoring other dressed states and using the sum rule ρ++ + ρ−− = 1, we get

ρ−− =12

(1 +1 − tqη

) ≥ 1 − tq/2 (23)

The above inequality means that we can use the transmission attenuation in Fig. 4(b) to get a lower boundof ρ−−, within the above assumptions. This method yields maximum ρ−− ≈ 86%. We quote the maximumfidelity (83%) in the main text as the average of these two methods.

Finally, we remark that pure dephasing in superconducting qubits can be greatly minimized and only re-laxation has to be taken into account. Additionally, the γ+/γ− ratio could easily be made orders of magnitudelarger by changing the coupling strength gk and the number of unit cells in the photonic crystal. In the future,it is very promising that complex many-body states can be prepared with high fidelity.

References[1] J. T. Shen and S. Fan, Coherent Single-Photon Transport in a One-Dimensional Waveguide Coupled with

Superconducting Quantum Bits, Phys. Rev. Lett. 95, 213001 (2005).

[2] Murch, K. W. et al. Cavity-assisted quantum bath engineering, Phys. Rev. Lett. 109, 183602 (2012).

[3] M. D. Reed, B. R. Johnson, A. A. Houck, L. Dicarlo, J. M. Chow, D. I. Schuster, L. Frunzio andR. J. Schoelkpf, Fast reset and suppressing spontaneous emission of a superconducting qubit, Appl.Phys. Lett. 96, 203110 (2010).

[4] Bronn, N. T. et al. Broadband filters for abatement of spontaneous emission for superconducting qubits,Appl. Phys. Lett. 107, 172601 (2015).

[5] Koshino, K. et al. Observation of the Three-State Dressed States in Circuit Quantum Electrodynamics,Phys. Rev. Lett. 110, 263601 (2013).

[6] Sajeev John and Tran Quang, Collective Switching and Inversion without Fluctuation of Two-LevelAtoms in Confined Photonic systems, Phys. Rev. Lett. 78, 1888 (1997).

[7] G. Calajo, F. Ciccarello, D. Chang and P. Rabl, Atom-field dressed states in slow-light waveguide QED,Phys. Rev. A 93, 033833 (2016)

[8] Tao Shi, Ying-Hai Wu, A. Gonzalez-Tudela, J. I. Cirac, Bound states in boson impurity models,arXiv:1512.07238.




quantum electrodynamics near a photonic bandgap ... · 5e ﬀective master equation in photonic...

Documents