Supplementary online material for the paper Ancillary qubit spectroscopy of cavity (circuit) QED vacua Jared Lolli1 , Alexandre Baksic1 , David Nagy1 , Vladimir E. Manucharyan2 , Cristiano Ciuti1 1
Laboratoire Mat´eriaux et Ph´enom`enes Quantiques, Universit´e Paris Diderot-Paris 7 and CNRS,
Bˆ atiment Condorcet, 10 rue Alice Domon et L´eonie Duquet, 75205 Paris Cedex 13, France and 2
Department of Physics, University of Maryland, College Park, MD 20742.
(March 3, 2015) In this Supplementary Material, we consider the effect of a finite temperature bath on the ancillary qubit spectroscopy. All the results shown in the Letter have been obtained considering reservoirs at zero temperature, because the focus is the physics of the ground state. For cavity QED systems with optical transitions and cryogenic conditions, experiments can easily fulfill the condition kB T ≪ ~ωC , which is equivalent to the zero-temperature case. For circuit QED systems with microwave resonators and dilution fridge cryogenic conditions, the thermal energy can be a fraction of the photon energy: to give an example, a temperature of 50 mK and a resonator transition frequency ωC /(2π) = 5 GHz corresponds kB T /(~ωc ) ≃ 0.21. We show that the in this range of temperatures, the Lamb-shift of the ancilla qubit is still resolved and the finite temperature produces a moderate thermal broadening.
MASTER EQUATION FOR FINITE TEMPERATURE
In order to include the effect of temperature, we need to consider the following master equation[1]: γc γ0 γM ′ ˆ ˆ ρ˙ = −i[H(t), ρ] + D ′ [Uˆc (ω)] + D ′ [Uˆ0 (ω)] + D [UM (ω)] 2 2 2 where the dissipative term D ′ are defined in the following energy conserving form D ′ [Uˆi (ω)] =
n o X J(ω) (1 + N(ω)) 2Uˆi (ω)ρUˆi† (ω) − ρUˆi† (ω)Uˆi (ω) − Uˆi† (ω)Uˆi (ω)ρ J(ωi ) ω≥0
(1)
2 n o X J(ω) † † † ˆ ˆ ˆ ˆ ˆ ˆ N(ω) 2Ui (ω)ρUi (ω) − ρUi (ω)Ui (ω) − Ui (ω)Ui (ω)ρ . + J(ωi ) ω≥0 where N(ω) is the bosonic thermal distribution and the jumps operator Uˆi (ω) are defined as Uˆi (ω) =
X
δω,ǫl′ −ǫl hl|Aˆi |l′ i|lihl′|.
ll′
The operators Aˆi are those involved in the coupling to the reservoir, namely Aˆc = a ˆ† + a ˆ for (M ) the bosonic mode, Aˆ0 = Jˆx for the two-level systems and AˆM = σ ˆx for the ancilla. The spectral function J(ω) depends on the density-of-states of the reservoir excitations. When the bath is a 3D electromagnetic field, we have J(ω) ∝ ω 3, hence it vanishes while ω → 0. An ohmic reservoir scales instead as J(ω) ∝ ω. In Fig. 1 , we show the ancillary transition spectrum as a function of the coupling λ for different values of temperature, namely kB T /~ωc = 0, 0.105, 0.21 and 0.42 (from bottom to top) for the Dicke (left panels) and Tavis-Cummings model (right panel). It is apparent that the Lamb shifts are still well measurable and that the main effects is a moderate broadening of the Lamb-shifted ancillary qubit resonances. We have also checked that the behavior at low frequency does not affect the results in a significant way. In Fig. 2, we show a typical ancilla spectrum with three kind of reservoirs with J(ω) ∝ ω α and α = 1 (ohmic), α = 2 and α = 3. It is apparent that the differences are negligible, indeed the broadening is dominated by the spectra dependence around the ancilla qubit transition. We have also tested the robustness of the ancillary spectroscopy under the effect of noise mechanisms different from flip errors. More precisely we considered the effect produced by (M ) jump operator σ ˆz and Jˆz , which correspond to pure dephasing respectively on the ancilla and on the intra-cavity two-level systems. We accounted for this kind of noise by adding to the master equation in Eq. 1 the corresponding term γd D ′ d [Uˆd (ω)], which is defined as 2
D ′ d [Uˆd (ω)] =
Xn
o 2Uˆd (ω)ρUˆd† (ω) − ρUˆd† (ω)Uˆd (ω) − Uˆd† (ω)Uˆd (ω)ρ
ω≥0
(M ) where the superoperator Uˆd (ω) is defined as above, using the jump operators Aˆd = σ ˆz or Jˆz . We have checked that the ancillary qubit spectroscopy is robust with respect to
pure dephasing, which gives a similar effect to what produced by dissipation (population finite lifetime). In Fig. 3 we show different spectra by varying the ratio between dissipation
3 and pure dephasing produced by the jump operator Jˆz . The broadening effect due to pure dephasing is suppressed for larger values of coupling λ between the cavity and the two-level systems. This kind of suppression is due to collective symmetry and is consistent with results obtained in other works on ultrastrongly coupled systems [2, 3]. We do not show the well (M )
known results about the effect of pure dephasing produced by the jump operator σ ˆz
, which
only causes an additional broadening of the ancilla transition. What is mainly relevant in the ancilla spectroscopy is the total broadening affecting the spectral linewidth more than the specific nature of its origin. We emphasize that in our Letter, we have conservatively considered spectral linewidths, which are considerably larger than what achievable in stateof-the-art circuit QED systems [3, 4].
[1] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Clarendon, Oxford, 2006). [2] P. Nataf and C. Ciuti, Phys. Rev. Lett. 107,190402 (2011). [3] F. Nissen, J. M. Fink, J. A. Mlynek, A. Wallraff, and J. Keeling, Phys. Rev. Lett. 110, 203602 (2013). [4] V. E. Manucharyan, J. Koch, L. I. Glazman, M. H. Devoret, Science 326, 113 (2009).
4 2.9
0.1 0.3 0.25
0.15 2.8
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ωp / ωc
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2.85
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0.1
2.76
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0.5 λ/ω
2.74 0
1
0.5
c
1 λ / ωc
1.5
2
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0.1 0.3 0.25
0.15 2.8
ωp / ωc
ωp / ωc
0.2
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0.04
0.1
2.76
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0.5 λ/ω
2.74 0
1
0.5
c
1 λ / ωc
1.5
2
2.9
0.1 0.3 0.25
0.15 2.8
ωp / ωc
ωp / ωc
0.2
0.08
2.8
2.85
0.06
2.78
0.04
0.1
2.76
0.05
0.02
2.75 0
0.5 λ/ω
2.74 0
1
0.5
c
2.9
1 λ / ωc
1.5
2
0.35
0.12
0.3
0.15
2.8
0.1
ωp / ωc
ωp / ωc
0.2
0.1
2.8
0.25
2.85
0.08 2.78
0.06 0.04
2.76
0.02
0.05 2.75 0
0.5 λ/ω
c
1
2.74 0
0.5
1 λ / ωc
1.5
2
FIG. 1: Excited state population of the ancilla M versus the coherent drive frequency ωp for different values of collective coupling λ at finite temperature. In the four rows from bottom to top, the value of the temperature is respectively kB T /(~ωc ) = 0, 0.105, 0.21 and 0.42. Left panels are for the Dicke model with Ωp = 0.5γM and dissipation parameters γM = γc = γ0 = 0.01ωc . Right panels are for the Tavis-Cummings model with Ωp = 0.2γM and dissipation parameters γM = γc = γ0 = 0.005ωc (nota bene: in some areas of the figures the color scale is saturated in order to improve the contrast). Other parameters: N = 3, ωc = ω0 , ωM = 2.75ωc , gM = 0.1ωc .
5 0.35
ω3 ω2 ω
0.3 0.25
Pee
0.2 0.15 0.1 0.05 0 2.78
2.8
2.82
2.84 ωp / ωc
2.86
2.88
2.9
FIG. 2: Example of ancilla spectrum dependence on reservoir density of states (J(ω) ∝ ω α with α = 1, 2, 3) for a finite temperature (kB T /(~ωc ) = 0.21). Dicke model, parameters: λ/ωc = 1,
0.35
0.35
0.3
0.3
0.25
0.25
0.2
Pee
Pee
γc = γM = γ0 = 0.01ωC . Other parameters: N = 3, ωc = ω0 , ωM = 2.75ωc , gM = 0.1ωc .
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0 2.7
2.72
2.74
2.76 2.78 ωp / ωc
2.8
2.82
2.84
0 2.78
2.8
2.82
2.84 ωp / ωc
2.86
2.88
2.9
FIG. 3: Ancilla spectrum of the Dicke model at zero temperature when the pure dephasing rate is tuned from γd = 0 (solid red), 0.5 (dashed green) to 1 γc (dotted blue). Left figure: λ = 0.5 ωC , right figure: λ = 1 ωC . Other parameters: γc = γM = γ0 = 0.006ωC , N = 3, ωc = ω0 , ωM = 2.75ωc , gM = 0.1ωc .
Laboratoire Mat´eriaux et Ph´enom`enes Quantiques, Universit´e Paris Diderot-Paris 7 and CNRS,
Bˆ atiment Condorcet, 10 rue Alice Domon et L´eonie Duquet, 75205 Paris Cedex 13, France and 2
Department of Physics, University of Maryland, College Park, MD 20742.
(March 3, 2015) In this Supplementary Material, we consider the effect of a finite temperature bath on the ancillary qubit spectroscopy. All the results shown in the Letter have been obtained considering reservoirs at zero temperature, because the focus is the physics of the ground state. For cavity QED systems with optical transitions and cryogenic conditions, experiments can easily fulfill the condition kB T ≪ ~ωC , which is equivalent to the zero-temperature case. For circuit QED systems with microwave resonators and dilution fridge cryogenic conditions, the thermal energy can be a fraction of the photon energy: to give an example, a temperature of 50 mK and a resonator transition frequency ωC /(2π) = 5 GHz corresponds kB T /(~ωc ) ≃ 0.21. We show that the in this range of temperatures, the Lamb-shift of the ancilla qubit is still resolved and the finite temperature produces a moderate thermal broadening.
MASTER EQUATION FOR FINITE TEMPERATURE
In order to include the effect of temperature, we need to consider the following master equation[1]: γc γ0 γM ′ ˆ ˆ ρ˙ = −i[H(t), ρ] + D ′ [Uˆc (ω)] + D ′ [Uˆ0 (ω)] + D [UM (ω)] 2 2 2 where the dissipative term D ′ are defined in the following energy conserving form D ′ [Uˆi (ω)] =
n o X J(ω) (1 + N(ω)) 2Uˆi (ω)ρUˆi† (ω) − ρUˆi† (ω)Uˆi (ω) − Uˆi† (ω)Uˆi (ω)ρ J(ωi ) ω≥0
(1)
2 n o X J(ω) † † † ˆ ˆ ˆ ˆ ˆ ˆ N(ω) 2Ui (ω)ρUi (ω) − ρUi (ω)Ui (ω) − Ui (ω)Ui (ω)ρ . + J(ωi ) ω≥0 where N(ω) is the bosonic thermal distribution and the jumps operator Uˆi (ω) are defined as Uˆi (ω) =
X
δω,ǫl′ −ǫl hl|Aˆi |l′ i|lihl′|.
ll′
The operators Aˆi are those involved in the coupling to the reservoir, namely Aˆc = a ˆ† + a ˆ for (M ) the bosonic mode, Aˆ0 = Jˆx for the two-level systems and AˆM = σ ˆx for the ancilla. The spectral function J(ω) depends on the density-of-states of the reservoir excitations. When the bath is a 3D electromagnetic field, we have J(ω) ∝ ω 3, hence it vanishes while ω → 0. An ohmic reservoir scales instead as J(ω) ∝ ω. In Fig. 1 , we show the ancillary transition spectrum as a function of the coupling λ for different values of temperature, namely kB T /~ωc = 0, 0.105, 0.21 and 0.42 (from bottom to top) for the Dicke (left panels) and Tavis-Cummings model (right panel). It is apparent that the Lamb shifts are still well measurable and that the main effects is a moderate broadening of the Lamb-shifted ancillary qubit resonances. We have also checked that the behavior at low frequency does not affect the results in a significant way. In Fig. 2, we show a typical ancilla spectrum with three kind of reservoirs with J(ω) ∝ ω α and α = 1 (ohmic), α = 2 and α = 3. It is apparent that the differences are negligible, indeed the broadening is dominated by the spectra dependence around the ancilla qubit transition. We have also tested the robustness of the ancillary spectroscopy under the effect of noise mechanisms different from flip errors. More precisely we considered the effect produced by (M ) jump operator σ ˆz and Jˆz , which correspond to pure dephasing respectively on the ancilla and on the intra-cavity two-level systems. We accounted for this kind of noise by adding to the master equation in Eq. 1 the corresponding term γd D ′ d [Uˆd (ω)], which is defined as 2
D ′ d [Uˆd (ω)] =
Xn
o 2Uˆd (ω)ρUˆd† (ω) − ρUˆd† (ω)Uˆd (ω) − Uˆd† (ω)Uˆd (ω)ρ
ω≥0
(M ) where the superoperator Uˆd (ω) is defined as above, using the jump operators Aˆd = σ ˆz or Jˆz . We have checked that the ancillary qubit spectroscopy is robust with respect to
pure dephasing, which gives a similar effect to what produced by dissipation (population finite lifetime). In Fig. 3 we show different spectra by varying the ratio between dissipation
3 and pure dephasing produced by the jump operator Jˆz . The broadening effect due to pure dephasing is suppressed for larger values of coupling λ between the cavity and the two-level systems. This kind of suppression is due to collective symmetry and is consistent with results obtained in other works on ultrastrongly coupled systems [2, 3]. We do not show the well (M )
known results about the effect of pure dephasing produced by the jump operator σ ˆz
, which
only causes an additional broadening of the ancilla transition. What is mainly relevant in the ancilla spectroscopy is the total broadening affecting the spectral linewidth more than the specific nature of its origin. We emphasize that in our Letter, we have conservatively considered spectral linewidths, which are considerably larger than what achievable in stateof-the-art circuit QED systems [3, 4].
[1] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Clarendon, Oxford, 2006). [2] P. Nataf and C. Ciuti, Phys. Rev. Lett. 107,190402 (2011). [3] F. Nissen, J. M. Fink, J. A. Mlynek, A. Wallraff, and J. Keeling, Phys. Rev. Lett. 110, 203602 (2013). [4] V. E. Manucharyan, J. Koch, L. I. Glazman, M. H. Devoret, Science 326, 113 (2009).
4 2.9
0.1 0.3 0.25
0.15 2.8
ωp / ωc
ωp / ωc
0.2
0.08
2.8
2.85
0.06
2.78
0.04
0.1
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0.02
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c
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ωp / ωc
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1 λ / ωc
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0.15 2.8
ωp / ωc
ωp / ωc
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c
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ωp / ωc
ωp / ωc
0.2
0.1
2.8
0.25
2.85
0.08 2.78
0.06 0.04
2.76
0.02
0.05 2.75 0
0.5 λ/ω
c
1
2.74 0
0.5
1 λ / ωc
1.5
2
FIG. 1: Excited state population of the ancilla M versus the coherent drive frequency ωp for different values of collective coupling λ at finite temperature. In the four rows from bottom to top, the value of the temperature is respectively kB T /(~ωc ) = 0, 0.105, 0.21 and 0.42. Left panels are for the Dicke model with Ωp = 0.5γM and dissipation parameters γM = γc = γ0 = 0.01ωc . Right panels are for the Tavis-Cummings model with Ωp = 0.2γM and dissipation parameters γM = γc = γ0 = 0.005ωc (nota bene: in some areas of the figures the color scale is saturated in order to improve the contrast). Other parameters: N = 3, ωc = ω0 , ωM = 2.75ωc , gM = 0.1ωc .
5 0.35
ω3 ω2 ω
0.3 0.25
Pee
0.2 0.15 0.1 0.05 0 2.78
2.8
2.82
2.84 ωp / ωc
2.86
2.88
2.9
FIG. 2: Example of ancilla spectrum dependence on reservoir density of states (J(ω) ∝ ω α with α = 1, 2, 3) for a finite temperature (kB T /(~ωc ) = 0.21). Dicke model, parameters: λ/ωc = 1,
0.35
0.35
0.3
0.3
0.25
0.25
0.2
Pee
Pee
γc = γM = γ0 = 0.01ωC . Other parameters: N = 3, ωc = ω0 , ωM = 2.75ωc , gM = 0.1ωc .
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0 2.7
2.72
2.74
2.76 2.78 ωp / ωc
2.8
2.82
2.84
0 2.78
2.8
2.82
2.84 ωp / ωc
2.86
2.88
2.9
FIG. 3: Ancilla spectrum of the Dicke model at zero temperature when the pure dephasing rate is tuned from γd = 0 (solid red), 0.5 (dashed green) to 1 γc (dotted blue). Left figure: λ = 0.5 ωC , right figure: λ = 1 ωC . Other parameters: γc = γM = γ0 = 0.006ωC , N = 3, ωc = ω0 , ωM = 2.75ωc , gM = 0.1ωc .
