Lamb shift enhancement and detection in strongly driven superconducting circuits

0.53 MB·

by  Vera Gramich

Lamb shift enhancement and detection in strongly driven superconducting circuits
Vera Gramich,1,2,∗ Simone Gasparinetti,2 Paolo Solinas,2,3 and Joachim Ankerhold1
1Institut fu¨r Theoretische Physik, Universita¨t Ulm, Albert-Einstein-Allee 11, 89069 Ulm, Germany
2Low Temperature Laboratory (OVLL) - Aalto University School of Science, P.O. Box 13500, 00076 Aalto, Finland
3SPIN-CNR - Via Dodecaneso 33, 16146 Genova, Italy
(Dated: May 7, 2014)
ItisshownthatstrongdrivingofaquantumsystemsubstantiallyenhancestheLambshiftinduced
by broadband reservoirs which are typical for solid-state devices. By varying drive parameters the
impact of environmental vacuum fluctuations with continuous spectral distribution onto system
4 observables can be tuned in a distinctive way. This provides experimentally feasible measurement
1 schemesfortheLambshiftinsuperconductingcircuitsbasedonCooperpairboxes,whereitcanbe
0 detected either in shifted dressed transition frequencies or in pumped charge currents.
2
PACSnumbers: 03.65.Yz,85.25.Dq,32.70.Jz,03.67.Lx
y
a
M
Introduction.- Quantum fluctuations of the electro- tention in superconducting circuits [20–22]. Under cer-
magnetic vacuum affect atomic spectra [1], a phe- tain resonant conditions, the system-environment cou-
6
nomenon termed Lamb shift (LS) which has triggered pling is substantially enhanced by the presence of the

] the development of modern quantum electrodynamics drive,yieldingadynamicsteadystatewhichislargelyde-
l
l (QED).Experimentally,cavityQED[2–4]hasopenedthe terminedbyenvironmentalfeaturesitself[23–25]. Within
a
doorforanunprecedentedlevelofaccuracyinthemanip- the same regime, the environment also induces a renor-
h
- ulation and measurement of atomic quantum states [5]. malization of the dressed-state energies (quasienergies).
s
Its most recent realization is circuit QED [6–8] based on This renormalization defines the LS of the driven sys-
e
m solid-state architectures, for example, consisting of a su- tem. For atomic states in a resonant radiation field a LS
perconducting Cooper pair box (CPB) embedded in a has been discussed in [26], however, here we consider a
.
at superconducting waveguide resonator. Circuit QED has generalized situation of open quantum systems subject
m been able to reproduce several quantum optics experi- to arbitrary periodic driving. We find that the relative
ments [9–11], with advantages in terms of design, fab- magnitude of the LS can exceed by far what is typically
-
d rication and scalability giving also access to parameter observed in static systems and exhibits specific scaling
n
ranges currently unreachable in optical setups [9–11]. trends as a function of the drive parameters. These two
o
c Cavity and circuit QED are based on strongly modi- features should make it easier to unambiguously identify
[ fied density of states of the electromagnetic environment the LS contribution.

seen by the atom compared to a continuum. This way, While enhanced LSs should be observable in other
2
v the LS has recently also been detected in a circuit QED driven solid-state systems as well [25, 27, 28], here we
6 setup [12] in form of zero-point fluctuations of a single focus on superconducting devices and consider circuits
9 harmonicmode. Eventhecreationofrealphotonsoutof containing CPBs. This provides direct contact to recent
3 the vacuum, known as the dynamical Casimir effect, has experiments on driven CPBs used to realize a Mach-
5
beenseen[13,14]. However,LSmodificationsshouldnat- Zehnder interferometer [20] and to operate as a charge
.
1 urally arise in electric circuits as the devices of interest pump [29–32]. Particularly, the latter situation reveals
0 can never be isolated from their surroundings, particu- that the LS may also induce clear signatures in coherent
4
1 larlyfromthosewithbroadbandspectraldensities.While charge currents.
: this class of environments constitutes the most common LS of a driven two-level system.- We start with a
v
one for solid-state systems, evidence for corresponding model, where a CPB subject to a general periodic drive
i
X LS effects has proven elusive yet. Engineered environ- is described by a driven two-level system (TLS) in terms
r mentssuchasthoserealizedinatomicsetups[15,16]and of Pauli matrices, i.e.,
a
studied for solid-state systems [17–19] may pose serious E
challenges in actual realizations though. HS(t)=−2σz+F(cid:126)(t)·(cid:126)σ. (1)
In this Letter, we propose a different scheme for the
Here, E is the level spacing of the bare system and
LS detection in a solid-state system weakly interacting F(cid:126)(t) = (F (t),F (t),F (t)) is a driving field obeying
x y z
with a broadband environment. Instead of engineering
F(cid:126)(t)=F(cid:126)(t+2π/Ω)withperiodΩ.ThisTLSinteractsvia
the environment, the system is subject to a strong and
H =S (cid:80) c (b†+b )withareservoirofbosonicmodes,
tunable driving field. This drive can not be treated as I j j j j
a perturbation and the interaction between driven sys- i.e. [bj,b†j] = 1, the distribution of which is character-
tem and its environment is best described in terms of izedbyaspectralfunctionJ(ω)=(π/(cid:126))(cid:80) c2δ(ω−ω ).
j j j
“dressed states” that recently have attracted much at- For typical solid-state aggregates, this distribution has
2
a broadband profile in contrast to single modes for high applyto any periodic driving, spectralfunction andcou-
quality cavities. The system operator plingoperator(2). Thermalcorrectionsto(3)andto(4),
alsoknownasac-Starkshift,playaroleif(cid:126)∆ iscom-
αβ,k
S(r)=sin(r)σ +cos(r)σ (2)
x z parabletothethermalenergy. Aswewilldiscussinmore
detail below, this imposes conditions on driving frequen-
is chosen such as to capture both decoherence (r =π/2)
cies and amplitudes to sufficiently exceed 1/β =k T.
or pure dephasing (r =0). B
Themasterequation(3)substantiallydiffersfromthat
A powerful approach to treat periodically driven
for undriven systems in that the effective system-bath
quantum dynamics is given by the Floquet formalism
coupling is given by ηΩ/|ω | [23] with a bare system-
[33]. One starts from a complete set of solutions of 12
bath coupling parameter η. Due to the tunability of
the time-dependent Schr¨odinger equation for H (t) =
S
ω viadrivingamplitudeorfrequency,environmentalin-
H (t + 2π/Ω) given by the Floquet states |Ψ (t)(cid:105) = 12
S α
e−i(cid:15)αt/(cid:126)|Φα(t)(cid:105), where the Floquet modes |Φα(cid:105) satisfy duced terms can thus even dominate the dynamics with-
out deteriorating the validity of (3) [23].
|Φ (t)(cid:105) = |Φ (t + 2π/Ω)(cid:105). The quasienergies (cid:15) play
α α α
the role of dressed state energies and are only defined
1.2 0.8
mod[(cid:126)Ω]. The Floquet description manifestly takes into
account the fundamental and all higher harmonics and 0.6
0.8
thusalsoappliestoarbitrary strong driving far from res- 0.4
onance. 0.4
0.2
Incaseofweakdissipation, theFloquetformalismcan
beconsistentlycombinedwithsecondorderperturbation 0.0 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.0 0.4 0.2 0.0 0.2 0.4
theory to arrive at a Born-Markov-type master equation
for the reduced dynamics of the system ρ(t) [33]. After
FIG. 1. Quasienergy gap (cid:126)ω /E with (solid) and without
performing a partial secular approximation (PSA) [23], 12
(dashed) LS according to (5) as a function of detuning ∆ for
this master equation becomes time-independent in the
driving amplitudes |A|/E =0.1(magenta)and0.3(blue), and
basis of the Floquet modes and takes in the Schr¨odinger different coupling mechanisms to the bath: (a) S = σ and
x
picture the form (b) S = σ . Damping parameters are η = 0.1 and cut-off
z
frequency (cid:126)ω /E =60.
(cid:88) c
ρ˙ (t)=−i(ω −δω )ρ (t)+ R ρ (t) (3)
αβ αβ αβ αβ αβγδ γδ
γ,δ Strongly driven CPB in an Ohmic reservoir.- To il-
with transition frequencies ω =((cid:15) −(cid:15) )/(cid:126). The Red- lustrate the above findings and to discuss their relevance
αβ α β
in actual experiments, we now consider a specific model.
field tensor R captures decoherence as well as de-
αβγδ
The bath is assumed to be Ohmic-like, i.e., J(ω) =
phasing and formally couples equations for the popu-
η(cid:126)ωexp(−ω/ω ) with coupling constant η and a large
lations (diagonal elements of the density) and the co- c
cut-offfrequencyω ,andtheexternaldrivein(1)istaken
herences (off-diagonal ones) (see Supplemental Material c
as F(cid:126)(t) = A(cos(Ωt),−sin(Ωt),0) with drive amplitude
[34]). In extension to previous treatments [23, 33, 35],
A. The Floquet states can then explicitly be calculated
Eq. (3) contains the reservoir induced renormalization
(see [34]) with quasienergies (cid:15) = (∆±(cid:126)ω )/2, where
δω of transition frequencies. At sufficiently low tem- 1,2 R
perαaβtures, it is dominated by environmental zero-point ∆ = E −(cid:126)Ω is the detuning and ωR = (cid:112)∆2+4|A|2/(cid:126)
fluctuations and then constitutes the LS for a driven the Rabi frequency. In typical realizations Ω (cid:29) ηωR
quantum system. This is the main focus of this work which justifies the PSA in (3). This model is also known
which, to our knowledge and despite of its relevance for as the semiclassical Rabi model, first used to describe
ongoing experiments, has not been addressed yet. optical transitions of atoms [36].
Specifically, in case of a TLS at zero temperature we Transparent expressions for the LS (4) with the cou-
find (see [34]) pling S(r) in (2) are obtained in the two limiting cases
with mixing angles r =π/2 (transversal coupling induc-
∞
δω = 1 (cid:88) |X(r) |2[G(∆ )−G(−∆ )] (4) ingdecoherence)andr =0(longitudinalcouplinginduc-
12 π(cid:126) 21,k 21,k 21,k ing dephasing). One gains δω12 =(ηωc/π)Λ(r), where
k=−∞
Λ(0) =−g(ω /ω )sin2(2θ)
which contains dressed transition frequencies ∆ = R c
αβ,k
((cid:15)α−(cid:15)β)/(cid:126)+kΩ and coupling matrix elements Λ(π/2) =g(ω−/ωc)sin4(θ)+g(ω+/ωc)cos4(θ) (5)
Ω (cid:90) 2π/Ω with ω± = ±Ω − ωR and tan(2θ) = 2|A|/∆. Reser-
Xα(rβ),k = 2π dte−ikΩt(cid:104)Φα(t)|S(r)|Φβ(t)(cid:105). voir zero-point fluctuations enter through g(x) =
0 x[Ei(x)e−x + Ei(−x)ex], the asymptotics of which
Reservoir properties are encoded in the principal value g(|x| (cid:28) 1) ≈ 2xln(|x|) reflects the logarithmic be-
(cid:82)∞
integral G(z) = −P dωJ(ω)/(ω −z). These results havior known from the atomic LS [1]. Apparently,
0
3
the LS (5) also carries information about the system 0.3 4
(CPB)-reservoir coupling mechanism. It turns out that 3
for (cid:126)ω /E (cid:29) 1, results discussed below depend on the 0.2
c
2
cut-off only very weakly.
0.1
Upontuningthedriveamplitudeand/orfrequency,the 1
LSthusdisplaysaqualitativelydifferentbehaviorincon-
0 0
0.0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.0 0.5 1.0 1.5 2.00
trast to the bare quasienergy gap (Fig. 1). Namely, for
transversal coupling (S = σ ), we find for the screened
x
transition frequency a pronounced asymmetry with re- FIG. 3. Same as in Fig. 1, but vs. driving amplitude |A|
specttonegativeandpositivedetuning(Fig.1a).Amea- closetotheresonancewith|∆|/E: 0.005(blue)and0.05(ma-
surement of this deviation from the linear scaling would genta). The corresponding curves in (b) cannot be resolved
on this scale.
be a clear signature of the presence of environmental
ground state fluctuations. On the contrary, longitudi-
nal coupling (S =σ ) leads to a smoother structure and
z
For longitudinal coupling (b) one finds a smooth behav-
less deviations from the bare gap, see Fig. 1b.
ior, where close to resonance the dependence on |∆| dis-
appears and (cid:126)δω ∼ |A|. The situation for transver-
12
0.2
0.3 sal coupling (a) is quite different. In this case, ac-
0.30.0 cording to Λ(π/2) in (5) one has δω12(|A|/|∆| (cid:28) 1) ≈
0.2 0.2 (2Ωη/π)ln(Ω/ω ), while then the bare gap (cid:126)ω ≈ |∆|.
c 12
0.20.4
0.1 This leads to a highly non-monotonic dependence of the
0.6
0.1 full gap on the driving amplitude and allows to distin-
0.8 0
0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.4 0.2 0.0 0.2 0.4 guish between the prevalence of either of the coupling
0
0.4 0.2 0.0 0.2 0.4 mechanisms. The discontinuity of the LS in the limits
|∆|=0,|A|/E >0 (seen in Fig. 2a) and |∆|/E >0,A=
FIG. 2. Relative magnitude of the LS δω /ω for (a)
12 12 0 (seen in Fig. 3a) is due to the θ-dependence of Λ(π/2).
transversal (σ ) and (b) longitudinal (σ ) coupling vs. de-
x z
tuning ∆ and for different values of the driving amplitude LS detection.- We now discuss how in recent ex-
|A|/E: 0.07 (magenta), 0.3 (blue). Other parameters are as periments with driven CPBs the model (1) together
in Fig. 1. with various system-bath couplings (2) is realized and
how the LS could be retrieved. In [20, 21, 37], a
To have a more quantitative estimate of the impact of CPB with tunable charging energy EC and Josephson
the LS and, in particular, to reveal optimal parameter energy EJ is subject to a microwave field and em-
regimes for detection with respect to signal strength and bedded in an environment which dominantly induces
suppressionofthermalfluctuations,inFig.2,weplotthe chargefluctuations.TheHamiltoniantakesforlargepho-
relative magnitude of the LS compared to the bare tran- ton fields the form HS,CPB(t) = −12ECτz − 12EJτx −
sitionfrequencywhenthedetuningisvaried. Infact,the λcos(ωt)τz with Pauli matrices {τi} and τz-coupling to
external driving increases the ratio |δω /ω | for both the bath. In the general situation, the eigenstate repre-
12 12 (cid:112)
coupling schemes far above the usual ratio of a few per- sentation of the CPB leads to (1) with E = E2 +E2
C J
cent [12]. The dressed system exchanges energy quanta and with both the system-bath coupling and the drive
with the bath with an effective coupling ηΩ/ωR which fixed by the angle r = arctan(EJ/EC) in (2) where
due to Ω (cid:29) ω can be strong compared to the static F(cid:126)(t) = λcos(ωt)(sin(r),0,−cos(r)). Simplifications are
R
situation. We mention that this enhancement also ap- achieved in two limiting cases. (i) At charge degeneracy
plies to the limiting situation of a single mode reservoir E = 0, one has E ≈ E and F(cid:126)(t) = λcos(ωt)(1,0,0)
C J
(cavity), see [34]. together with S(π/2) (transversal coupling). For |λ| (cid:28)
Experimentally,strongsignalsfortheLSinbothcases (cid:126)ω ∼ E , within the PSA, the drive even reduces to
J
of transversal and longitudinal noise can be expected for the model in (5). (ii) For E (cid:54)= 0 but strong driving
C
weak to moderate detuning |∆|/E (cid:46) 0.3 (Fig. 2). With (cid:126)ω (cid:29) λ (cid:29) E , effectively, longitudinal noise (r = 0) is
J
respect to low thermal fluctuations, the conditions are obtained. Then, instead of working in the CPB eigen-
different. For σ -coupling, Fig. 2a, thermal noise is sup- basis, one applies a dressed tunneling picture [20, 34].
x
pressed if Ω(cid:126)β (cid:29)1 (see [34]) which is easily fulfilled also Close to the n-th order photon resonance E = n(cid:126)ω,
C
for relatively weak driving |A|/E (cid:28) 1, where the LS is the PSA consistently reduces H (t) to the n-photon
S,CPB
pronounced. Instead,forσ -coupling,Fig.2b,duetothe sector and one arrives at (5) with E = E , A =
z C
constraint ω (cid:126)β (cid:29) 1 [34] stronger driving is required −(E /2)J (2λ/(cid:126)ω),Ω=nω, and r =0 (see [34]).
R J n
which, as shown, supports enhanced LS yet. Now, after preparation of the CPB, its steady state in
An alternative way to measure an enhanced LS by presenceofthemicrowavedrive[ρ˙ =0in(3)]isprobed
αβ
varying the drive amplitude A is presented in Fig. 3: with a weak pulse H =µ cos(ω t)σ ,µ (cid:28)A, to ac-
P P P z P
4
cesstheresonanceatω12−δω12 =ωR−(ηωc/π)Λ(r). The 0.4
LS then appears in the absorption spectrum upon vary-
ingeitherthedetuning(Fig.1)ortheamplitude(Fig.3)
of the pump field (cf. [38]). For typical experimental pa-
0.3
rametersfrom[37]E/(cid:126)≈10GHzandη ≈0.05,(cid:126)ω /E ≈
c
60, Eβ ≈ 7, one finds for σ -coupling |δω ((cid:126)Ω/E =
x 12
1.1,|A|/E =0.1)|/ω ≈0.2 with Ω(cid:126)β ≈8, while for σ -
R z
coupling|δω ((cid:126)Ω/E =1.3,|A|/E =0.3)|/ω ≈0.1with 0.2
12 R
still sufficiently weak thermal noise ω (cid:126)β ≈ 5. Already
R
theseLSvaluesareatleastoneorderofmagnitudelarger
than in the static case [12] and can further be enhanced 0.1
by optimized circuit designs and detection protocols.
Cooper pair pump.- The devices discussed up to this
point are tailored to externally manipulate their level
0
structure. However, the LS may have also profound im- -0.34 -0.3 -0.26 -0.22 -0.18
pactinsuperconductingcircuitswheretransportproper-
ties are addressed. A specific example is a charge pump
in form of the Cooper pair sluice [29, 39] sketched in FIG. 4. (a) Schematic circuit diagram for the ‘sluice’ and
(b) driving protocol with the three time-dependent control
Fig. 4a. It consists of a single superconducting island
parametersJ andn foronepumpingcycle. (c): Pumped
(CPB)separatedbytwoSQUIDswithtunableJosephson L,R g
chargeQ with(solid)andwithout(dashed)LScontribution
P
energies J (t). A third control parameter is provided
L,R vs. phase ϕ for different values of the system-bath coupling:
by the gate charge ng(t) capacitively coupled to the is- η = 0.001 (blue), η = 0.005 (green), η = 0.01 (red) and η =
land. Full control of the quantum system is guaranteed 0.05 (black). Parameters are chosen according to [29]: drive
via the three experimentally accessible parameters JL,R time τ = 1ns, EC =1K, ωc = 100GHz, ∆ng = ng −0.5 =
and ng which allow for charge pumping when steered in 0.2, Jmax =0.1EC, Jmin =10−3Jmax.
a periodic protocol (see Fig. 4b). In the charging regime
E (cid:29)max{J ,J }andclosetoahalfintegerofthegate
C L R
charge n (t), this device is described by a pseudospin- system properties prevail against decoherence. Noise in-
g
Hamiltonian [29, 40] ducedtransitionsbetweentheenergylevelsarethussup-
pressed and the pumped charge follows the bare one for
1
H (t)=− B(cid:126)(t)·(cid:126)σ, (6) η = 0.001. Within the domain of the bare crossing
S 2 ω ≈ 0 (range of the peak), the steady state of (3) is
12
where B (t) = J (t)cos(ϕ), B (t) = J (t)sin(ϕ), completely determined by reservoir quantities, i.e., the
x + 2 y − 2
B (t) = E [2n (t)−1] and J (t) = J (t)±J (t). The LS and the Redfield tensor. Since both are proportional
z C g ± L R
total superconducting phase difference across the sluice toη, thefrictionparameterdropsoutofthesteadystate
isdenotedbyϕ. Dominantnoisesourcesarechargefluc- equation. Apparently, within this latter range the pre-
tuationsimplyingaσz-couplingtoenvironmentalmodes. dictions for QP with LS qualitatively deviate from those
This system has a more complex structure than the without LS giving rise to a peak instead of a dip. This
Rabi model as it includes transversal and longitudinal verifies the pronounced impact of vacuum fluctuations
driving with many higher harmonics. Its main observ- also on transport properties of superconducting circuits.
able is the charge Q pumped through the sluice during Conclusion.- We have analyzed the impact of envi-
P
sequences of driving cycles. Numerical results in steady ronmental zero-point fluctuations with broadband spec-
state based on (3) are depicted in Fig. 4c with and with- tral densities in strongly driven quantum systems. De-
out the LS versus the phase ϕ across the sluice. The pending on the drive parameters, the relative strength
latter one can be adjusted by an external magnetic field. of the system-reservoir coupling is enhanced and, thus,
the induced LS increased to an extent unreachable in
The pumped charge displays a very sensitive depen- standard experiments. This LS displays distinctive sig-
dence on the phase difference which in turn deter- natures as a function of the driving amplitude and/or
mines the energy splitting. Close to a degeneracy of the frequency. The predicted effect should be accessible in
quasienergies at ϕ ≈ −0.26, one enters a regime where many solid-state systems, particularly in current super-
c
environmental effects on Q are strong and of order conductingdevices. Specificdetectionschemeshavebeen
P
ηΩ/|ω −δω | [23]. In contrast to the bare situation, discussedforcircuitswithdrivenartificialatomsandcon-
12 12
however, the pumped charge including the LS depends trolled Cooper pair charge flow. The proposed protocols
only very weakly on the coupling parameter η. Namely, would shed new light on the impact of broadband envi-
away from degeneracy (away from the peak), the LS en- ronmentsonquantumsystemsatcryogenictemperatures
hances the level splitting so that renormalized dressed which is completely absent in the classical regime.
5
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M.Gu¨ntherandS.Pugnettiforfruitfuldiscussions. This ett, D. Kielpinski, W. M. Itano, C. Monroe, and D. J.
work has been supported by the German Science Foun- Wineland, Nature 403, 269 (2000).
[16] D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett,
dation(DFG)withinSFB/TRR-21andAN336/6aswell
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as by the European Community’s Seventh Framework
291, 1013 (2001).
Programme(FP7/2007-2013)undergrantagreementNo.
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228464 (MICROKELVIN). We gratefully acknowledge
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also financial support from the DAAD (V. G.) and the [18] P.Solinas,M.Mo¨tto¨nen,J.Salmilehto, andJ.P.Pekola,
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6
Supplemental Material
Inthissupplementalmaterialtoourarticle‘Lambshiftenhancementanddetectioninstronglydrivensuperconductingcircuits’
we present further details about the derivation of the Lamb shift as well as the semiclassical Rabi model, its Lamb shift for a
single mode reservoir, and its equivalence to experimental realizations in limiting cases.
Derivation of the Lamb shift
To reveal the impact of the Lamb shift expression given in Eq. (4) in the main text, we present here its full
derivation, starting with the time-independent Floquet master equation (for further details follow the steps in, e.g.,
Refs. [1–3]) written in the basis of Floquet modes |Φ (t)(cid:105)=|Φ (t+ 2π)(cid:105) and in the Schr¨odinger picture
α α Ω
(cid:88)
ρ˙ (t)=−i(ω −δω )ρ (t)+ R ρ (t), (7)
αβ αβ αβ αβ αβγδ γδ
γ,δ
where
(cid:88) (cid:88)
R =Γ+ +Γ− −δ Γ+ −δ Γ− (8)
αβγδ αγβδ αγβδ δ,β µγµα γ,α µβµδ
µ µ
represents the Redfield tensor and
Γ+ = 1 (cid:88)S(∆ )X(r) (X(r) )∗, Γ− = 1 (cid:88)S(∆ )X(r) (X(r) )∗ (9)
αβγδ (cid:126) αβ,k αβ,k γδ,k αβγδ (cid:126) γδ,k αβ,k γδ,k
k k
the effective relaxation and dephasing rates, respectively. A crucial point in deriving the result (7) is that we have
performed the so-called partial secular approximation (PSA) [3]. It consists in retaining all terms which oscillate
with (cid:15) (cid:54)= (cid:15) (in contrast to the usual rotating wave approximation, RWA) and neglecting only those with multiple
α β
integers k ,k of (cid:126)Ω where k (cid:54)=k . The PSA imposes the constraint Ω(cid:29)η|ω | which is a much weaker condition
α β α β αβ
than that for a full RWA which requires |ω |(cid:29)Γ with a typical relaxation time scale 1/Γ. For details see [2, 3].
αβ
TheLambshiftcontributionsδω arisefromprincipalvaluetermsvia(cid:82)∞dteiωt =πδ(ω)+iP(1/ω)intheFloquet
αβ 0
master equation in the basis of Floquet states |Ψ (t)(cid:105) following the procedure outlined in Refs. [2, 3]. The matrix
α
elements
Ω (cid:90) 2π/Ω
X(r) = dte−ikΩt(cid:104)Φ (t)|S(r)|Φ (t)(cid:105) (10)
αβ,k 2π α β
0
withthedrivingfrequencyΩcontainthesystemoperatorS couplingtothereservoirandobeythesymmetryrelation
X(r) =X∗(r) . The latter one helps to simplify the Lamb shift terms. Transition energies are given by
αβ,k βα,−k
(cid:126)∆ =(cid:15) −(cid:15) +k(cid:126)Ω (11)
αβ,k α β
with (cid:15) (t),i = α,β playing the role of a dressed state energy. The quantity ω = ((cid:15) −(cid:15) )/(cid:126) thereby expresses the
i αβ α β
Floquetquasienergygapwhichisanon-dissipativecontributionofthedrivenquantumsystem. Intheaboveformulas
we have also used the abbreviation
S(ω)=θ(ω)J(ω)n (ω)+θ(−ω)J(−ω)[n (−ω)+1] (12)
th th
containingthespectralbathdensityJ(ω)inunitsofanenergy. Here,θ(ω)denotestheHeavysidefunctionandn (ω)
th
is the usual Bose-Einstein distribution.
IntheabovederivationprincipalvaluecorrectionstotheratesinEq.(9)areneglectedastheyprovidehigherorder
corrections only. In the singular coupling regime to be of main interest here, the Lamb shift provides the leading
order contribution. To evaluate it explicitly, one has to consider integrals of the form
(cid:90) ∞ J(ω)n (±ω)
G±(∆ )=P dω th . (13)
αβ,k ω−∆
0 αβ,k
7
This leads us to the following expression
δω = 1 (cid:88)(cid:2)|X(r) |2{G+(∆ )+G−(∆ )}
αβ π(cid:126) βµ,k µβ,−k βµ,k
µ,k
−|X(r) |2{G+(∆ )+G−(∆ )}(cid:3). (14)
µα,k µα,k αµ,−k
TheaboveresultsapplytoarbitraryspectraldensityofthebathJ(ω). Now,foraTLS(µ=1,2)atzerotemperature
(only zero-point fluctuations) n (ω > 0) → 0, the G+-contributions drop out. Applying the symmetry relations
β
X(r) =X∗(r) as well as X(r) =−X(r) for a traceless noise operator (which is true for all combinations of Pauli
αβ,k βα,−k 11,k 22,k
matrices)andorthogonalFloquetmodesatalltimes,onlytermswithdifferentindicesinthecouplingmatrixelements
survive in (14). This leads to Eq. (4) in the main text.
For an Ohmic-type distribution with exponential cut-off J(ω) = η(cid:126)ωexp(−ω/ω ) with a dimensionless coupling
c
constant η and a large cut-off frequency ω , one obtains
c
G(∆αβ,k)≡G−T=0(∆αβ,k)=−η(cid:126)ωc+η(cid:126)∆αβ,ke−∆αβ,k/ωcEi(∆αβ,k/ωc) (15)
with Ei(z)=−(cid:82)∞dye−y/y, where the integral is understood in the principal value sense. The terms linear in ω in
−z c
(15) describing the static effect of the bath do not contribute to the Lamb shift [4] as they cancel each other. Finally,
the Lamb shift (14) reduces for a TLS in the zero temperature limit to
δωT=0 = η (cid:88)(cid:2)∆ f(∆ /ω )|X(r) |2−∆ f(∆ /ω )|X(r) |2(cid:3), (16)
αβ π βµ,k βµ,k c βµ,k αµ,−k αµ,−k c µα,k
µ,k
where we have introduced the function f(x)=Ei(x)exp(−x) with the property f(x)→C +ln(x) for x(cid:28)1 with the
γ
Euler constant C . For the Rabi model discussed in the main text, this yields the expressions in (5) together with
γ
Λ(r) =(cid:80) g(∆ /ω )|X(r) |2, where g(x)=x[f(x)+f(−x)].
k 21,k c 21,k
Thermal corrections to these zero temperature results tend to play a role for dressed transition frequencies with
(cid:126)β∆ (cid:46) O(1), while they are suppressed exponentially for (cid:126)β∆ (cid:29) 1. We discuss details for the Rabi model
12,k 12,k
below. In principle, there is also a constraint on temperature corresponding to the Markov-approximation associated
with (7), namely, that bath correlation functions decay sufficiently fast compared to a typical system relaxation time
scale 1/Γ. However, in a steady state situation, the time-independence of the density guarantees that non-Markovian
effects are of no relevance. They may only play a role when one is interested in time-dependent correlation functions.
Semiclassical Rabi model
WeconsiderheredetailsofthesemiclassicalRabimodelwhich, despiteitssimplicity, describesrecentexperimental
realizations, see below and, e.g., [5–7]. The Hamiltonian is given by
(cid:18) −E A∗eiΩt(cid:19) E
H(t)= 2 =− σ +A{cos(Ωt)σ −sin(Ωt)σ }, (17)
Ae−iΩt E 2 z x y
2
where for simplicity the driving amplitude A is taken as real-valued and E is the bare energy level spacing. To find
the solution of the Schr¨odinger equation, we follow a standard procedure: First, moving to a rotating frame reveals
a time-independent Hamiltonian which we diagonalize. In a second step, we revert to the laboratory frame and cast
the solutions finally in Floquet form. The so-found quasienergies are (cid:15) =(∆±(cid:126)ω )/2 with detuning ∆=E−(cid:126)Ω
1,2 R
and ω = 1(cid:112)∆2+4|A|2 being the Rabi frequency.
R (cid:126)
The corresponding Floquet modes are:
(cid:18) (cid:19)
cosθ
|φ (t)(cid:105)= (18)
1 −e−iΩteiφsinθ
(cid:18) e−iφsinθ (cid:19)
|φ (t)(cid:105)= (19)
2 e−iΩtcosθ
where φ=−argA and tan(2θ)= 2|A|.
∆
8
As we have chosen A to be real, one has φ=0. The coupling matrix elements X(r) are obtained in the following
αβ,k
way: For σ -noise only terms with k =0 survive with
z
X(0) =cos2θ−sin2θ =cos2θ =−X(0)
11,0 22,0
X(0) =2sinθcosθ =sin2θ =X∗(0). (20)
21,0 12,0
TogetherwithΛ(r) specifiedintheprecedingsection, thisthenleadstoΛ(0). Forσ -noise, thenon-zerocontributions
x
k =±1 provide the coefficients
1
X(π/2) =− sin2θ =X(π/2) , X(π/2) =X(π/2) =−X(π/2)
11,−1 2 11,1 22,−1 22,1 11,1
X(π/2) =cos2θ =X∗(π/2) , X(π/2) =−sin2θ =X∗(π/2) , (21)
12,−1 21,1 21,−1 12,1
which yields Λ(π/2) [Eq. (5) in the main text].
Thermal fluctuations in the master equation (7) and the Lamb shift (16) are important if (−ω ±Ω)(cid:126)β (cid:46) 1 (σ -
R x
coupling) or ω (cid:126)β (cid:46) 1 (σ -coupling). In actual experiments [5–7], close to resonance one typically has Ω (cid:29) η|A| so
R z
that thermal fluctuations can be sufficiently suppressed for σ -noise, where this coincides with the optimal regime to
x
detect a Lamb shift (see main text). For σ -coupling the constraint is in general harder to fulfill. However, since a
z
correspondingLambshiftclosetoresonanceisparticularlypronouncedintherangeofstrongerdriving(|A|/E (cid:38)0.3),
for realistic values Eβ ≈10 the condition ω (cid:126)β ≈(2|A|/E)(Eβ)≈6 is sufficiently obeyed as well.
R
Single mode case
While it makes in general no sense to derive a master equation such as in (7) for a single mode reservoir, the result
derived in Eq. (4) of the main text for the Lamb shift can also be applied to analyze this limiting case. It is only
basedonzerotemperaturesecondorderperturbationtheory. ToillustratetheenhancedLambshiftfordressedstates
we thus consider here a TLS interacting with such a single mode reservoir with frequency ω . With the distribution
0
J (ω)=η(cid:126)ω2δ(ω−ω ) the integrals (13) are easily evaluated and based on the coupling matrix elements (20), (21),
1 0 0
one obtains for σ -noise
z
η ω2
δω(single,z) = 2ω sin(2θ) 0 . (22)
12 π R ω2−ω2
0 R
For σ -noise, we have
x
ηω2 (cid:20)(ω +Ω)cos4(θ) Ωsin2(θ)/4(cid:21)
δω(single,x) = 0 R + . (23)
12 π ω2−(ω +Ω)2 ω2−Ω2
0 R 0
In the static situation (no driving) the so-called dispersive limit (E and ω far detuned) has been considered in [8].
0
It yields
η ω2
δω(static) ≈ 0 . (24)
12 π(E/(cid:126))−ω
0
While this static Lamb shift leads to shifts in bare transition frequencies of a few % only, apparently, by properly
tuning A and Ω the dressed Lamb shift can be significantly enhanced.
Mapping to the Rabi model
As discussed in the main text, recent experiments with driven CPBs are described with
1 1
H (t)=− E τ − E τ −λcos(ωt)τ (25)
S,CPB 2 C z 2 J x z
with tunable charging energy E and Josephson energy E [5–7]. The coupling to the reservoir is dominated by
C J
charge noise and thus proportional to τ . Here, we show how in limiting cases this setup reduces to the Rabi model
z
(17) with either transversal or longitudinal coupling to the bath.
9
Transversalcoupling: ClosetochargedegeneracyE =0,intheeigenstaterepresentationoftheCPB,i.e.,τ →σ
C x z
andτ →−σ ,onehasE =E andthecouplingtothebathistransversalwithS(r =π/2)=σ . Closetoresonance
z x J x
(cid:126)ω ≈E aRWAinH canbeappliedwhichisconsistentwiththePSAusedtoderivetheFloquetmasterequation
J CPB
(7) if (cid:126)ω/|λ|(cid:29)1(cid:29)η. The drive parameters then read A=λ and Ω=−ω.
Longitudinal coupling: For E (cid:54)= 0 and for strong driving λ,(cid:126)ω (cid:29) E , a dressed tunneling picture applies.
C J
Accordingly, a unitary transformation U(t)=exp[−iφ(t)τ /(2(cid:126))] with φ˙(t)=2λcos(ωt) leads to
z
H˜ (t)=U†H (t)U(t)+i(cid:126)U†(t)U˙(t)
S,CPB S,CPB
1 1 (cid:104) (cid:105)
=− E τ − E eiφ(t)/(cid:126)τ +e−iφ(t)/(cid:126)τ (26)
2 C z 2 J + −
with τ = (τ ±iτ )/2. By decomposing the time-dependent phase factors in terms of Bessel functions using the
± x y
Jacobi-Anger expansion exp(izsinϕ) = (cid:80)∞ J (z)exp(inϕ), near the n-photon resonance E = n(cid:126)ω one may put
−∞ n C
eiφ(t)/(cid:126) ≈J (2λ/(cid:126)ω)einωt. This RWA is consistent with the PSA provided (cid:126)ω/E (cid:29)1 as assumed.
n J
Accordingly, one obtains the Rabi model with E = E and drive parameters A = −(E /2)J (2λ/(cid:126)ω),Ω =
C J n
nω. As the transformation U(t) commutes with the CPB-bath coupling operator τ , one arrives at pure dephasing
z
(longitudinal coupling).
∗ [email protected]
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